with the indication of former publication of the
articles, most in Spanish.
troubles with actual science history.
E.Neuenscwander & L. Bouquiaux, Science, philosophy and music
(Proceedings of the XXth International Congress of History of Science,
Brepols, Turnhout (Belgium), 2002.
On scientific presupposition: the
essays on metaphysics (1940) by
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
de la Real Sociedad Matemática Española (2006), v. 9, n. 1.
A chocolate mint? On a certain widely spread ideology
involved in the historiography of mathematics and in many other non trivial
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.
On an old discussion: pure or impure mathematics?
KUHN’S PHILOSOPHICAL TROUBLES WITH ACTUAL SCIENCE
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
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
"...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.
The consequences that Kuhn presents have even a larger
"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.)
Amazing ... Kuhn, as he says, is not far of the strong
"...facts are not prior to conclusions drawn from
them and those conclusions cannot claim truth" (ibid.).
A final confession, advanced earlier as a sketch, is
"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.).
Sensational, then history of real science, what for?
Towards the end of T Kuhn returns to its central
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
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
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
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
1 See Otero (1996).
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
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.
6 See.Otero (1996) and Solís (1994).
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,
Otero, M.H. (1975),
"Tres modalidades de inmanentismo", Diánoia.
Otero, M.H. (1996)
"Apuntes sobre el último Kuhn". Llull, v.19.
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,
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
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,
Zamora, F. (1994)
"El último Kuhn", Arbor, v.148.
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
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:
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
P 2. Every question involves a presupposition.
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.
4. To assume is to suppose by an act of free choice.
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.
P 4. A
presupposition is either relative or absolute...
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…
P 5. “Absolute
presuppositions are not propositions” (chapter 4).
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”
“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.
4. The second part of the final note of the chapter
“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”
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.
“Where there is no strain there is no history”
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
“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.)
The strains towards the breach are such that they
cannot be concealed.
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).
There has been a very clear specific case for a long
“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).
7. And Collingwood sums up his well-supported view as
“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
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
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.
And he goes on,
“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.”
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
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
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.
“Thus styles are in a certain sense
“…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.
ii. Fuller (1998/2002) says:
“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:
“…radical change can occur quite unradically”.
Unlike Hanson, Toulmin, Feyerabend and Shapere,
according to Fuller,
“…conspicuously absent from Kuhn’s account is any
discussion of how argumentation may facilitate this transition” (p.306).
And this would also oppose him to Collingwood
“...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.
(1) Jacobs, L.S. (2002) has also pointed at Polanyi.
(2) Rossi, already in 1986, stated the imaginary
character of the official history character of the so called meta-scientific
(3) Otero (1991)
(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
Collingwood, R. 1940. An essay on metaphysics,
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
(1985) Estado, legitimación y crisis. Siglo XXI, México.
(1988) Conocimiento, sociedad y realidad. Fondo de Cultura Económica, México.
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
(1982) Previsione e conoszenza; un’indagine suglo scopi della scienza. Original
Edition Foresight and understanding; an enquiry into the aims of science,
Toulmin (1967) “Conceptual revolutions in science”.
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.
“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
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.
1. Rewinding up to Pasch (1882)
Fano had already published by 1891 an article (3) where
“On the grounds of our study we include any variety
of entities, which we will call briefly points independent from their nature
And he used, in that article, other expressions of
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
“...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)
The observation is rightfully claimed in the case of
the axioms but
“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).
With this, Pasch confirms his strict but special
“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).
Just before he says:
“The fundamental concepts have not been defined,
because there is no definition whatsoever capable of replacing the observation
of the appropriate natural objects...”
Nevertheless, the deductive character of the discipline
is utterly saved:
“... 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
The fundamental concepts have not been defined. Still,
“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).
There, he also says:
“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
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.
2. Tables, chairs, beermugs. What for?
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:
“Let’s consider three different sets of
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
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
3. Rewinding even more.
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
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
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
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,
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).
The duality is expressed in another way in:
“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,
“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.
5. Historiographical consequences.
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
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.
Blumenthal, O. (1935) In Hilbert, D., Gesammelte Abhandlungen, Springer,
Berlin, v. 3, pages 402-3.
(1882) Freudenthal remarks, not groundlessly, the curious fact that neither
Klein nor Poincaré, although they repeatedly talk about axioms, present cases
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.
Palladino (1985), pages 253.
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
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.
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.
According to Gardiès (1997) there would be fourteen primitive terms in Hilbert
(1899), according to Hilbert, only eight.
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
expression is still used in the 1930s.
Freudenthal, among others.
16 Here the
Frege-Hilbert controversy is not under dispute.
17 We will
not tend to account for the outspread controversies on the precedence of those
who used it.
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.
19 See four
Gergonne’s texts in the References.
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. 
Introduction by Mario
words about the text
the book “Writing the history of mathematics: its
(2002) on the historiography of mathematics, Cristoph
Scriba and Joseph W. Dauben state:
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, (6th
to 5th century B. C.); the period of
elementary mathematics (up to the 16th century);
the establishment of mathematics of variables
(to the middle of 19th 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 19th 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).
long paragraph suffices to highlight the significance of the text we are
(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
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
1. Rolando Rebolledo summarised
an aspect of his view on Kolmogorov:
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
work deals with these subjects and the interrelations among them. From his
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.
