Penultimate draft of a paper published in Foundations of Science 18 (2013), pp. 93-106.
Top-Down and Bottom-Up Philosophy of Mathematics Carlo Cellucci Department of Philosophy, La Sapienza University of Rome
[email protected] +39 06 8558322
Abstract. The philosophy of mathematics of the last few decades is commonly distinguished into mainstream and maverick, to which a ‘third way’ has been recently added, the philosophy of mathematical practice. In this paper the limitations of these trends in the philosophy of mathematics are pointed out, and it is argued that they are due to the fact that all of them are based on a top-down approach, that is, an approach which explains the nature of mathematics in terms of some general unproven assumption. As an alternative, a bottom-up approach is proposed, which explains the nature of mathematics in terms of the activity of real individuals and interactions between them. This involves distinguishing between mathematics as a discipline and the mathematics embodied in organisms as a result of biological evolution, which however, while being distinguished, are not opposed. Moreover, it requires a view of mathematical proof, mathematical definition and mathematical objects which is alternative to the top-down approach.
Keywords: Mavericks, Proof, Definition, Evolution.
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1. Mainstream and Maverick Philosophy of Mathematics
The philosophy of mathematics of the last few decades is commonly classified into two categories: mainstream and maverick philosophy of mathematics (see Aspray and Kitcher 1988; Hersh 1997). The use of these terms does not seem entirely felicitous, because it suggests that mainstream philosophy of mathematics is the orthodoxy, maverick philosophy of mathematics the heresy, a heresy with no possibility of ever becoming the orthodoxy. As an alternative, it might be preferable to use ‘static’ and ‘dynamic,’ respectively (see Cellucci 2011). But the use of the terms ‘mainstream’ and ‘maverick’ is so well-established that there seems to be little chance of changing it. The main features of mainstream and maverick philosophy of mathematics may be summarized as follows. Mainstream philosophy of mathematics considers mathematics as a static body of knowledge; it is mainly concerned with the question of the justification of mathematical knowledge; it holds that there is an absolutely certain, or at least a fairly reliable, foundation for mathematics; it regards mathematical logic as a canon for the philosophy of mathematics; it holds that a detailed account of mathematical practice is desirable but not really essential; it generally sets itself within the framework of analytic philosophy. Maverick philosophy of mathematics considers mathematics as a dynamic body of knowledge; it is mainly concerned with the question of the growth of mathematical knowledge, including the dynamics of mathematical discovery; it holds that there is no absolutely certain foundation for mathematics; it regards mathematical logic as very useful to show the limitations of mainstream philosophy of mathematics through the limitative results, but inadequate to deal 2
with the question of the actual growth of mathematical knowledge; it holds that only a detailed analysis of mathematical practice could lead to a philosophy of mathematics worth its name; it generally sets itself outside the framework of analytic philosophy. Mainstream philosophy of mathematics includes the three big foundational schools of the first decades of the twentieth century, namely, logicism (Frege, Russell), formalism (Hilbert), intuitionism (Brouwer, Heyting). In addition, it includes the positions which ensued from such schools in the second half of the twentieth century: neo-logicism (Wright, Hale), platonism (Gödel, Penrose), neoformalism
(Curry,
Mac
Lane),
neo-intuitionism
(Bishop,
Dummett),
implicationism (Putnam), structuralism (Bourbaki, Shapiro, Resnik), fictionalism (Field), internalism (Maddy). Maverick philosophy of mathematics originated from Lakatos. Then Davis, Hersh, Kline, Kitcher, Tymoczko, Rota, Gillies, Grosholz, Rav, van Bengedem and others have contributed to it.
