Texto de M.H.Otero PHILOSOPHICAL PAPERS

In: E.Neuenscwander & L. Bouquiaux, Science, philosophy and music (Proceedings of the XXth International Congress of History of Science, Brepols, Turnhout (Belgium), 2002.

Tables, chairs, beermugs; or the prehilbertian use of the primitive concepts of David Hilbert’s Grundlagen der Geometrie.

In: Velamazán, Mariángeles et al. (eds., 2008) La historia de la ciencia y de la técnica; un arma cargada de futuro; ensayos de homenaje a Mariano Hormigón. Diputación de Cádiz, Cádiz.

A classic historiographic text: Andrey N. Kolmogorov’s article “mathematics” presented in the 1936 Soviet Encyclopaedia.

A chocolate mint? On a certain widely spread ideology involved in the historiography of mathematics and in many other non trivial discourses.

In: Algunos avatares de la llamada matemática pura. Universidad de Zaragoza, Zaragoza, 2003.

The so-called autonomy of mathematics; what really induces them in a significant proportion of their production.

Avatares…

KUHN’S PHILOSOPHICAL TROUBLES WITH ACTUAL SCIENCE HISTORY (1)

Almost everybody knows that Thomas S.Kuhn was both an historian of science and a philosopher of science. According to him he is not, nor was, both at once. It seems that is possible for him to distinguish when he is either one. Even more, he thinks that both enterprises should be separated (2) not only in his work but also in general notwithstanding the mutual fertilization between them.

Many of us estimate very doubtful that the separation be present even in Kuhn intellectual practice (3), in SSR (1970), his main and most famous work, we do not find such alleged distances.

Concerning this question we would like to compare here some passages of his recent paper on "The trouble with the historical philosophy of science" (1992, T from now on) with others of the former one on "The relations between the history and the philosophy of science" (1968, revised in 1976; from now on, R), included in the book The essential tension (1977).

We should remember the double autobiographical character of R, in aspects concerning Kuhn own formation and activities and his long experience in the teaching of both disciplines and in the orientation of doctoral theses. Even more, the last several Kuhn papers of the nineties (1991i, 1992, 1993) have also a strong autobiographical character.

It would seem that the original position in R - not totally exempt of ambiguities - could have been broken by his practice and that in T they would appear some very surprising theses.

1. As for T, philosophical construction seemed to be attained, in Kuhn's original generation, from observations of scientific actual behaviors. But for him that image is misleading because in that historical philosophy of science conclusions may be reached with scarse reference to real historical records. Even more, the historical perspective, following T, was in the beginning alien to the received and dominant philosophical tradition that was guided rather by the existence, or not, of a rational guarantee as a basis to affirm this or that. For Kuhn gradually the static image of the tradition became to be dynamic in the new philosophy and science began to be conceived as a developmental practice or enterprise. Even the attained new perspective could be derived from principles and not necessarily from historical records (4).

"Now I think we overemphasized the empirical aspect of our enterprise" (T, 6). ans so, because the point of departure were principles, one may explain for Kuhn the scarse contingence of consequences,

"...making them harder to dismiss as a product of muckraking investigation by those hostile to science" (T, 10).

2. The result of the historian activity would be a narrative that would include a description of the initial state of the process to be explained. It would include also a description of the beliefs at that moment and of the conceptual vacabulary in use. Those resulting considerable changes at the end of the process would come from intermediate and not too notorious gradual changes. What goes on in the process would be a change of beliefs within changes in the context. Concerning the former ones it would be necessary to investigate precisely why the actors decided those changes.

2.1 For the philosophers (5) the problem would be the same: that is, to understand small changes in beliefs. Rationality, objectivity and evidence would come to be subjects easier to deal with that with the referents of the corresponding beliefs. The static Archimedian platform required by the so called neutral observation in the former tradition was then unnecessary and it would have vanished.

First of all, as for Kuhn, the rationality in historical perspective needs a transitory rationality only in relation with the members of the group which produces each decision. Secondly the changes to evaluate are always relatively small even if they may seem gigantic in retrospect. Thirdly, in general truth would not come from of comparing beliefs with reality: the evaluation would be indirect. The criteria that intervene are secondary criteria: precision (only aproximate and often unattainable), consistence with other accepted beliefs (at most local), breadth of applicability (increasingly narrow when time goes on), simplicity (depending on the observing eye), among others. They are ambiguous values that anyway are not satisfied at once. But if those criteria are applied to belief changes they would get, for Kuhn, new relevance and sense, both relational ones: a set of beliefs may become more precise, more consistent, larger in applicability, more simple, without becoming truer (T, 13-14).

The expression 'truer' in sometimes interpreted as 'more probable' but that would carry, even in this Kuhn, what has received the name of 'disastrous metainduction' (as Kitcher baptised it):

"All past beliefs about nature have sooner or later turned out to be false...the probability that any currently proposed belief will fare better must be close to zero" (T, 14).

Chilling result…, and erroneous from my point of view; already discarded by Poincaré, not without good reasons, at the beginnings of the century. The disastrous metainduction would complement in this way, even radicalizing it, the so recurred underdetermination of theory.

"I am not suggesting, let me emphasize, that there is a reality which science fails to get at. My point is rather that no sense can be made of the notion of reality as it has ordinarily functioned in philosophy of science" (ibid.)

"...facts are not prior to conclusions drawn from them and those conclusions cannot claim truth" (ibid.).

"I've reached that position from principles that must govern all developmental processes, without, that is, needing to call upon actual examples of scientific behavior" (ibid.).

The trouble with the historical philosophy of science comes for him from the fact that its quasihistorical or perihistorical examples have questioned the authority of science itself. The pillars of that authority - 1. the priority of facts and its independence from the consequences and 2. the truths concerning an independent external world - would have melt. The option Kuhn faced was either to provide them a firm foundation or to eliminate them completely. But now he maintains that what matters are not observed facts concerning scientific practice but necessary characteristics owned by the evolutionary processes in general. Should we think that in such way Kuhn's difficulty - a quite persistent and enough annoying one - would be totally overcome?

3. From the early R - very rich and at the same time questionable text - we will take only one point, leaving for some other opportunity other very interesting aspects.

When Kuhn strongly doubts about the value of the covering law model for history (R, 15-16), his central criticism points to the triviality in some cases, or the non historical character in others (sociological aspects or belonging to social sciences), of the laws that would be assumed by the historian in that model (7). To suppose those laws would amount to force the historian to employ instruments totally alien and of doubtful validity for accomplishing his job.

Then we could demand ourselves if the principles and examples quasi- or perihistorical that Kuhn prefers for the historical philosopher of science would not be purely speculative, because, avowedly, they renounce both to empirical test and to actual historical records and explanations (we must remember that for Kuhn historical work needs not to be only descriptive.

4. Even if we have considered here only limited aspects of R and T, consistent with many other not alluded passages of those texts and of others, we may point the origin of our strong surprise concerning the central thesis included in T.

a. For Kuhn history of science and philosophy of science are different things even if the fertilize each other,

b. Kuhn's practice in his main works, and especially in SSR, seems to be different to the conception exposed in R (and obviously in T), with a strong overlapping if not integration of both supposed separate disciplines,

c. The independence - so it seems in the texts - of the historical theses belonging to philosophy of science (hypostatiated principles and examples) and opposed to the results of actual history of science, far from immunizing those theses extremely weakens them, and

d. Kuhn would not be situated in such way, from the0 comparison of his own words, far neither from the "deconstruction gone mad" of the strong program of the sociophilosophy of knowledge nor from the constructivist-idealist (8) theses that Edouard LeRoy exposed almost a hundred years ago.

2 Stuewer et al. (1970) deal extensively with the subject of the "distance" or "divorce" between history and philosophy of science.

3 Zamora 1994 discusses important aspects of historico-philosophical practice in Kuhn's last period though not specifically about his theory on the relations between them.

4 Nevertheless the reciprocal influence between history and philosophy of science is clear not only in The structure of scientific revolutions, but also in The copernican revolution. Still more, many other Kuhn books, papers, reviews and short notes on historical subjects, listed in Hoyningen-Huene (1989). are not alien to the theme of the referred reciprocal influence.

5 Not only "The trouble with the historical philosophy of science" raises the subject of the philosophical enterprise of those ocupied with science; also "Dubbing and redubbing..." and Kuhn (1989) raise it, in a somewhat but not essentially different version of the former. In both Kuhn elaborates on the natural class concept and on local holism. Kuhn (1991i) and (1993) - this written earlier than T -, also work on the subject of that philosophical enterprise.

7 It is enough evident that Kuhn alludes to the well know Hempel papaer "The function of general laws in history", The Journal of Philosophy, v.39, 1942. Shortly later Theodor Abel, presented a very intelligent contribution in "The operation called Verstehen" American Journal of Sociology, v.54, 1948. After a lapse of large domination of the covering law model, with its well known sequels, appeared often the criticisms that, in many cases, arrived to a notion very close to that of Verstehen, the very notion that Hempel had tried to supersede. Von Wright presented in his "Explanation and understanding" a new paradigmatical concept. But he didn't go back to the diltheyian and marburguian Verstehen. Kuhn was strongly influenced by this new orientation. Each time Kuhn used the renewals produced in the hardware of the ortodoxanalytic philosophy and then he produced the corresponding rectifications in his thought.

8 Constructivistas and even idealist modes appear in the niche idea at the end of T; see Hoyningen-Huene (1989) and Otero (1996).

Hoyningen-Huene, P. (1989) Thomas S. Kuhn's philosophy of science. The University of Chicago, Chicago*.

Kuhn, T.S. (1970) The structure of scientific revolutions, University of Chicago, Chicago. Second edition.

Kuhn, T.S. (1977) "The relations between the history and the philosophy of science", T.S.Kuhn. The essential tension. The University of Chicago, Chicago. /conference delivered in 1975/.

Kuhn, T.S. (1979) "History of science". P.D. Asquith & H.E.Kyburg (eds.), Current research in philosophy of science, Philosophy of Science Association, East Lansing, MI.

Kuhn, T.S. (1989), "Possible worlds in history of science". S.Allen (ed.) Possible worlds in humanities, arts and sciences. Walter de Gruyter, Berlin.

Kuhn, T.S. (1991i) The Road since Structure. A.Fine, M.Forbes & L.Wessels (eds.), Proceedings of the 1990 Biennial Meeting of the Philosophy of Science Association. PSA 1990, v.2. Philosophy of Science Association, East Lansing, MI.

Kuhn, T.S. (1991ii) "Tthe natural and the human sciences". D.H. Hiley, J.E.Bohman & R.Shusterman (eds.) The interpretive turn. Cornell University, Ithaca.

Kuhn, T.S. (1992) The trouble with the historical philosophy of science, Harvard University (Department of the History of Science), Cambridge, MA.

Kuhn, T.S. (1993) "Afterwords". P.Horwich (ed.) World changes; Thomas Kuhn and the nature of science. MIT, Cambridge, MA.

Peral, D., Estévez, P. & Pulgarín, A. (1997) "Presencia del pensamiento kuhniano en la literatura científica: 1966-1995", Llull, v.20.

Solis, C. (1994) Razones e intereses: la historia de la ciencia después de Kuhn. Paidós, Barcelona.

Stuewer, R. (ed.) Historical and philosophical perspectives of science. Gordon & Breach, New York. /First edition in Minnnesota Studies in the philosophy of science, v.5, 1970, University of Minnesota, Minneapolis/.

Wartofsky, M. (1976) "The relation between philosophy of science and history of science". R.S.Cohen, P.K.Feyerabend & M. W. Wartofsky (eds.) Essays in memory of Imre Lakatos. Reidel, Dordrecht.

ON THE PRESUPPOSITIONS OF SCIENCE: ROBIN COLLINGWOOD’S ESSAY ON METAPHYSICS (1940) AS ANTECEDENT OF A WHOLE ERA

Towards the end of the dominance age of neo-positivism, signs of a new conception of science start to emerge. Names such as Hanson, Toulmin, Feyerabend, among others, (1) have been remembered as the founders of a new era (2). Antecedents of Kuhn’s work have been pointed out, as we have done with Ludwig Fleck and Julio Rey Pastor’s work (3).

1. Robin Collingwood’s An Essay on metaphysics (1940) is a strange book with regard to its content. It is written with precision and wit, by an author that was both, at length a well-known archeologist and philosopher, and above all, an outstanding participant of a strong idealist trend, widespread in his time in the Anglo-Saxon world. Nevertheless, the book, although it may inherit from that trend (aspects which we do not intend to remark here), is something other than idealist.

It is not a metaphysics book but a book about metaphysics. Curiously, he rejects the idea of conceiving metaphysics -and he supports so (4)- as the science of the pure being. Instead, he conceives metaphysics as the science which deals with the presuppositions underlying ordinary science. By “ordinary” he understands science which is not a constituent part of metaphysics. (EM, page 11). It is, therefore, metaphysics without ontology as “science” of the pure being. For Collingwood “ontology” is just the name of an error.

Even though Collingwood does not share with Aristotle the conception of metaphysics as the science of the pure being – sharing instead that of the study of the presuppositions in science –, he agrees with Aristotle in some of the basic principles: 1. that all science deals with the universal or abstract, 2. that there is at least potentially a science of each universal, 3. that there are degrees of universality or abstraction and that there is a hierarchy of the universals and a hierarchy of the sciences, and 4. that A is not only the presupposition of B and C, but also its sufficient logic foundation.

Presuppositions, then, are not priorities in time; they are logic priorities (5). Only through a certain kind of analysis a presupposition can be known as such.

2. Then Collingwood raises the question of his basic meta-propositions (in our terminology) and also certain basic definitions:

P1. Every statement that anybody ever makes is made in answer to a question.

D1. Let that which is stated (i.e. that which can be true or false) be called a proposition, and let stating it be called propounding it.

D 2. To say that a question ‘does not arise’ is the ordinary English word of saying that it involves a presupposition which is not in fact being made.

D 3. The fact that something causes a certain question to arise I call the ‘logical efficacy of that thing.

P 3. The logical efficacy of a supposition does not depend upon the truth of what is supposed, or even on its being thought true, but only on its being supposed.

D 5. By a relative presupposition I mean one which stands relatively to one question as its presupposition and relatively to another question as its answer

D 6. An absolute presupposition is one which stands, relatively to all questions to which it is related, as a presupposition, never as an answer…

Thus, every question -starting point- involves an absolute or relative presupposition; relative, if it is posed in relation to a question as its presupposition and in relation to another question as its answer, while an absolute presupposition (6) is posed in relation to all the questions only as a presupposition.

3. Before drawing some conclusions, we will refer to Collingwood again -the fifth chapter of his book- and then especially in the following section, the note which closes that chapter (4).

“In this kind of thinking /by introspection/, absolute presuppositions are certainly at work; but they are doing their work in darkness, the light of consciousness never falling on them. It is only by analysis that any one can ever come to know either that he is making any absolute presuppositions at all or what absolute presuppositions is he making” (p. 43)

“In my own experience I have found that when natural scientists express hatred of ‘metaphysics’ they are usually expressing this dislike of having their absolute presuppositions touched” (p. 44)

“The purpose of the experiments is to find out what absolute presuppositions are as a matter of fact made on a certain occasion or on occasions of a certain kind… In ordinary science the relative presuppositions are put into a basket, and later on the question is raised when and how they shall be justified. The absolute presuppositions are thrown back. In metaphysics it is the relative presuppositions that are thrown back, and the absolute presuppositions that are put into the basket… in order to have them scientifically described” (p. 45-46)

“…it is a special characteristic of our modern European civilization that metaphysics is habitually frowned upon and the existence of absolute presuppositions denied” (p.46)

“To sum up. Metaphysics is the attempt to find out what absolute presuppositions have been made by this or that person or group of persons. On this or that occasion or groups of occasions, in the course of this or that piece of thinking” (p.47)

Therefore, only the analysis can determine if there are absolute presuppositions and which they are. The scientist does not want to accept that his absolute presuppositions are considered and hence his rejection to “metaphysics”. On the other hand, he sets apart the relative presuppositions and then tries to justify them.

While the metaphysician rejects the relative presuppositions and analyzes them to describe them scientifically. European civilization rejects metaphysics and denies absolute presuppositions. Metaphysics tries to see which presuppositions are made and how.

“But an absolute presupposition is not a ‘dodge’, and people who ‘start’ a new one do not start it because they ‘like’ to start it. People are not ordinarily aware of their absolute presuppositions” (p.43), and are not, therefore, thus aware of changes in them; such a change, therefore, cannot be a matter of choice. Nor is there anything superficial or frivolous about it. It is the most general change a man can undergo, and entails the abandonment of all his most firmly established habits and standards for thought and action. Why, asks my friend, do such changes happen? Briefly, because absolute presuppositions of any given society, at any given phase of its history, form a structure which is subject to ‘strains’ (pp.74,76) of greater or less intensity, which are ‘taken up’” (p.74)

in various ways, but never annihilated. If the strains are too great, the structure collapses and is replaced by another, which will be a modification of the old with the destructive strain removed; a modification not consciously devised but created by a process of unconscious thought” (p.48).

