M. OTTE

DOES MATHEMATICS HAVE OBJECTS? IN WHAT SENSE?

1. INTRODUCTION

Kant introduced two distinctions into the epistemology of mathematics. The first is the distinction between existence and identity (today we would perhaps characterize it as between real and systemic existence). The second distinction made by Kant is that between the factual and the possible. Modern philosophy of mathematics, or various brands of it like Platonism and Formalism, have on different accounts considered the first distinction as irrelevant to the philosophy of mathematics whereas the second distinction has gained more and more prominence since the days of Kant. When Bolzano and others characterized mathematics as the science of the possibility of things they were promoting an analytical ideal of mathematical knowledge. Rather than trying to construct a mathematical relationship, one first asks “whether such a relation is indeed possible”, as Abel stated in his memoir On the Algebraic Resolution of Equations of 1826, in which he presented one of the famous impossibility proofs of modern mathematics. Abel writes: “One of the most interesting problems of algebra is that of the algebraic solution of Equations. . . . But in spite of all the efforts of Lagrange and other distinguished mathematicians the proposed end was not reached. This led to the presumption that the solution of general equations was impossible algebraically; but this is what could not be decided, since the method followed could lead to decisive conclusions only in the case where the equations were solvable. . . . Instead of asking for a relation of which it is not known whether it exists or not, we must ask whether such a relation is indeed possible”. Abel’s theorem is not only paradigmatic for quite a number of impossibility proofs, which culminate in the work of Cantor and Gödel, but also expresses a general feature of modern mathematics, namely the iterative use of its basic concepts, like the notion of set or function. If one aims at possibility only, existence, which Synthese 134: 181–216, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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is fundamentally important in problem solving activity, seems to lose this importance. Both, Platonism as well as the thesis that the question of mathematical existence is reducible to that of consistency, an opinion strongly voiced by authors like Poincaré or Piaget and which can easily be interpreted as just a variant of Platonism, completely ignored the role of symbolization in the transformation of mathematics as a science of the possible. Impossibility proofs, which represent the birth certificate of modern pure mathematics, are not only indirect proofs but also depend on the choice of a certain representation. In order to prove, for instance, that the doubling of the cube is impossible, one has to represent the constructible numbers to show that the third root of 2 is not a constructible number. As Bochner has aptly stated, “The efficacy, in mathematics, of abstractions from possibility and, conjointly with this, of abstractions from abstractions, reveals itself most manifestly in so-called algebra, if, in the present context, we understand by algebra the topic of calculatory operations with symbols” (Bochner 1966, 54). The famous Cartesian “x” by which the yet unknown is made an object of mathematical activity and investigation marks the first essential step of the transformation of mathematics into a science of the possible and ideal. Although the algebraic diagram is necessary already in the representation of the first impossibility encountered in arithmetic, that is in representing the idea of incommensurability, the full scope of algebraic symbolization became visible only after mathematics had been transformed into a practice of relational thinking. In order to achieve this transformation, algebra had to be seen, however, also as a part of combinatorics (a view clearly expressed in Leibniz’s letters to l’Hospital of 1693) and had thereby to be transformed into systems of formal structures rather than merely as generalized arithmetic. In this sense, Descartes did not only algebraisize geometry but also geometrisized algebra, a fact that comes out perspicuously in linear algebra as a structural theory of projective geometry. The algebraic variable “x” as such is just an existence claim and thus an index, and this aspect of indexicality remains important for the algorithmic perspective on arithmetic. But within an algebraic diagram or formula iconicity prevails. A diagram is essentially an icon “even although there be no sensuous resemblance between it and its object, but only an analogy between the relations of the parts of each. . . . Thus, an algebraic formula is an icon, rendered such by the rules of commutation, association, and distribution of the symbols. It may seem at first glance that it is an arbitrary classification to call an algebraic expression an icon; . . . But it is not so. For a great distinguishing property of the icon is that by the

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direct observation of it other truths concerning its object can be discovered than those which suffice to determine its construction. . . . This capacity of revealing unexpected truth is precisely that wherein the utility of algebraical formulae consists, so that the iconic character is the prevailing one” (Peirce CP 2.279). The reason for this capacity of producing new insight and knowledge lies in the fact that not everything can be defined distinctly. An Icon is a form. The index is important to show on which object an idea or form should be applied. It makes no sense calling a diagram an icon, if we had not some unexpected application in mind and if this application were not essential to the further progress of knowledge. It is the interaction between genesis and application, which renders mathematics simultaneously synthetic and analytic. What do we mean by this? First that the possible (and consistency just means truth within a possible world) is nothing but a perspective on the real world and as such is a representation. Second that the distinction between real and systemic existence is to be conceived in representational terms, index and icon representing these types of existence. Finally that mathematical generalization consists in the construction of a suitable representation, which has the “capacity of revealing unexpected truth”, and thus contains elements of observation or perception. By the process of generalization an intuitive and rather vague idea is transformed into a mathematical concept proper and thus becomes an object of mathematical activity. The Pragmatism of Peirce as well as the Neo-Kantianism of Cassirer have compressed the two distinctions made by Kant into one by reformulating them in semiotic terms. Kant had, as was said already, characterized the human intellect, in distinction to Gods infinite mind, by the necessity of making a sharp distinction between the reality and the possibility of things. For Kant, this difference is not ontological, but epistemological. It does not denote any character of objective reality but applies only to our knowledge of things. Both, Cassirer and Peirce interpreted this distinction in terms of the symbolic character of human thinking. E. Cassirer, for instance, in his Essay on Man defines Man as a symbolic being, rather than as rational being, because it is “this character of human knowledge, which determines the place of man in the general chain of being. . . . Human knowledge is by its very nature symbolic knowledge. It is this feature which characterizes both its strength and its limitations. And for symbolic thought it is indispensable to make a sharp distinction between real and possible, between actual and ideal things. A symbol has no actual existence as a part of the physical world; it has a ‘meaning’. In primitive thought, it is still very difficult to differentiate between the

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two spheres of being and meaning. They are constantly being confused: a symbol is looked upon as if it were endowed with magical of physical powers. But in the further progress of human culture the difference between things and symbols becomes clearly felt, which means that the distinction between actuality and possibility also becomes more and more pronounced” (Cassirer 1962, 28th printing 1977, 56, 57). Nominalist philosophies have in general interpreted observations of this kind by characterizing our knowledge as merely representational, intending to indicate that symbolic meanings are merely relative or conventional and thereby denying that a sign could be intrinsically a sign, which then in a way determines its interpretation or use on this account. To be and to be represented appear in disjunctive opposition to one another, and the “Law of Mind” and the “Law of Nature” have nothing in common if we neglect the objectivity of the evolution of symbolic activity. Therefore, the majority of mathematicians, believing in the objectivity of mathematical truths, are skeptical with respect to the importance of the representational aspects of mathematics, and they tend more or less towards Platonism. Peirce, in contrast to Kant as well as to other types of nominalism, emphasizes the ontological foundations of the distinction between the factual and the possible and considers possibility and generality to be more or less synonymous, and actuality and particularity as well, rendering the former distinction just a variant of the distinction between facts and laws. “When I say”, Peirce writes, “that really to be is different from being represented, I mean that what really is, ultimately consists in what shall be forced upon us in experience, that there is an element of brute compulsion in fact and that fact is not a mere question of reasonableness” (CP 5.97); otherwise the application of knowledge could be part of that knowledge itself. A brute compulsion may be a habit or a tacit experience, which cannot be made completely explicit and thus cannot be generalized. And as far as reality is intelligible its laws and our ideas are one kind of entities, that is, are signs. Even if signs had been established conventionally or by apodictic observation or intuition of factual regularities, they take on meaning as soon as they enter into the foundations of successful behavior. This implies for instance that mathematical axioms and natural laws are to be classified as ontologically of the same nature. These general laws or possibilities have an interest only in relation to their factual applications. The application of a law or a theory, however, cannot be explained just in theoretical terms, but demand experience and rather vague and general principles. Every compulsion, “is something which takes place hic et nunc, that is on a particular occasion, and affects an individual person. It is essentially

