On Naturalism in Mathematics

Transcription

1 On Naturalism in Mathematics Alfred Lundberg Bachelor s Thesis, Spring 2007 Supervison: Christian Bennet Department of Philosophy Göteborg University 1

2 Contents Contents...2 Introduction... 3 Naïve Questions... 3 Contemporary concerns... 4 Naïve answers of naïve questions... 6 Traditional Schools of Mathematical Philosophy... 7 Point of Departure... 9 Quine s philosophy On Ontology An Empirical Dogma Naturalism Meaning according to Quine Quine on mathematics Maddy s philosophy Criticism: Quine...17 Mathematics in the web Meaning Ontology Cartesian doubt is not the way to begin Criticism: Maddy...22 Conclusion...25 References

3 Introduction Philosophy has aimed to ask and answer clever questions on a wide variety of subjects throughout the ages. These can be summarized in a single one:? Or, spelled out a bit: What is its nature? Turning towards mathematics, this can be differentiated and specified in several ways, each way resulting in one of the traditional philosophical questions of mathematics. These may, as in other philosophical fields, take simple or more technical form, but they originate, at least historically, in common sense questions of the nature of mathematics. Therefore, a common sense starting point for this exploration seems appropriate. Also, for people not familiar with the discipline this is supposedly the best approach. We will concentrate mainly on a philosopher who urged the leaving behind of far-fetched philosophical theorizing in favour of common sense questions in a scientific context. Thus, this approach will not oppose his manners, but will on the contrary meet central standards of consequence. Naïve Questions So, what is mathematics? What does it do? What does it tell us? Mathematics is a modern scientific field of research, but with roots of thousands of years. It is definitely among the oldest subjects found at universities of today. Taking a closer look, many questions arise: What is the nature of mathematics? This rather general question is seldom asked as it stands. It splits up into a variety of more precise questions. What does mathematics talk about? Mathematics seems to be about something. Mathematicians talk as if they talked about something. This, any grammatical analysis would consent to. Well, the answer seems quite simple: Numbers. Mathematics is about numbers, and their relations and their properties. It is also about functions and matrices, of course, and sets and points and surfaces and so on. But several of these entities are of a somewhat strange nature. In what ways are these entities? What are they, and where? The ontological status of mathematical objects has been the subject for discussion throughout the ages. These objects seem to be, in a way, independent of our thoughts and wishes, yet they have a rather clear mental status, which connects them to the mind, rather than the outer physical world. What is mathematical knowledge? Mathematics seems to deliver knowledge in some sense. First of all, mathematical studies have the result that the student, insofar as the studies are successful, feels that he knows more afterwards. Mathematics has been rigorously important in the development of the natural sciences. And science tells us something about our environment. Thus, we have all the usual philosophical questions connected to knowl- 3

4 edge, in their mathematical form: How do we acquire mathematical knowledge? Mathematical knowledge seems more secure than other kinds of knowledge, as it stands in no dependence upon empirical knowledge, or does it? Anyhow, if so, how can that be? And if mathematical knowledge is independent of sensory input, how is it possible to explain the remarkable success of mathematics in natural science? Epistemology has traditionally been a central concern of the philosophy of mathematics. Are mathematical statements meaningful, despite their scent of triviality? How is it possible for mathematical statements to be meaningful, as they only state what seems not to could have been otherwise, what seems to be trivially true? Ever since Pythagoras until our days, mathematics has been worshipped in strong romantic terms. The field is claimed to offer an unparalleled rational training, to exhibit artand music-like beauty. It has been claimed to provide knowledge of the most secure form possible and it has inspired religious tendencies ranging from Pythagorean number mysticism, via so-called sacred numbers appearing in almost all major world religions, to cabbalistic permutation and the like. What is the real ground for these reactions? What eternal virtues of mathematics does this witnessing really tell of? Is our mathematical theory the only (possible) one? Might another culture, another species, take a radically different line of mathematical development? Thus, in practice we might ask whether there are other axiomatisations that are equally good but that differs essentially from our present one. If so, we would consider there to be two possible mathematics, thus rejecting uniqueness. Contemporary concerns The connections of mathematics to logic seem obvious. The fields share a range of attributes due to their common axiomatic and deductive method, their security of results, etc. It is hence not a surprise that efforts were made in the late 1800 s to establish mathematics on the grounds of logic alone. The notion of a collection or a class of objects was already in the air, and mathematician Gottlob Frege used the concept in trying to provide a foundation for arithmetic. His aim was to establish the philosophical status of mathematics through giving a complete account for the natural numbers, and these he defined as classes of equinumerous sets. Unfortunately, his logical system was shown inconsistent 1 by Bertrand Russell not long after. To save the project one had to skip what was called the Axiom of Comprehension, and to rebuild the theory through specifying a number of new axioms. This was done in order to weaken the system, to avoid contradiction, while retaining most of its expressive powers, those necessary in expressing and proving the results of mathematics. This work was mainly accomplished by Richard Dedekind, Abraham Fraenkel, Ernst Zermelo and Thoralf Skolem, and the resulting theory is commonly known as Set Theory, or ZFC 2. 1 A system is inconsistent if it proves a contradiction, consistent otherwise. 2 ZFC denotes the Zermelo-Fraenkel axiomatisation of set theory, together with the Axiom of Choice. This system is widely accepted as the foundation of all of mathematics, though not the only set theory. 4