One is not aware of absolute presuppositions, or of the changes produced in them, changes that cannot be a matter of choice and that lead to the abandonment of habits and standards which seemed the most solid ones. These absolute presuppositions form, at each moment of history, a structure which undergoes strains that are not eliminated. Thus, the structure can collapse and be replaced by another which does not involve the destructive strain which shattered the original.

5. As a result, metaphysics is a historical science. It does not involve eternal, crucial or central problems (p. 72).

“The metaphysician’s business, therefore, when he has identified several different constellations of absolute presuppositions, is not only to study their likenesses and unlikenesses but also to find out on what occasions and by what processes one of them has turned into another” (p.73).

“One phase change into another because the first phase was in unstable equilibrium and had in itself the seeds of change. And indeed of that change” (p.74).

The different constellations of absolute presuppositions throughout history thus show that the unstable equilibrium has the seeds of change.

According to Collingwood, for Hegel the study of history was fundamentally a study of the internal strains and hence his influence on the 19th century historiography, and if Oswald Spengler is today rightly forgotten it is because he always described a constellation of historical facts – a culture –

“…he deliberately ironed all the strains out of it and presented a picture in which every detail fitted in to every other as placidly as the pieces of a jig-saw puzzle lying at rest on a table” (p.75).

The historical references lead Collingwood to support his conception of metaphysics in a new way.

“This is why the conception of metaphysics as a ‘deductive’ science is not only an error but a pernicious error; one with which a reformed metaphysics will have no truce” (p.76).

6. The second part of the book deals with the anti-metaphysics. Its general form and its positivist and irrationalist varieties, among others, and its totally erroneous orientations, are studied there.

“So the battle-cry of ‘Back to Kant’ expressed in philosophical terms the attempt of nineteenth century scientific orthodoxy to muster in its own support all the forces which could be conjured into reactionary activity by appeal to the name of a great and honored philosopher whose doctrines, understood in a pseudo-metaphysical sense, gave no support to the movements that threatened it” (p.95).

This argument turns out to be particularly convincing. However, Collingwood still remarks

“The new physics and the new geometry involved a definite breach with the Kantian system” (ibid.)

That is why the temptation to reject metaphysics (obviously in Collingwood’s sense) arises,

“Behind that cry /’No More Metaphysics’/ there lay a feeling that the constellation of absolute presuppositions made by this reactionary science was exposed to strains which could only be ‘taken up’ by keeping them in darkness” (p.96).

“Nature seemed to the eighteen century historian an absolute presupposition of all historical thinking”.

But the nineteenth century historiography dissolved that illusion of concealment in the sense that

“...what man makes of nature depends on man’s own historical achievements, such as the arts of agriculture and navigation, the so-called conditioning of history by nature is in reality a conditioning of history by itself” (p. 98).

“When once it is realized that the absolute presuppositions of eighteen-century science, far from being accepted, semper, ubique, ad omnibus, had only a quite short historical life, as we nowadays think of history, in only a quite limited part of the world, and that even inside Europe other systems of science worked before then and since then on different presuppositions, it becomes impossible for any one except the most irresponsible kind of thinker to maintain that out of all these and all the other possible sets of presuppositions there is one set and only one which consists of propositions accurately describing observable characteristics everywhere present in the world, while all the other sets represent more or less systematic hallucinations as to what these characteristics are” (p. 180).

8. More than sixty years after Collingwood’s Essay it is clear how a change in terminology when interpreting it – a certainly viable change – would lead to Kuhn’s theses, known for 30 years and widespread since then – although strongly criticized too – , theses which are more than hinted in that author, even if they are not sufficiently well expressed in today’s language.

By 1961, Toulmin (1961) briefly acknowledges Collingwood in Foresight and Understanding. But in “Conceptual revolutions in science” (1967) he already studies Collingwood quite thoroughly, comparing him with his own thought.

By 1972, he already uses and criticizes Collingwood extensively, in Human Understanding (1972).

In a lecture at the University of Indiana at the beginning of 1960, published the following year, Toulmin introduces the term “paradigm” in two different senses: as models of the natural order and as standard cases, examples chosen in order to illustrate what scientific explanations comprise. Some references, a few, to Collingwood show interesting aspects.

While Toulmin (1972) ponders Collingwood trying to show his alleged relativist tendencies, Toulmin (1967) provides, beyond the criticisms, an adequate analysis of Collingwood’s central theses. For that reason, we will consider Toulmin’s 1967 text as more relevant.

For Toulmin, although Collingwood’s examples are not convincing – specially the three-phase division (Newton, Kant and Einstein) which would be too rudimentary – these examples “do not spoil a valuable philosophical account” (Toulmin 1967).

The hierarchy in scientific matters and propositions is not deductive as it may be in mathematics, but it is based in the meaning and relevance to the general doctrines. And these are not linked as theorems are to axioms but as presuppositions are to concrete issues. Questions arise or do not arise depending on the general principles assumed.

“The relevance and acceptability of the narrower concepts depend on … the relevance and acceptability of the wider concepts … If the general axioms on Newton’s dynamics were to be abandoned, the specific statements about the forces and their effects in movements are not only falsified: they cease to be posed as they were before” (ibid pages 77-78).

Crucial intellectual decisions in science concern, thus, changes in basic assumptions. The historical background, which must be studied, shapes such decisions. It is important to determine how the basic assumptions are replaced and followed, namely the absolute presuppositions.

According to Toulmin, Collingwood deems important -as he clearly expresses- to determine the occasions and processes in which a constellation of presuppositions becomes another one. However, for Toulmin, this is precisely what Collingwood does not resolve. And this is Toulmin’s central problem (still in 1972). Yet, according to this commentator, there would not be a clear distinction between absolute and relative presuppositions.

From the note already cited on page 48 of the Essay, according to the commentator “a key footnote which is perhaps the most significant element in all the book” (page 79), Toulmin holds that to label the change of constellations as an unconscious thought process constitutes a new mistake. Furthermore it would not be clear how internal strains would become manifest or how they would be recognized after they have been removed from theconstellation resulting from the change.

“We should introduce now a “super-absolute” presupposition in order to decide if, in any specific case, the shift from the previous presuppositions to the new ones was a “rational” shift or not.” (p. 80)

The relative or absolute character would require these “super-absolute” presuppositions, which would go against previous theses. Additionally, the fact that Collingwood establishes a relation between strains and sociological and cultural crises shows a tendency towards Marxist theses – while mild – which the author of the Essay had recently acquired by then.

“After all, an acceptance of certain Marxist propositions was entirely consistent with the argument of Essay on metaphysics and was somehow a sequel of it” (p. 80)

According to Toulmin, it was a Marxism, which even if mild, Collingwood’s colleagues, suspected and feared.

9. Normal science, paradigm, scientific community, puzzle, anomaly, crisis, scientific revolution, conceptual incommensurability among subsequent paradigms, are not only expressions which appear repeatedly in Kuhn (1962); not only arms which are trying to grasp through a brilliant intuition phenomena such as stability and scientific changes, but are also expressions which gave way to countless criticisms and problems which appeared in more than, say, a thousand articles and books; the so-called Kuhn’s industry.

Just one of these topics – conceptual incommensurability – has generated an enormous amount of literature even if today is dying out (Kitcher dixit). We only mention these expressions here without considering them as topics.

As we have already advanced, Kuhn 1962, does not emerge ex nihilo; its antecedents are numerous and even if his conception seems to revolutionize the research front constituted by neo-positivism, elements in his work which challenge such a radical interpretation have been pointed out.

As we have seen, Collingwood’s terminology is totally different, apparently obsolete nowadays, at least the main concepts. While Fleck and Kuhn’s terminology can be related, Collingwood’s terminology cannot.

Yet, the Essay presents decisive elements to understand scientific change; I have remarked them. We have seen that Toulmin frowns on Collingwood for not giving one explanation of the radical alterations in the constellations of absolute presuppositions, but two insufficient and incompatible explanations. But both Toulmin and Kuhn – with his so rightly criticized conversion – commit the same omission. And it cannot be neglected that the reason for this is that they share the same basic explanation model.

If we were asked, at this stage, to re-baptize Collingwood’s meta-scientific concepts according to Kuhn’s usage, we could try it. Constellations of absolute presuppositions as paradigms, in one of the basic senses of this word; relative presuppositions would fall into normal science in which breach strains (anomalies) emerge and give way to Kuhn’s crises, and so on. However, conceptual incommensurability does not appear in Collingwood, even if the absence of an only explanation of radical change may be interpreted as a blind alley, as the extreme incommensurabilities which Kuhn remarked.

10. Before we finish, I would like to propose three brief remarks on some central aspects which are worthy of attention.

i. Hacking (1982, 1992) presents, following Alistair Crombie, the styles of scientific reasoning (he distinguishes six at present) which appear historically and then coexist. The styles become objectivity standards as they allow to attain the truth. A statement is true or false only in the context of a style.

“…admission of the historicity of our own styles in no way makes it less objective” (2002, p.164)

Although the constellation of absolute presuppositions in Collingwood do not have some of the features of Hacking’s styles, they are in a certain way comparable, because they appear historically and they are certainly self-authenticating, even if silently.

“A second telling feature of Foucault’s method is that, unlike Kuhn, who explains the need for paradigm shifts in terms of the old paradigm’s inability to resolve standing anomalies , Foucault notoriously offers no account of why and how one episteme (roughly, paradigm) replaces another” (p.151).

I greatly doubt that this is Kuhn’s explanation, considering the role conversion play as determinant non-explanation. But this remark about Foucault expresses, once again, Toulmin’s concern about Collingwood’s lack of explanation of the constellations shifts.

iii. On the other hand, according to Fuller (2000, p. 69-70) Collingwood’s presuppositions aim to the context of scientific activity rather than to direct content and that author refers us to Chapter 6 of his Social Epistemology on the inscrutability of silence. Moreover he says:

“…conspicuously absent from Kuhn’s account is any discussion of how argumentation may facilitate this transition” (p.306).

“...the period since 1980 has been marked by a slow but significant devaluation of the role of language, especially argumentation, in the constitution of scientific authority among historians, philosophers and even sociologists of science” (p.314).

Yet we should remember that Toulmin re-described Collingwood’s absolute presuppositions as “ideals of the natural order”, paradigms, in terms of which specific explanations are estimated. (p. 312)

“An interesting feature of Crombie’s account is the role that research for hidden presuppositions (i.e. unexpressed questions) of past scientists played in justifying a role for research – not merely teaching – in the history of science” (p.316).

On the other hand, I have doubts about Fuller’s assertion that Collingwood has played in Great Britain the role Koyré has played in the United States.

iv. Bourdieu (2001) once again attributes to Kuhn -but this is interesting in the context of his last book and the way he does so- a strictly internalist account of scientific change, even revolutionary change.

“Chaque paradigme atteint un point d’épuisement intellectuel... à la manière d’une essence hégélienne que s’est réalisée, selon sa logique même, sans intervention externe” (p.37).

We have seen before that, unlike Kuhn, Collingwood is far from an internalist position. The advances achieved by Kuhn are compensated by the retreats regarding Fleck and Collingwood.

(2) Rossi, already in 1986, stated the imaginary character of the official history character of the so called meta-scientific revolution.

(4) There is nothing to investigate in the pure being; there cannot be a science which deals with it.

(5) On presuppositions, see Olivé 1985 (sections 1.2 al 1.4) and Olivé 1988, p.287-291).

(6) “Unlike Kuhn, Rorty does not insist on a stage of “crisis” between those of normality and revolution. He seems to think that the new ways of speaking need not be motivated by active strains to the old ways” (Gutting, Gary. Pragmatic liberalism and the critique of modernity. Cambridge University, Cambridge, 1999). ‘Strains’ is precisely the term used by Collingwood in the work commented here.(6) The fact of using the word “absolute” which has more frequent connotations, may be disturbing. But it is no longer disturbing if we take the function this word has strictly in the meta-propositions and definitions of Collingwood. Donagan (1962) also criticizes the notion of absolute presupposition.

Bourdieu, P. (2001) Science de la science et réflexivité. Raisons d’Agir, Paris.

Fuller, S. 2000. Thomas Kuhn; a philosophical history of our times. University of Chicago, Chicago.

Hacking, I. 1982. “Language, truth and reason”, Hollis, M. & Luke, S. Rationality and relativism. Blackwell, Oxford.

Hacking, Ian. (1992)“’Style’ for historians and philosophers”. Philosophy of Science. Also in Hacking, I. (2002) Historical ontology, Harvard University, Cambridge MA.

Jacobs, Struan (2002) “Polanyi’s presagement of the incommensurability concept”. Studies in the History and Philosophy of science, v.33.

Olivé, León (1988) Conocimiento, sociedad y realidad. Fondo de Cultura Económica, México. /p.287-291/

Otero, Mario H. (1991) “¿Modelo Reyfleckuhn?”. In Valera, M. & López Fernández, C. (eds.) Actas del V Congreso de la Sociedad Española de Historia de las Ciencias y de las Técnicas, v.III. SEHCT, Murcia.

Rossi, Paolo (1986) “Fatti scientifici e stili di pnsiero; appunti in torno a una revoluzione immaginaria”. In I ragni e le formiche; un’apologia della storia della scienza. Il Mulino, Bologna

Toulmin, Stephen (1982) Previsione e conoszenza; un’indagine suglo scopi della scienza. Original Edition Foresight and understanding; an enquiry into the aims of science, 1961.

Toulmin (1972) Human understanding. V.I The collective use and evolution of concepts. Princeton University. Spanish version La comprensión humana; El uso colectivo y la evolución de los conceptos. Alianza, Madrid, 1977.

Toulmin (1977) “Della forma a la funzione: filosofia e storia della scienza nelli anni ’50 en el tempo presente”, also in Toulmin (1982).

TABLES, CHAIRS, BEERMUGS; OR THE PRE-HILBERTIAN USE OF THE PRIMITIVE CONCEPTS OF DAVID HILBERT’S GRUNDLAGEN DER GEOMETRIE.

In 1. we will point out the background concerning the subject of primitive concepts of geometry previous to the publication of Grundlagen der Geometrie (1899). We will not consider important developments following its publication; in particular, we will not consider Sommer’s report (1900), Poincaré’s (1902), the correspondence between Frege and Hilbert, Frege’s texts on this subject (1903, 1906) (1), not even the fundamental, only apparently tardy, of Tarski (1959) (2).

It will not cover other mathematical books of comments, critics or development later than 1899, as for example Pieri’s (1900 and 1908) or Schur’s (1909) contributions to the foundation of mathematics. We will take another direction; we will first go over only some milestones of the period starting with Pasch (1882) to 1899.

In 2. we will briefly consider some contributions to the notion of definition, and in particular, the notion of implicit “definition” of primitive concepts.

In 3. we will state some hypothesis related to quite former, sometimes explicit but at least always tacit, developments, which seem to me worthy of taking into account when dealing with the subject of primitive concepts (or, in another version, primitive entities).

As it is obvious this paper will not deal with the set of contributions of Hilbert’s Grundlagen to the foundation of geometry but will only deal with just a point, but certainly a strategic one.

“On the grounds of our study we include any variety of entities, which we will call briefly points independent from their nature itself”,

While commenting a H. Wiener’s lecture during that same year, Hilbert had seemingly said, “it must be possible to replace (in geometry statements) the words “points”, “straight lines” and “plane surfaces” by “tables”, “chairs” and “pints”. However, this resounding dictum will yield consequences only seven and eight year respectively later, in his lectures prior to the Grundlagen (winter 1898-99) and written in these (3).

The content of the first paragraph of this work - precisely referring to primitive concepts - will become one of the most commented and argued texts by historians and philosophers of mathematics for a long while, till the present day.

1.1 Moritz Pasch is justly considered the first producer of a modern axiomatic for geometry (4). His procedure later became characteristic. His conception of the origin of the axioms is clearly empiricist, but then his development of geometry excludes other procedures than demonstrative ones, considering that all the others, not unusual ones, are inconsequential (5).

“...The theorem is only really demonstrated when the demonstration is completely independent from the figure. The axioms cannot be conceived without the corresponding figure, they are the expression of what has been observed in certain very simple figures. The theorems are not found in observation, they are demonstrated, every conclusion that appears in the course of the demonstration must be confirmed in the figure, but it is not justified by it, but rather by a certain proposition (or definition) that precedes it... No matter how little we detach from this procedure, the spirit of the demonstration loses all precision” (Pasch, paragraph 32)

“Apart from the perception of the senses, it is not licit to refer to “intuition” or “imagination” as special sources of mathematical knowledge” (Op. Cit., added in 1912 to Chapter 23).

“Mathematics establish relations between mathematical concepts that must accord with the experimental facts, although they are mostly not taken directly from experience, yet they are “demonstrated”; the same knowledge needed for the demonstration (apart from the definitions of derived concepts) constitute part of those relations. If the propositions based on the demonstration -the theorems- are left out, it remains a group of propositions, from which all the rest can be deduced – the axioms...” (op. Cit., paragraph 12).