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anti-general. But the compulsion of rational assent is not merely an individual compulsion; . . . Such a general compulsion supposes a law. . . . The perception, or seeming perception, of a general compulsion, and so of a law, must enter into every inference” (Peirce MS 787 (1897)). Not everything in the world is reasonable and intelligible. There exist brute facts or particular experiences that seem to escape any reasonable explanation. We could express this metaphorically by saying that there are things and signs in the world and that they are relatively independent from each other. Signs or representations are what makes reality intelligible. “A law is in itself nothing but a general formula or symbol. An existing thing is simply a blind reacting thing, to which not merely all generality, but even all representation, is utterly foreign” (5.107). Therefore, the problem of knowledge is to be seen in the question of how particular and general interact, how objects and signs are connected. This question can only be answered from a genetic point of view by explaining how laws or representations arise and how meanings evolve. Peirce in his answers to this question emphasizes the importance of the continuity principle, because intellibility or meaningfulness presupposes continuity, whereas a new and yet inexplicable fact always marks a rupture or discontinuity. The continuity principle represents a top-down perspective on things. Individual objects are identified by the set of their relations and properties, like a point in geometry being characterized by the set of intersecting lines (linear functions). Leibniz thought this to be the only manner of specification and his principle of the identity of indiscernibles became just the complement or dual of the continuity principle. Objects would thus become just instantiations of general concepts and could be identified up to an isomorphism only. Kant, in contrast, argues that two objects being coexperienced in different spatial locations already suffices to secure their distinctness. But he agreed with Leibniz that the general precedes the particular and considered space as a mental rather than an empirical entity, calling it “a pure intuition”, which is indeed “the substratum of all intuitions determinable into particular objects (aller auf besondere Objekte bestimmbaren Anschauungen), and certainly the conditions of the possibility of manifoldness of the latter lie in it, but the unity of the objects is determined solely through the mind (Verstand), and in fact according to conditions that lie in mind’s own nature” (Proleg. §38). This conception does not only separate mathematics from its applications (the latter being characterized once and for all in general terms (KrV B 742), but also transforms it into a closed field, into an area where nothing radically new is to be expected (Comte very aptly calls Kant the father of positivism).

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Scientific discoveries always bring about surprise (sometimes as great as to overthrow a paradigm and cause a scientific revolution in the sense of Thomas Kuhn); and when somebody is surprised he knows that he is surprised by direct awareness, rather than by inference. An intuition, a perception or a quality of feeling, in itself, has no generality; “but it is susceptible of generalization”. Why is this so? Because an intuition which is linked to a perception depends essentially on irregularity and difference. We come to generalize as soon as this difference or this interruption of the original uniformity spreads or is re-encountered again and again. “And indeed all the qualities of feeling we are able to recognize are more or less generalized. In a mathematical hypothesis the qualities of feeling are so subordinate as to be scarcely noticeable” (Peirce 7.530). To summarize: there are three elements involved, an intuition or awareness, a difference or contradiction and a general idea or representation. Any scientific explanation is meant to moderate the surprise by integrating the newly discovered facts into the system of other facts by means of some ideas. Scientific analysis and explanation therefore try to break up a change in the situation or a difference between things into “infinitesimals”. Can the circle have an area although, it is so thoroughly different from the quadrangle (or any other unit of measurement)? It can if one accepts the continuity principle, which renders the difference between the circle and the polygons as small as we please. As was said already, the continuity principle gives relations priority over relata, thus representing the complement of Leibniz principle of the identity of indiscernibles. If a concept (like the concept “area”) or a symbol is to be applied to some particulars, continuity or similarity between particulars is required or a variation among them must be brought about or imagined. An absolutely isolated existent or particular can neither be explained, nor even perceived. A difference that does not persist cannot be real. But a difference or distinction, which remains just a difference, cannot be explained or represented. This implies that there is strictly speaking no such thing as intuitive knowledge and that an idea as such represents no knowledge although these are indispensable elements of all knowledge. By intuition or idea something is only given to us rather than being apprehended. In order to become knowledge such an idea has to be specified and transformed into a definition. To abandon oneself exclusively to intuition or experience is therefore to ignore or to disdain anything general. But everything new and any creative insight in particular results from the combination of intuitions or ideas, and experiences.

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Intuitions and ideas are as indispensable to enlarge our knowledge as are new facts, which surprise us. Now any perceptual judgment already presupposes these aspects, continuity or generality and a set of discontinuities. For, on the one hand, as Lewis Carroll had shown in his little piece on Achilles and the Tortoise (see Hofstadter: Gödel, Escher, Bach, Basic Books NY 1979, Chap. I), logic or general argumentation can never force on us the acceptance of anything. Such an acceptance occurs apodictically and intuitively or not at all. A perceptual judgment is always apodictic and hence it is something, we are unable to control and unable to criticize. The critical “analysis would be precisely analogous to that which the sophism of Achilles and the Tortoise applies to the chase of the Tortoise by Achilles, and it would fail to represent the real process for the same reason. Namely, just as Achilles does not have to make the series of distinct endeavors which he is represented as making, so this process of forming the perceptual judgment, because it is sub-conscious and so not amenable to logical criticism, does not have to make separate acts of inference, but performs its act in one continuous process” (Peirce CP 5.181). Thus the continuity principle serves not only to understand the interaction between the particular and existent on the one side and the general or possible on the other, but also confirms the view that in mathematics we encounter both of these aspects of reality, and that mathematics therefore has particular as well as ideal objects. This fact becomes better understandable if we take into account that mathematics is an activity – the activity of building theories on the one hand and of problem solving, on the other.

2. MATHEMATICAL GENERALIZATION

Scientific or mathematical research essentially aims at theoretical representations i.e., at generalization. Generalization is essential because it is this process that distinguishes mathematical creativity from mechanizable or algorithmic behavior. A characteristic of mathematical thought is, says Peirce “that it can have no success where it cannot generalize”. Mathematicians strive for the greatest possible generality, many times “exchanging a smaller problem that involves exceptions for a larger one free from them” (Peirce 1957, 264). And in a similar vein Poincaré states that “there is no science but the science of the general” (Poincaré 1902, 34). Both Peirce as well as Poincaré illustrate their views by comparing mathematics to a game of chess, where this essential element of generalization is missing. Chess is mathematics for machines, however, much ingenuity may be required to win a game of chess.

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Mathematics thus depends on generalization. But mathematical generalization cannot just consist either in the broadening of the extensions of pregiven mathematical concepts or in mere empirical induction. Neither scientific nor mathematical knowledge is on the average inductively generated, but most frequently depends on unexpected hypotheses or intuitions. A hypothesis or idea becomes effective on generalization only if it represents a relation between things given. Generalization therefore implies synthesis or construction. Synthesis as present in the discovery of the relation between electricity, magnetism and light, which were all found to be different aspects of the very same thing, which we call the electromagnetic field today. Or synthesis as exhibited by the greatest discovery of the Industrial Revolution, namely the relation between heat and motion, expressed by the theorem of energy-conservation. Or just think of Newton’s law of inertia as based on the identification of inert with heavy mass. Or, to mention one more example, think of Planck’s introduction of the quantum hypothesis into the theory of black body radiation. Synthesis thus means to establish equations A = B, on the basis of a construction; it means to represent something in a novel way, thereby shedding new light on the situation. A representational “equation” A = B is commonly interpreted saying that A and B are different intensions of the same extension. This extension, as in the examples of energy or the electro-magnetic field, is not necessarily an empirical object, but is rather a universal or ideal object, being constructively generated. Scientific and mathematical generalization is based on construction rather than just on empirical abstraction. As mathematical construction is really the claim of the possibility of construction, rather than construction itself, we could say that mathematical statements are abstracted from possibility. This is the reason why the notion of mathematical axiom historically preceded that of natural law. Natural philosophy began more geometrico in the 17th century. The concept of a complex number is an abstraction from possibility, inasmuch the possibility to extend the basic arithmetical operations of addition, subtraction, multiplication, and division, in a suitable manner is assumed. The consistency of this assumption needs to be confirmed by a suitable representational generalization. As long as the imaginary number had gained admission to arithmetics as a calculatory symbol only, √ it produced the most horrible confusion (cf. Paul Nahin, The Story of −1, Princeton University Press, Princeton 1998). Only after Gauss had given a relational interpretation to the imaginary unit in the frame of the model of the so-called Gaussian number-plane, it became a legitimate mathematical object, which subsequently assumed an important role in function theory