5 The notion of equinumerous sets was also the basis for Georg Cantor s theory of cardinal numbers. In it, Cantor established some nontrivial facts on the number of elements of sets, applicable on sets of numbers through their interpretation in ZFC. Now, one soon ran into further worries, as one of Cantor s hypotheses proved to be independent of the axioms of ZFC 3, and more examples have later been found. Thus, Frege s initial theory had proved erroneous, in allowing contradictions to be deduced, and so being to strong. And its replacing theory, ZFC, now proved to be too weak, as it was not able to settle natural questions appearing in the theory. This is the picture still today. Hence, the question arises, what are we to do about such results? For every such question, is it to be answered in the affirmative or in the negative, when the theory cannot advise? Or should we simply accept their independence? If mathematics is about real, somehow existing, objects, what moral is to be drawn from such a situation? And how will mathematics proceed? But more is to be said on logic and mathematics, and even more intriguing questions arise. Among the most important logical results to this day are the so called Incompleteness Theorems that were proved by Kurt Gödel in These results came as a mere chock to the contemporary mathematico-logical society. David Hilbert had in 1899 showed how the Euclidean geometry could be formalized in a first order logic, and Alfred Tarski proved some years later that Hilbert s theory was complete and decidable 4. Now, the hope was to repeat the feat for number theory. Through the works of Richard Dedekind, Giuseppe Peano and others, the Peano Arithmetic, PA, was formulated as a logical theory of first order, and the idea was now to provide rigorous ground for that also PA was complete and decidable. This project was completely overthrown by Gödels results. His first theorem shows that in any axiomatizable first order theory T that is consistent and includes PA, we can construct true sentences, in the language of T, that do not have a proof or a disproof. That is, any such theory, PA included, is not complete. In his second theorem, Gödel showed that this result has consequences for what can be said about consistency. Any logical theory aiming to found mathematics must account for number theory. In fact, with number theory at hand, most of classical mathematics can be constructed. So mathematics without number theory is simply not imaginable. Now, the second incompleteness theorem says that if we are to formalize mathematics in an axiomatizable theory T that includes PA, and we do this in first order logic, then T cannot prove its own consistency. Thus, we may not, in first order logic at least, account for the consistency of mathematics, using elementary mathematical methods. Now, the reason we would want to do this in first order logic, is that, while second order logic can account completely for number theory 5, it has no complete deductive system. On the other hand, any logic with such a system, first order or not, would again be subject to Gödels theorem. It appears thus that we cannot, as far as mathematical methods can guide, be certain of the consistency of mathematics. And also, it appears as number theory inevitably contains statements 3 The Continuum Hypothesis, that any set of real numbers either is equinumerous to the set of all real numbers, or to the set of natural numbers, or is finite, was proved independent of Set Theory by Paul Cohen in A theory is complete if every sentence in the language of the theory has a proof or a disproof from the axioms. It is further decidable if there is an algorithm for determining for any such sentence that it is a theorem, i.e. has a proof, or not. 5 Dedekind formalized number theory in second order logic in 1880, and subsequently showed it to have exactly all true statements of number theory as theorems. 5