2. The Philosophy of Mathematical Practice
In a recent book, edited by Mancosu, it is maintained that, in alternative to mainstream and maverick philosophy of mathematics, there is a ‘third way’ in the philosophy of mathematics: the philosophy of mathematical practice. The book is meant to be a sort of manifesto of that philosophy. According to Mancosu, both mainstream and maverick philosophy of mathematics have limitations, though of different importance. On the one hand, mainstream philosophy of mathematics has paid “no particular attention to mathematical practice” (Mancosu 2008, 1). On the other hand, maverick 3
philosophy of mathematics “has not managed to substantially redirect the course of philosophy of mathematics,” since “logically trained philosophers of mathematics and traditional epistemologists and ontologists of mathematics felt that the ‘mavericks’ were throwing away the baby with the bathwater” (ibid., 5−6). There is then need for a third way, paying “attention to mathematical practice” (ibid., 2). It is somewhat peculiar that Mancosu, on the one hand, presents the philosophy of mathematical practice as a new kind of philosophy of mathematics, and, on the other hand, characterizes it in terms of attention to mathematical practice. For such attention is a distinguishing feature of maverick philosophy of mathematics. As soon as one considers mathematical practice, one realizes that mathematics is a dynamic body of knowledge, in particular mathematical discovery is a dynamic process. Therefore, attention to mathematical practice naturally leads to maverick philosophy of mathematics. On the contrary, the philosophy of mathematical practice, in the sense of Mancosu’s book, does not treat mathematics as a dynamic body of knowledge, in particular it pays little or no attention to the dynamics of mathematical discovery. Moreover, Mancosu speaks of ‘mathematical practice’ as if it were a monolith. Actually, there are many kinds of mathematical practices, with contrasting features. What they have in common is only that mathematics is a dynamic body of knowledge, but Mancosu’s book does not account for that. To substantiate the claim that the philosophy of mathematical practice is a genuine third way in the philosophy of mathematics, Mancosu states that the maverick approach has been a failure. But the only evidence he provides for this is
that
logically trained
philosophers
of
mathematics
and
traditional
epistemologists and ontologists of mathematics, that is, mainstream philosophers 4
of mathematics, have not accepted the viewpoint of the ‘mavericks’ nor have been converted to it. This, of course, is a non-argument because, after all, mainstream philosophers of mathematics are not the arbiters of truth. Indeed, that their viewpoint is the right one is exactly what is in question. In fact, Mancosu’s manifesto book simply proposes an extension of mainstream philosophy of mathematics which pays attention to mathematical practice, but without changing the basic assumptions of mainstream philosophy of mathematics. Mancosu himself acknowledges that the philosophy of mathematical practice does not want to “dismiss the analytic tradition in philosophy of mathematics” but rather “to extend its tools to a variety of areas that have been, by and large, ignored” (Mancosu 2008, 18). Thus the philosophy of mathematical practice is not an alternative, but only a complement to mainstream philosophy of mathematics. This helps to put into perspective Mancosu’s claim that the maverick approach has been a failure. Contrary to his claim, there is now a widespread opinion that the mainstream approach has been a failure because it is inextricably connected with the original goal of the foundational schools of giving an absolutely certain foundation to mathematics, or at least a fairly reliable one. One cannot seriously deal with questions concerning mathematical practice without treating mathematics as a dynamic body of knowledge. The philosophy of mathematical practice is unable to do that, because it retains the main assumptions of the mainstream philosophy of mathematics. In particular, mathematical logic provides no tools to that purpose, since it does not deal with the non-deductive rules on which discovery processes are based. As to Mancosu’s argument that maverick philosophy of mathematics has not managed to substantially redirect the course of philosophy of mathematics, in 5
this connection it is worth repeating what Dewey wrote a century ago: “Old ideas give way slowly” since “they are habits, predispositions, deeply engrained attitudes of aversion and preference” (Dewey 1910, 19). Actually, “intellectual progress usually occurs through sheer abandonment of questions,” an “abandonment that results from the decreasing vitality and a change of urgent interest. We do not solve them: we get over them. Old questions are solved by disappearing, evaporating, while new questions corresponding to the changed attitude of endeavor and preference take their place” (ibid.).