“The fundamental concepts have not been defined, because there is no definition whatsoever capable of replacing the observation of the appropriate natural objects...”

“... From a purely mathematical point of view, (the conformity with its applications) may be left out and the definitions of a concept which have not relation with its applications may be accepted as good and still be preferred to the rest” (Op. Cit. added in 1912 to its Introduction).

“The point, the straight line and the plane surface (in the general sense), two elements being incident and pairs of elements being separated, perform in Position Geometry the role of primitive concepts, to which all the rest must be referred to”. (Op. Cit., paragraph 55).

Paragraph 77 focuses on Pasch’s conviction on the strict logical deductive character of the axiomatic:

“We have said before how much of Graphic Geometry exists as a consequence of theorems SS 7, 8 and 9, in these we can replace constantly the words point and plane surface and therefore, the consequences are also legitimate, without restrictions when these substitutions are done in them. Moreover, if Geometry is to be deductive, the procedure of deduction must be effectively, independent of the character of the geometrical concept, as it should be of the figures; we can only take into account the relations between the geometrical concepts established as definitions in the theorems used. In the case of deduction it is licit and useful, but in no way necessary, to think about the meaning of the geometrical concepts present. So precisely when this is necessary, the defects of the deduction and (if the defects do not disappear by modifying the reasoning) the insufficiency of the theorems, that had been put before as means of demonstration, arise. (...) It is clear that this discussion is not superfluous when we observe that the conditions set beforehand often remain unfulfilled, even in works concerning the foundations of Geometry or other mathematical disciplines. From a generalised viewpoint, the theorems must be logical consequences of the axioms. But there is not always a conscientious use of all the means of demonstration” (op. cit.)

The rigorously deductive method is not a useless hindrance, it excludes all arbitrariness and it gives to mathematics the character of absolute certainty that it is put down to it. (Ibid.).

We have quoted Pasch extensively because when his work is presented, it is not totally apprehended if the ideas exposed, usually very concisely, are not just the result of a very common presentism.

Thus, there is a strong sense of deductive demonstration without interference of intuitions, observations, applications or others in Pasch’s construction of Geometry. The non-defined character of primitive concepts and their independence of figures and particularly of meanings is clear.

1.2 Among the various papers that Peano dedicated to the foundation of geometry those of 1889 and 1894 stand out (6).

We will just indicate two traits that are revealed there: the tendency to use the fewest geometrical entities as primitives (point and segment, and congruence) (Freguglia, 1985) and the way, not in the least abstract, he considers them. It has been remarked that this is what makes the difference between Peano and not only Pasch but also Pieri (7) and Hilbert. Thus, Peano tell us:

“Each author can assume the experimental laws which pleases him, and he can adopt the hypothesis he likes best; this choice is carried out by induction, and does not belong to mathematics” (1891/243).

And he uses similar conceptions in other texts. Nevertheless, he occasionally talks about “any such entities”. In spite of all this, we will point out that it is Pasch who had great influence on Peano. And also did Grassman, to whose study Peano dedicates his 1888 work.

The proposal of a new series of postulates for Projective Geometry in Pieri (1894-5) conceives it as:

“...As a deductive science, independent form any other body of mathematical or physical disciplines (or, in particular, from the axioms and hypothesis of elementary geometry) and governed far and wide by certain fundamental laws, like the projection and duality principles, which, to say it somehow, shape and print its character ... amply using the symbols and ways of algebraic logic”

The expression “projective point” attempts, more that anything else, to detach the mind from the ordinary idea of point.

A complementary note introduced the following year (Pieri 1895-6) is dedicated to some consequences of the logic geometrical principles of the previous work:

“The arguments exposed in these two notes seem sufficient to show how it is possible to develop all the pure geometry of position, and even metric geometry, from our postulates on the primitive (projective) entities point, straight line and plane surface”.

Pieri (1896-99) states that Projective Geometry, as independent science of the hyperspaces, continues to be a controversial topic for those who do not think that all its principles have a degree of clarity and rigour pretended in other branches of exact sciences. He also states that the nineteen postulates he introduces are enough for his purpose: founding the standard Projective Geometry.

Pieri (1896-97) uses only two primitive entities (projective point and union -“congiungente”- of two projective points). He defines “projective segment” with these two categories, not defined in any other way but through the postulates.

If Projective Geometry was built up, as in other authors, out of the elemental, or out of observations of the outer world, it would be an aspect of the physics of extension (Pieri, 1897-98).

“A more modern criterion -whose natural development concerning the principles tend to a different aim from that- requires that position geometry (together with the abstract metric geometries derived from it) is considered as a pure deductive science, independent from any other body of mathematical or physical doctrines and even from the axioms or hypothesis of elementary geometry...”

From his viewpoint the matter is to build up a speculative and abstract geometry whose topics are mere creations of our spirit, with the rigour of algebraic logic. Projective Geometry is a hypothetical science with method and premises independent from intuition. Its primitive and simple (indecomposti) concepts, over which the postulates turn to, are the raw materials of all its propositions. And Pieri understands that it is in this sense that his contribution has no precedents. He even advances a distinction between axioms according to properties of configuration and connection. All the axioms introduced are sufficient for the development of each theory. Each one of the intervening properties is the result of the logical combination of its primitive propositions on primitive entities (8).

Pieri (1898) asserts the hypothetical deductive character of geometry and introduces the concepts of projective point and homography by means of postulates, with the help of the logical categories of individual, class, belonging, inclusion, negative representation and a few more. Although the resulting system may not be very intuitive, it is, compared to the usual scabrousness, built up in a rigorously hypothetical way.

Finally, Pieri (1898-99) –last work we will refer to- outcomes particularly polished. He insists that if we talk about definition as a simple imposition of names, the mother ideas would be not defined concepts. On the other hand, there he defines “hypothetical deductive system” bringing in the characteristics previously introduced individually. It is presented as the theoretic desideratum – and as the practical accomplishment – of having the minimum number of primitive ideas, recognising the antecedents of Pasch and Peano, but introducing a new conception of geometry (9) (10). Moreover, the introduction of homography as a primitive allows the elimination of such a movement, and therefore deductive simplicity is obtained with it.

We have shown that Pieri gradually develops, not only his geometry, but also his approach to the primitive concepts and to geometry as a science. He differs from Pasch when the latter holds the primitive character of concepts; but he agrees in conceiving geometry as strictly deductive, and Pieri adds, independent from the other sciences (although maybe not logic). He clearly presents geometry as a hypothetical deductive system. He develops Peano’s tendency to accept the minimum number of primitive geometric entities, even showing, in one of his geometries, that only two would be enough to build it up, apart from a few logical concepts.

The first chapter of Hilbert’s Grundlagen, on the five groups of axioms, and specifically, in his first paragraph on the elements of geometry and on those groups of axioms, starts with the word “Aufklärung” that would introduce the following topic:

This term -Aufklärung (11)- has been subject to a number of different interpretations and even translations. Just to give two examples, the last English version of this work use the word “definition” and the latest Spanish translation of Lecciones de Wussing puts in its place the term “clarification”. But these are not the only ways in which it is interpreted or translated, as early as the original publication, and many have induced to confusion in the interpretation of the mere beginning of the text.

Thus, the text of the Grundlagen on the primitive concepts (or at least on the first three introduced, point, straight line and plane surface (12) is interpreted in many different ways.

Among these, Aufklärung is interpreted as clarification or as definition, but also as implicit definition (13), or as determination, and in many more ways.

However, if we consider the defining, clarifying or determining block, this block could carry out these alternative functions not in regard to the primitive concepts in themselves, but in regard to the block, as for example, referred to space or to geometric spaces, even though it is not this the most popular interpretative version. We rather think that the function refers to the primitive concepts themselves and this is how we are going to understand it here.

2.1 Not every geometric statement is provable, nor every primitive concept is definable, so we can identify “primitive” and “indefinable”. Among the clearest texts, apart from some of Aristotle subject to more than one interpretation, are the texts of Pascal in L’esprit géométrique that do have it clear. Then the non-definability of all the concepts makes that some of them should be introduced in some other way. But it is often considered that this introduction must in some way determine them and hence certain curious inventions.

To start with, the largely outspread use, since the beginning of the century till at least the thirties (14), of the expression “implicit definition” of the primitive concepts does not fit into a strict theory of definition. And it has been often considered by its critics (15). as only a way of wrongly saying something different in the suitable sense – determination – but that would not fit at all in that theory.

It has been done and the notion of implicit definition dates back to J. D. Gergonne (1818). But it is not in the least applicable to “definition” by means of axioms. I believe to have proved a long time ago (Otero, 1970) (Torretti 1976 takes later the same demonstration) that the interpretation of numerous historiographers -among which someone as remarkable as Enriques, particularly in his history of Logic- applying the notion of Gergonne to the axiomatic systems is wrong. If it were correct, the notion would have covered four successive stages: the innovation of Gergonne, its generalised use referred to axiomatic systems, a wide discredit of the last and its very special “rehabilitation” by Quine. (1964)

Among other reasons the condition of eliminability that stands, even for the implicit definitions, in Gergonne’s broad theory of definition, would turn out to be inapplicable to the “definitions” of the primitive concepts by axiomatic systems.

Peano ponders the function carried out by the axioms in relation to the primitive concepts but, as he demands more strictness for the definitions, he does not use the term, so popular then, of implicit definition.

The primitive statements determine, or if you like “define” the primitive ideas, they behave to a certain extent as definitions. The primitive ideas are explained in ordinary language, the postulates act as definitions but do not have their form, the primitive concepts are the system that satisfies these postulates. These twists round a function, which requires to be characterised, reveal the variety of formulations of an elusive notion.

Pieri uses the expression “implicit definition” but not until 1900, beyond the period we are referring to. Before that, he moves more carefully. Hilbert does not use it in the Grundlagen and he does – also beyond that period – only towards 1902 in his first letter in reply to Frege (16), who does use it critically. And Pasch also does it in later writings to his classic work and beyond the referred period, too.

What is then the Aufklärung? We have remarked till now what it is not. We have pointed that in Hilbert, as well as in previous authors, it carries out a function that turns out to be obstinately elusive. We are not going to move forward much more here but everything seems to lead to make us think that it is a boundary mark of a system of primitive symbols, without any precise meaning, without content, a relational mark on behalf of the set of axioms. In interpreted systems it would be a system of entities that satisfy a certain axiomatic body. Tables, chairs and beer mugs – with all the difficulties that this can carry to common sense – would constitute a valid interpretation. These three amusing terms would then be relationally bounded by the axioms, and that is enough.

There is no way to know if Hilbert’s use of the expression Aufklärung in the Grundlagen has had a definite purpose, as for instance to bring about a limited clarification, or perhaps it has been the result of the somehow subtle difficulty of Hilbert himself to explain the function he attributes to this bare little word in the text following.

Let’s jump back now to a single case and refer briefly to the process following this case. Although we are not going to refer to the axiomatic structure, since by 1820-30 there is not one in the modern sense; we are going to consider anyway, the primitive concepts, in the broad sense, from which the rest of the concepts arise by definitions.

We are going to consider only the duality in the first development of Projective Geometry, beyond the introduction of elements in the infinite – improper points and straight lines -, beyond the special properties of Pascal’s hexagon or of meaningful theorems to that development.

The introduction, in a primitive but sufficiently useful fashion, of the principle of duality, in the 1820s, is exemplary to our topic. We are interested to specify its function in the development of geometry (17).

The fundamental propositions of this geometry admit the exchange of the expressions “point” and “straight line” in plane surface, and “point” and “plane surface” in space. These laws of conceptual symmetry are introduced in all the geometry of the plane surface and space, respectively. The duality appears in Poncelet's study of the polarity of the conicals (poles and polars mediate in the duality) while in Gergonne (18) appears as a general principle that establishes a relation between theorems (if in the Plane Projective Geometry the terms “point” and “straight line” are substituted respectively and in every instance by “straight line” and “point”, we obtain the so called dual theorems to the original ones and reciprocally) (19). Poncelet’s duality is particularised, in Gergonne generalised. On the other hand, duality does not appear there as being subject to demonstration.

The dual presentation itself introduces a new way of understanding geometry, and as a result, explaining that principle in a new way turns out unnecessary. Given the form of presentation in two columns, geometry can be conceived as build by a column alpha – o by another beta -, or rather, from a more reasonable point of view, by the complete presentation that contains both columns at the same time. In the first case, we deal with two “parallel” deductive systems in which “point” and “straight line” are determined by the statements that contain them.

Yet, we can wonder which the objects of these dual geometries, or of this global geometry, are and how they are susceptible to an interpretation, for instance, a physical one. Although the idea of magnitude – for instance length of a segment – has been eliminated (20), which is already a meaningful step as the only interest is exactly the projective properties, we can wonder if this geometry could be interpreted in such a way that the domain consisted of physical elements – positions or trajectories -, as the “primitive” concepts of the Euclidean and analytic geometry could be interpreted. Furthermore, we can wonder what things the expressions “point” and “straight line” in this particular use denote. This is what emerges as a meaningful innovation.

For instance, we can ask ourselves if both columns, read alternatively, constitute Projective Geometry. They are two isomorphic deductive systems, with the same structure, whose variables – let’s call them in this neutral way – take the value of straight lines and points. In the other interpretation it could be made out that geometry has a single structure with variables (alpha and beta) to be substituted adequately. In this way geometry is this structure, obviously supposing appropriate (or more explicit) rules for the substitution of variables. Hence, the variables obviously do not possess a fixed designation, that is, determined values, for instance Euclidean or others.

The leap produced by the physical Euclidean geometry, and even analytic, is noticeable, and this does no longer constitute a way of construing geometry itself, but a new way of conceiving geometry and its objects. The structure introduced by a set of properties (axioms and theorems no matter how loosely conceived compared to the modern axiomatic) overthrows the previously basic idea of intrinsic properties of points and straight lines.

Therefore, there is a clear shift concerning the substance of geometry. This shift has been traditionally attributed to later times and different to the case of the Projective Geometry. Nonetheless, there is no doubt that it already appears in it. The fashion in which deductive constructions are presented is a revolutionary step and not just another improvement or a development of pre-existing geometrical forms. That is why I insist, the distinctly expressed dual presentation makes explicit a new way of facing geometrical properties. The specificity introduced by duality turns it into something more than just the discovery of theorems. At any rate, the duality provides (1) unknown dual properties of other properties already known, and (2) a previously unknown liaison between known properties; for instance, in the case of famous theorems on hexagons.

The time to which I am referring to, portrays a shift that will not remain without consequences, although it was not evident then. As the duality is applied, its use is adjusted without any essential change. Möbius expresses himself in the following way( (21):

“Out of all propositions by which a system of points and straight lines arbitrary chosen in the plane surface, by subsequent unions and intersections, other points and straight lines are deduced, out of which three of the former lie on a straight line, or three of the latter are in a point, other can be deduced in which the points are interchanged with straight lines and “lie on a straight line” with “be in a point”.” (Möbius, 1885)

The subsequent versions introduce some changes. As an example we include von Staudt’s version:

“The first propositions of geometry allow to feel certain reciprocity or duality law by which the point and the plane surface are one in front of the other in space, and every proposition in which no distinction is made between proper and improper elements, has a complementary one that results from the first by exchanging point and plane surface ... Two such propositions are commonly placed one next to the other as two aspects of the same proposition” (von Staudt, 1847).

“To each system S it can be assigned a dual reciprocal system S’. A pair of collinear forms in S corresponds to every pair F and F’ of collinear forms. To each element of S joined for the first two forms, corresponds an element in S’ that is joined to the latter two” (von Staudt, 1847, taken from Pieri’s translation, 1889)

Finally Clebsch, in his Lessons on geometry, by conceiving the conical forms either as a set of points or by their tangents, says:

“Certain valid relations for sets of points can be driven to line figures”.

4. Some more: to be almost in the condition needed to switch off the silly box.

The enunciation and application of the duality principle imply throughout the 19th century, and even before Pasch (1882) the consequences I have remarked. But from the beginning (since the end of the 1820s) what we have called variables, the primitive concepts in the sense mentioned, are open in a way not conceived before. Points, straight lines and projective plane surfaces are limited in their meanings - although they do not possess fixed meanings – by the propositions to which they belong.

Certain logic structure, certain abstract character of geometry, the relational character of the entities handled as the aforementioned, have been performing for a long time; since much earlier than what is commonly admitted. Although there is still a long way to go to reach the possibility that tables, chairs and beer mugs might be imaginable substitutes for point, straight line and plane surface variables, there was something going on that should be acknowledged.

On the other hand, Hilbert’s dictum took long to be moulded in writing in the Grundlagen, in its consequences, and in those years between dictum and written text (or till his lectures shaping the book) the Italians, particularly Pieri, assumed modern geometry with beard and all.

Two fundamental contributions can be imputed to Hilbert in his Grundlagen:

The consideration and detailed and far reaching elaboration of a broad set of topics already entirely meta-mathematical as the independence and consistency of the axiomatic systems of geometry. Although some of them have antecedents, Hilbert’s approach ensues a decisive contribution; and having assembled the contributions of a whole century of research on the foundation of geometry; not only giving it an excellent structure but also in such a way that it started, beyond all question, to provide theoretical output. The set of results, and above all the geometric grounds worked on during almost a century through stages so well known I am not going to mention, is present in the Grundlagen.