6 that can be proved neither true nor false. This is of course a fact contradictory to what might be called an advanced naïve view on mathematics, as held by mathematicians throughout mathematical history. Thus, at a first philosophical glance at mathematics, the following themes of questioning arise: ~ What is its nature? ~ The ontology of mathematics ~ Applicability ~ Mathematical epistemology ~ Meaning and mathematics ~ Eternal virtues ~ Uniqueness ~ Incompleteness of the axioms of set theory ~ Incompleteness in the sense of Gödels theorems Surely, there is no limit to the complexity of philosophical questions on mathematics, this aims just to account for the major areas of interest. Naïve answers of naïve questions 6 Given a multitude of naïve philosophical questions on the nature of mathematics, we might as well try to answer first in a completely commonsensical spirit, just to get oriented, just to establish a common view point of reference in order to prevent any unnecessary taking off into too philosophical abstractions, or at least to make any such divergence evident and explicit. Little would indeed be gained in insight if we came up with fancy theories, but were unable to relate them to common sense and the every day meaning of words involved. Briefly, then the following is taken to be widely accepted first impressions of the science. In its everyday use, mathematics may well be used in the following senses: Mathematics is an activity, just as any other scientific field. Mathematics is a language, the mathematical theories have communicative uses, much seems to be possible to formulate in the language of mathematics. Further, Mathematics is symbols on papers, and Mathematics exists in some way in our minds. But while mathematics occurs in our minds, the mathematical results seem remarkably independent of this mind. It seems indeed to speak of objects of some kind, but no one has ever seen any, nor are we very well advised on the nature of their existence. Somehow, is the immediate conclusion, the nature of existence seems not very central to mathematics. Seemingly, it has nothing to do with the essence of mathematics. If mathematics is to take place, in equal extent, in the head, on a sheet of paper, and in computers, then one would assume some similar characteristics of these media and the 6 These statements are mainly drawn from [Bennet]. 6

7 mind. If the scientific understanding of the mind is the brain, then from what we know of the brain and the workings of computers, this would be a structural likeness. This seems perhaps more reasonable, looking from the common sense view point, than an existence and a related perception of unseen objects. Mathematics seems appropriate when revealing relations we had not foreseen. But once discovered, the results seem undeniable, and often trivial. So we experience a meaning through laying bare, formerly hidden, but trivial, and necessary results. The eternal virtues of mathematics seem to be the effect of its nonnegotiable results, its strict rule-governed behaviour. Its beauty due to the succulence of final insight of trivial truths, the art-likeness due to its pattern-modelling, its unlimited complexity potential, and so on. Finally the possibility of a similar realm of thoughts, of equal use, but of a distinguishable character seems unbelievable, but admittedly on vague grounds. Traditional Schools of Mathematical Philosophy 7 The traditional schools of philosophy of mathematics all arose, in their modern versions, in the late nineteenth century, but carry obvious residue from their philosophical ancestors. In addition the terms Logicism and Fictionalism will be explained: Realism This view considers the field of mathematics to be a science of a matter of fact, attributed to the world. For the realist distinguishes truth from knowledge. The mathematician becomes an explorer, rather than an inventor, for the mathematical landscape is already there, waiting for us to discover it. Therefore, independent questions of set theory are assumed to have their answers despite their logical independence of the axioms. We need just take a look in the realm of sets. Therefore, the axioms of the theory must be judged incomplete in some sense, they do not completely account for the universe of sets. It follows that the foundational problems concern the actual logical theories rather than the mathematical questions, or mathematics itself. Some philosophers (Gödel e.g.) claim that we acquire mathematical insight through some special faculty of mathematical perception. As there is no such imaginable thing as the simultaneous being and nonbeing of an eternal object, or the having and non-having of a given property, of a certain number, realism must be taken to imply the consistency of arithmetic and set theory, if not of the actual logical systems of arithmetic and set theory. As the mathematical theory models the relations in the one mathematical realm, uniqueness must follow too. The eternal virtues of mathematics are, it is proposed, best understood in connection to their mystical metaphysical status, and the related perceiving of such entities. The meaning of a statement of mathematics is something parallel to meaning in ordinary scientific or commonsensical context. Mathematics talks of a matter of fact as much as does science. Intuitionism was founded by L. E. J. Brouwer in an attempt to reconstruct mathematics in accordance with epistemological idealism and Kantian metaphysics. The natural numbers are the primary objects of mathematical knowledge. These spring from immediate 7 This section originates from, in addition to Maddy, the encyclopaedia articles listed under References. 7