3. Limitations of Maverick Philosophy of Mathematics: Lakatos
This does not mean that maverick philosophy of mathematics is without limitations. They are already apparent from Lakatos, the originator of the maverick tradition. Lakatos opposes the deductivist conception, typical of mainstream philosophy of mathematics, since it present mathematics “as an ever-increasing set of eternal, immutable truths,” deduced from “axioms and definitions” that “frequently look artificial and mystifyingly complicated” (Lakatos 1976, 142). The deductivist conception “hides the struggle, hides the adventure,” the “successive tentative formulation of the theorem in the course of the proofprocedure,” and exalts the end result “into sacred infallibility” (ibid.). To the deductivist conception Lakatos opposes his “Hegelian conception of heuristic,” according to which, while being a product of human activity, mathematics “ ‘alienates itself’ from the human activity which has been producing it. It becomes a living, growing organism, that acquires a certain autonomy from the activity which has produced it; it develops its own autonomous laws of 6
growth, its own dialectic” (ibid., 145). Heuristic deals with such autonomous laws of growth, “with the autonomous dialectic of mathematics” (ibid.). Thus Lakatos is not concerned with the activity of individual working mathematicians, in particular with their discovery processes. According to him, there are no genuine discovery rules, and the so-called “‘logics of discovery’ consist merely of a set” of “rules for the appraisal of ready, articulated theories” (Lakatos 1978, vol. 1, 103). It is “pathological” to attempt “to turn heuristics into a system of rules which claim to take account of the art of discovery” (Lakatos 1961, 75). Rather, Lakatos is concerned with the alleged objective laws of mathematical development, which are independent of the mind of individual working mathematicians. The latter are “just a personification, an incarnation of these laws,” though an incarnation which “is rarely perfect” (Lakatos 1976, 146). That Lakatos is not concerned with the activity of individual working mathematicians depends on his Hegelian approach to heuristics. It is constitutive of the latter that one should not consider the processes of individual human beings but rather the objective laws of development of rationality in history. For such reason, Lakatos does not deal with the processes through which individual working mathematicians discover new results, and only seeks a rational reconstruction of the objective development of mathematics. As a result, Lakatos’s heuristic rules are not genuine discovery rules. This is already apparent from his very first heuristic rule, which states: “If you have a conjecture, set out to prove it and to refute it. Inspect the proof carefully to prepare a list of non-trivial lemmas (proof-analysis); find counterexamples both to the conjecture (global counterexamples) and to the suspect lemmas (local counterexamples)” (ibid., 50). This rule does not tell you how to discover a 7
conjecture, it assumes that you already have one, and is only concerned with how to prove or to refute it. That Lakatos provides no genuine discovery rules, and generally no explanation of the actual growth processes of mathematical knowledge, as opposed to the alleged objective laws of development of mathematics, is a consequence of the fact that his approach to the philosophy of mathematics is topdown. There are two approaches to the philosophy of mathematics: top-down and bottom-up. The top-down approach explains the nature of mathematics in terms of some general, unproven assumption. The bottom-up approach explains the nature of mathematics in terms of the activity of real individuals and their interactions. Lakatos has something in common with mainstream philosophy of mathematics: the use of a top-down approach. For he explains the nature of mathematics in terms of the assumption that there is an autonomous dialectic of mathematics. Such assumption is unproven and has no connection with the activity of individual working mathematicians.
4. Limitations of Maverick Philosophy of Mathematics: Kitcher
The limitations of maverick philosophy of mathematics are also apparent from Kitcher, another seminal figure in the maverick tradition. Kitcher states that mathematics is “the constructive output of an ideal subject” (Kitcher 1983, 109). The latter “is an idealization of ourselves” (ibid.: 111). It is like ourselves except that, “given our biological limitations, the operations in which we actually engage are limited” (ibid., 109). To talk about the operations of an ideal subject “is not to suppose that there is a mysterious being 8
with superhuman powers,” but only that “mathematical truths are true in virtue of stipulations which we set down, specifying conditions” which “are approximately satisfied by operations we perform” (ibid., 110). The relation between mathematical statements and “our actual operations is like that between the laws of ideal gases and actual gases” (ibid., 116). Ideal gases do not exist in our world, nevertheless “actual gases approximately satisfy the condition set down in the Boyle-Charles Law” (ibid., 117). In fact, “if the molecules of gases were of negligible size and if there were no intermolecular forces then gases would obey the Boyle-Charles Law” (ibid.). This “prompts us to abstract from certain features of the actual situation, introducing the notion of an ideal gas to describe how actual gases would behave if complicating factors were removed” (ibid.). Similarly, the ideal subject does not exist in our world, and yet the operations we perform approximately satisfy the conditions of an ideal subject on such operations. In fact, if real subjects were not subject to biological limitations, their operations would exactly satisfy such conditions. This prompts us to “specify the capacities of the ideal agent by abstracting from the incidental limitations on” our “practice” (ibid.). The whole of mathematics is the constructive output of the ideal subject, including set theory and the cumulative hierarchy of sets, that is, the hierarchy “obtainable from the integers (or some other well-defined objects) by iterated application of the operation ‘set of’,” where “this phrase is meant to include transfinite iteration” (Gödel 1986–2002, vol. 2, 259 and footnote 13). Indeed, according to Kitcher, the cumulative hierarchy of sets “may be viewed as a literal account of the iterated constructive activity of the ideal mathematical subject” (Kitcher 1983, 133). This requires us “to ascribe to the ideal subject an ability to 9