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.
it is necessary to point out something else.
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
2. Which is the state of historiography of science in 1936 when Kolmogorov
published his article? I will just note two things.
S. Kuhn (1979)
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)
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.
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.
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
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.
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
3. The structure of the document starts with
of the object of mathematics, connection to other sciences and techniques.
History of mathematics
until the 19th century, and
II comprises a general section, subdivided in sub-sections on
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.
A period corresponding to the creation of variable magnitude mathematics,
subdivided in a general section, 17th century and 18th century.
III comprises the following subsections
extension of the object of mathematics
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.
document also has a Conclusion.
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.
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
– holding a kind of neo-humanism for which mathematics and philosophy are
united, which has been exposed and criticised brilliantly by Lewis Pyenson
on the other hand, adheres in 1936, avant la
lettre, to that critique, which is not negligible.
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.
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
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.
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.
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).
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.
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
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.
“… 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
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.
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
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:
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.
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
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
and he continues with a copious and explosive listing.
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.
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.
4. Vladimir Arnol’d worked
with Kolmogorov in
share, to a great extent, the same basic philosophy of mathematics. He states
is a part of physics. Physics is an experimental science, a part of natural
Mathematics is the part of physics where experiments
Jacobi identity (which forces the heights of a triangle to cross at one point)
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.
the middle of the twentieth century it was attempted to divide physics and
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,
first paragraph of this text only contradicts apparently with Arnol’d’s own
statements such as:
mathematics we always encounter mysterious analogies…” (Arnol’d,
the extent that he quotes Hilbert when he said in 1930:
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 have said something very different?
Arnol’d adds in the same work:
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”
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A CHOCOLATE COIN? ON A WIDESPREAD IDEOLOGY INTERVENING IN THE HISTORIOGRAPHY OF MATHEMATICS AS WELL AS IN MANY OTHER NON-TRIVIAL DISCOURSES .
Dieudonné (1987) remarks:
“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”.
We will consider:
i. the concept of pure mathematics he used.
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
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
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
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.
In this way, the periods to consider would be:
1927-1942, relatively basic stages of mathematical
education and production.
actually taking off in the sixties.
dictatorship, which stops the process when
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.
These statements are decisive:
“The educational systems of Europe have been
conceived, prepared, with the specific and fundamental aim of preserving and
conserving the existing order of things”.
“…politics is the mother of all sciences”
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
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.
We will distinguish, in Uruguay, in the earlier period:
1. Colonial period (to 1825, 1828 or 1830)
2. 1825-1839, from the declaration of Independence
3. 1839-1888, from the foundation of the University
4. 1888-1903, from the foundation of the Department of
Mathematics (actually, Engineering and Architecture)
5. 1903-1915, from García de Zúñiga in
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
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
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
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
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.
3. Applied mathematics?
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.).
Moreover, he states that
“… 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
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.
4. Research and teaching
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,
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).
and family resemblance.
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:
The material world seventy years ago is recognizably
“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
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,
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”
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:
- Research as the description of the dominant type
of scientific activity,
- A new concept of science (Wissenschaft),
- An idea of the improbability of the communication
- A critique of education as a normative ideal for
- Theoretical ideas on the academic lecture and on
the academic dialogue,
- A preference for unity or unities –on contradistinction to the segmentation or hierarchization of spheres of reality (Stichweh 1994).
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
“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”
“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”.
Hilbert would not escape idealism when, according to
“…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
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
“…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
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”
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
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.
7. Succinct comparison
“…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).
8. Mathematics in technical universities
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 , Dhombres [1989, 1998]
and Siegmund-Schultze . 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.
Four characters enter the play:
applied mathematics (concrete, factual, in act
mathematics) in physics;
mathematics for engineers, and
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
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.
Randall Collins, A global theory
of Intellectual change
In Germany, the situation of these issues is expressed
in these words:
“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”
the beginning of the 21st century, Mueller
“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”.
And Russell’s Problems of philosophy, much
“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.
In R. Tobies’ words:
“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”.
Klein himself remarked in 1872:
“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
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
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
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.
9. Little is known about mathematics during Colonial
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;
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
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,
massive transformation of the field that took place after the beginning of the
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:
Mathematics syllaby of 1915, when the Department of Engineering was founded;
establishment of a specialized library, and
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
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.
Monteverde, even with his experimentalist progressive
“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
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
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
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
16. Some limitations to the previous analysis, as
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
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
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.
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
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
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.
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.
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.
utilidad como presunta retórica en textos de matemáticas”, Revista
Brasileira de Historia da Matemática,.
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
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
"...neither the pure theory nor the applied theory are in
all cases epistemically prior".
He also claims
"...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
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
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
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.
6. It is interesting to quote here some recent
“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…”,
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…
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
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)
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”.
- The indispensability of mathematics for sciences
has been pointed out by many authors, but Quine and Putnam have hold that
this indispensability confirms the realism concerning mathematics. This last
thesis has been
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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