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during the 19th and 20th centuries. Still Leibniz had called the “imaginary root” a “monster of the ideal world”, and an “amphibium between being and not-being” (GM, V, 357). Gauss begins his “Theory of Biquadratic Residues” in the Göttingische Gelehrte Anzeigen of 1831 by assuming that to him who is less familiar with the “nature of imaginary quantities” the latter may appear scandalous and unnatural and could lead to an attitude in mathematics which “moves entirely away from intuition. Nothing would be more unfounded than such a view”, Gauss writes. As opposed to that view, “the arithmetics of complex numbers is capable of concrete visualization. . . . Just as the absolute whole numbers are represented by a series of points distributed on a straight line at equal distances, . . . the representation of complex numbers requires but the addition that series be considered as being situated in a determinate unbounded plane, and that parallel to it an unlimited number of similar series is assumed at equal distances from one another, resulting in our having before us, instead of a series of points, a system of points which can be aligned in a twofold way into series of series. . . . In this representation, the execution of the arithmetical operations becomes capable, with regard to the complex quantities, of a representation, which leaves nothing to be desired. Thereby the true metaphysics of if imaginary numbers is placed under a new light”. This metaphysics of imaginary numbers is based on two assumptions, namely that they are subject to all arithmetical operations and further that we can form an intuition of their objective meaning. Gauss conceives of this objective meaning of the imaginary numbers in terms of their possible applications. Whenever the elements of a field of application "are of such a kind that they cannot be aligned into one series, albeit unlimited, but only into series of series, or, which is the same, if they form a manifoldness of two dimensions”, a representation of all relations is required, of the whole numbers as well as of the imaginary numbers. It is thus the structure of relations, by which mathematical ideas are interpreted, and Gauss presented an iconic model or diagram of this relational structure. Interpreting this structure in terms of vector space theory we may state that the introduction of the complex number is the result of an inter-structural generalization. Another more advanced example of such a generalization, again involving vector-calculus as one of the initial structures, is provided by Hermann Grassmann in his “Neue Theorie der Elektrodynamik” published in Poggendorff’s Annalen der Physik und Chemie in 1845. A paper in which among other things he criticized Ampere’s Law of Electrodynamical Force because Ampere had transferred assumptions valid in the area of gravitational force to a very different area “auf welchem die Elemente

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mit bestimmten Richtungen begabt sind”; where the elements are vectors rather than points. Both Grassmann as well as Maxwell synthesized this new type of vectorial quantity and introduced it into geometry and natural science. Despite the fact that Ampère had taken great pains to derive his electrodynamical force law from the results of experiments, he nevertheless found it necessary, as Grassmann points out, to introduce the assumption that the force between two current elements acts along the line connecting their midpoints, in supposed analogy to the mode of action of gravitational forces. Ampère in his important “Memoire sur la theorie mathématique des phénomènes électrodynamiques uniquement déduite de l’expérience” of 1827 had in fact written: “Guided by the principles of Newtonian philosophy, I have reduced the phenomenon observed by Oerstedt to forces always acting along the line that joins the two particles between which such forces are exerted.” This statement appeared in the context of two other additional facts, more or less hidden. First Ampere more or less explicitly accused Oerstedt of introducing unnecessary and unwarranted hypotheses, having been influenced by German Naturphilosophie; and second that mathematical constructivity has no active role to play in finding the laws of nature, and mathematics is just an auxiliary means for expressing that which has been concluded from experience alone. Mathematics is just a language. Grassmann, in contrast, believes that in every new knowledge there is involved a construction and a hypothetical synthesis. Although Ampère always stressed that in establishing the laws of the new phenomena he “had consulted experience alone” he assumed hypothetically that the direction of the forces must be “necessarily that of the line joining the material points between which the forces are exerted”. Now Grassmann believes that this assumption could not be true. Grassmann, however, accepts Ampere’s view that the form of the law, which describes the mutual influence of two electrical currents, is to be the same as the form of Newton’s law of gravitational interaction between material points. For Grassmann as well as for Ampère it was very important that the very same form could represent electrodynamical attraction and gravitation. But Grassmann believed that this analogy between the laws of gravitation and of electrodynamics had to be based on a new interpretation of the mathematical objects and operations in question, taking into account that the entities between which electromagneti forces act are directed quantities, vectors, not points. Grassmann writes: “For this purpose (to preserve the form of the law, my insertion), I must, however, quote that concept here which I have introduced in a work recently published, doing so before I had any inkling of

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the new theory. I have indeed proved there that the line segment connecting them must be seen as the product of two points. . . . At the same time, I have shown that the area of a parallelogram must be conceived of as the product of the adjoining sides, taking into account these sides’ direction and length.” Grassmann had to generalize the idea of a product from numerical quantities to directed ones, to vectors, giving up commutativity. He defines the idea of a product in a completely new and structural way and by this is able to preserve analogy between gravitation and electro-magnetism. Because of the fact that the product is to be formed from one of the current vectors and the magnetic field vector of the other current, rather than from the two current vectors, the direction of the resulting force can be parallel to the line joining the infinitesimal elements of electrical currents, as Ampère had assumed. This holds true, for example, if the currents are parallel to each other. This means that in many cases, for instance, when one takes two electrical currents as lying in the same plane, empirical observation does not help to distinguish between Grassmann’s theory and that of Ampère. Grassmann therefore devotes the last quarter of his paper to describing experiments that should help to make the distinction. But the hypothetico-deductive approach adopted by Grassmann, as by all scientists influenced by Romantic Naturphilosophie, reversed the order of the relationship between theory and experiment. It is clear to them that theory must precede and orient experimentation, and that Ampère’s procedure, namely inductively searching for a formula that interpolates the given data, is insufficient. Mathematics has an important role to play in Natural Philosophy because all generalization depends on construction and abductive inference. It is worthwhile to ask why it is synthesis and abstraction rather than mere inductive extension that seem to lead to the important generalizations. The answer lies in the fact that knowledge is based on activity rather than being determined directly by its subject matter. From this results what Quine has called underdetermination of theory by observation and experience. If we have, for example, a set of observations represented by a scatter diagram in a two-dimensional Cartesian coordinate system, these data alone do not compel the choice of the interpolating function which represents the regularity that is meant to explain these data. Thus we have to introduce an additional hypothesis about the form or character of the function. This form is not a product of induction, but of abduction, as Peirce called it. The formation of an idea or an intuition by abductive inference is not suggested by the appearance of the givens or data but by their “essence”.

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We know – among others – from Goodman’s analysis of the problem of induction that inductive inference is plausible as soon as we assume that the observed characteristics of the phenomena and properties of things are essential to them. But this would make any knowledge analytical knowledge. We no longer believe that knowledge is the result of a direct and unmediated access to reality and we have learned from Hume that relations are external, that they represent nothing of the essence of the relata. What in the nature of Paul should cause his being taller than Peter? Why should Peter catch a certain infectious disease, rather than Paul? Continuity we find, according to Hume and Kant, only in the realm of phenomena, as they are synthesized by mental activity. This is, however, too one-sided a view. Although we argue, in fact, that a distinctive trait of real in contrast to ideal existence (the mode of existence of an ideal object) is the possession of accidental properties by the former, we do not conclude from this, however, that all ideal objects or general ideas are just invented and thus are merely subjective. Relations may be external, but they may nevertheless be objective, objective possibilities, for instance. The notion of possibility, intended here, is not a concept but is rather a heuristic idea or regulative principle. “The idea”, Kant says, “is only a heuristic, not an ostensive concept. It does not show us how an object is constituted, but how, under its guidance, we should seek to determine the constitution and connection of the objects of experience” (KrV B 699). In contrast to Kant, one would add that (transcendental) ideas and heuristic principles are to be derived from experience, although, on the other hand, they determine possible experience. Thus a genuinely genetic epistemology is needed to deal with the problem of generalization. Even so the interpretation of a certan area of phenomena in terms of a structural relationship must be based on constructive or hypothetical assumptions, the success and the fertility of these assumptions show that they are not just subjective and arbitrary. The success is, however, never guaranteed in advance. The objective is not to be identified with the objectual and there is objectivity without appeal to objects. It is the objectivity underlying activity, practice or experience and intuition. If all the properties of an entity are essential, all we can conclude is that it has structural identity as defined by a set of axioms and definitions which are internally consistent but claiming no independent status. If an axiomatized theory determines the extensions of its theoretical terms in this manner it is an intensional theory and as such analytic. From the point of view of application such an axiomatized theory is just a sign or a representation of the things to be explored. Although we might believe that it captures the essential aspect of these, we acknowledge that what