However, the way he understands the primitive concepts dates back to the development already present in different stages of formulation in Pasch and the Italians, particularly in Pieri.

We insist that, from 1882 to 1899, we deal only – though not of little account– with the formulation of a new way of understanding primitive concepts because, as we have seen, the use of the primitive concepts in this “new fashion”, although not conceived explicitly yet, dates back to, at least, the duality principle and its use which have already possessed since the 1820s extremely meaningful theoretical revenues. Although we have taken only one case of the use of the primitive concepts, and, in addition, we have presented it briefly, it is sufficient to show the efficiency of things as tables, chairs and beer mugs instead of point, straight line and plane surface. All the same, these three items suggested amusingly by Hilbert, represent for him interpretations of undetermined entities that perform a new function, in this new way of conceiving primitive concepts of geometry used long before.

1 Mehrtens... can help as a first guide of the subject with an extensive valid bibliography.

2 Sui postulati fondamentali della geometria projettiva in uno spazio lineare a un numero qualunque di dimensioni”, Giornale di Matematiche, V. 30, 1891.

3. See Blumenthal, O. (1935) In Hilbert, D., Gesammelte Abhandlungen, Springer, Berlin, v. 3, pages 402-3.

4 Pasch (1882) Freudenthal remarks, not groundlessly, the curious fact that neither Klein nor Poincaré, although they repeatedly talk about axioms, present cases of them.

5. This procedure is, to a certain extent; comparable to the procedure used by Hilbert for his finitism.

6 See Peano on Grassman (1888), who includes his first version of his logic.

8 Pieri does not use Peano’s notation “etc” in many of his writings. He had sent one of his papers to a first category German journal for its publication, but it had been rejected for using Peano’s notation (!). Since then, Pieri apologetically uses a “simple language”. As a matter of fact, he does not publish in Germany

9 For Pieri, the primitive ideas are forms analogous to raw materials for industry; Gergonne established a parallelism between the material for the construction of ideas to the raw materials for the industry. Pieri quotes antecedents extensively, while Hilbert does not.

10 Wussing, H. The genesis of the abstract group concept, Cambridge (MA), MIT, 1984.

11 Wussing, H. Lecciones de historia de las matemáticas, Madrid, Siglo XXI, 1998.

12 According to Gardiès (1997) there would be fourteen primitive terms in Hilbert (1899), according to Hilbert, only eight.

13 Hilbert uses the expression “implicit definition” for the first time in 1902 in a letter to Frege; is it just an allowance to fashion or something more significant?

17 We will not tend to account for the outspread controversies on the precedence of those who used it.

18 Otero, M.H., “Structure déductive et ontologie des théories; un cas de protomathématique: La dualité dans la géométrie projective du début du XIXème siècle”, Galileo, Second Age, number 15, 1997.

20 It will be almost complete in Staudt; but certain imperfections endured; for instance, not taking into account the subject of continuity.

21 We have followed the broad outlines of L. Nuvoli (1960) on this procedure.

A CLASSIC HISTORIOGRAPHIC TEXT: ANDREY N. KOLMOGOROV’S ARTICLE “MATHEMATICS” PRESENTED IN THE 1936 SOVIET ENCYCLOPAEDIA. [1]

Introduction by Mario H. Otero

Preliminary words about the text

In the book “Writing the history of mathematics: its historical development” (2002) on the historiography of mathematics, Cristoph Scriba and Joseph W. Dauben [2] state:

One final work of note from the pre-Wold War II period was written by Andrei Kolmogorov (1903-1987, B), one of the leading Russia’s mathematicians. His article on mathematics for the first edition of the Bol’shaya Sovetskaya Entsiklopedja (Great Soviet Encyclopaedia) (1936) contained a well known periodization of the history of mathematics with very concise descriptions of each period. Kolmogorov divided the history of mathematics into four periods: the birth of mathematics, (6^th to 5^th century B. C.); the period of elementary mathematics (up to the 16^th century); the establishment of mathematics of variables (to the middle of 19^th century); and the period of modern mathematics. This periodization has been widely discussed by historians of mathematics (see for example Youshkevich 1994) and became universally adopted in Soviet historiography. During all of his creative life Kolmogorov was actively interested in history of mathematics. As a result, he wrote several remarkable works (for example on Newton’s researches (Kolmogorov 1946)). In 1978-79 together with Youskevich he edited three volumes on the history of mathematics of the 19^th century (Kolmogorov/Youshkevich 1978-1987) He also devoted considerable energy to improving mathematical education in secondary schools, and was especially active in the reform of mathematical education (Petrova 1996)” (op. cit., p. 186-187).

This long paragraph suffices to highlight the significance of the text we are presenting.

Incidentally (but not irrelevantly) we should say that some of the greatest Russian mathematicians have showed a great love for the history of mathematics, for the philosophy of mathematics and … for history itself. Kolmogorov and Arnol’d[3] have proved this repeatedly and it can be claimed assertively that the knowledge of history of mathematics is an efficient tool for the learning of mathematics itself.

“This century’s mathematicians were used to finding their names in relation to many different theories, always scoring significant contributions. The theory of Trigonometric Series, Measure, Set theory, Constructive Logic, Topology, Approximation theory, Probability theory, Stochastic Processes theory, Information theory, Mathematical Statistics, Dynamic Systems, Finite Automatons, Algorithmic theory, Mathematical Linguistics, Turbulence theory, Celestial Mechanics, Differential Equations, Hilbert’s Thirteenth problem, Ballistics and the application of mathematics to problems of biology, geology, metal crystallizations, poetic creation by studying mathematical linguistics and many others”.

Kolmogorov’s work deals with these subjects and the interrelations among them. From his article Une série de Fourier-Lebesgue divergente presque partout, published in 1923 in Fundamenta Mathematicae to the book Probability theory and mathematical statistics (Nauka, Moscow, 1986) there is a cascade of extremely valuable published texts.

It would not be an exaggeration to claim that if all historical records of mathematics in the twentieth century were to be lost, but for the large and profound work of Kolmogorov, we would still be able to have a clear view of the development of these disciplines during that century[4]^.

Jan von Plato (1994) has pointed out that the constitution of the mathematical theory of probability – an extremely important contribution – did not emerge spontaneously and wholly from Kolmogorov’s head. His book is an exceptional text in two senses: it was the first book to cover the contemporary period – of the mathematical theory of Probability – and it takes into accounts especially the lapse from Lebesgue’s Measure theory and Borel’s Enumerable Probabilities to Kolmogorov. The path towards modern probability – Hilbert not being alien to this process – is followed by considerations of probability in statistical physics, Quantic Mechanics, von Mises’s Frequency Probability to Kolmogorov’s probability based on Measure Theory. The period covering 1919 to 1933 is particularly clarifying. He compares two contributions of de Finetti’s Subjective Probability and in an appendix, the surprising precursor Nicola Oresme.

2. Which is the state of historiography of science in 1936 when Kolmogorov published his article? I will just note two things.

“Until twenty five years ago /until circa 1954/, only half a dozen people were employed in the United States and Canada as historians of science. Three or four times that number published occasionally in the field or attended meetings of the History of Science Society. But their primary association was with other academic fields, mostly sciences, or else they had been drawn to history of science by a vocational or a vocational concern with book collecting. The last quarter century has seen that situation transformed. Though the field is still very small, its professional practitioners number two or three hundred, an increase by a factor of close to fifty. Amateurs have disappeared from its meetings and mostly from its journals as well, the latter having meanwhile more than doubled in number. Many of the newer journals are devoted to specialized subject matters”. (p.121)

We must not forget that between 1954 and 1979 the Cold War was in its heyday. And in addition, Europe and other places were not alien to history of science as the Kuhn-USA self-centeredness seems to believe. Aldo Mieli – not to mention Rodolfo Mondolfo in a bordering field – and others in the “savage” regions of Rio de la Plata were making their contributions. And, obviously, in Europe and in other continents there was already a production of serious historiography of mathematics at least since the late nineteenth century. Let us remember Tannery just to mention one of these authors.

2.1. It is said that Boris Hessen’s text (1931) on the socio-economic conditions of Newton’s mechanics is well known. However, an extremely externalist character is attributed to it. Pablo Huerga (2005) has established that Hessen is far from being that. In addition, his 1999 book presents a collection of enlightening previous texts.

What is in fact known is the effect produced by the Soviet delegation in the Congress of History of Science which took place in London in 1931.[5] In that occasion it was perceived again that not only philosophy is the class struggle in theory but also history of science is. It could be perceived that Hessen’s presentation and other lectures of the same delegation produced such an effect and also that they promoted in Great Britain a mostly favourable effect – Needham’s work is an example of this – but also an adverse effect there and in other places. Moreover, those events constituted a significant incentive for the development of the history of science as a profession, even though Kuhn did not want to remember this episode prior to the dates of his account[6]^.

Therefore, 1931 is a milestone to take into account; it must not be neglected either intentionally or unintentionally. We will see later that 1936 becomes the milestone but probable 1931 – or its preparation, the research work prior to this date – is not alien to the fact that an outstanding mathematician, with renowned studies of history, had dedicated himself to make a significant contribution to the history of mathematics. Although the title of his work is Mathematics its contents refer to a period and especially a major contribution to the history of this discipline.

It may be surprising that during the 1930s there were works of this magnitude in the Soviet Union. However, we must remember that the highest political censor of scientific work – and of others probably – wrote crucial comments on the margins of the works examined which meant recognition of the objective character of mathematics.[7]

I. Definition of the object of mathematics, connection to other sciences and techniques.

2. A period of elemental mathematics, subdivided in the Hellenistic and Roman period, China, India, Central Asia and Middle East, Western Europe until the 16th century, Russia until the 18th century.

3. A period corresponding to the creation of variable magnitude mathematics, subdivided in a general section, 17th century and 18th century.

2. History of mathematics in the 19th century and the beginning of the 20th century, subdivided in The First Half of the 19th century, The Late 19th century and the Beginning of the 20th century.

Thus, it is evident that historical consideration covers most of the text, the systematic part is covered – without detriment to the systematic pieces in the historical part – in part I, III-1 and in the Conclusion. These systematic portions explain certain recurrence in the text.

3.1 In relation to Engel’s claim that research in pure mathematics requires separating forms from content, Kolmogorov states the richness of the content that sciences and techniques need to study – growingly – and to which mathematics are inextricably integrated.

Engels clearly follows the conception of mathematics developed from Humboldt in 1810 – and all the German Mathematics as our Eduardo García de Zúñiga know in 1903 to 1905 in Berlin-Charlottengurg[8] – holding a kind of neo-humanism for which mathematics and philosophy are united, which has been exposed and criticised brilliantly by Lewis Pyenson (1983).

Kolmogorov, on the other hand, adheres in 1936, avant la lettre, to that critique, which is not negligible.

That the field of application of mathematics is not limited, that in any case the models do not exhaust reality – confer Lenin, Materialism and Empirio-Criticism – nor does the concretion of real phenomena, that logic attempts to isolate these from their form, are all elements to take into consideration to show that only a dialectic analysis can take them into account and consider particularly their depth.

The examples presented by Kolmogorov show this. Beyond the valid criticism that can be objected to dialectic, we must admit that the neo-humanist distinction of pure form, or of pure mathematics, does not hold in 1936, even though the “pure” mathematicians of that time regretted it. Kolmogorov cleverly detects this.

When our author considers the problem of the relation of the function of mathematics in the relation between sciences and reality, he specifically refers to the application of mathematics to these sciences. We can mention incidentally that Mario Bunge holds that mathematics is not applied directly to reality but to the sciences which study it.

He presents the topic of the oscillation between the empirical case studied and its mathematical schematization, the recurrence to the directly empirical and the opposed resource to too abstract models. Thus, when Celestial Mechanics conceives bodies as point mass – even though this is not a frequent case – it yields, for example, the case of the Moon.

Kolmogorov chooses the examples very well and he multiplies them. Just to take one example, we have to say that his treatment – mostly in physics– of the relation between the macroscopic Theory of Diffusion and the corresponding Statistics Theory is analysed profoundly and gracefully. There is another oscillation in biology and in the social and human between the application of mathematics in relation to physics and an ulterior development as research advances, but then

...in the final analysis of the social phenomena the moments of qualitative uniqueness of each stage acquire a position so dominant that the mathematical method recedes again (I).

The direct demands of the technical practices gave way to elemental mathematics – as Herodotus classically described – other more advanced practices have depended on either the already created mathematical theories or on the creation of new ones. As it will result clear, Kolmogorov acknowledges a phenomenon that has increased in our times – as the repeated example of the strings has shown – when physicists produce mathematics which requires the development of their theories.

Under the heading The extension of the object of mathematics Kolmogorov considers the change in the objects of mathematics. The accumulation of mathematical materials required their logical analysis. The collection of new findings, listed and also analyzed in the text, rendered it necessary. Let us recall that Lagrange prompted a new interpretation of infinitesimal calculus so that it avoided certain ontological problems which had appeared.

Apart form the connection of mathematics with the development of natural sciences, beyond the answers to direct requirements of other sciences, internal needs emerge. It could be said that these needs derived gradually to a purist conception of mathematics that neo-humanism adopted.

And “… it turned to much generalized viewpoints”. According to Kolmogorov the structure of the crystals and, later, Quantum Physics took Group Theory as a tool. Likewise, Vectorial and Tensorial Calculi emerged as tools of mechanics and physics.

Therefore, the expansion of the object of mathematics emerged from external and internal needs, leading to a different way of conceiving expressions such as “spatial form” and “quantitative relations” which become a significant object of attention. New algebras and Non-Euclidean geometries arise. These are, according to Kolmogorov, among the most important discoveries of the 19th century. In our opinion, we think we have proved that place should be granted to Projective Geometry with its duality principle, not only because it appeared earlier, but also due to its theoretical significance[9].

As a consequence of that process there is a need to rigorously justify the change in mathematics. Our author states the delay of a rigorous, mathematical presentation of the probability calculus which was one of the main contributions. Perhaps it was this need of rigorous justification of the “illegitimate” that led Weisrstras, and much later Robinson, to their non-standard logic.

The axiomatic method – exposed by Kolmogorov in a modern fashion, in Hilbert’s Grundlagen der Geometrie fashion – answered to wider needs of foundation, which, in turn, gave way to important developments in mathematical logic applied to mathematics, and to Kurt Gödel’s central result to which our author refers as follows: “…no one deductive theory can exhaust the variety of problems of Number Theory. More precisely, once in the limits of the Natural Number Theory, it is possible to formulate the sequence of problems … of this kind, that for any deductive theory it would be insoluble within the limits of that theory”. And from that, Kolmogorov draws important consequences:

Thus the concept of mathematical theory discovered in the sense of a theory included in a theoretical-systematic axiomatic system, is substantially broader than the logical concept of deductive theory.

This consequence is enough, it is not necessary to point out, as a result of logical rigour, a general theory of algorithms and of “algorithmic solubility” of mathematical problems, which is what Kolmogorov considers in Part III of his work.

The Conclusion does not refer to the achievements of all the work, but to the contemporary situation, in the years 1936 to 1954. Essentially, it points out the diversification, the emergence of new branches of mathematics and he starts by listing the theory of Algorithms, Information theory, Game theory, Operations study… Perhaps this expression covers the research on operations developed simultaneously in the Soviet Union and in the United States[10]and he continues with a copious and explosive listing.

The work strongly underlines pre-Soviet and Soviet mathematical production – each with its own characteristics – and the production abroad, stating how the institutionalization of mathematical societies in the 17th and 18th centuries, of congresses and international congresses, contributed to the development of contemporary mathematics. Perhaps the omission to mention periodical mathematical publications is due to the fact that it is taken for granted.

In this presentation we have opted to remark the philosophical lines of the historiographic construction, more than its historical part, mainly for two reasons: (1) the interpretative lines, let us say philosophical, of his work presuppose that historiographic construction; and (2) because it is best to suggest the reading of the text presented.

They share, to a great extent, the same basic philosophy of mathematics. He states unmistakably:

“Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap.

The Jacobi identity (which forces the heights of a triangle to cross at one point) is an experimental fact in the same way as that the Earth is round (that is, homeomorphic to a ball). But it can be discovered with less expense.

In the middle of the twentieth century it was attempted to divide physics and mathematics.

The consequence turned to be catastrophic. Whole generations of mathematicians grew up without knowing half of their science and, of course in total ignorance of any other sciences. They first began teaching their ugly scholastic pseudo-mathematics to their students, then to school children (forgetting Hardy’s warning that ugly mathematics has no permanent place under the sun)” (On teaching mathematics, Paris, 1997).