perform iterative collective activity through an infinite sequence of stages” (ibid., 147). Since “the sequence of stages at which sets are formed is highly superdenumerable,” it follows “that each of the stages corresponds to an instant in the life of the constructive subject, and that the subject’s activity is carried out in a medium analogous to time, but far richer than time. (Call it ‘supertime.’)” (ibid., 146). Thus Kitcher is not concerned with the activity of individual working mathematicians, in particular with their discovery processes. This is a consequence of the fact that his approach to the philosophy of mathematics is topdown. For he explains the nature of mathematics in terms of the assumption that mathematics is modelled by the activity of an ideal subject. This assumption is not only unproven but also hardly justified. For although ideal gases do not exist in our world, actual gases approximately satisfy the conditions set down by the laws of ideal gases, and often the degree of approximation can be computed. On the contrary, Kitcher assumes the ideal subject to be able to perform operations that cannot be approximately carried out by any individual working mathematician. For example, in the case of set theory, he assumes the ideal subject to be able to perform uncountably many collecting operations, for example, the operation of taking the set of all sets of natural numbers. As regards the cumulative hierarchy of sets, Kitcher even assumes the ideal subject to be able to perform an iterated collective activity through the highly super-denumerable sequence of stages of the cumulative hierarchy of sets, and to perform it in a super-time. While actual gases approximately satisfy the conditions set down by the laws of ideal gases, the structure of human time by no means approximates the structure of such super-time. Therefore, it is unjustified to say that the relation between mathematical statements and our actual operations 10
is like that between the laws of ideal gases and actual gases. Indeed, an ideal subject capable of performing super-operations in a supertime would be rather godlike than human. Thus to talk about the ideal operations of an ideal subject is really to suppose that there is a mysterious being with superhuman powers. This conflicts with the claim that mathematical truths are true in virtue of stipulations which are approximately satisfied by operations we perform. The ideal subject has nothing in common with a real working mathematician, so one cannot account for human mathematics in terms of it. It might even be argued that the capability of performing super-operations in a super-time would be super-godlike rather than godlike. For, in philosophical theology, the divine mind is thought of either in time or in eternity, not in a supertime. The trouble with Kitcher’s position is that mathematical concepts, including the cumulative hierarchy of sets, are human-made, and properties of mathematical concepts are discovered by concrete human beings, not by an ideal subject. Kitcher is unable to explain how concrete human beings actually make mathematical concepts and discover their properties.
5. Natural and Artificial Mathematics
What has been said above about Lakatos and Kitcher suggests that a proper approach to the philosophy of mathematics cannot be top-down. A top-down approach pretends to explain a complex phenomenon, such as mathematics, in terms of assumptions that are not only unproven but also more problematic than the phenomenon they purport to explain, so it is not really explanatory. The only proper approach to the philosophy of mathematics is bottom-up, since only 11
starting from the activities of individual working mathematicians one may hope to give an explanation of all aspects of mathematics. More precisely, a proper bottom-up approach to the philosophy of mathematics should start not only from the high level activities of individual working mathematicians, but also from their low level activities. In other words, it should start not only from artificial mathematics, that is, mathematics as a discipline, but also from natural mathematics, that is, the mathematics embodies in organisms as a result of biological evolution. For artificial mathematics depends on capacities belonging to natural mathematics. As we have just said, natural mathematics is embodied in organisms as a result of biological evolution. All organisms survive by making hypotheses about the environment. Some of these hypotheses are embodied in the biological structure of organisms, and some of them concern mathematical properties of the environment. Biological evolution has hardwired organisms to perform certain mathematical operations, building mathematics in several aspects of their biological makeup, such as locomotion and vision, which require some sophisticated embodied mathematics. These mathematical operations are essential to escape from danger, to search for food, to seek out a mate. As a result, some organisms have an innate sense of space, number, size, shape, order, and most of them have at least one of these senses (see Rav 1987, Devlin 2005, De Cruz 2009, De Cruz and De Smedt 2010). These innate senses are mathematical in kind. They have a biological function and are a result of biological evolution, which has selected and embodied them in organisms as innate knowledge. In fact, experiments show that “geometrical knowledge arises in humans independently of instruction, experience with maps or measurement devices, or 12