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appears as essential depends on our goals and intuitions. An axiomatic system provides just a particular perspective on some object field. And every theory may be axiomatized in a variety of ways. An axiomatized theory on such an account is not supposed to provide complete identifications or descriptions of its intended applications. The axiomatic approach aims at exploring everything that is compatible with the axiomatic descriptions, and hence everything that has not been explicitly excluded must be admitted. Only if we take such an attitude, the intended applications of a theory will contribute to theoretical development itself. The process of applying a theory would then not be conceived as just being based on straightforward reification of its terms. The “reflective abstractions” (Piaget) by which we form an axiomatic description or an idea of a situation represent mere possibility. The possible, as given by the possibility of some activity has an objective reality, even if I cannot reify it. If I say, for instance, “the rose is red”, then this redness possesses an existence in my mind relatively independently of the rose. The rose might well have another color, and the redness, for its part, might be connected with many other things, not with roses alone. If roses were essentially red, that is if the sentence “roses are red” were analytical, perception of the roses would not be necessary, and there would not even be a basis for it. The same is true of an intended application of axiomatized theory. If one says that a concept like redness operates on things, what is meant is that a relation to things is given to us via this concept, and that this relation is necessary to set our thinking and judging into motion in the first place. In fact, what we perceive when we see something only vaguely is a general: a rose, a horse, a wolf. Any rose, not: this rose or that wolf. Possibility, however, always refers to a possible perspective on or a representation of some reality. And if I act in a certain context this reality basically consists of particular objects. When I see it here and now and act towards it the rose appears as a particular existent rose. Kant already stated that every particular experience or thought must be an instantiation of a general possibility, based on some subjective faculty of ours. But he could not really explain how general and particular work together in fostering knowledge because he believed the transcendental ideas to be determined a priori and with absolute necessity. Genetic epistemology a la Piaget shares this defect of its Kantian origins. On occasion of the examination of the epistemological ideas of another Kantian, the biologist Konrad Lorenz, Piaget said, “As a biologist, Lorenz is well aware that . . . specific heredity varies from one species to another. . . . As a consequence, Lorenz while believing as a precondition that our major cat-

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egories of thought are basically inborn, cannot for the very reason, assert their generality, hence his very enlightening formula according to which the a prioris of reason consist simply of innate working hypotheses. In other words, Lorenz while retaining the point of departure of the a priori sets aside necessity . . . whereas we are doing exactly the opposite” (Piattelli-Palmarini (ed.) 1980, 30). 3. GODEL VS . PIAGET ON IDEAL OBJECTS

Mathematics, although dealing with intensional objects is interested in truths about real objects, and therefore is fundamentally interested in extensions. There is, in fact, a systematic study of extensions, namely set theory. Mathematics as based on set theory becomes a breeding bed for Platonism. In his paper on Russell’s logic Gödel accordingly proposed to introduce a version of the axiom of extensionality for concepts which is in fact a genuine counterpart to Leibniz principle of indiscernibles, claiming that “no two different properties belong to exactly the same things”. And he illustrates this proposal: “Two, for example, is the notion under which fall all pairs and nothing else. There is certainly more than one notion in the constructivistic sense satisfying this condition, but there might be one common ‘form’ or ’nature’ of all pairs” (Gödel in: PA. Schilpp (ed.) The Philosophy of Bertrand Russell, La Salle 1944, 138). In order to avoid the difficulties into which Frege had run one had to abandon his conception of set, known as “collection-as-many” and would have to conceive of sets as ideal entities in their own right (collection-as-one). From the point of view of mathematical activity ideas or universals are mere possibles. This possibility may be interpreted objectively – in the sense that an idea may or may not apply to a particular situation or thing, and hence is a possible –; or it may be interpreted subjectively, meaning that the ideas identity is derived from the identity of a certain person having it. In the modal Platonist conception there is a complete symmetry between possible worlds and possible perspectives on the one real world. This symmetry is familiar to every student of linear algebra as we cannot distinguish between linear mappings (change of world) and coordinate transformations (change of perspective). The difficulties of this view again are due to problems of existence rather than of identity or essence and thus cannot be dealt with neither within Platonism nor within structuralism à la Piaget (Grayling 1982). Piaget is led by his opposition against empiricism to exclusively emphasize the subject’s constructivity, as well as the resulting mathematical structures as the source of all possibility and of all reality. From Piaget’s

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perspective mathematical structures can very well get along without existence as they are constructions several times removed from their genetic base in concrete actions. He considered it one of the major tasks, or perhaps the main task, of his genetic epistemology to explain how necessary knowledge, knowledge that does not rely on observation, comes about. The possible becomes a derivative from the necessary only, “since the new combinations are not free but were within the framework of possibilities determined by the structure from which reflective abstraction started out”. Can we then limit ourselves, Piaget asks, “to saying that any structure, however elementary, provided that it be of a logico-mathematical nature . . . involves a whole system of possible developments, and that the novelty of later structures consists, merely of actualising some of them? This is our hypothesis, and as we see, it does not differ in all respects from Platonism, since it is sufficient to confer existence on these possibilities to be a Platonist or even to assume an intelligence which is infinite or at least superior, which comprehends them in a single intuition. But what we object to for genetic reasons, is the transition from the possible to the real entity so long as there has been no actualization by an effective construction” (Beth and Piaget 1966, 300, 301). Being a constructivist Piaget believes that “the set of all possibilities is as antinomic as the set of all sets” and he thus justifies the importance of an operative approach to mathematics and a genetic epistemology because from a constructivist point of view “to believe in the existence of possibles in the form of ideal Platonic entities, is to take as given in advance, operations capable of actualizing the former, before knowing these operations or these possibles” (loc. cit.). Gödel certainly would take such a stance, at least with respect to the possibles. Piaget’s description of logico-mathematical thought relies on a separation he makes between Aristotelian abstraction and what he calls “reflective abstraction”, which is drawn not from objects at all but from the coordinations of actions or operations. Structures seem to arise quite naturally out of the combinations of actions. Piaget started from the fundamental observation that operations on any set of objects can be combined to form structures in a very natural manner, whereas the objects themselves seem completely isolated from one another. For instance, the transformations of any set of objects onto itself form a mathematical group structure. Hence structures of actions, for instance, the structure of a group of transformations, are general, whereas the actions themselves remain individual, concrete entities. Piaget called this “self-organization” of actions into structured wholes, by which they are generalized, reflective abstraction. Piaget believes that “in order to build a more abstract and general structure from a more concrete and particular structure, it is first of all necessary

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to abstract certain operational relationships from the antecedent structure, so as to generalize them in the later one; (b) but both this abstraction and generalization presuppose that the relationships thus abstracted should be ‘reflected’ (in the true sense) on a new plane of thought, so as to form the generalized answer to it” (Beth and Piaget 1966, 242). This new plane of thought consists of a system of ‘laws’, that governs the totality of the given entities. Thus reflective abstraction is not a discovery, “because it implies a construction and because the structure or ‘reflected’ entity are not the same as those from which they are derived” (Beth and Piaget 1966, 206). Reflective abstraction always implies a reconstruction of an ensemble of actions within a more general structure and on a different plane of action. In as much as a system of actions forms an autonomous whole only on the condition of its coherence”, Gödel’s incompleteness theorem signifies a relation of dependence of the original ensemble from its generalized reconstruction (Beth and Piaget 1966, 273). But reflective abstraction, Piaget believes, may neither be interpreted as invention in the sense of “a new and free combination, for the new elements which then enter in and above those which are discovered are never free in the sense in which they might have been different” (Beth and Piaget 1966, 207). The development of mathematical structure, according to Piaget, is governed by strict logical necessity. The possible has only a structural existence. The freedom concerns the methods of demonstration and formalization. As mathematics is based exclusively on structural identity (that means, up to isomorphism) “the essential property of mathematical construction and creation seems thus neither reducible to discoveries nor to inventions, but to an indefinite succession of combinations at once new and yet within a well determined system of possibilities”. The problem then is to know whether we have the right to talk of this system of possibilities, whether we can reason and say something valid about it before it has been realized in effective operations, that is before the possibilities have ceased to be mere possibilities. “All that we can say, and verify, about relations between the possibilities and their realization in a new logicomathematical construction is that, genetically speaking, a structure observed at a given level of development always contains more possible generalizations (for example, by raising a restriction or abstracting a new transformation etc.) than the subject perceives” (Beth and Piaget 1966, 207 f). The hierarchy of formal structures (established in accordance with the limitations which Gödel’s incompleteness theorems imposes on formal mathematics: loc. cit. 274 f) appears to be the true epistemic subject in Piaget’s theory in as much as “this gradation in the strength of the sys-

