Fréchet and the Logic of the Constitution of Abstract Spaces from Concrete Reality

LUIS CARLOS ARBOLEDA and LUIS CORNELIO RECALDE

FRÉCHET AND THE LOGIC OF THE CONSTITUTION OFABSTRACT SPACES FROM CONCRETE REALITY

1. FRÉCHET AND POINCARÉ’S CONVENTIONALISM

In several of his works Fréchet made important commentaries on Poin-caré’s ideas on such topics as conventionalism, mathematical language,intuition and logic, and geometry and experience. In this part of our studywe will refer to the relation between these ideas, how both of them havebeen conceptualised, mainly with regard to the nature and function of con-ventions in mathematics. Nevertheless, beyond the record of a similarity ofideas that becomes evident, what interests us most at the present time, is toidentify the difficulties encountered (at least those faced by Fréchet), whentrying to defend the conventionality position, researchers were graduallyled to elucidate more fundamental problems, particularly the problem ofunderstanding those acts of reasoning that allow the cognitive subject toformulate and to study mathematical entities as conventions. They werealso led to attempt to clarify the genetic relation between mathematicalconventions and empirical reality. We will begin by a quick revision ofsome publications which throw light on the subject in order to locate thisissue within the philosophical and intellectual context of the time.

1.1. The Practical Value of Science Down the Centuries

Poincaré belongs to the generation of philosophers and scientists that, atthe end of the XIX century and the beginning of the new millennium,openly participated in the debates about the new philosophical and epi-stemological tendencies in science and mathematics, in particular, on theproblem of the genesis of knowledge from experience, and the debateson the practical value of science. They also reflected on the relationshipsbetween intuition and logic that were gaining such importance at a timein which the movement of the arithmetisation of analysis was becomingconsolidated. We will begin by referring to the first two of these tendencies.

Synthese 134: 245–272, 2003.© 2003 Kluwer Academic Publishers. Printed in the Netherlands.

246 LUIS CARLOS ARBOLEDA AND LUIS CORNELIO RECALDE

Around the year 1900, in contrast to ideas of a static reason, onlycommitted to the rigorous study of eternal truths, intellectuals such asNietzsche, Boutroux and Bergson, as also the so-called empiricist critics(Avenarius and Mach)1 were promoting different intellectual values, whichwere more closely related to action and to life as primary realities. As LeoFreuler2 has recalled, at the turn of the century science could no longer es-cape from practical impositions. In contrast to neo-Kantian intellectualism,expressed for instance by Leon Brunschvicg, philosophers like EdouardLe Roy argued that “thought should be lived to be fruitful”, that “it isonly known by its action” and that “knowledge is less the contemplationof a clarity than the effort and movement to descend down to the intimatedarkness of things and to fit into the rhythm of its original life”3.

For this intellectual view it is not possible to continue defending thethesis that the laws of logic and mathematics, like the laws of nature, arenatural and eternal truths. On the contrary, the former would constituteinstead of a language – a type of “intellectual shorthand” (James) – in-vented by man to register his observations of natural phenomena. On theother hand, the knowledge of the laws of nature is not a natural agreementbetween human understanding and nature, as man thinks and registerssuch observations according to his practical needs. It is better to considerthe correspondence between this knowledge and natural phenomena fromthe perspective of an agreement among human beings. We are then here– according to Freuler – before one of the original sources of that con-ventionalism of which Poincaré figures as one of its most characteristicdefenders.

1.2. The Conventional Character of Geometry in Poincaré

Poincaré’s position with respect to the nature of the geometry’s axioms4

is well known. These are not synthetic a priori judgements, as Kant re-quired from arithmetic propositions (whose true example is the principleof induction). If they were, they would impose on our understanding insuch a way, that they would make it impossible to conceive an axiomaticin an opposite sense, on base of which to erect a theoretical constructionlike non-Euclidean geometry.5 Furthermore, geometrical axioms are notexperimental facts. Ideal straight lines or circumferences cannot be exper-imented with in the same way as material objects. In his 1895 publicationabout space and geometry,6 Poincaré would make this approach moreexplicit establishing the distinction between geometrical space and repres-entative space. This is based on sensorial experiences (visual, tactile, andmotor) and on associations among them. Geometrical space on the otherhand, corresponds to a more complex level of conceptualisation and of

FRECHET AND THE LOGIC OF THE CONSTITUTION OF ABSTRACT SPACES 247

organisation of the sensations, compared with representative space. Whatgeometry borrows from experience, are the properties of (ideal) bodiesthat inhabit representative space. This is the aim of geometry: the laws ofdisplacements of bodies, not the bodies themselves. The concept of a groupof transformations to which the study of geometry is applied,7 pre-existsin our understanding. It is presented to us rather as a form of our under-standing than as a form of our sensitivity. As it is not possible to considergeometry as experimental science, its axioms are therefore conventions:

Our choice among all possible conventions is guided by experimental facts but remainsfree and only responds to the necessity of avoiding all contradiction. For this reason itspostulates can be rigorously valid, even though the experimental laws that have determinedits adoption are only approximations.8

It is convenient to note the historical character of this affirmation. Poincarésays that once they are adopted with the help of experience, geometricalpropositions are subject to a demand for freedom which can only beconditioned by the principle of non-contradiction. Thus, Poincaré makesgeometry comply with one of the most important epistemological canonsregarding the rigour and arithmetisation of the mathematics movementin the second part of the XIX century. Hankel, Dedekind, but above allCantor, have insisted that mathematics can be defined in relation to othersciences, precisely because is a free creation of our understanding. Cantorstated that the essence of mathematics is found in its freedom, and that onthis aspect, its simple and inhibited character depends precisely:

Mathematics is completely free in its development, and only knows of one obligation (. . . ):its concepts must be non-contradictory in themselves and, moreover, they must relate toconcepts which have been previously formed, already present and secured, with fixed rulesregulated by definitions.9

Given its conventional character, one geometry can derive from another(regardless of how contradictory its axioms can be) providing the defini-tions are adequately chosen. Therefore, for Poincaré there is not sense inasking oneself if Euclidean geometry is truer than other. It is – and willcontinue to be – the most comfortable one. In the first place because it isthe most simple. No in terms of the intuition we have of Euclidean space,but as far as a first grade polynomial is simpler than a second grade one, orstraight line trigonometry is simpler than spherical trigonometry. Second,the Euclidean geometry is the most comfortable as it agrees quite wellwith the properties of natural solids of the sensitive experience.10 MarcoPanza11 has observed that Poincaré does not have a logical explanation ofthe notion of comfort. That maybe owing to the fact that Poincaré’s thoughtshows difficulties in satisfactorily explaining the process from which the“geometrical hypothesis” are constructed from experience. Poincaré does


not tell us how it is that – from the experience of our physical displacement– our understanding arrives at the mathematical expression of Euclideandistance as a characteristic notion of the Euclidean group of transform-ations. This type of explanation is necessary in order to understand (bymeans other than intuition, as required by Poincaré himself), that Euc-lidean geometry is more comfortable than other groups such as those ofRiemann’s or Lobatschewski’s. Another difficulty related with the previousone is Poincaré’s refusal to accept that the geometrical hypothesis can beplaced within the Kantian classification of analytical or synthetic a priorior a posteriori judgements. According to Panza it is due to the fact thatfor Poincaré the nature of conventions, escapes logical explanation for allacts of knowledge. Poincaré’s conception would state that individuals havethe mental capacity to develop conventions; the intervention of experiencewould consist in offering the signal and providing the opportunity for thiscapacity to materialise.