The first paragraph of this text only contradicts apparently with Arnol’d’s own statements such as:

“In mathematics we always encounter mysterious analogies…” (Arnol’d, 2000)

“Geometry is nothing more than a branch of physics; the geometrical truths are not essentially different than physical ones in any aspect and are established in the same way”

“Kolmogorov in return expressed to Hilbert his own worries that our culture would probably not survive for such a long period: the united burocrats of all countries would soon be able to stop all kind of creativity making further mathematical discoveries impossible, as are geographical discoveries today”

[1] The text we present here is the 1936 one with later additions presumably from 1954. Pablo Huerga Melcón has done the valuable translation from Russian and a small team from Oviedo has frankly lessen our work, or better, has performed the main work, as we do not know any other translation to a “western” language. In that sense, they are the main authors of this work.

[2] Beno Eckmann has said in his “Kolmogorov and contemporary mathematics”, EMS, 2003: “The better I got to know Kolmogorov the more I realized that his cultural universality went far beyond mathematics, into logic and foundations, into arts, poetry, history and education. His human and humanistic universality enabled him to be an extraordinary teacher”.

[3] See, for example, Arnold 2000, which an earnest and at the same time delightful text.

[4] There are many studies on his work which would exceed our purposes here. The bibliography only contains some titles, only a few, which we include for special reasons.

[5] There are many descriptions, among them Otero, Mario. Historia de la ciencia e ideología, in Ideología y ciencias sociales, México, UNAM (Coordinación de Humanidades), 1979.

[6] Steve Fuller 2000 gave an extensive account of the political conditions in which Kuhn’s work was developed.

[8] Scriba, Christoph & Dauben, Joseph W. 2002, text on Eduardo García de Zúñiga

[9] Otero, 1983 and 1997.

[10] The study presented by Sonia Brentjes about these parallel developments is especially interesting. We must mention that this outstanding researcher of the University of Leipzig in RDA, won, after the Re-Unification of Germany the academic audition to substitute Christoph Scriba in the Hamburg University when he retired, and later on it was decided that the budget was not enough for her to fill the position.

[11] In 1956, Kolmogorov reached the conclusion that each continuous function, of any number of variables, can be represented as the composition of continuous functions of three variables. He reduced the 13th problem to a problem of representation of functions on three-dimensional trees. Later, that problem was solved by Arnol’d in 1957 under the direction of Kolmogorov, giving a negative answer to Hilbert’s conjecture: every continuous variable can be represented as the composition of continuous functions of two variables (Rolando Rebolledo 1993).

ARNOL’D, V. Polymathematics: is mathematics a single science or a set of arts?

ARNOL’D, V. et al.(2000) Mathematics: frontiers and perspectives, International Mathematical Union-American Mathematical Society /conferencia/.

ARNOL’D, V. (2000) “On A. N. Kolmogorov”, in A. N. SHIRYAEV et al. Kolmogorov in perspective. London, London Mathematical Society.

BRENTIES, S. (1985) “Zur Herausbildung der linearen Optimierung”, in L. V. KANTOROVIC et al. Ökonomie und Optimierung. Berlin, Akademie.

FULLER, S. (2000) Thomas Kuhn; a philosophical history for our times. Chicago, University of Chicago.

HACKING, I. (1975) The emergence of probability. Cambridge, Cambridge University.

HACKING, I. (1992) “Statistical language, statistical truth and statistical reason: the self-authentification of a style of scientific reasoning”, in E. Mc MULLIN (ed.) The social dimensions of science, Notre Dame, Notre Dame University.

CHERONI, A. (ed.) (1988) Newton, el hombre y su época. Montevideo, EUBCA. /en prensa una versión francesa de Serge Guérout, Paris, Vuibert /.

FRÉCHET, M. (1925) “L’analyse générales et les ensembles abstrites”. Revue de Métaphysique et de Morale. v.32.

HESSEN, B. (1931) “Las raíces socioeconómicas de la mecánica de Newton”, in N. BUJARIN (ed.) Science at the crossroads. London.

HUERGA, P. (2005) “Raíces filosóficas de Boris Mihailovich Hessen; crítica al mito del externalismo del histórico informe presentado al congreso de Londres de 1931”. http://galileo.fcien.edu.uy, sección Textos G.

/traducción de Grundbegriffe der Wahrscheiblickkeitsrechnung, Berlin, 1933/

KOLMOGOROV, A. (2000) “Newton and contemporary mathematical thought”. A. N. SHIRYAEV et al. Kolmogorov in perspective. London, London Mathematical Society.

KOLMOGOROV, A. ANDREI. N. & YUSHKEVICH, A. P. (1996) Mathematics of the 19^th century. Basel, Birkhäuser.

KUHN, T. S. (1979) “History of science”. P. D. ASQUITH & H. E. KYBURG, Current research in philosophy of science. East Lansing MI, Philosophy of Science Association,.

LEVY, P. (1925) “La probabilité dans des ensembles abstraites”. Revue de Métaphysique et de Morale, v. 32.

OTERO, M. H. (1979) “Historia de la ciencia e ideología” in M. H. OTERO (ed.) Ideología y ciencias sociales. México DF, Universidad Nacional Autónoma de México.

OTERO, M. H. (1983) “El escándalo de los beocios: historia de un caso de adulteración filosófica de la historia de la geometría”. Llull, v.6.

OTERO, M. H (1995) “Sobre un tema para nada filológico; de cómo una vieja discusión sigue; aporte de nuevos documentos para elucidar un punto de interés sobre don Pepe y el carácter de clase de las matemáticas”. Llull, v. 34. Hoja informativa de Galileo, 1995.

OTERO, M. H (2003) Sobre ciertos avatares de las llamadas matemáticas puras. Zaragoza, Universidad de Zaragoza.

von PLATO, J. (1994) Creating modern probability; its mathematics, physics, and philosophy in historical perspective. Cambridge, Cambridge University.

PYENSON, L. (1983) Neohumanism and the persistence of pure mathematics in the Wilhelmian Germany. Philadelphia, American Philosophical Society.

REBOLLEDO, R. (1996) Kolmogorov y su época: el pensamiento y la acción. Leeds, University of Leeds.

ROSENFELD, B. A. (1988) A history of non-euclidean geometry; evolution of the concept of a geometric space. New York, Springer, New York.

SCHAPPACHER, N. (2004) “Lo político en matemáticas”. La Gaceta de la RSME, v.8.

SCRIBA, CHRISTOPH & DAUBEN, J. W. (2002) Writing the history of mathematics: his historical development. Basel, Birkhäuser.

SHIRYAEV, A. N. et al. (2000) Kolmogorov in perspective. London, London Mathematical Society.

VON WRIGHT, G. H. (1957) The logical problem of induction. New York, MacMillan.

VON WRIGHT, G. H. (1960) A treatise on induction and probability. Paterson NJ, Littlefield Adams.

YUSHKIEVICH, A., KOLMOGOROV, A. N. (1994) “O sushchnosti matematiki I periodizacii ee istorii”. MIT, v.35.

A CHOCOLATE COIN? ON A WIDESPREAD IDEOLOGY INTERVENING IN THE HISTORIOGRAPHY OF MATHEMATICS AS WELL AS IN MANY OTHER NON-TRIVIAL DISCOURSES .

“To whom ever explains to me by why the social background of the small German courts of the 18th century where Gauss lived would inevitably lead him to occupy himself with the construction of the regular polygon of seventeen sides, well, I would give a chocolate coin”.

ii. his ideological intervention in the generation of the historiography of mathematics.

iii. how this would lead to a radical presentism that would eliminate the best part of the history of mathematics, and

iv. how in this way the history of the Uruguayan school of mathematics would also be altered.

This paper is a second and significantly modified version of the original text.

Even in the case of the reception and diffusion of mathematical knowledge in any country, the stages of a particular national development can obviously span a range that does not necessarily have to coincide with the development of Western thought at that time. In any case, it is possible to make certain comparisons. The introduction of mathematics and later of modern, professional mathematics in the U.S., Spain, Japan, Argentina or Mexico is not just out of step with the European case but differs significantly from it.

In each case, there is a combination of circumstances that depends on the global development of societies, levels of education, degree of industrialization, within a complex which we will not exhaust here.

This notwithstanding, it is usually held that the history of mathematics should begin with the introduction of modern, professional mathematics in the contemporary sense of the expression. In this view, some events are considered irrelevant or at most pre-historic, as is the case of the introduction of the metric decimal system and its teaching, not to speak of the different modalities of this introduction, which frequently turn out to be significant in one sense or another.

Failing to acknowledge that applied mathematics is mainly the outpost of mathematical knowledge in new countries and, again, that the modalities of its use are not trivial – even for mathematics as a whole, or for the original appearance of this discipline in Europe-- is a serious mistake.

All of thi -and much more- leads scholars to construct impoverished histories of mathematics, on the ideological basis that whatever is not professional mathematics al uso nostro (that of present-day professional mathematicians) is not mathematics or, more frequently to condemn serious histories that do not follow such a peregrine conception. This leads to a radical cultural schism and to the distortion not only of what actually happened in those new countries which are the object of study but also of the development of mathematical thinking almost at its European beginnings.

But doing this means that a certain use of the expression “pure mathematics” has been hypostasized through a series of arbitrary breaks, each different from the next but still ideologically identified. A foundational myth is actually operating as elimination criterion, selecting in historiography what is desirable – as pure—and what is undesirable – as impure. An old myth, and no more than a myth.

Far from being trivial, the ideologization of the history of mathematics is of present interest and not just for the construction of a new local historiography. Besides the arguments that contribute to this ideologization – and we have pointed out a central one, that of the presumed fecundity of pure, isolated mathematics, which would suddenly and unexplainably produce applications – appear as fallacious, product of an obliteration of real complexities by means of an extremely doubtful procedure of purification.

The historiographer must therefore strive to avoid the surreptitious filtering of ideologies for the resulting history of mathematics -even if and especially because it is a social science- to be truly scientific.

To conclude this section, let us consider some aspects of a concrete process of diffusion of mathematical knowledge. It is usually held that 1) its reception results from unaltered transmissions unaffected by the climate in which knowledge is received, and 2) what is fundamental is determining when and how a band of modernity appears in the recipient country or region, whereas the previous periods are considered irrelevant. These elements acquire the determining carácter of how the little historiography that is done is carried out and particularly the kind of historiography that is supposed to be carried out.

What has been called Uruguayan school of mathematics developed in an extremely fertile way after the 1960s. At that time, a small group of mathematicians started to publish in international refereed journals, which was an unexpected thrust given the dimensions of the productive community and the country itself. The creation of the Institute of Mathematics and Statistics, in 1942, would mean the beginning of Uruguayan professional mathematics. The process that prepared that entrance into the international community can be traced back at best to 1927.

1927-1942, relatively basic stages of mathematical education and production.

mathematicians are expelled from University and all subscriptions to mathematical journals are cancelled.

1985 up to now, the community of mathematicians is rebuild and strongly expanded with the return to the country of trained mathematicians and implementation of a Doctorate program.

In the 30s there was a serious but frustrated attempt to establish at University regular mathematical studies. A Certificate in Mathematics program was implemented -following the model of French universities. This program was quite comprehensive - it even had a History of Science course- but for several reasons it was cancelled.

The appalling curriculum for the degree. during the dictatorship was modified. Thanks mainly to these professors- that taught before the dictatorship, the M. A. and Ph.D. in Mathematics are implemented now.

The branch of mathematics developed was basically what we have called “pure”, and it persisted until very recently. This fact is not trivial, and it should not be attributed only to the prestige of this branch of mathematics. The prevailing trends on in international scale were not significantly different, even though to a lesser extent.

Not only was José Pedro Varela -as is commonly acknowledged- the champion of compulsory, non-denominational and compulsory primary education, but he is also to be credited with an extremely modern scientific policy for the country.

“The educational systems of Europe have been conceived, prepared, with the specific and fundamental aim of preserving and conserving the existing order of things”.

Three dates can be mentioned as the beginning of the reception of diverse external influences: in 1903 Eduardo García de Zúñiga attends Berlin Charlottenburg; in 1927 Rafael Laguardia attends the Sorbonne, shortly before the Bourbak outburst (with the “Goursat Bible”) and the forties and fifties when Laguardia and José Luis Massera work at different North-American universities.

However, a not always explicit belief dates the existence of mathematics to the creation of the Institute (1942) and at most to the process that gave rise to it (starting in 1927). In this way, professional mathematics is confused with mathematics tout court. This kind of attitude in the context of the development of mathematical knowledge on a global scale would dismiss many of the developments previous to the 19th century or to Hilbert-99 while still keeping well-isolated poles -for instance the Euclidean Elements- something frankly absurd.

4. 1888-1903, from the foundation of the Department of Mathematics (actually, Engineering and Architecture)

6. 1915-1927, from the very modern Mathematics syllabi for University and College education

Periods 5 and 6 -from 1903 to 1915 and from 1915 to 1927- are already dominated by the introduction of pure mathematics, fundamentally after the reception of the German conception arising, as is known, under the influence of Neo-Humanism. But beyond the conception and the knowledge received for the first time from the rich 19th and 20th century sources, it has to do with the material base, in the shape of a wide bibliography and advanced syllabi, which we have described in earlier works.

What happened before that extraordinary advance towards modern, professional mathematics of the sixties? I think that suppressing periods 1 to 4 (before 1903) is the result of a naive acceptance of the elitist ideology described earlier. We will not lay aside these first periods but rather point out what underlies their denial.

Navigation, different kinds of measuring, and other applications, give rise to a teaching which at the beginning rarely goes beyond an elementary level but which towards the end of the 19th century reaches, several centuries too late, the mathematics of engineering in its traditional version of calculus – clearly with its own ideology.

But to understand that teaching at different levels, however backward in those days, and the application of more or less traditional but by not mean crude techniques, does not belong to the local history of the reception of mathematical ideas, amounts to suppressing the awareness of the needs of a young country undergoing a process of development that would not become clear until the beginning of this century. It amounts to thinking that the requirements for the production of goods and services do not demand the know-how of relatively simple but adequate mathematical knowledge.

Studies on the introduction of the decimal metric system in France and almost every European country have not turned out to be at all superfluous. Much less can this be said about the numerous manuals about the new measurement system in Uruguay, published around 1870. They gave place to a publication boom of sorts in Latin America, not only with regards to this topic, but also about other mathematical knowledge.

To provide a more detailed idea of this whole process during periods 1 to 4 would be in fact to produce the corresponding historiography, and a small group of researchers has undertaken this task.

The purpose of this section, has been to illustrate with this example the way in which the ideology underlying professional mathematics can lead to a self-complacent conception which at the same time suppresses elements which can be valuable even for the reception of modern mathematics itself (period 1903-1927).

It might be said that, on a general level and beyond local histories, this phenomenon does not take place or it does in a very small measure.

However, as we have remarked, beyond local and to some extent marginal histories, it is enough to browse a certain type of historiographic texts to notice that when the “dirty” origins of mathematical knowledge are not summarily suppressed, the scene is dominated by a purism that could only be worthy of a non-existing history. This is because any rational reconstruction is indebted to the real history of mathematics which only successive approaches will be able to provide.

2. Did Uruguayan mathematics have, in their origin and development, a well-defined ideological agenda?

The following paragraphs are only an outline of issues related to the question posed in the title. I believe that at a time when the relationship between science, technology and innovation are discussed in Uruguay in an apparently intense and profound manner, it might be interesting to take a look at what happened with the origins of the professional and professorial history to which the question refers.

We will use the term “mathematics” in plural, even though the mathematicians of the school of Montevideo prefer to use the singular, “la matemática”. We will do so for the reasons we have developed in the book Sobre ciertos avatares de las llamadas matemáticas puras [On certain avatars of the so-called pure mathematics) which do not need to be developed here.

Were perhaps the Marxists the first cultivators of this discipline among us? Two in fact were – not García de Zúñiga—but that ideology was not expressed in mathematics, not even as a moderate materialism such as Chandler Davis’ (1974, 1994). In any case, these mathematics were serious and rigorous, and they took part in the international community of mathematicians.

The international production was -in relation to modern mathematics- several decades and perhaps even a century old when a mathematician from Montevideo -no less than the author of the project for the Port of Montevideo- approached it, to our knowledge either in or before 1903.

Peressini (1999) addresses the very core of so-called pure mathematics. He wishes to distinguish mathematics from applications. According to this author, there is a strongly important way of regarding things: to say that mathematical applications do not involve more than the replacement of mathematical with physical terminology.

It has frequently been said that only after a theory has been developed would it be applied to real problems. This is not the case. Neither does it happen that the progress in pure mathematics be due only to developments in the use of mathematics in the other sciences. For Peressini both statements are wrong, and he remarks that

“…neither the pure theory nor the applied theory are in all cases epistemically prior” (ibid.).

“… not every mathematized scientific theory is also an application of a (pure) mathematical theory. There are mathematized scientific theories that do not bear the ‘applied’ relationship to any pure mathematical theory and so, strictly speaking, should not be considered applied mathematical theories […] In such cases in which the mathematized scientific theory is worked out first, and then only later, if ever, a pure mathematical theory is worked out, we have the inverse of the operation of application –call it abstraction” (Ibíd.).