8 temporal experience. Therefore, the objects of mathematics stand in ultimate dependence on an experiencing mind. In this sense, the mathematician invents his mathematical objects. Thus mathematics is not an attribute of the world, but rather of our experience. This accounts for the applicability of mathematics on the experienced world. According to Brouwer, there are no unexperienced truths, therefore mathematical results must be based on intuitive clarity rather then logical proof, and so a mathematical claim is true if one intuitively understands every stage of the proof process. In that, mathematical knowledge is a priori and intuitive. Also, we may read out a theory of meaning where constructive proof replaces the ordinary correspondence notion of truth. The mathematical knowledge is thus profoundly linked to our experience of the world, and this attributes to it its glossy guise. Brouwer accepted the formalisation of intuitionistic logic, and this would suggest that mathematics describes an aspect common to many, and not completely subjective. Therefore mathematics must be the only of its kind. Formalism is a mathematical relative of the much older doctrine of nominalism. The nominalist view is that concepts at large do not have separate being in any abstract way. They are but symbols. So, the formalist rejects any type of abstract mathematical entities, mathematics is no more than a rule-governed game of insignificant notation. The game might well be of great practical utility, but this implies in no way a significance of the literal linguistic kind. David Hilbert, the central advocate of this view, held that finitary mathematics describes indubitable facts about real objects, but that one introduces the ideal objects that feature elsewhere just in order to facilitate research about the former. The reference to real objects establishes uniqueness of arithmetic, and thereby of all of mathematics. This relation to the physical world would explain some of the virtues attributed to mathematics. Mathematical knowledge amounts to knowledge of the behaviour of a game of symbols, and of some aspects of the physical world. Logicism was the attempt to found all of mathematics on logic, and thus to provide both ontological and epistemological clarity. Gottlob Frege was the pioneer through his attempts in constructing mathematical objects from logical concepts alone. Bertrand Russell further continued the efforts in his Type Theory, so to avoid the paradoxes discovered in Frege s system, but he was forced to use ad hoc axioms to be able to account for the real numbers. The Hilbert Program aimed to secure mathematics through reducing mathematical theory to formal systems, and then prove the consistency of these. As we ve seen, these efforts have not been successful so far. One first thought that this was possible using elementary, mathematical methods, involving number theory, but this project was dealt a lethal blow from the results of Kurt Gödel. It is still up to discussion whether or not his results put an end to Logicism. Fictionalism, or instrumentalism, is the view that rather than aiming at literal truth, sentences or the use of objects in science or mathematics is to be seen as a fiction, a way of speech, which may have various pragmatic advantages. Thus statements are not true or false, but convenient or inconvenient. In mathematics, we talk as if numbers exist, because doing so permits us to construct various things in technology, take various associative steps in science or reasoning, but by saying 2 we do not succeed in actually referring to anything in particular. 8