1.3. Intuition and Logic in Poincaré

More elements could be found in favour of this last interpretation of Poin-caré’s ideas on logic and intuition expressed in his famous communicationto the second international congress of mathematicians (Paris 1900).12 ForPoincaré logic and intuition each fulfill a necessary role. As instrument ofdemonstration, it corresponds to logic to ensure certainty. Intuition is, forits part, the instrument of invention. There are different types of intuition:“in the first place, the type that is based on senses and imagination; next,generalisation by induction, copied in a certain sense from the proceduresof experimental science; finally, we have the intuition of the pure number”.This last is the foundation of the principle of mathematical induction (inPoincaré’s opinion the true synthetic a priori judgement), and breeds “truemathematical reasoning”.13 In front of the two forms of intuition, arith-metic intuition is the only one capable of giving us certainty. Apparentlyin agreement with the intellectual tendencies of the time, Poincaré does notrefer to the synthesis criterion that Kant developed in the Introduction tothe Critique of Pure Reason. Although it is not clear what his criterion is,Poincaré at least wanted to show when he stated that the arithmetic propos-itions are synthetic a priori, that they are based on something that we couldunderstand as a certain assumption a priori to which human beings are ledby reason of its intrinsic nature.14 Thus, this type of intuition is based ona purely intellectual certainty. The principle of the human mind that rulesit is mathematical intuition. Otte has insisted that for Poincaré logic andnumber intuition are functionally related only to the subject and not to ob-jective reality. Poincaré proceeded in a similar way to Dedekind, for whom


the certainty of arithmetic depends totally on our (mental) ability to provethe existence of an infinite system, by infinite repetition of mental acts.15

It means, that the safest intuitions from the mathematical point of view arenot related to an object of concrete reality, but to the mind of the individual.This position, is to some extent contrary to the Kantian tradition, in thesense that all knowledge is knowledge of something. Knowledge is, aboveall related functionally to a reality. According to Kant, the subject’s activerole in his perception of the world, makes knowledge both an activityof human reason and a function of inputs coming from external reality.Therefore:

Intuition depends on experience and experience emerges when the subject is confrontedwith something external to his own mind (. . . ) Poincaré, on the contrary, started frommathematical psychology leaving aside the objective character of the cognitive. As wehave already pointed out, cognition always attempts to understand and explain the natureof mathematics in terms of mathematicians’ activities, without assuming the existence of(mathematical) objects.16

1.4. Fréchet’s Opposition to Poincaré’s Ideas on Intuition

For his part, Fréchet strongly opposed to the conception of Poincaré onnumber intuition, although without mentioning indirectly and without en-tering into a polemic discussion with him. In fact, the ideas to which weare going now to refer, and that according to him were presented in theEntretiens de Zurich do not appeared in the proceedings of that meeting.Fréchet argues that not because in an axiomatized theory arithmetization isimposed, and that intuitive representations or the referents of the physicalworld are excluded in their logical development, that we may affirm thattheory has been exclusively developed for our understanding. Arithmeticcould not be “a town impenetrable by the external noises, where the purespirit reigns”. The integer number is not an spontaneous creation of a logicspirit away from contingencies. On the contrary, is a schematic expressionof a common characteristic of several collections. In the same way as theconcept of mass is a common characteristic of certain collections of differ-ent bodies. Furthermore, the integer number is the “fundamental scientificnotion that was first separated from the complications of human bargainingnot because necessarily it was the simplest, but because it was the mostuseful”.17 In respect to the notion of countably finite, Fréchet says thatthe notion of countably infinite sequence does not appeared in the mindthrough the intervention of the pure intuition of mathematical induction.Is precisely the opposite. Initially the infinite successions of integer num-bers were accepted in arithmetic for the same reasons as the Euclideanstraight lines are accepted: both are comfortable schematisations of con-


crete objects. Later, “from the moment we introduced the considerationof integer numbers, the principle of complete induction is legitimised”.18

Fréchet thought he had thus eliminated the radical separation establishedby Poincaré between the three categories of intuition. To each of these,the character of conformable conventions can be applied which geometryhypothesis have when they are based on experience. The rules of logicthemselves are for Fréchet a product of our experience. An axiomatic is aschematisation which is essentially revisable from the practical rules ofreasoning. If we accept these rules from our predecessors, is preciselybecause our daily experience teach us that if we applied them correctly,we would never make mistakes. Later, we will study Fréchet’s ideas oninductive synthesis that led mathematical thought from the concrete to theabstract. This concept, allows Fréchet to elude the concept of synthetica priori judgement. Somehow in this inductive synthesis the reasoningabstracts, gradually, from the complexity of things, the simple principlesthat constitute the basis of a deductive theory. Therefore, deductive theoryis not a spontaneous creation. It is only because of a persistent delusionin the mentality of mathematicians throughout history, that it was believedthat:

The immediate data of consciousness’, the synthetic a priori judgement, themselves leadto the formulation of the axioms that are the starting point of the deductive theory.19

1.5. “Natural” Conventions and “Comfortable” Conventions in Fréchet

We will now follow Fréchet in his attempt to scrutinise “from a closer pointof view”, according to his words, Poincaré’s thought on conventionalism.20

We will point out in passing that this scrutiny is necessary for him, asin his opinion there has been attempts to relegate Poincaré to a simpledefender of nominalistic positions that give definitive and absolute value tothe scientific constructions. Fréchet probably refers to those like FederigoEnriques, who strongly reacted at the beginning of this century to nomin-alistic conceptions.21 Enriques would easily agree with Poincaré in that thepostulates of pure geometry symbolise positional relationships of bodies.Nevertheless, agreement ceased when Poincaré – according to Enriques –went so far as to state that such geometrical properties do not correspondsto true facts. Instead, they constitute a simple system of conventions thatexpresses physical facts, in the same way as magnitudes are related in ameasurement system. The system can be comfortable, agrees Enriques, butthere is nothing to stop us from changing it. “To ask oneself if a phenomenais possible in a certain geometric system and impossible in the oppositesystem, is to ask oneself if there are lengths that are expressed by metres


and not by English feet”. Enriques can not accept this type of reasoning,as, in his opinion, the conceptual validity of pure geometrical proposi-tions, – that at the same time represent (even though only approximately)certain entities of the physical world – is not a sufficient condition forthese propositions to be the subject of arbitrary choice in relation to realitywhich is represented. For Enriques geometry can not be separated fromthe experience of space, as this is the first representation of the physicalworld.22

For his part, Fréchet has no difficulty in subscribing to the notion thatPoincaré’s conventionalism is based on a criteria of selection of geomet-rical propositions, not so much in that one would be more valid than theother, but in that it is more comfortable. He limits himself to checking thatalthough it is true that there is an arbitrary limit in such criteria, this disap-pears in the development of the theory when convention presents itself tous with an absolute meaning. Fréchet adds the following example to thosealready presented by Poincaré: It is about comparing the income distribu-tion of two populations (English and Italians) {xi} and {yi}, 1 ≤ i ≤ n,comparing their respective “typical value”, a unique number that undercertain conditions represents their order of magnitude. The problem is ini-tially reduced to define this representative or typical value. X is the typicalvalue of {xi}, if X is as near as we want of all xi (therefore X ∈ [x1, xn]).But that is not enough; it is necessary to assign some sense to the notion ofproximity between the points of the space. It is where it is indispensable toestablish one or the other convention:

• Lets define X is as near as we want of {xi}, if and only if the followingcondition is verified:

X = sup{max{|X − xi |}}, where X ∈ [x1, xn], (1 ≤ i ≤ n)

In this case the arithmetic mean is chosen between the minimum andmaximum of the ordered set.

• But the condition could as well be the following:

sup{∑ |X − xi |}, where X ∈ [x1, xn], (1 ≤ i ≤ n)

In this case, as natural as the proceeding one, it must be verified that X isthe median of the xi .