mastery of a sophisticated geometrical language” (Dehaene at al. 2006, 384). Some “core geometrical knowledge” is “a universal constituent of the human mind” (ibid.). Similarly, “in many animal species, as well as in young infants before they acquire number words,” experiments “have revealed the rudiments of arithmetic,” although “infants and animals appear to represent only the first three numbers exactly” (Pica et al. 2004, 499). Moreover, simple mathematical operations, such as counting, addition, subtraction, on small numbers, are innate in infants and in other animals as well. Experiments show that “numerical approximation is a basic competence, independent of language, and available even to preverbal infants and many animal species,” so “sophisticated numerical competence can be present in the absence of a well-developed lexicon of number words” (ibid., 503). Natural mathematics consists of the innate mathematical senses that biological evolution has embodied in organisms. That some mathematical capacities are innate was already asserted by Roger Bacon, who stated that “the knowledge of mathematical things is almost innate in us,” since “laymen and people who are utterly illiterate know how to count and reckon”.(Bacon 1900, vol. 1, 103). The innate mathematical capacities in question did not arise only once, and then were inherited by all species of animals which possess them. They have independently arisen several times in the course of biological evolution. Similar mathematical innate skills occur in species from widely diverging families, and capacities such as the senses of space and number are common to various animals. This is not surprising, since animals all live in the same world, where they have to face similar problems, and the number of solutions accessible to them is limited. In particular, all mobile organisms have to deal with problems of a similar nature, 13
such as to keep track of predators or prey, to distinguish amounts of food items in their environment, and so on. Senses such as those of space, number or size are present in several species because they are essential for survival. Human beings, however, have more complex mathematical capacities than those found in other species. Such capacities are the result of adaptations to the niche Homo occupied in the Lower Paleolithic, that of stone tool technology. Stone tool-making abilities shaped geometrical abilities, because they exerted selective pressures on modules for assessing two- and three-dimensional shapes. On the other hand, counting arose from such disparate needs as to keep track of cyclical phenomena like lunar phases or menstrual cycle, to have a calendar system, to count animal group size, or to keep track of animal migratory events. Counting was first assisted by using body parts, most notably fingers. Counting aids other than body parts appeared in the Upper Paleolithic, in the form of notched bones, pebbles, and so on. Such aids external to the brain extended the capacity of the body, since they can be used to remember, calculate and argue much better than an individual brain can. (This can be explained in terms of the theory of the extended mind; see Cellucci 2008b, ch. 21). Use of external aids provided the basis for the rise of artificial mathematics, namely mathematics as a discipline. While biological evolution can explain the rise of basic human and animal mathematical capacities, it cannot explain the scope and power of artificial mathematics. The latter has arisen too recently to be a direct product of biological evolution, it is rather a product of cultural evolution. Nevertheless, artificial mathematics is based on the innate capacities of natural mathematics and is strictly dependent on them. As computers can only run the software their hardware permits them to run, human
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mathematicians can only do the artificial mathematics their cognitive architecture permits them to do. A terminological remark may be in order. Calling mathematics as discipline ‘artificial mathematics’ may appear awkward, but it is not really so. For the first dictionary meaning of ‘artificial’ is, ‘produced by human art or effort rather than originating naturally.’ This is exactly what is meant here. Indeed, the expression ‘artificial mathematics’ is meant to suggest that it is not a direct natural product, since it is not a direct result of biological evolution, but rather a human product, so only an indirect natural product. For such reason, the expression ‘artificial mathematics’ seems preferable to the expression ‘abstract mathematics’ which is sometimes used (for example, see Devlin 2005, 249).
6. Limitations of the Empiricist View of the Mind
In the previous section we have stated that some mathematical knowledge is innate. This contrasts with the empiricist view of the mind as a tabula rasa, which assumes that the mind is originally “white paper, void of all characters, without any ideas” (Locke 1975, 104). According to this view, children are born without knowledge, and gradually learn to deal with the world as they are confronted with its properties, since the responses a brain produces are shaped by experience alone. In particular, there is no natural mathematics, all of mathematics is a cultural product. The empiricist view has been very influential until recently. Examples of its influence are Piaget and Kitcher. Piaget claims that “the operatory structures of intelligence are not innate,” they “are not preformed within the nervous system, neither are they in the 15
physical worlds” (Piaget 1972, 211). Rather, they “are constructed by the interaction between the individual’s activities and the object’s reactions” (Bringuier 1980, 37). In particular, the ability to count arises gradually at the age of four or five, because it requires the prior development of logical skills. The “construction of number goes hand-in-hand with the development of logic” (Piaget 1952, viii). The logical capacities required include “the gradual elaboration of systems of inclusion” and systems of “qualitative seriations” (ibid.). The “sequence of whole numbers” results from “the fusion of inclusion and seriation of the elements into a single operational totality” (ibid.). Similarly, Kitcher claims that “children come to learn the meanings of ‘set,’ ‘number,’ ‘addition’ and to accept basic truths of arithmetic by engaging in activities of collecting and segregating” (Kitcher 1983, 107−108). Therefore, they are able to engage in them not earlier than two or three years of age. The concepts of set, number, addition were formed by our remotest ancestors through these very same operations. Thus “we can view the activities of contemporary children as indicating the ways in which our ancestors, unaided by authority, began the mathematical tradition” (ibid., 108, footnote 9). Piaget’s and Kitcher’s claims, however, contrast with the experiments mentioned in the previous section.