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tems implies unlimited construction” (Piaget 1972, 42). Piaget’s ‘epistemic subject’, “whose cognitive structures derive from the most general mechanisms of the coordination of actions” (Beth and Piaget 1966, 308), is a dynamical version of Kant’s transcendental subject in as much as this structural hierarchy does not exist as a completed and actualized “structure of structures”. Piaget was a Kantian, he adhered to Kantianism, as he often affirmed, but to a Kantianism “that is not static, that is, the categories are not there at the outset, it is rather a Kantianism that is dynamic that is, with each category raising new possibilities, which is something completely different. I agree that the previous structure by its very existence opens up possibilities, and what development and construction do in the history of mathematics is to make the most of these possibilities, to convert them into realities, to actualize them” (Piattelli-Palmarini (ed.) 1980, 150). But Piaget’s dynamism collapses, I believe, under the emphasis he puts on necessity. All ideas and all possibility remain totally subordinated to structural constraints. There are no contingent facts, no particularity that could influence the development of structural knowledge. For Piaget everything that is new is constrained by the possibilities a given structure opens up. And Piaget’s genetic epistemology becomes with respect to mathematics just a variant of Platonism. The Platonist believes that each thing has an individual essence, a set of properties, which are essential to it. From the structuralist point of view these properties are just structural relations. Whatever brand of essentialism we adopt the distinction between existence and identity is eliminated. The only reservation Piaget has with respect to Platonism and to Platonic ideas concerns the “methods of knowing these ideal entities”. Piaget rightly emphasized the fundamental importance of human activity as a means of mediation between the subject and the object of knowledge but he conceives of mathematical activity in formal terms. Piaget like Kant acknowledges the fundamental epistemic role of human activity and construction. But, Piaget unlike Kant or Peirce makes a radical distinction between acting and perceiving and between empirical and reflective abstraction. The reason for that resides in his own structuralism and in his devaluation of intuition and continuity. Piaget believes that even the notion of space should be interpreted as completely subordinate to the concept of structure. Referring to the axiomatics of group theory he writes, for instance: “The associative law of the transformation groups is fundamental for the coherence of space, because if termini in group theory did vary with the paths traversed to reach them, space would lose its coherence; what we

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would have instead would be a perpetual flux, like the river of Heraclitus” (Piaget 1970, 20). This is true enough, but it is the coherence and continuity of space and the possibility it offers of employing indexical representations which provides mathematics with the idea of objective knowledge, rather than the other way around. Without the “Here” and “Now” of spatio-temporal locations relational geometrical thinking would not have developed towards the “transfigural” stage, as Piaget called it, because it would have been difficult to conceive of one’s own actions as objective entities in order to be able to refer to them in an unambiguous manner, rather than just experiencing them as mere processes in time. In contrast to what Piaget seems to believe, there are no relations without relata; and the relata represent some external influence on the relational structure. Piaget rightly emphasizes the importance of a constructive actualization of the structurally possible but he seems not to perceive that this actualization is influenced by contingent factors which are independent of the actual structure itself. Piaget’s combination of structuralism and constructionism, as well as Gödel’s Platonism in the end appear as the two faces of the very same coin. They deal with identity but neglect the relation between identity and existence and thus render (cognitive) activity a secondary phenomenon. Both, Piaget like Gödel, would, as it seems so far, not think of conceiving possibles in terms of something in its own right (rather than something which has at the moment not yet been actualized) and simultaneously only describable in relation to human activity and practice (like our general regulative ideas or general concepts). About 30 years after the publication of his essay on Russell, Gödel himself no longer believed “that generally sameness of range is sufficient to exclude the distinctness of two concepts” (see: Hao Wang, A Logical Journey, The MIT Press 1996, 275). Gödel was now no longer convinced that the range of applicability of a concept generally forms a set. “Only concepts having the same meaning (intension) would be identical”, he now said. Ideas or concepts seem entities, whose mode of being consists in that they are universals and that at the same time they depend on instances of concrete applications. 4. “ MODERN ” MATHEMATICS AS BASED ON HYPOSTATIC ABSTRACTION

The topologist Salomon Bochner considers the iteration of abstraction as the distinctive feature of the mathematics since the Scientific Revolution of the 17th century. “In Greek mathematics, whatever its originality and

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reputation, symbolization . . . did not advance beyond a first stage, namely, beyond the process of idealization, which is a process of abstraction from direct actuality. . . . However . . . full-scale symbolization is much more than mere idealization. It involves, in particular, untrammeled escalation of abstraction, that is, abstraction from abstraction, abstraction from abstraction from abstraction, and so forth; and, all importantly, the general abstract objects thus arising, if viewed as instances of symbols, must be eligible for the exercise of certain productive manipulations and operations, if they are mathematically meaningful. . . . On the face of it, modern mathematics, that is, mathematics of the 16th century and after, began to undertake abstractions from possibility only in the 19th century; but effectively it did so from the outset” (Bochner 1966, 18, 57). In a similar vein Peirce writes: “One extremely important grade of thinking about thought, which my logical analyses have shown to be one of the chief, if not the chief, explanation of the power of mathematical reasoning, is a stock topic of ridicule among the wits. This operation is performed when something, that one has thought about any subject, is itself made a subject of thought” (NEM IV, 49). In this way even the means and conditions of thought become an object of it. A predicative or attributive use of some concept is transformed into a referential use in order to incorporate the entity thus synthesized into new relational structures. The above mentioned example of the introduction of the imaginary numbers provides a case in point. At first after having been introduced to generalize certain algebraic operations, these “numbers” seemed the paradigmatic model of an artificial invention, whilst the subsequent history of complex functiontheory would tend to provide this invention with the characteristics of something indubitably objective. In all necessary reasoning, Peirce continues “the greatest point of art consists in the introduction of suitable abstractions. By this I mean such a transformation of our diagrams that characters of one diagram may appear in another as things. A familiar example is where in analysis we treat operations as themselves the subject of operations” (CP 5.162). Piaget, as we have seen, entertains a similar view of mathematical generalization but misses the fact that perception and observation will necessarily play a role throughout the process. In the discussion of the so-called fundamental theorem of algebra, it is said, that Lagrange had tacitly and implicitly used the intermediate value theorem for continuous functions to give a proof of this theorem. In actual fact Lagrange did nothing but provide algorithms for attaining approximate solutions of algebraic equations. It was Cauchy who read the intermediate value theorem for continuous functions into Lagrange’s argument, trying

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to make this presupposition explicit. What Cauchy did was nothing but a hypostatization or reification of algorithmic procedures, transforming them into new abstract objects. He turned an algorithmic rule for instance, into a mathematical function or a converging process into the notion of limit. Abstraction from abstraction is certainly facilitated by the grammatical structure of European languages, which makes it simplest for users of those languages to speak of abstract entities as if they existed. Hypostatic abstraction thus is achieved, for instance, by hypostazising a predicate or a quality, thereby turning it into a subject capable of further predication. We transform, for instance, the proposition, “honey is sweet,” into “honey possesses sweetness.” This may sound trivial, although it facilitates such thoughts as that the sweetness of honey is particularly cloying; that the sweetness of honey is something like the sweetness of a honeymoon; etc. But language appears to be a flat game in this respect when compared to mathematics or computer science. That abstractions of this kind are particularly congenial to mathematics is a fact exemplified particularly well by Cantor’s set theory. Or think of the idea of a space of arbitrary dimension invented by Grassmann in 1844: A point moves: it is by abstraction that the geometer says that it “describes a line.” This line, though an abstraction, itself moves; and this is regarded as generating a surface; and so on. Always and again a construction or an algorithmic procedure is taken as an object to be incorporated into another construction or procedure. But in order to reify operational concepts it might be necessary to employ spatial intuition, because mathematical intuition and activity do not operate on singular objects but on “spaces” of all kind. Piaget himself provides some hints of how to cure the defects of his own mathematical epistemology when speaking about the heuristic function of spatial intuition. The latter, he believes, “is limited by two fundamental considerations: (1) spatial imagery only progresses when directed and molded by the subject’s active operations, in such a way that its figural aspect is more and more subordinated to the operational aspect of thought, and only provides information which is relatively adequate in terms of this subordination; (2) in its most adequate forms spatial intuition is never anything but symbolic, that is, it expresses by symbols which are always imperfect (however much they may be improved) a system of things symbolized which, although spatial or geometrical, consist of abstract concepts and operational concepts” (Beth and Piaget 1966, 219). Both of these claims seem to point into the right direction but still appear somewhat defective in that they miss the importance of individual existence and indexical thoughts.

