La lettre de la Preuve

       

ISSN 1292-8763

Juillet/Août 2000

 
Proof and perception IV

 

M. Otte
Institut für Didaktik der Mathematik
Bielefeld, Germany

  

All mathematical knowledge we might have seems to fall into two categories.

1. Knowledge that certain mathematical claims follow from certain other mathematical claims, or bodies of claims.
2. Knowledge of the consistency of certain mathematical claims, or bodies of claims.

This view leads to the impression that mathematical knowledge is simply logical knowledge. Today, however, our logical investigations rely on mathematics and because of this, logic appears as a chapter of mathematics. Therefore in the end, mathematics appears as discipline which is supposed to justify itself. Such a "naturalism" is rather unusual. The core of this circularity seems to come down to the following. Ever since Euclid and until the 19th century, as a rule, axiomatics was used as an instrument for dealing with the complexities of some realm of intended applications (think of Euclid's axiomatization of geometry and of Newton's axiomatization of mechanics). Ever since Hamilton and Boole, and particularly after Hilbert and Gödel, axiomatics began to be used as an instrument for dealing with the complexities of mathematics itself. It became a means of mathematizing or modeling mathematical theories themselves!
  While the layman, as a rule, is instructed that absolute "what-is"-questions make no sense in the realm of a scientific treatment of objective phenomena, and that man must rather be content with the structural representations of nature in mathematical language, in the case of mathematical knowledge the relational point of view is constantly challenged by another opposite position which tries to give explicit definitions of every mathematical term. In this manner attempts are made to justify the abstract mathematical language through another less abstract language. A typical illustration of this attitude can be found in Bertrand Russell's Introduction to Mathematical Philosophy (Routledge London 1998/1st ed. 1919).
  While Russell emphasizes the concepts of "relation" and "relational structure" with great care because this is conductive to an illustration of the general principle "that what matters in mathematics, and to a very great extent in physical science, is not the intrinsic nature of our terms, but the logical nature of their interrelations", for "we know much more about the form of nature than about the matter", he takes much trouble in explaining what the term "number" means.
  According to the previous principle, what we should be asking for is not "what" numbers "are," but only what the forms used to describe the relationships between them are. This is, indeed, the core of the pragmatic maxim which stands behind Peano and Hilbert's axiomatic approach to arithmetic and according to which, ultimately, mathematical concepts must be defined through the operative possibilities opened up by the forms used to represent these concepts, that is, via the differences in behavior made possible or triggered by them. This can be done without any reference whatever to any absolute arithmetical ontology. Yet, as was said already, an axiomatized theory is not taken as an abstract symbolic game carried out on meaningless signs, but is rather developed in connection with certain sets of possible applications or intended interpretations. Within the historical development of abstract group theory, for example, a theory presented in axiomatic terms from the very beginning, group representations of various kinds (as linear transformations or permutations, for instance) have played an essential role.
  Still, to try to secure the applicability of an axiomtized theory in an a priori fashion by providing final interpretations of the undefined terms is completely wrongheaded. The question, in particular, what numbers are, in the sense of first looking for objects with which numbers can be identified, is ill posed. Not what numbers are is the question, but rather to which purpose numbers are constructed, and which function within the process of cognition they fulfill. Numbers do not exist outside the theory of arithmetic (as points and lines do not exist outside axiomatized geometry; here lies the difference between Euclid's and Hilbert's conception of an axiomatized theory). Hence it is problematical to try, as Russell did, to first obtain some arbitrary objects, say in the frame of set theory, in order to pass these off subsequently as the numbers. Russell claims that Peano´s axiomatic characterization of arithmetic is insufficient if "we want our numbers not merely to verify mathematical formulae but to apply in the right way to common objects". To this one could reply by pointing out first of all, that theories are applied as structured wholes and not in a piecemeal fashion, and second that it is impossible to specify before hand all the possible applications of a theory. Indeed, the applications are often uncovered in time, alongside the elaboration of the abstract theory itself; as Lebesgue once remarked: arithmetic applies when it applies.