7. The Top-Down vs. Bottom-Up Approach to Mathematics
As there are two approaches to the philosophy of mathematics, top-down and bottom-up, so there are two approaches to mathematics itself: again, top-down and bottom-up.
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The top-down approach to mathematics develops mathematics from within, starting from given axioms and deducing theorems from them, then it tries to apply the mathematics thus developed to non-mathematical fields. Since, however, it develops mathematics from within, not in view of some specific nonmathematical field of application, the resulting mathematics will reflect only roughly the features of the non-mathematical field to which it is applied. An example of the top-down approach is Bourbaki’s development of mathematics in his Éléments de mathématique. According to Bourbaki, mathematics is “a storehouse of abstract forms – the mathematical structures” (Bourbaki 1996, 1276). The development of the “theory of a given structure, amounts to the deduction of the logical consequences of the axioms of the structure” (ibid., 1271). For example, to develop the theory of the group structures, or, briefly, groups, one first fixes certain conditions “which are explicitly stated and which are the axioms of the structure under consideration” (ibid., 1270−1271). Then one deduces logical consequences from these axioms. Although mathematics deals with abstract forms, namely, the mathematical structures, “it so happens – without our knowing why – that,” by giving a suitable interpretation of the terms occurring in the axioms, “certain aspects of empirical reality fit themselves into these forms, as if through a kind of preadaptation” (ibid., 1276). One might ask “why do such applications ever succeed” but, “fortunately for us, the mathematician does not feel called upon to answer such questions” (Bourbaki 1949, 2). The bottom-up approach to mathematics develops mathematics from without, starting from arguments used in some non-mathematical field, and trying to transform them into mathematical arguments, so into new mathematics. Since it
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develops mathematics starting from arguments of some non-mathematical field, the resulting mathematics will reflect features of that field. An example of the bottom-up approach is Newton’s development of calculus. Newton states: “I consider mathematical quantities” as “described by a continuous motion. Lines are described and by describing generated” through “the continuous motion of points, surface-areas through the motion of lines, solids through the motion of surface-areas, angles through the rotation of sides, times through continuous flux, and the like in other cases. These geneses really take place in physical nature” (Newton 1967-81, vol. 8, 122). On such ground, Newton develops “a method of determining quantities out of the velocities of motion or increment by which they are generated,” and calls “these velocities of motion or increment ‘fluxions’” (ibid.). Thus he transforms physical arguments into new mathematics. To the top-down and bottom-up approaches to mathematics there correspond two different views of the method of mathematics. To the top-down approach there corresponds the axiomatic method. This is the method according to which one assumes as starting points axioms which are true in some sense, and deduces logical consequences from them. A weak sense of the axioms being true is that they are consistent, namely, no contradiction can be deduced from them. That to the top-down approach there corresponds the axiomatic method is apparent, for example, from Bourbaki, who states that “‘mathematical truth’ resides uniquely in logical deduction starting from premises arbitrarily set by axioms” (Bourbaki 1994, 17). Although premises are arbitrarily set by axioms, the latter must be consistent since, if they are inconsistent, “every theorem is both true and false in the theory, and the theory is consequently of no interest” (Bourbaki 18
2004, 12). To the bottom-up approach there corresponds the analytic method. This is the method according to which, to solve a problem, one starts from the problem, one finds a hypothesis that is a sufficient condition for its solution by means of some non-deductive inference (inductive, analogical, metaphorical, metonymic, diagrammatic, etc.), and one checks that the hypothesis is plausible, by deducing consequences from it and testing that they are compatible with the available data. (On different kinds of non-deductive inferences to find hypotheses, see Cellucci 2002, 235−295). The hypothesis, in turn, is a problem that must be solved, and is solved in the same way, namely, finding a new hypothesis by means of some non-deductive inference and checking that it is plausible. And so on, with a potentially infinite process. (On the analytic method, see Cellucci 2002, 2008a, 2008b). That to the bottom-up approach there corresponds the analytic method is apparent, for example, from Newton, who states that the propositions of Principia Mathematica “were invented by Analysis” (Newton 1971, 294). Analysis, that is, the analytic method, is the method of “mathematicians of the last age,” who “have much improved Analysis” and “stop there” for they “think they have solved a Problem when they have only resolved it,” namely solved it by Analysis, and “by this means the method of Synthesis,” that is, the axiomatic method, “is almost laid aside” (ibid.). Notice that the downward movement, which is characteristic of the topdown approach, occurs also in the bottom-up approach, though subordinately to the upward movement. For, as we have said, in the analytic method, to establish that a hypothesis is plausible, one deduces consequences from it and tests that they are compatible with the existing data. Thus the analytic method actually 19
involves a double movement, upward, to discover hypotheses, and downward, to test them. Between these two movements, however, there is a difference. While the downward movement may be finite, the upward movement is potentially infinite. For in the analytic method the process of finding hypotheses for solving a problem is a potentially infinite one. Davis raises the important question: “When is a problem solved?” (Davis 2005, 163). He states that “discovering a sense in which a solved problem is still not completely solved but leads to new and profound challenges, is one important direction that mathematical research takes” (ibid., 177). By describing the process of solving a problem as a potentially infinite one, the analytic method suggests an answer to this question.