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With respect to the first we may exploit one of Piaget’s own examples. Piaget (see Piaget 1977) at various occasions describes the following experiment: A child is constructing two parallel walls, a blue one and a red one out of “bricks”, that have equal size but different colors, by simultaneously taking one blue and one red piece with both hands and placing them in the proper place. A child in the stage of empirical abstraction can answer questions with respect to the comparative length of the blue and the red wall, but is unable to predict what will be the case “if we continue indefinitely in this manner”. An older child, being capable of reflective abstraction (abstraction from actions) will have no difficulty in predicting that the two walls will always have the same length But how is this judgment to be justified? It certainly depends on empirical ideas as well as on reflective abstraction. Reflective abstraction gives us only the possibility of an action, not a concrete action itself and one has to decide how to instantiate such an action in a particular case, how to apply it. One also has to make sure that the general circumstances of such an iterated application do not change in course of the matter. Otherwise we could prove the parallell axiom of Euclidean geometry and related things from mental activity alone. With respect to the second point we only should like to emphasize once again the importance of indices, which serve to refer to existents determined by ostension within a given context. A complementarist perspective seems more promising with regard to the relationship between the figural and operational aspects of thought than an effort to try and subordinate perception to logical operation. The fundamental problem of science and mathematics is complexity, and complexity results from the combination of innumerous aspects, which in themselves seem rather simple and transparent. P. Bernays, Hilbert’s collaborator, once indicated that a mathematical proof has two essential elements: “The clarification of the concepts, . . . and the mathematical aspect of combination” (Bernays 1976, 25). Bernays then draws our attention to the second aspect because its impact is generally underestimated, as he believes, by giving an example: “Suppose that in a formal inference the rule of modus ponens has to be employed and suppose that neither A nor “A implies B” are among the formulas to begin with. Rather we have a chain of deductions S resulting in A and another one T which leads to “A implies B”. Than this formulas together provide the formula B according to the rule of modus ponens. If we want analyze what is going on here we have to be careful not to anticipate the essential point by our symbolization already. The formula which stands at the end of the chain of reasoning T is given to us only by this very chain of reasoning and to see that this formula is identical with the other one

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namely “A implies B” represents new knowledge. The verification of an identity is by no means always a tautological act” (Bernays: Abhandlungen zur Philosophie der Mathematik, Wissenschaftliche Buchgesellschaft, Darmstadt 1976, 26). And it depends, I believe, on indexical thought. The establishment of such equations X = Y or A = B depend on abductive reasoning and creative synthesis and these in turn are in general based on intuition and experienced observation. Conversely, the essential features of an act of imaginative creation may be summarized by stating that they consist in seeing an A as a B: A = B, or “all A are B”, or “A represents B” etc. Important, however, is the fact that there is nothing in the world that will a priori guarantee the success of such an act of creative imagination. It is just a hypothesis and as such it is based on some sort of idea introduced by the interpreter (the creative subject), who perceives some proximity or relation between two phenomena and tries to explain it. The equation A = B is based on associations of contiguity. That is, it expresses a synthetical judgment. Synthetical reasoning necessarily contains a fact of experience which is forced on us without our will or control, as in perception. Perception is a constructive process; it is an activity that certainly depends on the skill and experience of the perceiver in constructing a representation of the situation, which reflects its “essence”. Yet it is “a characteristic of perceptual judgments that each of them relates to some singular to which no other proposition relates directly, but, if it relates to it at all, does so by relating to that perceptual judgment” (Peirce CP 5.153). In mathematics, perception relates to a symbol and is symbolic itself. Symbolization has become the profoundly distinguishing mark of “modern” mathematics since the 16th century. It is on the basis of this characteristic property that we want to answer to the question of the title of this note. A mathematical concept, such as the concept of number or function, does not exist independently of the totality of its possible representations, but must not be confused with any such representation, either. It is a general that cannot as such be exhausted by any number of its representations. And it is a possible that has no meaning apart from its applications. An idea is not to be conceived as a completely isolated and distinct entity in Platonic heaven, but must not on the other hand be confused with any set of intended applications. Primarily for the reasons Gödel had enunciated, namely that the range of possible applications is no definite set at all meanings are generals in the sense of referring to an indefinite and undetermined collection of possible applications. Further, two predicates or concepts or functions (or functions of functions) are to be considered as different even if they apply to exactly the same class of objects because they influence mental activity differently and may lead to

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different developments. We shall provide some examples to illustrate these claims. And what has been said with respect to mathematical concepts applies as well to other types of mathematical representations. A mathematical theory does not exist independently of the entirety of its axiomatic characterizations, and it must not be confused with one of them. A formal system can be represented in various ways, and still the theorems have to be invariant in their truth content with regard to changes of the representation. However, here, too, this does not mean that “there is a hypostatized entity called a formal system which exists independently of any representation” (Curry 1970, 30).

5. TWO CONCEPTIONS OF MATHEMATICAL ACTIVITY

Two features characterize mathematics. Firstly, the acknowledgement that all knowledge is of formal nature. Theories are realities sui generis which find an application only via the form, which they convey on our thoughts or actions. “All supremacy of mind is of the nature of Form” (Peirce CP 4.611). The second idea consists in the insight, complementary to the first, that cognition is an activity and that we solve our problems by behaving towards them in the proper way. The mathematician solves his problems by juxtaposing various representations of one and the same situation. As all cognition is tied to representation, it can be said that mathematical reasoning consists in organizing suitable diagrams. Every conclusion, on the other hand, concerns a certain object which is the outcome of a mathematical construction. Hence, both icons and indices are required for any mathematical diagram judgment and any proposition which results from its analysis. “A proposition is the meaning of a sign which represents that an icon is applicable to what is indicated by an index” (Peirce MS 599). Considering mathematics mainly as hypothetico-deductive reasoning it is easily possible to overlook the significance of the index and thus of the various objects. Mathematics, however, is also problem solving, and its two aspects, establishing theories on the one hand, and solving problems and tasks on the other, both posses a relatively independent existence in mathematics. This results in two types of approach: A: From X, however it may be given, follows Y . B: The point is not X in itself, but I consider the mode in which X is given, and then I try to infer Y .

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A considers the relations of things independent of the mode of their being given, and independent of a chosen method while B makes just this intensional aspect, or in other terms the perspective on the objects, the basis and the means of study. I should like to illustrate the difference presented here by an example: A: Let us assume that the real numbers were enumerated in some way and organized into a list. Cantor then constructs, by means of his own diagonal procedure, a number which does not occur in the list, and from this results an objection to assuming the enumerability of the real numbers. This, however, is a purely extensional way of consideration in which we are informed about impossibility rather than about any new property of a real number. B: We know that the computable numbers (in Turing’s sense) represent a countable infinite set. If we assume that we had enumerated the computable real numbers and organized them into a list we are able to construct, again by means of Cantor’s diagonal method, a number which is not contained in the list. As such, this is not a very interesting proposition. If we assume, however, that we had computed this list itself by means of one of Turing’s machines, this new number would obviously be a computable real number, and contradiction would result. If we concede, however, that the set of computable numbers is not effectively denumerable, we gain a representation of a non-computable number in this way, rather than merely affirming the mere existence of non-computable numbers. We have thus found out about two things. Firstly, that while the totality of the computable numbers is countably infinite, it cannot be effectively enumerated, and secondly our procedure has lead to the determination of a non-computable number. This in itself is already something interesting, for all the numbers that come to our minds spontaneously are as a rule computable. On the other hand, we know that the majority of real numbers are non-computable ones, as the set of computable real numbers is indeed countably infinite. There is a further conclusion one can draw from the argument B, namely that even in a world in which everything is completely calculable and appears determined there can nevertheless be something unanticipated or non-determined. Or, in other terms, that laws on the one hand, and the things which are ruled by them in their behavior on the other, possess modes of existence which are relatively independent of one another. The laws do not determine things completely. A further difference between A and B seems to lie in that one starts from a concrete task or problem in case B. I intend to find an algorithm

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or a program which organizes the totality of all computable numbers into a list. I do not ask whether it is possible to accomplish this task. To ask whether such a task is possible requires quite a different consideration. In case A I am not faced with a concrete task, at least not if one considers concepts like “the totality of all possible methods” or set of all possibilities to be ambiguous and just as antinomic as the concept of the set of all sets. Conception A proves its value in particular in proofs of impossibility, that is for the question whether things may accomplished at all. If one tries to escape for example from a maze by means of the Pledge Algorithm, this will work out only if the maze truly has an exit. As the algorithmic approach does not cognize and map the world, a robot will be unable to know whether its own calculations will not extend endlessly and without result into the future. Whenever an algorithm does not reach its goal, one will not know what is the cause. Whether one has been too impatient, overestimating efficiency, or whether one is in the presence of a basic impossibility of finding such a solution. Conception B in contrast deals with relative possibility. That non-computable real numbers do exist we know by means of A after Turing had made the algorithm, or the process of calculation itself, the object of study and analysis. How to define a non-computable number is shown by B. We are in this section not aiming at the usual distinction between constructive vs indirect approaches in mathematics, but rather have in mind what W. T. Gowers has called “two cultures of mathematics” (Arnold, Atiyah a.o. (eds.), Mathematics: Frontiers and Perspectives, AMS 2000). Gowers addresses “the distinction between mathematicians who regard their central aim as being to solve problems, and those who are more concerned with building and understanding theories” (p.65). Theories are build from theorems whereas problems are solved on the basis of appropriate methods and heuristic principles. A theorem should explain a wide range of facts whereas a general idea or a principle should be applicable to a great variety of problem situations. Mathematics obviously depends on both, on propositions and theories as well as on cognitive means and principles. If communicability and generalization is the goal of mathematics concepts and theories might become more important than principles and thought experiments, or so, one might think. If, however one wants to avoid “empty concepts” (Kant) and “general abstract nonsense” one would have to conceive of mathematics as an activity and as a unity of knowledge and experience. Many deep intuitions and experiences cannot be made explicit as precisely stated theorems, but have to be represented by artifacts or problem situations and thought experiments. Theory building in contrast,