However, as we were saying, Peressini does not ignore that pure mathematical theory is often applied within pure mathematics itself, of which he provides numerous, perhaps unnecessary, examples from the present. Finally, with regards to the issue we are dealing with here, he mentions historical cases which he divides into two kinds:

First type: late in history there are clear examples of the application of pure theories, since frequently pure theories appear much later. Euclidean geometry is clearly a physical geometry, insofar as it moves from a scientific mathematized theory to a pure theory. Hence it is not accurate to say that Kepler received a much earlier pure mathematics (Massera, 1986). When Newton developed calculus he did not so do in a pure way but rather in connection to things which existed, according to Newton himself, in nature.

An instance of the second type is when Einstein develops his general theory of relativity, appealing, for the effects of gravity, to the structural features of the curved space-time of Riemann’s geometry, through the tensorial calculus developed by Ricci and Levi-Civita, who are much earlier, and who had not been noticed by the physicists at that time. The same happened with the Galois’ set theory, much later applied to physical symmetries.

Finally, Peressini remarks that, albeit with certain rather minor shades, the distinction between pure and applied mathematics is a logical distinction, which in our view posits some consequential doubts.

A further characteristic of higher scientific institutions is that they never consider science as a perfectly solved problema, and consequently they continue researching; the opposite happens in schools, where only acquired and consecrated knowledge is taught and learned. (Humboldt, 1959);

This text was written in 1809-1810, and published only in 1896 – which is surprising, given its influence, and it deserves a close reading.

Everything the teacher says has to be presented by him in front of his listeners in its process of development; he may not narrate what he knows but has to reproduce his own cognition, the act of cognizing itself (Schleiermacher, cited by Stichweh 1994).

Instead of a long exposition on Neohumanism, we will offer a series of text with the intention of apprehending in a significant way how the term is understood.

The most important work about Neo-humanism is perhaps Lewis Pyenson’s book published in 1983 by the American Philosophical Society, even though Ferreirós’ Del neohumanismo al organicirmo: Gauss, Cantor y la matemática pura, which deals with a somewhat more limited topic, also deserves an attentive reading.

5.1 In the first sentence of the Introduction to this work, Pyenson tells us:

“Air and earth form an ant hill, veined by channels of traffic, raising storey upon storey”, Robert Musil noted about Viennese impression of American cities on the eve of first world war.

“Questions and answers click into each other like cogs of a machine” (Pyenson 1983).

According to Pyenson, a Friedrich Poske said that “The essence of the world is not captured by any formula”. Kart Heinrich von Stein and Poske belonged to a circle that also included Richard Wagner, Joseph Arthur Gobineau y el self-proclaimed anti-Semite Houston Chamberlain. Stein was obsessed with the identification of the Arian-Germanic personality in art, contrasting it with the materialist qualities of the Latin Semitic personality (Pyenson 1983).

These mathematicians had received in Germany an intensive education on the Greek classics. Penyson remarks:

“They /mathematicians/ argued that, in the scientific age, exact science were the pedagogical equal of ancient languages…. hastened to suggest how, without fundamentally changing its character,.. pure mathematics could be turned to solve problems in the real world. Classical revival are the signal reaction of a ruling elite in trouble…”

Fritz Ringer (1969) has emphasized that cultural activity in nineteenth century Germany was controlled by a learned meritocracy, a mandarin class to which any aspiring youth could in principle belong. The neohumanist Friedrich August Wolf saw mathematics as a school of thought.

“…the mathematical part of exact sciences did not pose a problem of assimilation, for mathematical manipulation, much like cooking chemical or collecting butterflies, was a technique that could be mastered. On the contrary, it was the implicit picture of the world firing the imagination of researchers that found so few receptive minds. While technology easily leaps over cultural boundaries, science remained cultural bound” (Pyenson, 1982).

“Among native aspirants to learned discourse the culture produced fantastic imitations of metropolitan practice” (Pyenson, ibid.).

In the same work Pyenson, as befits a careful anatomist, studies the relationship between external - dominant and colonized scientific groups. The German headquarters in La Plata at the beginning of the 20th century constitutes one of the many situations he carefully analyses.

Some places that would deserve this kind of attention today would be the Instituto Pasteur in Montevideo, one of the almost two dozen branches worldwide. It is therefore just one century after a macro-case of scientific imperialism, as Pyenson calls it, very similar to that studied by this author where

“…power would have been seen to reside in unintelligible reports forwarded to Germany” (Pyenson, ibid).

“The ideological roots of science lies elsewhere. Semiotics is still in its enfancy, but we may hazard that ideology enters scientific discourse at the level of prejudice and predilections that motivate and guide general direction of research…After Greek and Latin, mathematics form a third language in the secondary schools of central Europe” (Pyenson, ibid).

In a more recent work Pyenson (2002) rigorously analyses a thesis which might seem trivial, namely that strictly national science, local knowledge, no longer makes sense.

“Towards the end of the 19th century there were two schools in Germany: one which was adopting a mathematics more engineering-oriented, and another in which pure mathematics reigned, despite the pressures from ardorous discussion of the topic, almost always among Secondary teachers and University professors of the subject”.

From 1790 to 1850 German neohumanism, romanticism and idealism formulate the new idea of a “unity of teaching and research”, This idea presupposes at least six fundamentally new concepts and ideas:

It is a good characterization, which will have to be compared with the apparently similar assumptions of Uruguayan mathematics.

5.2 Del neohumanismo al organicismo…by José Ferreirós is a rigorous study and the remains of the title specifies the task: it is about no other than Gauss, Cantor and pure mathematics. And all that in two significant moments. It was hard to believe in the Romanticism of those two mathematicians.

In fact my most immediate aim will be to contribute to a better understanding of the Romantic period by means of a reflexivity… (it is) a period suffused with idealism

“… In the heyday of idealism there were very influential authors such as Fries and Herbart, who explicitly detached themselves from absolute idealism… later, there appear several tendencies that can be called “late Romanticism”, amongst them several reactions to materialism” (Ferrreirós 2003).

“Mathematicians were not in general receptive to the speculative ideas of the Natürphilosophie… The cultural phenomenon of the neo-humanist movement is fairly unknown among historians of philosophy” (ibid).

“A beautiful perspective (was) the quintessence of academic purism which characterized German professors of the last decades of the 19th century and the first of the 20th. These were also the years of Hitler’s rise to power, which should also remind us of the dangers of this aristocratic love of contemplation and its concomitant disdain for the stuff of daily life, politics, social problems”. (ibid.)

“The rise of pure mathematics in Germany was not the product of change, but yet another aspect of the new cultural and educational trends generated by the neohumanism of the late 18th century. … Mathematicians had to be equal to the Platonizing expectations, they had to prove that their science deserved the dignity of counting among the contemplative disciplines of the Faculty of Philosophy …. Neohumanism is a cultural trend which not only preceded absolute idealism, but also followed it…” (ibid).

It is quite meaningful that Ferreirós should take up Cantor’s speculative ideas in that

“…he fiercely criticises Haeckel’s attacks (the notorious evolutionist who formulates a ‘monist’ doctrine with materialistic undertones—to metaphysics and traditional religiousness”. (ibid.)

“…with regards to mathematical objects, in order to consider them legitímate and existing, it suffices for them to be well defined, and to form a logically consistent system (i.e., insofar as their immanent or ideal “reality”… The physical hypotheses that we have mentioned and their biological applications were left undeveloped and sterile”.

“The bourgeois University professors, educated in the religious tradition and the romantic cult of the spirit, and well established as “intellectual officials”

in the society of that time, spoke of thought as a “secretion of the brain… Cantor defended in 1883 a combination of “idealism and realism”,… Cantor wanted to be the Newton of organicism. He defended organicism and attacked mechanicism”.

“…philosophers such as Kant left a very deep trace in the conception of science. Their role was none other that redefine the ethos of science”.

“…we have found a new cultural, intellectual and educational tendency, as was neohumanism, daughter of the Enlightenment in a certain sense, but mother of Romanticism in another”.

5.3 Randall Collins (1998) has published a thick volumen where he works with very special techniques: the networks of intellectuals, and in particular of philosophers. He attempt to aim at the great (and not so great) historical processes, offering also the relevant contexts. Among several topics of great interest he studies the one we are dealing with here and tells us that

“…The battle first fought in Germany recurred as the old religious schools were reformed in one country after another. Along the lines founded by the University of Berlin in 1810. Variants of idealism appeared several generations later in Britain, the United States, Italy, Sweden, and elsewhere, when the general academic model was imported” (Collins 1998)

He successively studies other important issues: the German Idealist movement, its networks and conflicts, the controversy over pantheism towards the end of the 18th, the proliferation of schools within the Idealist network, the way in which philosophy takes over the University, the rebellion of the Department of Philosophy, the Idealists in the reform of the University, and as ideologues, the diffusion of the University revolution in England and the United States, in Italy, Scandinavia and Japan, and after the secularist rejection, of Idealism.

“… It was Fichte’s program, stripped of his utopian politics, that Wilhelm von Humboldt, one of the Fichte audience in 1808, put into effect” (Collins 1998)).

I have not yet explained why the Kantian movement should have appeared at the time it did, nor indeed why it should appeared at all. We see the older networks transforming and taking on a new context; for a time its content stirred enormous enthusiasm and generates a panoply of opportunities for creativity. To understand this, we must move to the underlying material base which supports the networks.

During the times of the idealists, this base was expanding and transforming in Germany in a change that was laying down the conditions for the modern intellectual” (Collins, 1998).

“…the point is not that idealism alone brought about the university reform, but rather that the strains of the old university system –above all the plight of young aspirants in theology and hence in its traditional feeder discipline, philosophy- motivated idealism” (Collins, ibid.).

While the aforementioned techniques run perhaps the risk of oversimplification, it is still a greatly informative corpus that has to be made the most of. However, this is not the place to study the foundations of such techniques.

6. After 1810, Neohumanism was the revolutionary trend at University, started by Wilhelm von Humboldt (1959) and spreading from Berlin to many universities founded in Germany and several European and North American universities. Neohumanism, characterized for positing the unity of teaching and research – but focusing on the latter—around seminars, was a decisive influence on university life, not only in Germany but also in Europe and the United States. For us it took the shape of a strange link between mathematics and philology, particularly as a multifold study of Antiquity.

“…It is not in the 19thcentury nor in Germany that linguistics and modern philology were born –they are much, much older. What did come about in the 19th century in Germany, however was the evolution of a certain kind of linguistics, that is to say comparative linguistics, and the establishment of modern philology, comprising both language and literature, as a university discipline, respectively as various university disciplines” (Christmann 1994).

In other passages of his work, Christmann insists on that rare company, and he is not the only author to do so. Several others proceed in the same way, locating mathematics in the Department of Philosophy. I have insisted on this conjunction of mathematics and classical philology because it lies at the core of Neohumanism.

By ‘Neohumanism’ we do not mean a “bad” ideology but -what is more important- an ideology rooted in many researchers, and therefore in several research institutions.

There were close bonds between neohumanism, idealism, spiritualism and romanticism. We account for these bonds in a work that runs parallel to this one, to avoid extending ourselves here.

“…while mathematics received relatively little attention in the idealistic systems of most of the German Romantics, it served as the foundation in the thought of the neokantians philosopher-mathematician Jakob Friedrich Fries (1773-1843). It fell to Fries to work out in detail the implications of Kant declaration that all the mathematical knowledge was synthetic a priori. In the process Fries called for a new science of the philosophy of mathematics, which he worked out in greatest dertail in his Mathematische Naturphilosophie of 1922. In this work he analyzed the foundations with an eye to clearing up the historical controversy over the Euclid’s theory of parallels. Contrary to what might be expected, Fries’ kantian perspective provoques rather than inhibited a re-examination of Euclid´s axioms” (Gregory, 1983).

Towards the end of the 19th and the beginning of the 20th centuries, it is insistently discussed in Uruguayan academic circles the kind of mathematical education that engineering students had to receive. This debate appeared in Montevideo in the daily press and also in specialized mathematical journals in European technical universities.

In order to understand the development of mathematics in Uruguay beyond its initial conditions, and on the issue of mathematics for engineers in Montevideo, it is worth consulting – among a wide bibliography—at least Schubring [1981b], Tobies [1989], Dhombres [1989, 1998] and Siegmund-Schultze [1995]. These works offer a wide coverage of the development of mathematics in technical universities in Europe and the United States, and they provide information that will unknowingly reveal – as we will see below—an intense and strong treatment of the subject in Uruguay.

iii. the mathematics proposed by Klein from Göttingen, which is a strange and at the same time extremely valuable combination.

From Wilhelm von Humboldt onwards, the applied mathematics of the 18th century is gradually but increasingly accompanied by a mathematics imbued with Neohumanism with its seminars and its imperative of research.

In the first half of the 19th century, the Industrial Revolution had not reached Prussia and therefore the pressing needs of industry gave leeway to a budding pure mathematics. Jacobi stated his dictum: the object of mathematics is the honor of the human spirit. In France, on the other hand, the École Polytechnique gave a strongly applied tone to its mathematics, and Fourier did pure mathematics but through his theory of heat. The Germans, on the other hand, were proud to work for the honor of that human spirit so multifaceted and still weak. Then, in the first quarter of the 19th century, the situation was complicated. It varied with time and geography; there in France it was applied mathematics (regardless of Poncelet and his Projective Geometry and, somewhat later, Chasles with his Higher Geometry, who did other things), mathematics which resulted in the engineer’s important public works. There in Prussia with its prevailing pure mathematics it was a different matter altogether. It’s true I’m oversimplifying, of course.

“Discussion of modern scientific research’s organization point to the 19th century emergence of German research universities as evidence that state investment in non directed academic research, when coupled with beneficial relations between academic research and industry, and when stimulates by appropriate incentive such as protection of intellectual property, in an open competitive system, can lead to explosive growth, in scientific knowledge and rapid improvement of industry” (Lenoir 1998).

Also we refer to Randall Colllins, A global theory of intellectual change, and in the United States the situation has been thoroughly studied by Pyenson (1982,1983).

“The idea of a “research university” did not emerge in the United States until the end of nineteenth century. Looking to Europe again, mathematic departments found inspiration in the German system, which was then promoting the construction and use of mathematical models in graduate education” (Mueller 2001).

“Indeed, up until world war I, Germany provided the model for unflattering comparison of Americas’s ‘deficient achievements, a role to which Japan has been assigned in more recent years” (ibid.).

“The mathematical teaching of the last ten years indicates a “rupture” with antiquated traditional methods, and an “alignment with the march of modern thought” (Peabody 1888).

“If we could hear history’s lessons, we might be more willing to see the current “crisis” as just another episode in a long historical “discussion” about the nature of mathematics itself. The issues haven’t been settled in a century -many centuries in fact- and they will not be settled in this one”.

“Philosophy is to be studied, not for the sake of any definite answers to its questions, since no definite answers can, as a rule, be known to be true, but rather for the sake of the questions themselves; because these questions enlarge our conceptions of what is possible, enrich our intellectual imagination and diminish the dogmatic assurance which closes the mind against speculation” (1912).

Mueller himself discusses in extenso the prevalence of idealism in the mathematics of the United States.

Gradually, French mathematics were yielding their previously undisputed prevalence perhaps due to this exaggerated applied focus. And the global center was displaced eastwards. The lengthy process of generation of a Polytechnic University in Berlin had some twists which were not unrelated either to the prussian Kulturministerium, who wanted to attend to the needs of a nascent or foreseen industry, or to the already partly globalized (it reached only a few countries) mathematics which included Galois, Cayley, and Riemann, and which was to produce the Erlangen program of the seventies. The debates around the creation of the Berlin Polytechnic still seem interminable, in filigree, and evade any attempt at a simple historical description.

The key to the process lies in Göttingen with Felix Klein who, beyond his Erlangen program, understands that he has a decisive and very delicate role to play.

“Klein was one of the few who realized the new demands that would soon be made on mathematics by industry, the natural sciences, and technology as well as by modern financial institutions, particularly the insurance industry ... (in 1872) while expressing an appreciation for practical applications subtly implied that it was something inappropriate for university mathematicians to tackle problems that went beyond the realm of the purely theoretical”.

“By the word 'applications' I am thinking much more of the theoretical services performed by mathematics in the development of other sciences. In any case, this is not what Jacobi intended but something quite different.

Things were not easy for Klein. He had to balance rigor with intuition, theoretical application with the demands of the Kulturministerium. But still that he would confront the Berliner mathematicians, whose inclinations were determined by Weierstrass (with his arithmetization of analysis) and Kronecker, among others.

Klein overcomes the sprouts of neo-humanism but not so much, luckily, the wishes of ministerial applicationists. Göttingen becomes the place for North American scholarship students, several of whom will become prominent mathematiciants, since the mathematics of this institution was particularly adapted to their needs.

Between 1900 and 1914, an important lapse, the climax of the debate about the kind of mathematics to produce and teach to engineers is reached, and this debate becomes globalized. By then, the French and German mathematic societies had reached a relative maturity, and the international meeting of 1914 is focuses its discussion and its contribution – before the so called Great War comes to divide, for quite a while, those mathematicians.