• Finally, in any case, the condition could be defined as a variant of thesame property although in a “less natural” manner:

sup{∑

(X − xi)2}, where X ∈ [x1, xn], (1 ≤ i ≤ n)


In this case, X it is verified as the arithmetic mean of the xi , (1 ≤ i ≤ n).If this third convention has been chosen in the study of the problem of thedistribution of income typical in populations, it is not because is the mostnatural, but because it is the most comfortable. Fréchet adds:

The most comfortable in algebraic calculations [not in the numeric ones] where mathem-aticians know that the sum of the squares have simpler properties that the sum of absolutevalues. Applying this principle has become so common that frequently it is assigned anabsolute value that does not possess.23

1.6. Legitimising Theoretical Conventions and Agreement withExperience

In consequence, Fréchet supports the first two characteristics used by Poin-caré, in his work on non-Euclidean geometry. When is one conventionmore comfortable than other? According to Poincaré when it is presentedto us as the simpler from the formal and analytical points of view. However,once the convention has been established at his formal level, it excludes –for Poincaré in an explicit manner, for Fréchet in an in-explicit manner,and in not at all for Enriques24 – any intervention, at least on this level ofreasoning, of any kind of intuition by faculties of our psychological appar-atus or of conditioning factors of our social and cultural behaviour. Formaland analytical thought is the final instance of conceptual legitimisation25

of the conventions agreed by mathematicians in their activity. At the sametime, it is here where they are able to reduce what constitutes their intrinsiclimitation: the arbitrary of its choice. The method of the sum of the squaresis the simplest convention from the point of view of analysis and operationsin which it is made to intervene in theory, and no mathematician – Fréchetadds – would agree with using any other method in which, for example,the notion of typical value of the {xi} is defined as the products of itssignificant digits.

With respect to second aspect that allows us to distinguish when a con-vention is more comfortable than other – in other words, when it agreesbetter with the characteristics of the phenomena revealed by objectiveperception-, Fréchet takes care to show in his comments that his ideas coin-cide with those of Poincaré. The careful choice of the following quotationdemonstrates his interest in highlighting this coincidence:

Conventions yes; arbitrary not. They would be if the experiences that led the founders ofthe science to adopt them, were not taken into account.26

But Fréchet does not hide the fact that he can not justify a convention, onlyor mainly because of easy or (theoretical) simplification. If he did so, itwould imply that a science is “complete” when it limits itself to postulating


axioms and deriving consequences. For this reason, he reminds Enriques –for whom the arbitrary of the person who defines, does not differ from thearchitect who built a house based on a harmonious project – that exactitudeor even the beauty of propositions in an axiomatic-deductive system, arenot the result of an “arbitrary” building plan (arbitrary in the sense ofbeing a free creation). In the same way as the architect is constrained tohis aesthetic project by the requirements of the solidity of his building,the mathematician is obliged by the choice of his conventions to remain inagreement with nature.

In this way, it is clear that Fréchet is expressing his doubts that math-ematical reasoning can by itself, without resorting to experience, guaranteethe exactitude (the simple character or inclusive beauty) of mathematicalpropositions. In fact, the central aim of his presentation in the Entretiensde Zurich is “to restore the experimental origin of basic mathematicalnotions”. Nevertheless, Fréchet does not offer a (reasonably) satisfact-ory explanation of how the analytical form of a convention is derivedfrom experience, a form in which mathematicians compare conventionsby means of the criteria and properties of the specific theory, finally doesrecognising it as the most comfortable and natural one (the expressions arefrom Poincaré). In the sum of the squares example, Fréchet limits himselfto reiterating the principle of relation between the analytical form andits corresponding empirical problem: to “compare the pecuniary mediabetween the “English” and “Italian” populations through the process ofreplacing (“approximating”) the two populations by the typical value ofeach one of them. Later, he seems to indicate that in the establishing actof convention (the typical value), our intuitions of magnitude or order ofapproximation among the elements of the populations intervene. Fréchetis not interested in providing any explanation about the cognitive nature ofthese intuitions and how their intervention is carried out in the constitutionof the convention. He goes on immediately to compare the different (con-ventional) definitions of typical value as a limit, in relation to the topologyassigned to the space of the population focused.

1.7. Cognitive Subject and Mathematical Objectivity

In fact, for Fréchet the mental capacities and abilities fulfill an importantrole in the subject’s reasoning activity, and allow him to play an importantrole vis-à-vis the object in the process of its constitution. Probably thisdoes not extend to the point that mental ability allows the subject to exhibitthe object directly (for example, to formulate the convention in a finishedmanner). However, it does allow him to “prefigure it” in a certain way, froma class of possible objects. Thus the gifted mathematician has the ability


to discard within the family of propositions that motivates his interestin a particular problem, the class of propositions that are false, withoutneeding to employ inductive mathematical processes. This mental capacityoperates unconsciously in the subject through intuitions, evidences, sub-jective experiences, inventions or faculties such as the researcher’s “senseof smell”.

In a publication during the last period of his life, dedicated to the studyof the life and works of Emile Borel,27 Fréchet would make a more rad-ical stance in the position he had adopted almost thirty years before inhis exposé in the Entretiens de Zurich. He underlines Borel’s conceptionthat mathematics should have a solid basis in concrete reality and in hu-man nature. With regard to the first idea, he reminds us that he expressedthe point of view before that mathematical notions which are “truly newand important” have been suggested by problems thrown up by nature. In”addition to these notions, mathematicians have developed others, throughindependent processes of (“artificials”) experience, in order to harmonise,generalise and simplify the results already obtained. This autonomous de-velopment of mathematics produces results which are “comfortable butnot absolutely necessary”. With regard to Borel’s second idea, Fréchetthinks that the mathematician’s personality or mental characteristics (notthe affective neither the cultural o social aspects) are “more or less inde-pendent of the studied domain”. Nevertheless they imprint a deep mark onhis works, being responsible for the choice of his research problems andhis epistemological focuses. It is thanks to them that:

Some mathematicians concentrate to discover amazing paradoxical situations; to diagnose“pathological cases”. (That) others, on the contrary, only study different cases in orderto modify the definitions and in this way, to present such cases as particular cases thatcould have been predicted. (That) others, who are talented analysts search, for a particularmathematical quantity, to establish their properties, the more precise and useful formulae.(That) others, compare different mathematical quantities, establish their common proper-ties and formulate a theory that allows for the immediate postulation of all their commonproperties.28

Fréchet never thought it to be necessary to go further than the declaration,to the logical explanation of the privilege that he assigns to the cognitivesubject in the production of mathematical objectivity. One possible explan-ation maybe found examining his conceptions on experience as a conditionof possibility of mathematical reasoning, the autonomy he concedes tothe development of certain pure concepts, although they are auxiliariesto the fundamental ones (those obtained by experience), and his idea ofconstruction of mathematics by the process of successive schematising orinductive synthesis, to which we will dedicate the second part of this study.


2. FRÉCHET AND THE MATHEMATICAL KNOWLEDGE AS INDUCTIVE

SYNTHESIS

Fréchet’s ideas on the origin of mathematical notions in experience, aretightly related to his critical postures towards the difficulties of using theaxiomatic-deductive method in the research and teaching of mathemat-ics. This opposition marks practically all his philosophical reflection, bothhistorical or educational on mathematics. Perhaps the first publication inwhich Fréchet expressed himself openly on this subject was in the open-ing conference of the analysis course in the university of Strasbourg, 17November 1919.29 Fréchet was part of the group of professors whose offi-cial responsibility was to higher studies in the province of Alsace, once theFrench political control was established, as a consequence of the territorialdistributions after the great war.