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does not think “so much about the intrinsic interest of a mathematical result as about how effectively that result can be communicated to other mathematicians, both present and future” (Gowers, loc. cit. 68). The ideas of existence and possibility are also conceived and understood in a very different manner by these “two cultures”. The problemsolving approach, for instance, assumes a rather pragmatic attitude defining both concepts relatively to the development of mathematical activity focussing on ability rather than knowledge, whereas theory building is interested primarily in objective definitions and knowledge. The point of view of the problem solving approach is anti-positivistic and anti-nominalistic in that it considers concepts or ideas to be real, whereas logical positivism claims that theoretical concepts are either unnecessary or at least mere facon de parler (see: R. Tuomela, Theoretical Concepts, Springer N.Y. 1973, 3).

6. PROBLEM SOLVING : SOME EXAMPLES

We have at several occasions in the course of our argument pronounced the conviction that mathematical epistemology has to understand mathematics also as an activity, as the activity of problem solving, rather than conceiving it merely as a set of propositions or theories. We shall now provide some examples, to illustrate this point of view. 6.1. In his famous essay on “Sinn und Bedeutung”, Frege writes: “It is natural to think of there being connected with the sign besides that which the sign designates, which may be called the meaning (or reference) of the sign, also what I should like call the sense of the sign, wherein the mode of presentation is contained.” In a diagram X = Y the meaning of the expressions X and Y would be the same, but not their sense. How do I know, or how can I convey, that X and Y designate the very same object? For this, space and the ostensive indication of points in space are important. A letter in geometry, like a variable in algebra, is an index that implies a sort of existence claim conceming the object indicated, without providing any description of that object. These letters in geometry or algebra indicate places within a diagram. There will always be different paths leading to the same place. And once we have arrived there, further paths and possibilities will always appear. This is the reason why it is critically important to note exactly those features of objects or concepts, which will be subsequently needed in reasoning. Space in this way be-

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comes the condition for the possibility of cognition, iconic representations of relationships as well as indices playing essential roles in this. I should like to present an example taken from “Cabri” computer geometry to illustrate these facts. Let a triangle with the vertices 1, 2 and 3 be given. Now make the following construction. Choose an arbitrary point G, constructing first the point E, which is symmetrical to G with regard to the vertex 1, then the point F , which is symmetrical to E with respect to 2, and finally the point Z, which is symmetrical to F with regard to 3. Now the midpoint of the line segment GZ is marked. The “Cabri” system permits to pull or move points and lines across the monitor. If the initial point G is moved on the monitor, it is surprising to see that the midpoint M of the line segment GZ remains fixed despite the fact that its definition resp. construction obviously depends on G and thus ought to vary with G. This means that this point is overdetermined and there must be a different manner to construct it N, a manner, which is independent of the point G. And the fact of the invariance of the midpoint may then be expressed by the equation M = N. If we now repeat the same procedure with regard to an initial constellation of 4 or 6 points, or more generally of (2n) points, we do not encounter any fixed point, but rather observe that the distance between the initial point G and the end point Z remains invariant. While we may select the point G, in the case of the triangle or of any odd number of initial points, such as to close the broken line drawn from G across the symmetrical points to Z, or in other terms to make G and Z coincide, and thus turn the initial constellation of the points 1, 2, 3 into a system of side midpoints of a (non-necessarily convex) polygon, this is not the case for an even number of initial points. Whether the first and end points of the construction will coincide here depends exclusively on the constellation of the initially selected points 1, 2, 3, and 4. In the case of 4 vertices, these will have to form a parallelogram, for instance. This is a fact well-known from school geometry, which also explains the triangle situation, for the further determination of the midpoint represented by N simply consists in selecting this point such as to complete the vertices 1, 2, 3 of the initial triangle into a parallelogram. It may now be asked which condition 2n points must satisfy in order to form a system of midpoints belonging to a polygon. Every regular hexagon can obviously be included within another regular hexagon in such a way that its corners are precisely the side midpoints of the larger hexagon. Pulling even a little bit only on any vertex of the hexagon, however, immediately destroys the entire constellation. It is possible to conduct ever-new experiments of this kind, “Cabri” permitting both to construct and to vary

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constructions in a very simple way and offering rich possibilities for inductive reasoning. But even after an entire week of experimenting, neither the teachers nor the students had come across any hypothesis concerning the constellation of 6 points which adequately generalizes the parallelogram condition for the case of the quadrangle, the monitor only showing the constructions in their results. All the objects appear like empirical objects of perception and nobody was able to offer an inductive hypothesis. In the given situation, algebra now re-establishes the connection between object and activity. What distinguishes algebraic calculation from observing a situation in itself? Algebra shows something different from the fact in itself, that is a point’s invariance, or the invariance of the distance between the initial point and the end point of the even numbered n-polygon. Algebra shows how the objects (points) have been constructed. It exhibits a relational fact, because it maps activity itself. And mathematics has to do with relations between object-related activities. Everything general becomes effective only via activity. Even a natural law will not apply itself. A stone will drop only if I let it fall, and only then will I be able to perceive the law of gravity. Only as soon as we actively reflect on the conditions of the experiment of falling bodies, however, rather than just observing them as they are falling down, may we hope to find the form of the law. In mathematics this shift of viewpoint is accomplished by algebraization, algebra being the activity and method of formal construction. To make matters concrete, let us begin in the present case of any even number of initial points by applying vector calculus, i.e., by using linear algebra to obtain the conditions, which the coordinates of the initial points must satisfy. After some calculation, it is easily seen that the sums of coordinates of the points with even numeration, that is 2, 4, 6 . . . etc. must be equal to the sums of the coordinates of the points with odd numeration, that is 1, 3, 5 . . . etc. In geometrico-physicaI interpretation, this means that the centers of gravity of the even-numbered points constellation and of the odd-numbered one will coincide. In the case of n = 6, this means that the triangle 1, 3, 5 has the same center of gravity as the triangle 2, 4, 6. This now seems to be an entirely contingent fact, and there are no higher grounds, which make the fact predictable, or might represent how it evolved. The conditions of the result have simply been calculated. Knowledge will always refer to things factual as long as I do not know the conditions of possibility of this knowledge. In the case of algebra, however, this fact is of a nature different from that of empirical observation. It is indeed a relational fact for it co-represents, as has been said, activity itself. In calculating, I have done nothing but

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map the geometrico-empirical process of construction quite verbally into a vectorial-algebraic mode. In order to arrive at any kind of insight or meaning, I must now return to the geometrico-intuitive level and to the semantic resources it offers. Why do the centers of gravity of the two triangles coincide in the case of n = 6? Quite simply because they are nothing but different intensions of the same extension, or in other terms, because they represent different modes of representing the overall center of gravity. This is true, as has been said, only if the system of the original 6 points is a system of side midpoints belonging to another, larger hexagonal polygon. If we assume that the points of this hexagonal polygon represent a distribution of mass, and if we want to determine this mass distribution’s center of gravity, we may do this in different ways. We may select the points 1, 3, 5 Each of these points represents as midpoint the center of gravity of the two corners of the respective side. That is, the point system 1, 3, 5 replaces the original system of 6 points and is again replaced by the 1, 3, 5 triangle’s center of gravity. We may now carry out the same procedure of determining the center of gravity for the initial constellation of six points with regard to the points 2, 4, 6, having of course to arrive at the same overall center of gravity. In other words: the 6 points 1, 2, 3 . . . represent a system of midpoints of a hexagonal polygon only if the center of gravity of the two triangles 1, 3, 5 and 2, 4, 6 are nothing but two different ways of determining one and the same point, that is if they are two intensionally different and extensionally identical objects. This is quite easily observable in the case of the quadrangle. In this case, I may replace the 4 points, for example, by the two midpoints 1 and 3, and these midpoints again by the midpoint of their connecting line. We can do the same with midpoints 2 and 4. Now we are able to understand why the midpoint quadrangle within any quadrangle must needs be a parallelogram: diagonal lines will halve one another only within the parallelogram. 6.2. Cognitive activity should, I believe, be described as a system of means and objects and the dialectic of means and objects may briefly be summarized as follows: – As in any other cognitive activity, object and means of cognition are linked in mathematical activity. Mathematics cannot proceed in an exclusive orientation towards universal, formal methods. Mathematics, too, forms specific concepts intended to help us understand mathematical facts.