8. Asymmetry Between the Top-Down and the Bottom-Up Approach
The top-down and the bottom-up approaches to mathematics are not on a par because the top-down approach is incompatible with Gödel’s incompleteness theorems, only the bottom-up approach is compatible with them. Indeed, by Gödel’s first incompleteness theorem, for any consistent axiom system for any mathematics field satisfying certain minimal conditions, there are true sentences of that field which cannot be deduced from the axioms. Therefore, mathematics cannot consist, as the top-down approach assumes, in the deduction of logical consequences from axioms which are true, not even in the weak sense that they are consistent. Moreover, by Gödel’s second incompleteness theorem, for any consistent axiom system for any mathematics field satisfying certain minimal conditions, it 20
is impossible to establish by reliable means whether the axioms are true, not even in the weak sense that they are consistent. Thus one cannot recognize, as the top-down approach requires, that the axioms are true in some sense. Therefore, there is no guarantee that the top-down approach does not lead to falsities. Actually, consistency is an inadequate criterion of truth because, by a corollary of Gödel’s first incompleteness theorem, for any consistent axiom system for any mathematics field satisfying certain minimal conditions, there is a consistent extension of the axioms from which a false statement can be deduced. Thus, not only there is no guarantee that the top-down approach does not lead to falsities, but one can actually prove that, even when axioms are consistent, it can lead to falsities. Mathematicians tend to dismiss the relevance of Gödel’s incompleteness theorems. For example, Davies states that they have “almost no relevance to the work of most mathematicians” (Davies 2008, 88). However, mathematicians holding this view do not explain how they may reconcile it with their belief that the method of mathematics is the axiomatic method, a belief which is incompatible with Gödel’s first incompleteness theorem. Actually, they dismiss the relevance of Gödel’s incompleteness results to their work because their viewpoint is confined to local features of mathematics, ignoring the global question of the nature of mathematics as a whole. Since the top-down approach to mathematics is incompatible with Gödel’s incompleteness results, such an approach may be useful for pedagogical purposes but does not express the nature of mathematics. (Actually, one may rise doubts even about its usefulness for pedagogical purposes; see Cellucci 2011). Only the bottom-up approach is compatible with Gödel’s incompleteness results, so the method of artificial mathematics can be identified only with the analytic 21
method. The analytic method is the method not only of artificial mathematics, but also of natural mathematics. For example, it is by that method that our remotest ancestors kept track of predators or prey, making hypotheses concerning their location by means of non-deductive inferences from crushed or bent grass and vegetation, bent or broken branches or twigs, mud displaced from streams, and so on. Between natural and artificial mathematics there is continuity, not in the sense that the latter is reducible to the former, but rather in the sense that they both depend on the very same capacities, which are a result of biological evolution, and use the same basic procedures.