2.1. Fréchet Reader of Arbogast: Critic to Formalism in Teaching

His inaugural talk was a carefully prepared piece; up to what was ex-pected of the strategic mission that was confided to him in a politicallydifficult situation. Fréchet chooses the mathematical work and intellectualbiography of an eminent Alsacian – Louis-Francoise-Antoine Arbogast– in order to revindicate before his audience a set of themes that wereundoubtedly of a central interest in such a context: (a) the originalityof Arbogast’s contribution to mathematics (formal calculus of operat-ors, discontinuity of real functions, the algebraic study of holomorphicfunctions,30 series theory). These helped Fréchet to emphasize that theboundaries of knowledge in mathematics can effectively pass to the dif-ferent regions; not only to central France; (b) his academic developmentand his public performance (particularly in educational institutions and inthe National Convention), shows to what extent, Arbogast knew how tocombine Alsacian tradition with the French esprit; and (c) his intellectualformation in the German culture and the regional influence that charac-terised him, were always far beyond any narrow sectarianism. Alborgastadvocated the establishment of a unified regime throughout the countryfor the teaching of science in the French language, which should be car-ried out in agreement with the requirements of French pedagogy, whichwere notoriously superior, in his judgement to those used in the Germaneducational institutions. We will now comment on this last point.

Fréchet underlines Arbogast’s opinion in the general plan of publicteaching adopted by the National Convention. He believed that the discov-ery method was also the most adequate way of communicating knowledge.Any person, independently of his/her capacities is able to understand the


chaining of ideas used to reach the invented object. For this, he/she onlyneeds to be given, according to his/her intelligence, “the procedure ne-cessary to develop all intermediate ideas between what is already known,which is the starting point, and the unknown point we wish to arrive at”.31

It is not about using the historical method at all costs, observes Fréchet.If in the present state of the theory there is a more direct procedure tointroduce a particular piece of knowledge, compared with the one initiallyused, it would be useless to make the student take an indirect route. Arbo-gast considered this procedure fell within the analysis method. ApparentlyFréchet represented it to himself, at least from the teaching point of view,in the following way: begin by introducing the problem in question in afew words; next step is to establish where the main difficulty lies, and toteach the student how to overcome it through a series of successive ap-proximations. For Fréchet, this method of presentation is most appropriatefor the purposes of public instruction in the first Republic, in the same wayas they were announced by Arbogast:

What he wanted above all, was to proscribe the method that presents science as a kind ofdivine revelation, through a succession of lemmas, theorems, and corollaries, each of whichis perfectly demonstrated, but which succession develops according to a inaccessible andmysterious law.32

Although comfortable, this dogmatism seems narrow to Fréchet, andbesides, does not correspond to the French method of teaching. He re-commends helping to remove from the minds of those then training ascandidates for the mathematics’ aggregation in France, the idea that whatwas important was comfort in the logic of presentation. The origin of thisdogmatism is found in formalist conceptions of institutional organisationand the teaching of German mathematics. “The Germans try to adorn sci-ence purposely with certain mystery and do not mind being obscure as longas they appeared to be profound”. With respect to his teaching of analysisat Strasbourg university, Fréchet describes his orientation in his inaugurallecture:

Our ideal is completely the opposite; we would like to be so clear and simple that whenthe lesson is over the student can say to himself: “how is it that I did not think like thisbefore myself?”. Probably it would go against our immediate prestige, but would be to ouraudience’s benefit.33

2.2. Fréchet’s conceptions on axiomatisation and “désaxiomatisation”in abstract spaces

In a conference given in Berna in 1925,34 Fréchet returns to these consider-ations about the type of exposition which would be more convenient in the


teaching of mathematics, setting aside, for the time being, the chauvinistictreatment of the question he had used 5 years back. Without disowningthe importance that the axiomatic-deductive method had then attained inmathematical activity, thanks to Hilbert’s school in Göttingen, he set outto explain in his article, the interest that a program of “désaxiomatisation”carried out in a parallel fashion, would have in teaching and research.He makes it clear that he is not an adversary of the ever more dominanttendency of basing science on the smallest possible number of simpleprinciples. He himself had used this method with the “utmost persever-ance” in a considerable part of his work, between 1904–1925.35 His mainconcern in this research has been to separate and extract from the linealset theory (parts of R and Rn), and from the theory of real functions aswell, those properties that do not depend on the nature of the objects inquestion. Without Fréchet using this terminology in any way, we may agreein that the formal class of these properties or mathematical propositions isprecisely what he proposed to call the abstract spaces theory (spaces inwhich certain topology is defined).

Based on this first class, Fréchet built another category of propositionsand theories, which was more complex in formal ways: general analysis,one of whose chapters is precisely functional analysis. The interventionof the deductive method consisted, according to Fréchet, in making avail-able an adequate choice of axioms to establish increasingly general kindsof propositions and mathematical entities. For instance, the axiom of themetric and the topological properties associated with this notion, allowedhim to structure the theory of metric spaces. This theory made possiblethe study of several kinds of functional spaces, (continuous functions, ana-lytical functions, curve spaces, etc.) which from then onwards, share thefact of having that same structure. The axiomatic-formal procedure wasused again by Fréchet to characterise the topology of space not in termsof metric, but of convergence of numerable sequences or of neighborhoodfamilies. From this, more general theories or classes of topological spaceswere delivered.36

What is it then according to Fréchet the function of the “désaxiomatisa-tion”? We need first to remember what representation Fréchet constructs ofthe functioning of the axiomatic method? This implies a double interven-tion: (a) to constitute mathematical objects from empirical objects; or, inFréchet’s terms to deduct the definitions from notions introduced from ex-perience, according to logical procedures; and (b) a second operation thatcan be understood as the formulation and demonstration of propositionsthat, through convenient hypothesis and certain modalities of reasoning(observation), affirm properties of such objects. In Fréchet own words, this


second intervention would consist of “trying to prove logically the lawsof observation from convenient hypothesis”.37 Then “désaxiomatisation”consists of carrying out with the sciences that have reached a high degreeof axiomatisation, an inverse procedure from the one carried out by theunderstanding when constituting mathematical objects from empirical ob-jects (non-elementals). He considers that if the main task of the scientificresearcher to contribute to the development and (formal) perfection of sci-ence, it is not forbidden for him to look back the road travelled and tryto determine the results of individual efforts, with the added purpose oftrying to control the negative effects of tradition and fashion. Althoughthis perspective was frequently applied in mathematical activity, it did notconstitute as yet an established doctrine. Then, Fréchet dedicates his 1925conference to specify and justify his ideas on “désaxiomatisation” with afew elemental examples: the definition of the length of a circumference,the geometrical definition of the tangent to a curve and the definition ofthe differential of a real value function.

2.3. Experience as the Founding Instance of Mathematical Entities

We will now consider the example of the definition of the length of thecircumference. We will try to decipher Fréchet’s conceptions about themodalities of reasoning that may guide the subject, through acts of repres-entation of a given reality, which is external to him, to construct the objectto which this definition may be applied. Without specifically referring tothis in any part of the paper, Fréchet establishes a distinction betweenmathematics as a theory or class of propositions and as a human activityof reasoning with different modalities which are adjusted to certain logicalprocedures. He thinks that the subject is faced with a world of objects inperceptible reality, towards which he has formed representations of a cer-tain kind, mainly about their space-temporal characteristics. Through hisexperience with these objects, the subject is confronted with the practicalproblem of determining the length of the iron plaque with which to repairhis carriage. He disposes of a serial of concepts and forms of assignationof concepts to objects, that allows him to determine an “experimental no-tion” of the length of the plaque. This notion is the following: it is abouta deformable non-elastic longitudinal plaque, that is applied exactly to thewheel’s contour. This is the first moment of the axiomatisation procedural,as Fréchet understands it.