  Finally, it must be said that Russell's project cannot be accomplished. Indeed, numbers are neither objects nor simply properties of objects. That numbers cannot be concrete or set-theoretical objects was shown by Paul Benacerraf by providing different set theoretical interpretations of the number concept. Benacerraf's argument is basically quite simple: "It wa pointed out ... that any system of objects, whether sets or not, that forms a recursive progression is adequate. But this is odd, for any recursive set can be arranged in a recursive progression. So what matters, really, is not any condition on the objects (that is, on the set) but rather a condition on the relation under which they form a progression. To put the point differently - and this is the crux of the matter - that any recursive sequence whatever would do suggests that what is important is not the individuality of each element but structure which they jointly exhibit." ((cf. P.Benacerraf, What numbers could not be, in: Benacerraf/Putnam (eds.), Philosophy of Mathematics, Cambridge Univ. Press 1965, 290). Numbers are no sets, but can be identified in a variety of ways with sets. This causes no problem, because what is important is not the individuality of each element, but the arithmetical structure. Numbers, by the way, cannot be identified with whatever concepts either, for this would amount to, establishing the identity of these concepts in the usual manner by means of the axiom of extensionality, only one further set theoretical reduction of the number concept. Russell, when having discovered the set theoretical paradoxes wished not only to reduce number theory to set theory but wanted also to interprete the latter in conceptual terms, in terms of propositional functions. We return to this below.
  The structuralistic philosophy of science (Quine, Sneed, Moulines) generally accepts that the identity of a concept or of a whole theory depends among other things on certain pragmatic criteria by which the possible applications of a theory and the values of the bound variables are identified. But mathematics being justified through mathematical practice ceases to be paradoxical as soon as it is understood in terms of a theoretical practice on systems of signs. This is relational in a double sense: the objects of which a system S of signs are made are defined only in relation to the system as a whole, and reference relationships between S and another system of entities S' (in terms of other systems of signs or of sensible objects) are no longer invoked to explain S through S' or the other way around in an absolute way, but rather as functional relationships in an almost mathematical sense, and these relationships are not, as it were, inherent in the individual objects of the system or in the systems themselves but they depend on the intervention of the human subject as a mediating agent. Without morphisms there is no group theory, and without monotonic functions there is no arithmetic.
  Pragmatism understands reality as a process rather than as something statically given, and therefore searches for fundamental epistemic processes, basing our cognition essentially on two irreducible mechanisms: on the development of language (Quine), and on perception (Peirce). While we may talk about certain things like "society" or "France", we are unable to perceive them. Other things closer to our corporeal existence we do perceive. As Gödel once said: "We perceive objects and understand concepts". And he considered, contrary to Russell, sets as quasi-spatial objects, not concepts, commenting on Russell's argumentation against this extensional view of mathematics that "these arguments could, if anything, at most prove that the null class and the unit classes (as distinct from their only element) are fictions (introduced to simplify the calculus like the points at infinity in geometry), not that all classes are fictions" (K. Gödel, Russel's Mathematical Logic, in Schillpp (ed)., The Philosophy of Bertrand Russell, Open Court, La Salle 1971, 141). Leibniz already had made this latter point quite clear in relation to the ontological status of the infinitesimals of (non-standard) analysis: these may be fictions established by an infinite process, but the relations leading to them are real and perceivable. Cabri Geometre offers plenty of opportunities to observe relational facts, using something like Leibniz' continuity principle.
  All mathematical objects arise from such perceptual contexts. What we perceive are relational sign structures. The notion of perception is to be understood in the sense of semiotics or of model theory, the representation taking place quite possibly in analogy with the medium of our own bodily movements.