As can be seen, it is not at all simple to describe the development of the debate about what mathematicians have to do in turn for the honor of the human spirit, to understand the material world, and last but not least, to contribute to public happiness. If anyone can be singled out for his knowledge and political skills it must be Klein, who was represented among us – in texts used much later—by his disciple Courant, who was linked to Hilbert to a considerable lapse.

Thus we have almost got to the point of answering the question about why mathematics in Montevideo was almost to this day – somewhat more precisely almost until 1972—extremely pure mathematics. In 1903 our García de Zúñiga arrives at Charlottenburg (Polytechnic University in a suburb of Berlin). And so begins the story of modern Montevidean mathematics. During this two-year stay, García de Zúñiga absorbs, in the mathematical circles of Berlin – dominated by pure mathematics—a Neohumanism that comes from Wilhelm von Humboldt, he collects an extremely wide mathematical bibliography for years and he will later donate it to the library of the Department of Mathematics (later Department of Engineering), publishing incomparable catalogues, and in so doing he will provide the material basis of the mathematical school of Montevideo.

The influence of Neohumanism will reach -not consciously, I believe- even Laguardia and Massera and his disciples, until the 1960s. A Neohumanistic origin and the absence of any serious industry -after Viera’s suddenly slowed it down in 1917- resulted in Montevidean mathematics becoming isolated from productive activities. Which did not prevent, however, that in due time these mathematics would reach an unusual level of quality in Latin America. The former made up for the latter; just that. If this is a risky hypothesis on our part, we have given reasons that support it. That is part of the story. Complementary texts could be consulted, thus opening up a field of research.

At present mathematical research in Uruguay is at the cutting edge of Latin American mathematics. The volume of the research in the region is not heterogeneous and it depends, among other factors, on the size of each country.

How can we explain that a country as small as Uruguay -with 3 million inhabitants -has had a production of such magnitude? The relatively recent institutional growth of the Uruguayan school of mathematics (foundation in 1942 of the Institute of Mathematics and Statistics -IME and later, in 1993, IMERL by Rafael Laguardia- of the Department of Engineering), and the research carried out after 1929, have not prevented its evolution into the present situation. The shape of this development and its antecedents provide to a large extent reasons that explain the aforementioned development.

Moreover, it must be taken into account that during the military dictatorship of 1973-1984, the IME was all but closed and even its best qualified researchers were expelled.

Even so, during this period researchers received solid qualifications abroad –doctorates form important universities, obviously without government funding- and, towards the end of the military dictatorship, a significant group of young people had Access to B.A. studies even though the curriculum was at the time appalling. The restitution of highly qualified researchers to the University as of 1985 (the beginning of the democratic transition) was a substantial contribution to research.

Nowadays, Uruguayan mathematics still covers a few fields, but it is very vigorous. Dynamic systems, probability and statistics, algebra and geometry, topology and functional analysis, are the main areas of work.

Applied mathematics, on the other hand, has not been considerably developed beyond a recent boost.

This is as far as we will go in our description of the latest stages of the Uruguayan school of mathematics, once it was constituted. There is bibliography about this topic. Suffice it to say that towards 1966, as few as eight researchers were already publishing several articles in international journals.

Of the other face of history - i.e., of what did not significantly affect the appearance of the Uruguayan school of mathematics- we will give no more than a brief outline.

During the first stages of education, reading and writing is accompanied by the teaching of arithmetic. The influence of the Encyclopedia, promptly condemned, and the hesitant access to Galilean and Copernican ideas only affect in those days the teaching of elementary mathematics. An observatory is established in Montevideo, where some elaborate if inconsequential studies take place.

After the Independence (1825) it is worthy of notice that, after the formal foundation of the University (1839), re-established some years later, the first Chair of Physics and Mathematics is created in 1850 in the Baccalaurate (based on the French model) and a second Chair of Mathematics in 1864. Towards 1855 the first book of elementary mathematics is published in Montevideo.

10. After 1867 and for some ten years, the Sociedad de Ciencias y Artes (técnicas) publishes bimonthly an important bulletin; numerous works for the diffusion of mathematics as well as other sciences are published, be it in a special section or in the body of the bulletin, which can be considered as a serious attempt of diffusion of the sciences even though among its articles there is a well-known one about the quadrature of the circle.

In those same years (1870s) there is a considerable production of school texts of elementary mathematics related to the use of the metric decimal system and the reform of Public Education by J.P. Varela, a process described by J.A. Grompone in a series of articles and in a book of limited circulation on the economic bases of the history of science in Uruguay. 7

Towards 1900 there is again a moderate publishing boom of elementary mathematics texts. The following peaks coincide with the extension of secondary education to the whole country (circa 1918) and with the publication by Rey Pastor y Pereira of a series, as well as other similar textbooks for Secondary education after the 30s. Afterwards, the publication of textbooks became more stable and, in general, less innovative. But let us not stray from our subject.

11. In 1888 the Department of Mathematics (actually Engineering, Land-surveying and Architecture). One of its first three graduates is, in 1892, Eduardo García de Zúñiga, whose contribution in the field of mathematics is, in our opinion, decisive. In 1915 that Department of Mathematics becomes two Departments, one of Land-surveying and another of Arquitecture.

The mathematics syllabi in higher education in the period 1888-1900 deals with 18th century mathematics, i.e. mathematics to be applied in the professions offered by the University, and it seems to ignore, especially at the beginning of this period,

the massive transformation of the field that took place after the beginning of the 19th century.

Higher education (as the expression was understood in those days) was completely out of step with the research front of that science and fundamentally directed to make calculus manageable on the basis of European teaching of the by then obsolete infinitesimals.

12. However a new teacher, García de Zúñiga, not at stranger to the professional practice of engineering (project of the Port of Montevideo, intervention in the railway network and the construction of viaducts and bridges) introduces the extremely renovated mathematics of the 19th century and beginnings of the 20th. His stay at Charlottenburg towards 1903 was decisive in this respect.

Guido Hauck, professor at the Charlottenburg Institute of Technology, elaborated Weber’s report, Hauck noted that the new regulations for applied mathematics required mastery of three fields -descriptive geometry, technical mechanics, and geodesy- any one of which could absorb the efforts of a lifetime (Pyenson 1983).

The turning point for the creation of the future Uruguayan school of mathematics took place early on through the work of García de Zúñiga, who introduced three elements:

i. The Mathematics syllaby of 1915, when the Department of Engineering was founded;

iii. the production of a few timid but rigorous research contributions which despite their modesty, and in conjunction with works by other scholars, showed the possibilities of mathematical work. A band of modernity9 appeared with the creation of infrastructures that would become the later development of that research.

From the University, which administered not only the courses of the Department of Engineering but also the two-year Baccalaureate in Engineering (otherwise controlled by Secondary Education), a fourth element was added: the implementation – for students of Engineering and Land-surveying, of extremely modern syllabi, outlined by García de Zúñiga, 10 which dealt with recent mathematics. As we have already said, the previous syllabi was dominated by the 18th century, and we could with some exaggeration characterize it as responding to the Lagrangian paradigm. This resulted in the paradox that the previous Department of Mathematics taught practicist syllabi for Engineering, whereas the new Department of Engineering, its successor, as well as the Engineering Baccalaureate, teach after 1915 modern Mathematics syllabi, albeit after a strong public debate. 11 Moreover, the introduction of higher mathematics at the Baccalaureate level allowed University courses to be not only more intense but also to cover topics of real significance. Delta and epsilon reign since 1915, even at Baccalaureate level. This is just an index of a set of surprising issues – from a local perspective—for the time. It means the true gestation of a truly modern mathematical culture, extremely advanced in comparison to many other countries.

At the same time, despite some resistance, it becomes necessary for teachers to deal not only with French bibliography (as was usual then) but also with British and German works. Once this change takes place, it is possible to gain access to mathematics in process, to the cutting edge of research. At the beginning there are no mathematicians, but curiously there is an available modern mathematical culture. Even in the absence of researchers in the proper sense of the term, there is a rigorous treatment of the subject, which was the dominant character in the mathematical metropolis.

Not only is a library of mathematical classics created but there is also a considerable investment in international mathematical journals, 12 transmitted not only through courses but also accessible on the library shelves. 13 Even though those who would later use that available arsenal had not appeared on the scene yet, when that time came the necessary bibliographical baggage was already available.

13. García de Zúñiga, later a member of the Spanish Mathematical Society and its representative in Montevideo, member of the Spanish Academy of Science, was faced with a titanic organizational task. When Rey Pastor, and later other European mathematicians came to Uruguay to give lectures,

they were not introducing a completely different kind of mathematics; what was being done on an international scale was understood.

It is often said that García de Zúñiga was not a full researcher judging by his few investigative works, and this is true. But the absorbing organizational tasks he faced was an objective impediment. In any case, towards the mid-20s the stage was set for the appearance of works that come to the forefront of research: mathematical rigor was the norm, the creativity of others was acknowledged, and their production was becoming knows, that is to say the forefront of research. In fact Laguardia, another hard-working and successful organizer (a time-consuming task) was already publishing, in the Journal of the Center of Engeneering and Land-surveying Students, articles which would much later be taken up, after the Publicaciones del Instituto (IME), particularly after 1942.

To sum up, not only did García de Zúñiga’s few works of mathematical research appear relatively early on, but the works of several other authors was also published, and even though they were not large-scale investigations, they were a correct and many times lucid exposition of modern mathematics.

The great initial task of a single man, García de Zúñiga, had a gradual impact on the first steps of what would later be the constitution of a true mathematical community, a very small but solid one.

14. Another relevant aspect of the debate between García de Zúñiga and J. Monteverde (1915), about the Mathematics syllabi for Baccalaureate and University, must be recalled. The (by no means negligible) vindication by the latter scholar of a close contact between the science to be taught and the actual practice of the Engineer led him to reject a fine mathematics like the one promoted by García de Zúñiga. The debate of what mathematics should be taught to engineers is a timeless subject, but it became particularly acute at the time. García de Zúñiga, himself a successful engineer, promoted a rigorous and thrusting mathematics.

“But all of this does not justify that the engineers our Department must teach should be taught more higher mathematics that those taught to the same professionals at Polytechnic Universities in Germany, Austria, England, United States, and so on; it would be an absurd pretension on our part to want to prepare specialized engineers or learned researchers, capable of advancing pure or applied sciences or to study the perfecting of industrial machinery and engines, competing with the most advanced nations in the world. The task of the learned researcher, whether or not an engineer, who studies scientific theories and helps advance them with his own works, must not be confused with that of the specialized engineer who applies these theories to the construction and improvement of the industrial machinery used in factories and their thousand applications; and none of them should be confused with the engineer that neither builds nor improves those machines, and who just studies their installation and their most economical and appropriate application. The conditions of our country and its needs only demand, and will do so for many years, the work of the engineer that applies procedures, engines and machines as is done in the countries that are our teachers and our guides”.

I underline Monteverde’s last sentence. I think it is accurate to remark that García de Zúñiga, beyond his contribution to important public buildings (during the golden period of the foundation of modern Uruguay), and the material and cognitive infrastructure of our mathematics, also sustained in general and in an advanced way -without neglecting the necessary experimentalism- the scientific ethos which will only be openly expressed in this country over thirty years later, on the occasion of the creation of the Uruguayan Association for the Progress of Science (of ephemeral existence and long-lasting effects).

15. On the other hand, with relation to substantive institutional aspects, it must be said that the trend promoted and executed by García de Zúñiga resulted in a Department of Engineering of polyvalent preparation, very far from a school that would only prepare engineers for installation and maintenance, as Monteverde wanted. Hence, for instance, during the military dictatorship (1973-1984) Uruguayan engineers stood out abroad because of their excellent preparation, even though they only had graduate qualifications.

It must be therefore understood in a strong sense that García de Zúñiga’s struggle for the establishment of a modern mathematical culture both at Baccalaureate and University levels, with an up-to-date library and the beginning of modest but valuable research work, provided the necessary bases for the gestation, between the end of the 1920s and 1942 (foundation of the IME), even in the absence of specific institutions, the Uruguayan school of mathematics.

The years of military dictatorship saw both the emigration of mathematicians (Chiancone, 1997) and the cancellation of subscriptions. But these circumstances have already been described in other works.

With the end of the dictatorship, the purist mathematical trend that was dominant until the mid-20th century starts to give way to intermediate forms, and this is because researchers understood the needs of the country. It is not that these were not present before as arrière pensées, but that even so they were not translated into projects of national interest or if they were they did not obtain funding. Government sources were nowhere to be seen, and the private sector did not contribute one penny, and what is worse expected to obtain beneficial results free of charge.

The return to the country of several mathematicians with doctorates and experience abroad was a significant contribution, since it aids the restitution of the 1960s syllabi of Mathematics in the Department of Humanities and Sciences.

The new trends apply nowadays mathematics to several problems of production (e.g., the managements of the available and desirable balance between dams and thermal plants) paid non-teaching responsibilities.

A) In Uruguay, mathematical research is frankly separated from research in the philological sciences.

B) The intellectual ideology in mathematics, namely purism, was dominant for a long period, both in content and in style.

C) The social ideology of mathematicians is extremely varied, and generally progressive. Social conditions are very different from those that prevailed during the European Neo-humanist period.

D) Therefore, of that Neo-humanism in Uruguay during that period all that is left is purism.

E) Is there possibly another ideology in Uruguay that might fulfill the functions of its Neo-humanism? This seems to be the case, with fashionable fury, of innovation. he reasons for this will be dealt with elsewhere.

THE SO-CALLED AUTONOMY OF MATHEMATICS; WHAT REALLY INDUCES THEM IN A SIGNIFICANT PROPORTION OF THEIR PRODUCTION

Before linking our conclusions of previous texts we must remember that they do not try to cover all the topics referred to pure mathematics. Neither do they try to historically cover the emergence of pure mathematics on the background of modern types, including mixed mathematics. Nor do they try to analyze the different meanings which the expression “pure mathematics” has been acquiring or losing. If this were the case, not only would this be a much thicker text, but also it would overlook the good accounts which have been made on the topic

On the other hand, we have barely skipped the philosophical problems faced by the three well-known positions during the 20th century, logicism, formalism, intuitionism -and whose solutions have obviously failed-, because it was not the objective of our work to do this. We thus want to remember some of the limits of our work. We have tried to cast some light on certain relevant aspects of the elucidation of the relation between mathematics and applied mathematics.

1. The view mathematicians have had for centuries about the utility of the discipline is not wrong. Yet, the view which considers this opinion as mere rhetoric is wrong. Only if we consider rhetoric the desire and the persuasiveness surrounding their work, we could apply this term, in a sense … But the strong sense of “rhetoric” applied by Catherine Neal (1999) is not applicable at all. Furthermore, her viewpoint follows a tendency to consider every scientific or philosophical act as rhetoric, which leads us to that night when all cats are black. Utility is a proper feature of knowledge; it is not an arbitrary, rhetorically added element[1].

2. Another approach to deny the utility of mathematical knowledge and confirm its purity, is opposed to sociologist interpretations, sometimes very exaggerated but also sometimes legitimate. We have characterized this approach as the sketch of all assertion tending to include the context of knowledge as a licit explanatory element. Jean Dieudonné (1987) has produced it in honor of the human spirit. Or in honor of a very restricted community. He has done this for his own purposes but it has harmful historiographic reach. It is easy for him to make a dummy[2] out of an example of sociological studies, but it is nothing more than that. Nevertheless his idea would result in a pruning of millenniums of real mathematical research to only remain in the scope of the present “professional” mathematics.

From 1 and 2 it results that they are two different historiographic tendencies, both wrong and also harmful. And, at heart, both positions insist on the uselessness of mathematics. The first would turn mathematicians into players of a rhetorical game on the character of the mathematics, self-consuming them. The second, by making irrelevant any application of mathematics, it would also make it at heart, a self consuming enterprise. All the history of the more advanced sciences and finer applications (for example, the theory of the queues at arrivals in the busiest airports) or lesser applications of life (for the queues at supermarkets, paradigms of massive consumption of those who can afford it), deny both tendencies. The “playful doings” of mathematicians are not so coarse. The real production of mathematics is not only socially conditioned – even if only because they require funding – but also by all which is required by any human activity, scientific or not.

3. As we have seen, two texts (Russell (1912) and Wigner (1960) respectively raise important subjects and are intertwined. The first one tries to date the emergence of pure mathematics and the second one raises the question of the incredible or at least unexplained, efficacy of mathematics. Authentic problems emerge from there which are impossible to avoid.

On the other hand, it is shown that the statement that all application of mathematics emerges as the specification of one or several systems of pure mathematics is unfounded. A mathematical account which emerges directly from physical sciences could be called “applied”, and is commonly called so, but it is not applied in the sense that it uses pre-existent pure mathematical theory. This theory may be generated later by abstraction but it does not pre-exist, it will post-exist. That is where many confusions result from the use of the term “applied”.

The meta-conception of pure mathematics arose from neo-humanism and got to be dominant independently from its origin. It is relevant to see how it developed in the period after its generation and how it took part locally in important centers of mathematical development and also in our research centers.