In vain we may look (at least in this paper) for some philosophicalinterpretation on the relation of the object-subject relationship that allowsus to understand the development of this experimental notion as an act ofreasoning. We will consider this absence more precisely. Let us suppose,


against all evidence, that Fréchet understands the representation of thiscircumference-object, which is still not know in terms, for example, of theKantian thesis on objective perception. But this elemental knowledge (thelength of the plaque that fixes exactly around the wheel) would only bepossible as the first of a series of subjective a priori constitutive acts.38

Later, we will see that Fréchet would never agree with an interpretationlike this. In reality, what Fréchet thinks is that that notion is “imposed”on the subject by experience, where experience – non-definable instance;information prior to all conceptualisation – would have, at the same time,a kind of intrinsic capacity to project this experimental notion on to thesubject. We would then have, up to now, two characteristics of Fréchet’sconception on experience.39

The philosophical explanation is not less absent in what refers to thesecond moment of the axiomatic method. In other words, the act of thesubject from which the “logic definition” results, “which is found in allgeometry books”: the length of the circumference as the limit of the totallength of a regular convex polygon inscribed in the circumference, whenthe length of the side tends to zero. The geometrical (or logical) definitionis different from the physical or experimental type, in that it is a com-bination (of course logical) of preceding notions. Fréchet does not think,naturally, this combination in terms of judgements that connect concepts ofobjects, and concepts of properties and relationships, even less of the dif-ferent acts involved in the exhibition of those objects and class of objects,and in the connection among them. This would have forced him to think ofthe criteria that the cognitive subject should mobilise in his consciousnessin order to individualise objects, and to produce the definition of length as asynthesis of reasoning. And, therefore to take into account the interventionof pure intuition that secures the unity of objective consciousness in thesynthesis. Fréchet escapes once again the a priori issue, having recourseto his idea of experience as original instance. The only objective of thegeometrical definition – according to Fréchet – , is to allow for the pre-diction of the physical evaluation of length. This is because for him, thereis no logical guarantee that the number corresponding to the geometricaldefinition agrees with the number that expresses the physical definition.The concordance is only probable (vraisemblable). It would originate in:

A series of experimental observations unconsciously stored in the understanding. The geo-meter already knew that when he placed a piece of string on wheels which are slightlyirregular, but of the same diameter, he would find the same length.40

We then find a third idea relating to Fréchet’s conception of experience:this would be the only irrefutable guarantee41 of the validity of math-ematical propositions. Because of this, it is necessary for him to do the


opposite thing, to return to experience to exam the correspondence ofthe geometrical definition with the experimental one. In other words, itrequires what Fréchet calls “désaxiomatisation”, which aims at the director indirect verification of the mathematical result of a physical hypothesis.The verification of the concordance between the two definitions is morenecessary than the teaching, as it is essential to make the student under-stand that all our science only give us an approximated idea of reality;and that an inductive theory can not by itself, explain the world of sens.42

When the presentation method begins with the announcement of a systemof axioms, the student refuses to accept as simple or intuitive the notionsintroduced by way of concepts, or the laws introduced as postulates. Thestudent must be helped in the reconstruction of the abstraction work carriedout by the author of the theory, in order for him no to deny his trust tothe theory. The teaching of mathematics has to take into account that theaxiomatic utterance and the deductive part of the theory are the result of aprevious work. This previous work constitutes the moment of justificationof all axioms. Jean-Louis Destouches had studied it as inductive synthesis:the first of the three parts of the construction of all physical theory. In hiscommunication for the Entretiens de Zurich Fréchet revisits Destouches’ideas and proposes to add a forth section to his classification, preciselyrelated to the verification of the agreement between what is abstracted,with concrete reality. This is the operation which he calls “désaxiomatisa-tion” and he assigns it an important role in the teaching of science andmathematics (see note 15).43 We will now examine, in general terms, whatDestouches-Fréchet’s classification consists of.

2.4. The Inductive Synthesis and the Kantian Position on the Synthetic apriori

Destouches formulates these ideas in the context of his thesis on the formof physical theories.44 In any (physical) theory, there is a preliminary partcalled inductive synthesis, which contains all reasoning that make up thegeneral ideas at the base of the theory and its presentation as axiomaticutterances. The other part of the theory is deductive: it is formed by a setof results from which the axiomatic utterance is deducted as the applicationof rules of reasoning. According to Destouches, then, in a physical theorythree parts must be considered: inductive synthesis, axiomatic utterance,and a deductive part.45 The axiomatic utterance marks the end of the in-ductive synthesis and the beginning of the deductive part. All theoreticalnotion, (for instance geometry) are not purely arbitrary construction ofreasoning. They are the result of a mental process of schematisation andabstraction from the physical reality. Besides, Destouches emphasises, it


is a schematization that proves to be efficient and useful in its summaryapplications and realisations where it was originally understood. As far asthe rules and laws of reasoning that intervene in the constitution of thetheoretical notion (form), they are neither a priori reality nor posses anabsolute formative character a priori. They are separated (dégaées) fromphysical reality and are one of the first theoretical constructions by thesame process of schematization and construction that gave rise to conceptsand other theoretical notions. The theoretical form appears at momentsin which the rules of reasoning and definition relate both terms and ut-terances. From certain notions it is then possible to move to others bymeans of definitions. Equally, it is possible to pass from one proposition toanother using the rules of reasoning and the original utterances. As theseutterances (concepts and postulates) are no evident in themselves, it isnecessary for the understanding to intervene in their acceptance througha process of inductive synthesis.

Among the different modalities of reasoning that constitute the induct-ive synthesis, Destouches stresses the following: experimentation, intuitiveknowledge, previous theory considered as a summary of the theory thatneeds to be created, the combination of partial deductive theories, anddiverse forms of induction and analogy. Fréchet undoubtedly adopts thisclassification in his conference of Entretiens de Zurich and, based on someexamples, he shows how some of these forms of thought associate withdifferent types of mathematical theories. It is not only the developmentof fundamental mathematical objects, that as we have already mention,originates, according to Fréchet, in experience. It is also the more abstracttheories and notions, those that are “imagined by the mathematicians asuseful devices that have not been imposed from outside”.46 But what isabsent both in Destouches as in Fréchet, is a satisfactory philosophicalexplanation of the modalities of reasoning that made up the classificationschema. In this necessarily global view of Destouches’ thesis on inductivesynthesis, we will focus on one idea that shows that although inductivesynthesis aims at interpreting knowledge as a human activity of conceptualschematisation from reality, it reflects a convincing philosophical explana-tion of mathematical reasoning, as the Kantian thesis of a priori synthesis.At least in one important point: the no acceptation of the subject’s capacity(a priori) to produce concepts of pure forms through a synthesis of thought.Destouches states that in the different types or forms of thought of hisschema, there is always a “subjective element”.

But he understands it as: “a reasoning that some accept as justifyingan axiomatic and, in consequence self-evident, would not be accepted byothers”.47 Nevertheless, in the Kantian thesis of the ‘Introduction’ to the


Critic to Pure Reason, as Panza remind us, from the moment an objectbecomes a subjective act, its relation to a pure form is seen as possible.It is pure intuition that will guarantee the availability of these pure forms(for example, circles and straight lines) and its compositions with otherswhich are more complex (triangles and polygons). Equally, there is thepure intuition that guarantees the unity of subjective consciousness thatconnects concepts to pure forms through real judgements, in the sense thatthese synthetic a priori judgements express the conditions of a discursiveknowledge a posteriori. These judgements are the dynamic principle of thepure understanding “analogies of experience” and “postulates of empiricalthought in general”; they are rules “according to which a unity of exper-ience of perception can emerge”.48 To sum up, if on one hand Fréchetagrees in understanding mathematics as acts of reasoning that lead fromempirical definitions to mathematical definitions, on the other, he doesnot accept that synthetic judgements, through which such definitions areformulated and discerned, have the character of a priori judgements. Thisnegation would lead him to maintain ambiguous stands, as some of hiscontemporaries pointed out to him on several occasions.