9. The Nature of Mathematical Proof
That the method of mathematics can be identified with the analytic method implies that mathematical proof is not, as the top-down approach assumes, only a means of justification. According to that approach, the aim of proof is to give an answer to the question: For any given mathematical proposition, “whence do we derive the justification for its assertion?” (Frege 1959, 3). But, if mathematical proof were only a means of justification, its role would be somewhat empty. For no axiomatic proof can be more reliable than the axioms on which it depends, and, by Gödel’s second incompleteness theorem, it is generally impossible to establish by reliable means whether the axioms are consistent. Mathematical proof is also, and foremost, a means of discovery. Its aim is to discover hypotheses capable of establishing the result. As Hamming says, 22
in mathematics “you start with some of the things you want and you try to find postulates to support them” (Hamming 1998, 645). The idea that you simply lay down some arbitrary postulates and then make deductions from them “does not correspond to simple observation” (Hamming 1980, 87). In addition to discovering hypotheses capable of establishing the result, proof has also another important role. Since, by Gödel’s first incompleteness theorem, for any consistent axiom system for any mathematics field satisfying certain minimal conditions, there are true sentences of that field which cannot be deduced from the axioms, to prove those true sentences one will generally need hypotheses from other mathematics fields. Now, by using hypotheses for mathematics fields other than the one to which the result belongs, a proof may establish connections between different mathematics fields. This may reveal unexpected relations between them, which may suggest new perspectives and new problems and so may be very fruitful for the development of mathematics. This important additional role of proof explains why, from ancient times to the present, a salient feature of mathematical practice has been to look for new proofs of results for which proofs were already known. For example, there are over 400 proofs of the Pythagorean Theorem. A Field Medal has been awarded to Selberg for giving a new proof of the primenumber theorem for which a proof was already known. This cannot be explained if, as the top-down approach assumes, the aim of proof is to give a justification of a result, thus placing “the truth of a proposition beyond all doubt” (Frege 1959, 2). If a proof placed the truth of a proposition beyond all doubt, what would be the point of proving the Pythagorean Theorem over 400 times, or of awarding a Field Medal for giving another proof of the prime-number theorem? The first proof which was found would have already placed the truth of the proposition beyond 23
all doubt. Conversely, to look for new proofs of results for which a proof is already known, is natural if one of the aims of proof is to establish connections between different branches of mathematics. Then the result is viewed from several different perspectives, which may open new possibilities for mathematics and lead to new developments.
10. The Nature of Mathematical Definitions
That the method of mathematics can be identified with the analytic method also implies that mathematical definitions are not, as the top-down approach assumes, mere typographical abbreviations. According to that approach, “no definition extends our knowledge. It is only a means for collecting a manifold content intro a brief word or sign, therefore making it easier to handle” (Frege 1984, 274). But, if mathematical definitions were mere typographical abbreviations, this would not explain why mathematicians often spend much time in finding suitable definitions to justify statements that they already know to hold. Actually, mathematicians often establish results a long time before an adequate definition of the concepts involved in them is found. Such is the case of Euler’s formula for polyhedra, which was known long before an adequate general definition of polyhedron was found (see Lakatos 1976). Rather, mathematical definitions contain an analysis of concepts which had previously been more or less vague. Such analysis is non-trivial, it is the result of an investigation, so finding an adequate mathematical definition is the result of a discovery, it is an advance in knowledge which may lead to fruitful 24
developments. Therefore, a mathematical definition is not a starting point but rather an arrival point of the investigation. That finding an adequate mathematical definition is an advance in knowledge, explains why mathematicians often spend much time in finding suitable definitions to justify statements that they already know to hold. Mathematical definitions are hypotheses and, like all other hypotheses, are tentatively introduced to solve problems.
11. The Nature of Mathematical Objects
That the method of mathematics can be identified with the analytic method also implies that mathematical objects themselves are hypotheses tentatively introduced to solve problems. A mathematical object is the hypothesis that a certain condition is satisfiable. Thus, for Euclid, a circle is the hypothesis that there can be a plane figure contained by one line such that all the straight lines falling upon it from one point of those lying inside the figure are equal to one another. If, in the course of reasoning, the condition turns out to be satisfiable, we say that the object ‘exists,’ if it turns out to be unsatisfiable, we say that it ‘does not exist.’ Thus speaking of existence is just a metaphor. There is no more to mathematical existence than the fact that mathematical objects are hypotheses tentatively introduced to solve problems. Mathematical objects have no existence outside our mind, since they are simply mental tools which we use to solve problems. They are bound to our biological makeup and do not exist independently of it. They are the way we make problems understandable, and their solution accessible, to ourselves. Fixing properties of mathematical objects, the hypotheses through which 25
mathematical objects are introduced characterize their identity. The identity of a mathematical object can be characterized differently by different hypotheses. For example, Euclid characterizes the identity of the real line by the hypothesis that it is a breadthless length which lies evenly with the points on itself, Cantor and Dedekind characterize it by the hypothesis that it is the set of real numbers. In any case, hypotheses do not characterize the identity of mathematical objects completely and conclusively, but only partially and provisionally. For the identity of mathematical objects is always open to receiving new determinations through interactions between hypotheses and experience. From such interactions new properties of mathematical objects may emerge since, by putting the hypotheses which introduce them in relation with other things, mathematical objects may get new determinations. Such new properties may also suggest to modify or completely replace the hypothesis through which the identity of mathematical objects has been characterized. This is a potentially infinite process, so mathematical objects are inexhaustible. This is the lesson of Gödel’s first incompleteness theorem. Acknowledgements
I am grateful to Reuben Hersh and Robert Thomas for comments on an earlier draft of this paper. References
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