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– Object and means are not only linked, but also stand in opposition to one another. Objects or problems are resistant to cognition. They do not produce the means to their solutions out of themselves. Modern mathematics even draws its own dynamics in no small part from applying theorems and methods which at first glance have nothing to do with the problems at hand. In this, we understand by “object” any problem and by “means” anything that seems appropriate to achieve mediation between the subject and the object of cognition, any idea that might help solving the problem and any representation of that idea. Now, two different ideas may be decisive in solving a particular problem and thus appear as equivalent in this respect. Another problem may elucidate their difference and may in turn itself be illuminated by this difference. Fundamental ideas and theoretical concepts are self-referential, that is, they themselves, at least in part, organize the process of their own deployment and articulation. These ideas are what the development of an entire theory is devoted to unraveling and to explicating. In mathematics, to understand an idea or a concept means to apply it and to develop a theory. These ideas are, however, at the same time the beginning and the base of the development. This means they have to be intuitively impressive, must motivate and guide activity and orient representation. PROBLEM 1. Our first problem is taken from S. Papert’s book “Mindstorms” (Basic Books 1980, 146): Imagine a string around the circumference of the earth, which for this purpose we shall consider to be a perfectly smooth sphere, four thousand’ miles in radius. Someone make a proposal to place the string on six-foot-high poles. Obviously this implies that the string will have to be longer. A discussion arises about how much longer it would have to be. Most people who have been through high school know how to calculate the answer. But before doing so or reading on try to guess: Is it about one thousand miles longer; about a hundred, or about ten?

Figure 1.

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Papert then suggests looking first for a similar but simpler version: A good general rule for simplification is to look for a linear version. Thus we pose the same problem on the assumption of a ‘square earth’.

Figure 2.

Increasing the size of the square does not change the quarter-circle slices, such that the extra string needed to raise a string from the ground to height h is the same for a very small square earth as for a very large one. This solves the problem. The amount of extra string needed is (2π.h). Papert himself says that the purpose in working on the problem is not “to get the right answer”, but to “look sensitively for conflict between different ways of thinking about the problem”. In a different way and perhaps, more consistently Leibniz might suggest as a linear version.

Figure 3.

But Papert wants to continue thus:

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Figure 4.

Nevertheless, both may point to the fact that their idea shows that the size of the “earth” makes no difference to how much extra string is needed. That eventually solves the problem. Thus both ideas, let us call them Cur; (the idea of curvature) and Lin (the idea of linearity) respectively appear equivalent with respect to the problem at hand.

Figure 5.

But both ideas yield more. Lets start with the Leibniz version and lets for every polygon (square, octagon . . . ) call the shortest distance from the center to the perimeter the radius of that regular polygon. Increasing the radius by h increases the length of the perimeter by “the perimeter of a similar polygon of radius h”. This is exactly the linearity of the function represented by the geometric form itself. The whole is the sum of its parts: If the radius grows from x to (x + h) the perimeter increases from f (x) to f (x) + f (h).

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Therefore we have f (x + h) = f (x) + f (h) and f (0) = 0. The perimeter of a polygon is a linear function of its radius. And the principle of continuity yields the same result for the circle (remember that Leibniz was the first to systematically employ this principle). The important thing is that you can read this fact directly off from the geometric figures. You can see with your own eyes that the enlargement of the perimeter is again represented by a polygon of the same shape. To derive another, more constructive than descriptive representation of the linear function, Leibniz would point to the fact that the shape of the polygon is not changed by putting the string around on poles (i.e., enlarging the radius by h). f (x) f (x + b) = = const. and y = f (x) = c.x x+b x Both ways give an illustration of the idea of a formbased conceptualization. The potentialities of the Papert’s idea Cur can only be appreciated as soon as one considers the algebraic expression y = c.x as the new and relevant shape. In contrast to Leibniz we then get one and the same linear function y = 2π.x for all types of “earth’s” (in fact the perimeter may be any smooth closed curve without self-intersection). The proportionality factor 1.(2π ) yields the number of full circles which the heads of the poles run through when following the curve. Therefore alternative values of c would be n.(2π.) (n being any integer) and from the discreteness of the range we understand immediately that small deformations of the curve cannot change the value of c. The geometric object in question is no longer the curve, but rather a vector field along a curve. The integer n is usually called the index of that vector field with respect to the curve. The index will not change as long as deformations of the curve do not pass through a zero of the vector field. We might derive from this an easily understandable argument justifying Brouwer’s Fixed-Point Theorem. Both approaches solve problem 1 and from this point of view appear as equivalent. We shall now present a second problem with respect to which the ideas Cur and Lin will prove inequivalent. We take this problem (a problem about the epicycloid) from Robert Davis’ book “Learning Mathematics” (Croom Helm, London 1984, 216pp.). PROBLEM 2. “In a recent ETS test, one question dealt with a small circle that rolls, without slipping, around the outside of a larger circle. (Thinking of them, if you wish, as gears.) The radius of the large circle is three times the size of the radius of the small circle. . . . How many times do we see the small circle revolve. . . . ”

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Bob Davis continues: “The experts apparently reasoned essentially as follows: if the radius of the larger circle is three times as great, then the perimeter is three times as great. ’Rolling without slipping’ means that the arc-lengths are equal. Since the arc-length s is the product of radius times central angle, the angle for the little circle must be three times as great as the angle for the larger circle. But the angle on the large circle must increase by 2π , therefore, the angle for the small circle must increase by 3 × 2π = 6π , and the small circle revolves (or ‘turns’, or ‘rotates’) three times. This answer is wrong”. Obviously the experts tried to use idea Lin. Let us therefore see which results idea Cur will produce. We thus again replace the larger circle by a quadrangle. We see immediately (Figure 6) that at the four corners the smaller circle rotates without making progress along the circumference of the larger figure. By replacing the larger circle by the quadrangle we are enabled to perceive that two different rotations are superimposed in the motion of the smaller circle. The quadrangle separates, so to speak, these two movements of the smaller circle. As it rotates by an angle of 90◦ at each of the 4 corners, we immediately understand that the correct answer must be, four times!

Figure 6.

The midpoint of the smaller circle describes an epicycloid, which has 3 “leaves” (Figure 7). This is what idea Lin tells us. That the experts must have used this idea and misapplied it because they neglected the curvature or rather the two different rotations of the smaller circle is also suggested by Bob Davis. He writes: “The experts apparently reasoned essentially as follows: if the radius of the larger circle is three times as great, then the perimeter is three times as great. ‘Rolling without slipping’ means that the arclengths are equal. Since the arc-length s is the product of radius times central angle, the angle for the little circle must be three times as great as the angle for the larger circle. But the angle on the large circle must increase by 2π , therefore, the angle for the small circle must increase by

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3 × 2π = 6π , and the small circle revolves (or ‘turns’, or ‘rotates’) three times”. This answer is wrong, as has been said already.

Figure 7.

For the sake of completeness we present Davis’ solution: In Figure 8, C is the center of the small circle in the starting position. A moment after the motion starts, the center has moved to C  . Because of non-slipping, arc BA on the small circle has the same arc-length as arc AB on the large circle. Therefore angle BC  A is three times the size of angle AOB. But, the moment we have available to us a nonrotating reference line, P C  , that translates so as always to pass through the center of the smaller circle, we see easily that angle BC  A is NOT the angle through which the small circle has rotated. Instead, angle P C  A is. The rest of the solution in now routine. . . . The small circle revolves – or rotates – or turns – exactly four times. . . . Notice that the wrong representation comes close to being right. If a bicycle wheel that has a perimeter of 87 inches rolls without slipping along a flat sidewalk, and covers a distance of 3 × 87 = 261 inches, then the wheel will have rotated exactly three times. Presumably some cognitive representation of this phenomenon was retrieved, or constructed, by the experts, so that they all agreed – on the wrong answer” because of using idea Lin only.

Figure 8.

Like Papert’s analysis of problem 1, this presentation of problem 2 is very strongly influenced by the traditional concern for formulas. The Leib-

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nizian alternatives, in contrast, are more interested in relational structure and intuitive objects. Universität Bielefeld Institut für Didaktik der Mathematik Postfach 100131 33501 Bielefeld Germany E-mail: [email protected]