2.5. The Dialectics of Gonseth: Mediation Between Fréchet’s“Empiricism” and Enriques’ Idealism

In the Entretiens de Zurich, the positions advocated by Fréchet on theorigin of mathematical notions and the inductive synthesis gave rise tocomments and criticisms from several of the participants: Gonseth, En-riques, Bernays, Lukasiewicz and Lebesgue. We will now look moreclosely at what the first two of these said about the subject we have beenconsidering.49 Enriques agrees with Fréchet that mathematics can not bereduced to its formal or logic aspect. However, he criticises him for falleninto excessive empiricism when trying to abandon pure logic. If it is truethat this was a current philosophical position of that time, its explanationseemed insufficient to him. Essentially Enriques argues for the need tocharacterise the mathematical object as an idealised object, not as an objectof direct experience. He accepts that Fréchet is right when he proposes adidactic strategy of geometry as the science of reality, but not at the cost ofcreating for the student grater difficulties in his interpretation of the natureof the act of reasoning. Certainly, for some ends within this strategy, it ispossible to suggest that the students should represent geometrical objectsas solid objects. But there is need to solve the obstacle of understanding,for example, the abstract rectangle as the measure of all possible materialrectangles. The measure of all possible cubs is something very near to acub, but it is not the cub itself. Nevertheless, Enriques goes further when


he states that “the idea of cub pre-exits in the understanding of each one ofus. (. . . ) the mathematical objects are not material objects, but ideals de-veloped in human understanding by the intimate laws of the structure of thespirit”. Fréchet answers that “the intervention of the understanding doesnot consist in creating the fundamental concepts of human thought in acompletely developed form, but only in separating the essential charactersof certain class of concrete objects leaving aside the secondary particular-ities. In this way, certain concrete objects are made to correspond with anideal object, a simpler one, and therefore one showing better the imprint oflogical reasoning, but one that seems the most possible one to the concreteobject”.50

At the end of the debate, Gonseth acknowledges the coincidence ofthe two points of view, that he has stated in several of his works, withthe ideas of Fréchet; particularly in two aspects: (a) the empirical rootsof the fundamental mathematical notions, and (b) the commitment of allour conceptual apparatus to experience in its most ample sense (the lat-ter should include mental experience, according to Bernays’ suggestion,as well as experience of the physical world). But he objects to Fréchet’sformer statement: “When Mr. Fréchet talks, for example, of the creation ofa simplify image by the elimination of secondary qualities, Mr. Enriquescould respond that none of those eliminations that do not already supposethe idea of the notion to abstract”. Further on, in the ‘Conclusions’ (seeNote 20), Gonseth reiterates that Enriques’ objection is a serious one, andthat the geometrical idea of the cub is not an experimental measure amongall possible realisations of the cub; that mathematical beings are idealobjects, created by the understanding with a certain independence fromthe immediate and conscious experimentation. And given that, previously,Bernays had pointed out that the debate could not be confined within tra-ditional philosophical stands, which are extremely schematic and simple,he takes on himself the task of reviewing such positions; especially as therelationships between intuition and experimentation, had been, if that werethe case, suggested in the interventions. From this perspective, Gonsethaligns himself with Fréchet against the idealist conception expressed byEnriques, that creation is a product which is completely independent fromunderstanding, and was simply under the immanent demands of reason.He proposes to interpret the question through the idea of dialect inspiredby some of the philosophical comments of Lebesgue’s presentation to theEntretiens.51

Lesbegue had said that mathematical activity develops as a double ten-sion between the study’s theme (object and aim to be reached) and theappropriate mode of reasoning. Gonseth proves that this idea has already


been validated by historical experience, and that it is consistent with thefact that scientific activity is not informed automatically and exclusivelyby the formal logic. Then, he suggests an interpretation of Lebesgue’sposition, based on a three level dialectic: (a) the meaning of things whichare been talked about; (b) the purposes of thought when talking aboutwhat is being talked about; and (c) the already accepted ways of talk-ing about the subject with good sense and efficiency. On the horizon ofdialectic, Gonseth adds, logic is found: the order of the symbolic, at thelimit, eliminates the dialectic’s three moments; or, in other words, in logic,the intentional dialect’s order disappears. Now, mathematical activity inits widest sense should incorporate, besides knowledge of a psychologicalorder, the knowledge of the world of natural and physical reality. Then,Gonseth proposes a classification of mathematical activities according tothe different essential intentions they are aiming at: (a) Dialectics of sen-sation: Intentional activities addressed to the real world and controlledby sensitive intuition (geometry, cinematic, the mathematical theory ofcolours); (b) Dialectics of the systematic experience: Intentional activitieslike the former ones but using systematic experience to intervene (rationalmechanics, classical or relativist mechanics and other mathematisation ofphenomena); and (c) Dialectics of our elemental behaviours: Intentionalactivities and elemental behaviour related to the most primitive aspects ofthe physical and mental world (elemental arithmetic and logic).

From this schema, Gonseth believes it possible to establish a comprom-ise between Fréchet’s realism and the idealism of which Enriques was thespokesman, at least on one occasion. In sensation dialectics informationis exercise through a priori forms of intuition. These forms are unques-tionably normative in character. The ideas that they develop are reasonentities in Enriques’ sense. But it is necessary not to forget that contactwith the real is established precisely through the forms of the intuition andthe senses. Therefore, the content of such forms should be thought of asschematic representation of reality. The same can be said about all math-ematical activity that points, directly or through intuition of the senses, toknowledge of the real. This experiment does not exhaust all possibilities ofmathematical activities. There are the dialectics which are directed towardsthe infinite and the formal. The mathematical objects that intervene hereare not schematic images of a certain reality obtained immediately, fromthe “natural and not specifically mathematical play of intuitive forms andthe “previous” categories that all human adults have at their disposition.They are abstract entities created by the mathematical imagination, or im-mediate intuitive representations, or by analogy and extension”. They arethe objects of general analysis and the theory of abstract spaces to which


Fréchet refers in the second part of his presentation. As Lebesgue had in-sisted that the objects in this new dialectic should be thought of at the sametime as the logic that is convenient to them, Gonseth adds that these lastdialectics are not a formalism. In the same way as “first degree” dialecticsare based on the intuitive knowledge of the objects to which they refer, theknowledge of “second grade dialectics” also involve their own intuitionand their own evidence, even if they are not explicitly formulated. Theguarantee of their coherence and legitimacy is given by the fact that activityat this level develops in a symbolic universe which is more or less conven-tional, influenced by intuitions and evidences. With this articulated schemaof evidences, Gonseth has attempted to contribute to the characterisationof the intentions that underlie the different manifestations of mathematicalactivity. This is an attempt at explanation that tries to overcome the levelof general formulae, devoid of content. It is part of a mathematical philo-sophy, yet to be built, says Gonseth, that should be rigorous and adequate.An “utilitarian philosophy” destined above all, to examine and explain howthe mathematical thought operates; how mathematics are built in realityand in a precise fashion.

NOTES

1 Analysing the impact of these conceptions on young Einstein, Michel Paty has donean interesting panoramic revision on the essential aspects that characterises “the nov-elty” of these philosophical stands. Particularly in regard to the experience’s problem andthe foundations of the modern physic-mathematical sciences: Paty, M.: 1993. EinsteinPhilosophe. Paris P.U.F. The last two parts of this work Parcours épistémologiques andConstruction théorique et réalite, are very informative on this question. In the present paperand in others on the philosophical problematic of mathematics and experience, we havegained from this and other Paty’s studies, from his generous personal talks in Colombiaand France, as well as from his conferences and courses as a visiting professor at Univer-sidad del Valle (Cali). Particularly important, for clarifying several punctual questions onPoincaré, was his December 1997 course in our Ph.D academic program on mathematicaleducation.2 Freuler, L.: 1995. Les tendances majeures de la philosophie autour de 1900, in M.Panza and J.-C. Pont (eds.), Les savants et l’épistémologie vers la fin du XIXe siècle. Paris,Blanchard, pp. 1–15.3 Le Roy, E. (1901): ‘Sur quelques objections adressées à la nouvelle philosophie’, Revuede métaphysique et de Morale, IX, pp. 292–327, 407–432; cited in: Freuler (1995), p. 9.4 Poincaré, H.: 1891, ‘Les géométries non-euclidiennes’, Revue générale des sciencespures et appliquées 2, 769–774. in Poincaré (ed.), (1902): La science et l’hypothèse. Paris,Flammarion; Chap. 3. In this and other relating parts to Poincaré’s conceptions on spaceand geometry, we have profited from Paty’s readings of this part of Poincaré’s studies. SeePaty (1995, pp. 250–264).