  The difference between these two sources of cognition has been clarified by Russell's theory of description. Russell distinguishes between acquaintance with a thing and knowledge about it. It is the second kind, which is stated by a denoting phrase; the first comes by an indication of the thing. Thus acquaintance comes from perception, knowledge through conceptual thought. The Kantian distinction between synthetic and analytic knowledge reappears at this point. In synthetic knowledge an object is ‚directly' given, in analytic knowledge it is indirectly presented, namely by means of some properties or relations. A brief look at the history of axiomatics suggests that the shift from Euclid to Hilbert represented a change from a synthetic to an analytic understanding of mathematics, from a paradigm of evidence to a paradigm of consistency. This is in a certain sense correct, yet it only tells half the story. In order to get rid once and for all of the evidence paradigm in mathematics, Hilbert was led sometimes to say that existence means consistency. But as outlined by Hilbert himself, finite consistency proofs are an attempt to justify the possible through the actual and therefore depend on concrete, actual sign manipulations. Thus, consistency proofs still rely on existence, on indexical signs, that is, ultimately, on knowledge by acquaintance.
  Knowledge by acquaintance is important in mathematics because mathematics is an activity and all its objects are either constructed or are to be understood at least in relation to a system of means of mathematical construction.
  One might object that what is at stake in pure mathematics and with respect to its proof procedures is identity rather than existence. Now with respect the claims of type 1 above identity is assumed to begin with, but with respect to the second type it has to be established. Identity of functions (of propositional functions in particular as well as of concepts) is commonly established by means of the axiom of extensionality: F=G if F(x)=G(x) for all x. But keeping completely silent about the nature of the universe of discourse to which the quantified variables belong makes no sense. From a logical point of view already there exist type-theoretical restrictions on the bound variables. Logical considerations aside, it remains true that our mathematical theories do not refer to a pre-given and pre-structured world as such. Therefore, identity statements make sense only in relation to certain practices and in contexts where there exist individuating conditions, like they are provided, for instance, in the classroom by the blackboard or the computer screen. Proof and Perception part II (Otte, 1998) gives a good illustration as to what this means.
  One might claim, as Russell did at times, that mathematics is essentially intensional, just being a part of language and conceptual logic and that mathematical activity consists in operating with concepts or predicates. Letting aside all logical problems (we do after all have only countably many linguistic expressions), we believe that adopting a certain theory or language implies not only certain modes of expression, but also a conception of the world, or an ontology. The objects to which our bound variables refer thus are not just given. Every access to the world is theory-laden and every theory is characterized by its own system of dissecting the world as well as by the type of meanings ascribed to the objects. On the other hand a theory in general does not fully determine its ontology, as the example of the different number concepts shows. Finally it is a fact that the human subject is capable to single out objects of the external world before she masters language and theory by a combination of perception and activity. Thus the question arises of how perception or intuition on one hand and language and communication on the other work together. We certainly cannot treat this problem exhaustively here. Suffice if to say that mathematical theory always has to employ perception.
  What we perceive are, on the one hand, particular things such as points on the computer screen or on the blackboard. We ascertain the identity of the latter by ostentation, representing them by indices. All the letters we use in geometry and algebra are such indices for singular things. This kind of perception has an important role in the transition from an intensional to an extensional point of view.
  Even if we believe that consistency were sufficient for existence we shall have to secure unambiguous relations of reference. Kant, however, believing that things have essential properties, claims that a different kind of perception is necessary in addition and concludes from this that existential assertions of a more elaborate kind are essential to deal with questions of consistency. Kant's argument is that a contradiction essentially comes from imaging an object and making various assumptions about the postulated object that are inconsistent with the nature of that object. But a contradiction could never arise if we rejected the object. And as conceptual thinking and logic can never categorically assert the existence of anything, existence in the end depends on intuition and perception. But having established the possibility of something by intuition we cannot just reject it no more than we could establish it just by postulation. Our intuitions do not necessarily obey conceptual postulation. Kant gives the following example: Postulating a triangle and then saying that it does not have three angles is a contradiction but "the proposition above-mentioned does not enounce that three angles necessarily exist, but, upon the condition that a triangle exists, three angles must necessarily exist in it" (Kant, B 622). There is, however, never a contradiction if there were no triangle, there is nothing there about which we have made contradictory assumptions. Triangles have however a possible existence within our intuition and proofs of consistency thus provide real knowledge. Kant wants mathematics first of all to be applicable to the real world and therefore emphasizes the importance of an intuition that reaches beyond mere ostentation. "It is a commonplace of newer Kant scholarship that he already knew about non-euclidean geometry and that in fact its very possibility reinforces Kants doctrine that euclidean geometry is synthetic a priori because only its concepts are constructible in intuition" (J. Webb, Tracking Contradictions in Geometry, in: J. Hintikka (ed) Essays on the Development of the Foundations of Mathematics Kluwer 1995).