4. The discussion about which mathematics engineers should be taught, even though it is still present today, reached its climax towards 1915 within the framework of mathematics with international connections and within institutions dedicated to the promotion of those disciplines.

In Uruguay that discussion happened almost simultaneously to that climax and García de Zuñiga’s programs for higher education and for pre-college engineering education reflected, with their predominance during decades, that neo-humanist conception, whether conscious or not. The rigor of those programs gave predominance to pure mathematics although there were Calculists working in the Institute of Mathematics and Statistics, positions more proper to applied mathematics. Only much later did the culture of applied mathematics or branches of mathematics with direct projections in the other sciences become significant.

5. The philosophy of mathematics experienced during the 20th century a quite unavoidable process. The search of foundations in the three classic stances: Logicism, Formalism, Intuitionism -, even with their properly mathematical results, gave rise to tints and shades when facing problems of very difficult solution. Yet, the obstacles were to a great extent insurmountable. But this does not prevent the subject of the foundations from being sill important in mathematics itself. Nevertheless, other ways of facing problems were found, beyond Formalism and Platonism, ways which have granted a central place to the study of practical mathematics. We should not forget the role played by computer sciences in the properly mathematical research, it is the practice considered in all its dimensions that opened new doors for the philosophical and sociological analysis and which has given a wider framework than the previous one to determine more rigorously the relation between mathematics and applied mathematics.

6, The inducers. Two types of inducers on the production of mathematics. Those two types vary their importance as the production of mathematics advances, but both are decisive.

a. The practical, productive activity, beyond what Herodotus relates, is co-determining of the production of mathematics even today (we have already seen the example of the String Theory). It is not true that mathematics only arise from mathematics.

b. The mathematical tradition and the pure mathematical research front are conclusive: also from creative mathematics more mathematics comes out. Thus things would be, but if there were no human practices, no institutions, no financing, no prestige derived from applications and so on, we could hardly say that mathematics would be in full swing in a modern sense. The two types of inducers and their co-presence happen simultaneously back and forth in mathematical research.

The classic debates of the beginning of the 20th century extend till today. The elaboration of this topic made some of the classic positions to be discarded for being unfeasible. (for example Mehlberg) The materiality of the objects of mathematics is defended by Chandler Davis (19, 19, 19) with strong arguments.

But, in addition, the materiality can be defended in the undertaking itself of mathematical research due to the strong participation of the aforementioned inducers in impure mathematics.

[1] Otero, M.H. (2002) “La utilidad como presunta retórica en textos de matemáticas”, Revista Brasileira de Historia da Matemática,.

[2] Otero, M. H. (2003) “A chocolate mint? On a certain widely spread ideology involved in the historiography of mathematics and in many other non trivial discourses, in this volume.

ON AN OLD DICUSSION: PURE OR IMPURE MATHEMATICS

The long philosophical debate on the applicability of mathematics restarts with Wigner’s intervention in a famous 1960 article:

"The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and there is no rational explanation for it...The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve"

1. In a very well-known passage of one of his articles Bertrand Russell said:

“Pure mathematics was discovered by Boole, in a work which he called the Laws of Thought (1854). This work abounds in asseverations that it is not mathematical, the fact being that Boole was too modest to suppose his book the first ever written on mathematics. He was also mistaken in supposing that he was dealing with the laws of thought: the question how people actually think was quite irrelevant to him, and if his book had really contained the laws of thought, it was curious that no one should ever have thought in such a way before. His book was in fact concerned with formal logic, and this is the same thing as mathematics” (in Misticismo y lógica y otros ensayos, Paidós, Buenos Aires 1951).

Let us disregard the logicism stated at the end and also the correct thesis that what Boole was doing was Logic -or we should say today, Algebraic Logic- and not studying real thinking. In the decades immediate to Boole the ideas (psychological elements) which were the bones of contention of English Philosophy and also of previous non-English philosophy, were being exterminated. However, I want to point out that when Russell talked of the christening of that monster -pure mathematics- he was completely wrong in dating the birth with Boole. Talking about the first appearance of an entity so often called pure mathematics, was the consequence of disregarding the historicity of the concept itself. While mathematics grew, the concept slipped over and over again. We do not even know if mathematicians in the late 21st century will call pure mathematics to something of what is developed nowadays under that name. The movement towards abstraction is beyond question but the doorway to the so called pure mathematics is hazy and, slippery anyway. When Leibniz and then Boole thought they have reached the doorway of a polyvalent instrument which would allow to solve all theoretical problems, they were also wrong.

S. B. Diagne says, in his delectable little book, Boole, l'oiseau de nuit en plein jour (Paris, Belin, 1989) that Boole falls into an intemperate symbolic optimism. And he is right. Boole produced a tremendous revolution, but he did not create, in any way, an organon which would allow later to solve all the theoretical problems, not even those of mathematized but non-mathematical sciences which were developing in the 19th century.

2. Peressini (1999, section 3 and 4) reaches the crux of the matter of the so called pure mathematics. And he wants to delimit application mathematics. According to Peressini there is a strongly confusing way of looking at things: saying that the applications of mathematics involve nothing else than replacing mathematical terminology with physics terminology.

It is often said that only after the theory has been developed it will be applied to real problems. This is not so. Neither is the opposite case, that the progress in pure mathematics is due only to developments in the use of mathematics in other sciences. For Peressini, it is neither. He says:

"...neither the pure theory nor the applied theory are in all cases epistemically prior".

"...not every mathematized scientific theory is also an application of a /pure/ mathematical theory. There are mathematized scientific theories that do not bear the "applied" relationship to any pure mathematical theory and so, strictly speaking, should not be considered applied mathematical theories... In such cases in which the mathematized scientific theory is worked out first, and then only later, if ever, a pure mathematical theory is worked out, we have the inverse of the operation of application - call it abstraction"(ibid.).

As we held before, Peressini does not disregard either that the pure mathematical theory is frequently applied in pure mathematics itself, for which he presents several, maybe redundant, current examples.

Finally, for our interests here, he points out historical cases, which he divides in two types.

First type: late in history there are clear examples of application of pure theories, because frequently very late there are pure theories. Euclidean geometry is clearly a physical geometry, in which from a mathematized scientific theory we get to pure theory. That is why it is not adequate to say that Kepler received a pure mathematics a long time before (as Massera 1986, tell us). When Newton developed calculus he did not do it purely but rather the calculus had to do with things existent, according to him, in nature.

A second type of case is when Einstein develops his general theory of relativity using for gravity the structural features of a curved space-time of Riemann geometry, through the tensorial calculus developed by Ricci and Levi-Civita, which are prior and had passed unnoticed by the physicists of that time. The same happened with Galois theory of groups applied much later to physical symmetries.

Finally, Peressini holds with some shades of meaning not very relevant, that the distinction between pure and applied mathematics is a logic distinction, which from our point of view, presents unavoidable reservations.

3. Since 1810, the first journal specialized in mathematics was called Annales de Mathématiques Pures et Appliquées. Then, the Journal de Crelle and the Journal de Liouville -which are still published today- contained even in their title this distinction between mathématiques pures et apliquées, which for many is obsolete today. In addition, one of the worldwide most prestigious institutes of mathematics at present – the institute of Rio de Janeiro, refer in its name to pure and applied mathematics. Is it perhaps just historical sensibility?

3.1. One of the thrusts of so called pure mathematics was started by Humboldt or even before; it came from neo-humanism, an ideology which L. Pyenson (1983) has studied in depth.

"Educated German were bound together in an aristocracy of learning based on classical precepts. Radicals from Karl Marx to Karl Liebknecht, statesmen like Otto von Bismarck and Wilhelm II, philosophers from Hegel to Lagarde, and mathematicians from Gauss to Hilbert, all at age eighteen would have been prepared to translate Greek poetry, compose Latin prose, and recite parts of Euclid's original writings...As its goal neo-humanism elaborated the image of humanity conveyed by Greek literature and culture and used the resulting interpretation of classical Greek antiquity as the foundation on which contemporary German-speaking culture was to be based and the standard by which German-speaking society was to be judged. In its purest form, the neo-humanist interpretation of classical antiquity centered around a simple and direct search for the absolute values of Greek philosophy and ideals"(ibid.).

Philology was a cultural paradigm for almost all of them; philology seminars in Gottingen congregated more than three hundred students. Mathematics resembled philology, even though the public was not comparable. The demand for research was prevalent and teaching played a lesser role, although the research-teaching unit preformed later ways which survived in the development of mathematics.

4. Naming pure mathematics is, for some, to name a process which start during the French Revolution, as an important part of the production resulting from the Ecole Polytechnique (or its German and other nation’s replications) and which comprises, in a non trivial way, professionalization of mathematicians, specialized journals, mathematical societies -a whole macro-industry- and in general the establishment of mathematical communities which gradually become international, and spread wider networks. They were gradual but irreversible phenomena and processes (5). This did not happen without a macro debate among pure mathematicians and engineers on which mathematics they should teach in gymnasiums, universities and polytechnic schools.

5. The meta-conception of pure mathematics (at this point we follow Schubring, 1981) goes along with the gradual imposition of the demand to research. Besides, the development of a methodology for mathematical research pressed for the professionalization (multiple phenomenon with multiple interpretations which included full-timing) and which was inclined to give autonomy and modernity and strong specialization to the main mathematical activities. The long debate prior to the creation of the Polytechnic university of Berlin was linked till the end to the fact that Berlin constituted a centre of pure pure mathematics, and Göttingen, instead, urged by Felix Klein, was trying to develop another mathematics, focused on the problems of industrial development. (See CHART). In opposition to the excesses of autonomy, of pure mathematics as an end in itself, with its rejection to utility, with the imposition of intrinsic values, Gottingen presented a quite difficult alternative model. Moreover, in Prussia the exaggeration of the values of research for the Gymnasium teachers led to demand them to write about topics of foundation or development of mathematics for their promotion, which in the end served for the personal prestige. Crelle reached the point of saying, with more exaggeration than Jacobi that “Everyone, without exception, need pure mathematics”.

5.1. Kant and Fries appeared as prominent figures. Kant for the acknowledgement that novelty is present even in mathematics and because he separates pure and applied mathematics. For Fries the reality of mathematical concepts and the certainty of results are beyond all question. His semiotic conception of mathematics emerges at the same time as the boom of combinatorials. In some way this represents a paradigmatic example of pure mathematics. Fries does not belong to institutionalized mathematics nor does he share the Berliner meta-conception of pure mathematics but he still has influence.

The resulting tendencies of the French Illustration are empirists, they do not share the Weltanschaung of German mathematicians, and French mathematics -pinned to multi-employment and without the benefit of full timing- do not have a system of autonomous mathematical values. Napoleon, with his dislike for ideologues, propelled spiritualism as the state ideology and changed the prevalent situation.

1. “The pure/applied dichotomy is a historical phenomenon which has not always existed and which probable will not exist forever. The notion of “pure science” was construed in the ideal of science in the historical context of the second half of the nineteenth century, while in mathematics the gravity centre was shifting from Paris to Berlin…”,

2. The Second World War has had a crucial significance in the development of applied mathematics in the USA, not only from the point of view of the incentive of special branches but also in the redefinition of the limits of the discipline and the change of image of the mathematician…

3. The pure/applied mathematics dichotomy acquired the character of rivalry and war of images -mathematics for the honor of the human spirit versus mathematics for the world and human issues- in the second half of the 20th century. The influence of the prestige of the Bourbaki group has determined the face of this opposition…

4. In the 80s of the XXth, the economic and cultural context favors a change in the values and the mathematicians present mainly an open image of mathematics, in multiple interactions with the other disciplines, with the world and with human needs. (Dahan-Dalmedico in her “Pure versus appliqué”: un point de vue d’historie sur une “guerre d’images”, page 199)

“All the mathematical community pursued the German heritage of the 19th century, at least in appearance. Mathematicians, except for the moments of great social significance, cared about their world. A world increasingly separated from reality, more involved in interiorized ideal questions. A world vigorously claimed by its researchers and considered with more skepticism by the rest. Once in a while, the amazing technological and scientific developments reveal a very efficient mathematical support. Also, organizations and very concrete entities, which do not have an aesthetic concern for the prince of sciences, continually bid projects for the mathematical community. And from these developments and projects emerge ideas that penetrate the intrinsic world of Pure Mathematics and give employment for a new season”. (Hormigon, 1990)

On the other hand, the problem of the applicability of mathematics attracts more and more attention. Mark Steiner, in his 1998 book, The applicability of mathematics as a philosophical problem, Harvard University, Cambridge, MA has raised again general questions and has studied the case of the application to quantum mechanics. Let this allusion highlight the significance of the problem of applicability in the current debate.

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Ferreirós, José (2003) Del neohumanismo al organicismo: Gauss, Cantor y la matemática pura. In J. Montesinos, J. Ordóñez y S. Toledo (eds.) Ciencia y Romanticismo. Fundación Orotava de Historia de la Ciencia, Tenerife.

García de Zùñiga, Eduardo (circa 1937) Curriculum vitae, Archivo de la Facultad de Ingeniería, Montevideo.

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PHILOSOPHICAL PAPERS ON MATHEMATICS

CONTENTS

Avatares…

Avatares…

KUHN’S PHILOSOPHICAL TROUBLES WITH ACTUAL SCIENCE HISTORY (1)

ON THE PRESUPPOSITIONS OF SCIENCE: ROBIN COLLINGWOOD’S ESSAY ON METAPHYSICS (1940) AS ANTECEDENT OF A WHOLE ERA

**TABLES, CHAIRS, BEERMUGS; OR THE PRE-HILBERTIAN USE OF THE PRIMITIVE CONCEPTS OF DAVID HILBERT’S GRUNDLAGEN DER GEOMETRIE.**

A CLASSIC HISTORIOGRAPHIC TEXT: ANDREY N. KOLMOGOROV’S ARTICLE “MATHEMATICS” PRESENTED IN THE 1936 SOVIET ENCYCLOPAEDIA. [1]

Introduction by Mario H. Otero

Preliminary words about the text

A CHOCOLATE COIN? ON A WIDESPREAD IDEOLOGY INTERVENING IN THE HISTORIOGRAPHY OF MATHEMATICS AS WELL AS IN MANY OTHER NON-TRIVIAL DISCOURSES .

THE SO-CALLED AUTONOMY OF MATHEMATICS; WHAT REALLY INDUCES THEM IN A SIGNIFICANT PROPORTION OF THEIR PRODUCTION

ON AN OLD DICUSSION: PURE OR IMPURE MATHEMATICS

Enviar correo electrónico a mhotero@adinet.com.uy con preguntas o comentarios sobre este sitio Web. Página en la red http://galileo.fcien.edu.uy
Copyright © 2001-2011 Galileo: Publicación dedicada a problemas metacientíficos. ISSN 07979533
Última Modificación: 26 de febrero de 2012

PHILOSOPHICAL PAPERS ON MATHEMATICS

CONTENTS

Avatares…

Avatares…

KUHN’S PHILOSOPHICAL TROUBLES WITH ACTUAL SCIENCE HISTORY (1)

ON THE PRESUPPOSITIONS OF SCIENCE: ROBIN COLLINGWOOD’S ESSAY ON METAPHYSICS (1940) AS ANTECEDENT OF A WHOLE ERA

TABLES, CHAIRS, BEERMUGS; OR THE PRE-HILBERTIAN USE OF THE PRIMITIVE CONCEPTS OF DAVID HILBERT’S GRUNDLAGEN DER GEOMETRIE.

A CLASSIC HISTORIOGRAPHIC TEXT: ANDREY N. KOLMOGOROV’S ARTICLE “MATHEMATICS” PRESENTED IN THE 1936 SOVIET ENCYCLOPAEDIA. [1]

Introduction by Mario H. Otero

Preliminary words about the text

A CHOCOLATE COIN? ON A WIDESPREAD IDEOLOGY INTERVENING IN THE HISTORIOGRAPHY OF MATHEMATICS AS WELL AS IN MANY OTHER NON-TRIVIAL DISCOURSES .

THE SO-CALLED AUTONOMY OF MATHEMATICS; WHAT REALLY INDUCES THEM IN A SIGNIFICANT PROPORTION OF THEIR PRODUCTION

ON AN OLD DICUSSION: PURE OR IMPURE MATHEMATICS

Enviar correo electrónico a mhotero@adinet.com.uy con preguntas o comentarios sobre este sitio Web. Página en la red http://galileo.fcien.edu.uy Copyright © 2001-2011 Galileo: Publicación dedicada a problemas metacientíficos. ISSN 07979533 Última Modificación: 26 de febrero de 2012

**TABLES, CHAIRS, BEERMUGS; OR THE PRE-HILBERTIAN USE OF THE PRIMITIVE CONCEPTS OF DAVID HILBERT’S GRUNDLAGEN DER GEOMETRIE.**

Enviar correo electrónico a mhotero@adinet.com.uy con preguntas o comentarios sobre este sitio Web. Página en la red http://galileo.fcien.edu.uy
Copyright © 2001-2011 Galileo: Publicación dedicada a problemas metacientíficos. ISSN 07979533
Última Modificación: 26 de febrero de 2012