5 For the same reasons he considers they are not analytical judgements. See H. Poincaré,(1886–1887): ‘Sur les hypothèses fondamentales de la géométrie’, Bulletin de la SociétéMathématique de France XV, 203–216. At the beginning of his studies on non-Euclideangeometries, he affirms too, that the characterisation of geometrical propositions will notconcern itself with analytical a priori judgements. These are not geometrical propositions,but belong to analysis. They are axioms of the type “two quantities equal to a third one areequal between them”, on which all educational science are based: Poincaré (1902, p. 63).6 Poincaré, H.: 1895. ‘L’espace et la géométrie’, in H. Poincaré (ed.), (1902, Chap. 4),Revue de métaphysique et de morale 3, 631–646.7 For Poincaré geometry is the study of the formal properties of a certain group of trans-formations. Almost ten years before his 1895 work on space and geometry, Poincaré hadset himself to determine the conditions that a group of transformations should be con-sidered a geometry. In particular an Euclidean geometry in two dimensions. The criteriafor the selection of such conditions, are precisely the simplicity and comfort regardingthe representation of physical phenomena. See Poincaré, H.: 1886–1887. ‘Sur les hypo-thèses fondamentales de la géométrie’, Bulletin de la Société Mathématique de FranceXV, 203–216.8 Poincaré (1902, p. 75).9 Cantor, G.: 1883. Über unendliche lineare Punktmannigfaltifkeiten, 5, Math. Annalen21, 546–586. French translation (1883) in Acta Mathematica 2, 381–408.10 Poincaré (1902, p. 76).11 Panza, M.: 1995a. L’intuition et l’évidence. La philosophie kantienne et les géométriesnon euclidiennes: relecture d’une discussion. In Panza and Pont (1995; pp. 39–87). Seeparagraph 4.4 Poincaré: Géométrie et groupes de transformations, pp. 65–68.12 Poincaré, H.: 1905. La valeur de la science. Paris, Flammarion. See 1970 edition,L’intuition et la logique en mathématiques, Chap. 1, pp. 27–40.13 Poincaré (1970, pp. 33, 39).14 Otte and Panza: 1997a. ‘Mathematics as an Activity and the Analytic-Synthetic Distin-tion’, in M. Otte and M. Panza (eds.), Analysis and Synthesis in Mathematics. History andPhilosophy, Kluwer, Dordrecht, pp. 261–271.15 Otte, M.: 1991. O formal, o social e o subjetivo. Uma Introdução à Filosofia e àDidáctica da Matemática. São Paulo, Unesp. Translation from the original in German: M.Otte (1994) Das Formale, das Soziale und Subjektive. EineEinführung in die Philosophieund Didaktik der Mathematik, Suhrkamp, Frankfurt a. M. See particularly Chap. 15: ‘In-tuição e lógica em matemática’, pp. 301–318. Of interest on the same subject is: Otte,M.: 1994a. ‘Intuition and Logic in Mathematics’, in D. F. Robitaille, D.H. Wheeler andC. Kieran (eds.), Selected Lectures from the 7th International Congress on MathematicalEducation, Les Presses de l’Université de Laval, Sainte-Foy, Québec, pp. 271–284.16 Otte (1991, pp. 310–311).17 Fréchet: 1955. Mathematiques et le concret, Paris, P.U.F., p. 18.18 Fréchet, op. cit., p. 21. Further on page 32, when examining Poincaré’s ideas on con-ventions, Fréchet agrees with those that object to Poincaré for having placed the completeinduction principle outside experience.19 Fréchet, op. cit., p. 28.20 Fréchet’s comments on Poincaré’s conventionalism were made in his 1938 lecture inthe Entretiens de Zurich, but was not published in the memoirs of that meeting and editedby Gonseth and published three years later: Fréchet, M.: 1941, ‘L’Analyse générale et laquestion des fondements’, Les Entretiens de Zurich sur les fondements et la méthode des


sciences mathématiques, Zurich, Leemann, pp. 53–81. Fréchet only published the com-plete text of this conference fifteen years later, with a title that emphasises the geneticcharacter of the question: “Les origines des notions mathématiques”, in M. Fréchet: 1955.Mathématiques et le concret, Paris, P.U.F., pp. 11–51.21 Fréchet was an assiduous reader of Enriques, and cites him on several occasions in hislecture of the Entretiens, and with whom he kept an interesting discussion to which we willrefer later. In this conference, Fréchet mentions his publications in the list of cited authors:Enriques, F.: 1912. ‘La critique des principes et son rôle dans le développement des math-ématiques’, Scientia 12, 59–79. The critical positions on Poincaré and conventionalismwere originally published in the second part of: Enriques, F.: 1906. Problemi della Scienza,Zanichelli, Bologna. This part was translated into French in F. Enriques (1919), Les con-cepts fondamentaux de la science: leur signification réelle et leur acquisition psycologique,Paris, Flammarion.22 Enriques (1919, pp. 11–12).23 Fréchet (1955, pp. 32–34).24 “Enriques finds the justification of the conventions either in the critically evaluateddata by the psychology of senses and the analysis of sensations, or in the general lawsof association of ideas”, in L. Rougier (trans.), ‘Avertissement’, in Enriques (1919, pp.1–2).25 This terminology is not foreign to Fréchet’s discourse when he expresses his ideas. Inthis context he includes the following reference to Poincaré to back up his argumentation:‘(. . . ) All probability problem presents two study periods: The first one is, the so called,metaphysical: legitimates such or such convention. The second, mathematics, applies tosuch conventions the rules of the calculus’, Fréchet (1955, p. 32).26 Poincaré, H.: 1902. La science et l’hipothèse. Flammarion, Paris. See 1968 edition(Chap. 6, p. 128). It is interesting to complete here Poincaré’s quotation: “Conventions,yes; arbitrary, no”; They would be so if it was not taken into account those experiences thatled the founders of the science to adapt them, even if they were imperfect, it is enough toadapt them. It would be good from time to time, to focus our attention on the experimentalorigin of such conventions.27 Fréchet, M.: 1965. La vie et l’œuvre d’Émile Borel. L’Enseignement mathématique,Genève; second part ‘Les tendances générales de l’oeuvre scientifique d’Emile Borel’ (seeparticularly pp. 39–42).28 Fréchet (1965, p. 40).29 Fréchet, M.: 1920. ‘Les mathématiques à l’université de Strasbourg’, La Revue du Mois21, 337–362. Reproduced almost totally in: Fréchet (1955, pp. 368–388), under the title‘Biographie du mathématicien Alsacien Arbogast’.30 A function f is holomorphic in a point z0 over an open domain D of C, if and only if, f

is defined on a neighborhood of z0 and f is derivable in z0. The function f is then infinitlyderivable in z0. If z is holomorphic in all points of D, f is analytic on D. The holomorphicconcept was introduced by Cauchy in 1851, under the name “synectique”. Briot andBouquet substituted this word by the word “holomorphic” in their 1859 study on doubleperiodical functions. See J. Dieudonné (ed.): 1978. Abrége d’histoire des mathématiques.1700–1900 (2 volumes). Paris, Hermann, Vol. 1, p. 147 and ss.31 Quoted in Fréchet (1955), op. cit., p. 383.32 Fréchet (1955), op. cit., p. 383.33 Fréchet (1955), op. cit., p. 383. Immediately Fréchet backs himself up in the followingopinion – that coming from Émile Picard, then life time secretary of the Paris Academy