  The argument of post-Kantian pure mathematics suggests an asymmetry, an asymmetry as it appears in Frege's analysis of A=B in terms of sense and reference (see Proof and Perception II). The reference, according to Frege, being an object, is defined quite precisely; but the sense is not so defined. All that is really provided is a criterion for when the senses of two sentences are the same or are different. The lack of symmetry is due to the difference between existence and identity or object and concept. Concepts or ideas might have an identity, but do not exist in the same manner as particular objects exist. According to Kant, however, they have a possible existence in intuition.
  Taking into account our observations with respect to the application of the axiom of extensionality, we might claim, as Kant did, that our world is always conceptually structured to begin with. In his recent Mind and World (Harvard UP 1994), J. McDowell has claimed that in order to resolve the problems of modern analytical theories of truth, we have, with Kant, to recognize that concepts and intuitions, understanding and sensibility must be integrated even in the mere intake of experiental content characteristic of sense perception. Even when making perceptual judgments we have to employ conceptual ideas. We thus realize that the distinction between ideas and objects is only a relative one, and that all we deal with in our cognitive activity or in our perception are signs rather than objects. In this way the difference between the two types of knowledge is present already in our perceptions. And it is represented there by two different types of sign, namely by indices and icons.
  An image of a unicorn is an icon which does not contain any existence claim, but gives only an image, or, if you wish, a description of this beast. The same is true of a general triangle or of an axiomatic system, etc. But if in the course of a geometrical proof I say "triangle A" is congruent to "triangle B", then I will have to indicate these different triangles and my representation will change character. As has been said before an icon is very often required to resolve confusion. An index is always necessary if I want to solve a problem of reference. An index, however, does not tell me anything about the object indicated. Kant was convinced that all mathematical judgments are intuitive. He distinguishes, however, between pure intuition and empirical intuition. In empirical intuition, only particulars are given to us. It is the kind of intuition we have just spoken of. It is very meaningful in the context of the Cabri-Géomètre as it leads us to new hypotheses via the perception of fixed points and other invariants. We do not obtain mathematical proofs, however, in this way.
  A mathematical proof is tied to a comprehension and thus to a generalization, and this is where another kind of intuition takes effect, that is the perception of ideas or of general concepts. In Part II, for instance, the concept of the center of gravity has led to a proof, which yields a solution of the problem quoted for all polygons. That the midpoint M remained fixed had to be understood as an indication of the invariance of the center of gravity of the whole constellation. The center of gravity is not an empirical thing, but it is immediately accessible to our experience, and thus to our intuition. The idea of number, which Russell so absolutely wanted to see fixed, also becomes perceptible to us through its effects within the system of our activities. We really can perceive the effects, which our ways of using the number concept have, and algebra is nothing but a means of representing these. Although we cannot provide a non-circular definition of the term 'number', we shall certainly recognize a number if we meet one.
  As has been said, such intuitions of ideas are necessary to obtain the mathematical knowledge of the first type in our above classification. Observing a mere fact does not yield a proof yet. A proof requires an interpretative idea, and that means a generalized representation. Such an idea has no reality beyond leading to the ordering and clarification of a confusing situation comparable, say, to that of the person shown in the picture in part I is confronted with. With regard to Nicolas Balacheff sending me this image together with the request to write something about "Proof and Perception", this is the moment where I may indeed consider my task completed. This fourth part "Proof and Perception IV" thus is the last.

  

© Michael Otte

 

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