of Sciences – shows to what extent it was shared by a group of personalities in the Frenchmathematical community of the time, as opposed to their German colleagues: “Usually,among the most illustrious ones, the guiding ideas remain obscure, maybe intentionally(. . . ) the reader walks with difficulty without knowing where to go”.34 Fréchet, M. (1925): ‘Sur une désaxiomatisation de la science’, conference published forthe first time in: Fréchet (1955), pp. 1–10.35 In this year (1925), Fréchet had already finished writing his study on abstract spaces[Fréchet 1928], which would be circulated among a selected group of his colleagues atinternational level. In the book and in Fréchet’s correspondence of the Paris Academyof Sciences, numerous traces can be found of the objectives reorganisation process, re-organisation of propositions, legitimation of criteria, to which the manuscript had to besubmitted to finally adopt the formal organisation of the published text. See: Arboleda(1979, 1981, 1984) (compacidity); Gispert (1980) (dimension); Taylor (1982, 1985, 1987).This too is the year in which Fréchet published an article on which he will elaborate theinteresting ‘Introduction’ to Abstracts Spaces: Fréchet, M.: 1925. ‘L’Analyse générale etles Ensembles abstraits’, Revue de Métaphysique et de Morale 32, 1–30.36 Fréchet (1955), op. cit., p. 2.37 Fréchet (1955), op. cit., p. 3. Fréchet does not explicitly use philosophical categories(Kantians or of other types) in his argumentation. But as he is approaching as a matter offact a philosophical problematic on the act of knowledge, we consider relevant here the useof these type of categories. This has made possible: (a) a more systematic reading of articlespublished at different times and of different purposes; (b) to decipher the meaning of itsargumentation; and (c) finally, to identify and characterise ambiguities in his conceptions.Obviously, if our interpretation of such categories and their application to the study ofFréchet’s ideas are correct.38 As in other parts of this study, we use here Panza’s study dedicated in several of hispublications on Kant’s philosophical thesis on mathematics. Most important for us hasbeen his reading of the Critic to the Pure Reason in Panza (1997a). We have also benefittedfrom his courses on history and philosophy of mathematics in May 1998 and April 1999,in the PhD program on Mathematical Education at Universidad del Valle. We have alsoclarified doubts consulting: Friedman, M.: 1992. Kant and the Exact Sciences, HarvardUniversity Press, Cambridge, MA.39 In an interesting study of the philosophy of Ferdinand Gonseth, Panza criticises hisfrequent use of the experience notion as a first step of the relationship of man with theworld, and judgment of reason, without going further into the formulation of philosophicalhypothesis on the forms of the relationships and the sanctions implied in these statements.“Therefore, as “manifest” experience, all “experience” needs a previous step to formulatethe problem (the gnoseological problem per excellency)”. Panza, M.: 1992a. ‘Gonseth etles prolégomènes d’une logique de la connaissance’, in M. Panza and J.-C. Pont (eds.),Espace et horizons de réalité. Masson, Paris, pp. 23–45.40 Fréchet (1955; p. 4).41 It is important to mention here Panza’s opinion, at the end of his study (Panza, 1997a, p.323): “Therefore, the philosophy of mathematics’ objective to me, is to provide powerfulcategories that allow the characterisation and understanding of mathematics as a typicalhuman activity, and not founded or legitimised on an irrefutable guarantee –although to un-derstand a mathematical theory is also to return to its origins and to clarify (and eventuallyto discuss) its reasons”.42 Fréchet (1955; p. 16).


43 The apart of the Communication of the Entretiens in which Fréchet mentionsDestouches’ ideas, is titled ‘Les quatre parties de chaque science mathématique’ [Fréchet1941, pp. 55–60]. Let’s remember that this is a summary version. It does not include severalexamples in which Fréchet analyses the inductive synthesis study that has allowed to builtin a deductive way, several correlated mathematical notions: the infinite, the numerablesuccessions, and the principle of complete induction.44 Destouches, J.-L.: 1938, Essai sur la forme générale des théories physiques. Thèseprincipale pour le Doctorat-ès-Lettres. Université de Paris (see Chap. IV). On this ques-tion, we have used a sample of the following manuscript that is found in Biblioteca L. A.Arango, Bogotá: Destouches, J.-L, (1955): Cours de Logique et Philosophie générale, 4ed., Centre de Documentation Universitaire, Paris (see Chap. 3). Lets remember here thatDestouches had an intense intellectual relationship with Fréchet. The second facsimile ofthe series ‘Exposés d’Analyse générale’ directed by Fréchet in Hermann, was highlightedby Destouches when he pointed out the importance, that around 1930, was initially re-cognised of the abstract spaces theory (particularly metric spaces) in quantic physics andwave mechanics was mentioned. There, Destouches expresses the following opinion: “It ispossible to say that the abstract spaces theory constitutes the geometrical bases that quantaphysicians should know, in the same way in which the Riemannian spaces constitute thegeometrical base of the relativity theory”: Destouches, J.-L.: 1935. Le rôle des espacesabstraits en physique nouvelle, Paris, Hermann.45 Destouches (1955, pp. 25–26). Fréchet prefers to talk of four parts in each mathematicaltheory: inductive synthesis, separation from them of the axiomatic enunciations, deductivetheory and verification of its consequences. This last part precisely results from the “désax-iomatización” function. Fréchet defines it as follows: ‘verification of the consequences ofthe theory when the abstract notions that figure in it are substituted by the concrete notionsto which they try to represent schematically’, Fréchet (1955, p. 22).46 Fréchet (1955, p. 37). These notions are not schemas of reality. They are mathem-aticians’ abstract creations through reasoning formed by analogy, extension, scaling ofdeductive processes from preliminary theories etc.47 Destouches (1955). Maybe this relativisation of the capacity of the subjective syn-thesis act to produce objectivity explains Fréchet’s determination to add a fourth item toDestouches’ classification: The verification – through the “désaxiomatisation” – of themathematical propositions obtained in the inductive-axiomatisation-deduction chain. It issomething more than objectivity’s natural pulsing, that he demands for the research spirit,to go back to the intentional act that originated reasoning, in order to verify that its purposewas not abandoned half way.48 Panza (1997a, p. 290).49 The outline of this debate was organised by Gonseth as co-ordinator of the Entretiensand as the debates’ chairman. The complete content was published both in Fréchet (1941)as in Fréchet (1955, pp. 45–51). Gonseth’s general comments, in which he analysis theseand other discussions that took place in the Entretiens, are only to be found in the pro-ceedings of the meeting: Gonseth (1941); ‘Conclusions. Sur le role unificateur de l’idée dedialectique’, pp. 188–209.50 Frechet (1955, pp. 47–48). Fréchet had supported a similar idea in his communication,which he identifies with Bacon’s method: “(this) consists of separating, gradually, from theregularities, from the approximated permanencies that we see around us in a multitude ofsimilar phenomena, permanencies more general each time, and gives schematic represent-ations, each time more simple of the sensitive world, but making sure, at each step, that the


approximations obtained remain within their admissible limits (limits conditioned by thesuccessive states of our instruments and measurement methods)” (op. cit., p. 17).51 Lesbegue, H.: “Les controverses sur la théorie des ensembles et la question desfondaments”, in Gonseth (1941, pp. 109–124).

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Universidad del ValleGroup of Mathematical Educ.Carrera 100 no. 1300 A.A.25360 305 Columbia

Fréchet and the Logic of the Constitution of Abstract Spaces from Concrete Reality

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