9 Point of Departure Given the questions introduced, the aim is now to approach the philosophy of Willard Van Orman Quine, and take a closer look at its consequences in the philosophy of mathematics. Penelope Maddy offers an account that has clear mathematical sympathies and her book, [Maddy], is thus a major source of the account at hand. There, she sets out to investigate the possibilities of a modern, mathematical method of research, firmly based on an aimed-for, accurate view of what have been basic conditions for the massive mathematical development over the last two hundred years. Her interests are chiefly methodological, but she grounds her conclusions in a variation of the naturalistic outcomes of Quinean thought. One that takes actual mathematics as central, that extends the same respect to mathematical practice that the Quinean naturalist extends to scientific practice 8. Despite methodological concerns, her investigation also provides an account of the metaphysical results of the stance. These pages, though, will overlook methodological intents. The objective is to ask the ordinary questions, and to answer them in a Quinean spirit, noting in what ways Maddy s variation of his thoughts brings her to differing conclusions. Further, a criticizing part will scrutinize their ideas, establishing the relations of dependence, looking for misconceptions and overly optimistic conclusions, and ask questions on their success in answering the naïve questions of the philosophy of mathematics in a satisfying way. Quine is chosen on the basis of the still central role of his philosophy in the contemporary debate. His thoughts dissolve, or so aim to, the traditional dispute on the metaphysics of mathematics, introduced above, and offer a modern, sound and sane view of the role of science, and, thereby, of mathematics. This commonsensical spirit motivates our outset in naïve questions on mathematics. 8 [Maddy], p185 (henceforth referred to just by p185 ) 9

10 Quine s philosophy The philosophy of Quine emerges from a deep conviction to empiricism, a conviction that scientific evidence is what evidence there is for any given statement about the world. Quine s interactions with Rudolph Carnap, who stressed the importance of the linguistic framework of statements for their meaningfulness, are present through his understanding of scientific research of all branches as parts of an over-all scientific project, the unique scientific framework. Although influenced to a certain degree, Quine would not follow Carnap in his thus implied views on mathematical meaning and ontology. A few themes will here be accounted for, that are central to Quine s view on mathematics, and which appear in the version given by Penelope Maddy. On Ontology Questions gain their intelligibility from the linguistic framework in which they occur. That is Carnap s point. Carnap took ontological questions to be external to any linguistic framework, he took the acceptance of them to be part of the very adoption of such a framework, and thus to be a pragmatic, rather then a theoretical question. Posed outside the framework, ontological questions were but pseudo-questions, with nothing to provide standards by which they were to be evaluated inside the framework they were trivial. For instance, in discussing number theory the existence of 2 is a trivial, since presupposed, matter. If, on the other hand, we take a step back and ask whether 2 really exists, then we are outside the explanatory framework that specifies what is meant by 2 so that our question becomes unintelligible, meaningless, as a theoretical question. Thus, the issue of the real existence of 2 is settled indirectly through our adoption of a framework that includes 2 exists as a part. And whether to adopt or not a given framework is a practical, rather than theoretical, matter. With the over-all scientific theory in view, also Quine would take ontological questions for meaningless outside this scientific context, but, opposing Carnap, he considers these to have meaningful counterparts inside science 9. These outer questions should be adopted on pragmatic grounds, such as simplicity, economy, efficiency, just as Carnap argued. But Quine notes that this is really the ground for adoption of any theory, or any theoretic part, of science. Thus in the framework of overall science, pragmatic standards are an integral part of the very framework. Thus ontological questions, those who would be settled on pragmatic grounds, have their intelligible, internal counterparts in the scientific context. So, how are we thus to understand a scientific statement of existence? There is nothing about real existence for Quine; electrons don t really exist. But the assertion of existence of electrons is central to the scientific field. It is an important component without 9 Rather, Quine would say that ontological questions are internal to science, while Carnap would consider them external, and therefore meaningless. 10

11 which science would definitely not be the same. The observation is thus, that science, as it is, includes the existence of electrons as a façon de parler, an assertion that has meaning in relation to all other assertions made by science, and this extends to all scientific claims: Scientific knowledge is a man-made fabric which touches perceptions only along the edges. 10 Quine holds that this is the only meaning ontological questions can acquire. Now Quine performs a logical trick: In forming our over-all science, we can judge its ontology by looking at what it claims there is. So, by adopting a given theory, we commit ourselves to its ontology, that is, to the things that exist according to the theory. This is Quine s Ontological Commitment. An Empirical Dogma Opposing Carnaps attempt to distinguish the mathematical from the physical portion of the over-all scientific framework, of which they agree in large 11, Quine directs a severe attack on an old, as he says, metaphysical article of faith 12. His attack concerns the concept of analyticity. This notion has been around in philosophical debate for quite a while. Kant considered, as an important part of his metaphysics, the possibility of synthetic truths known a priori. The whole idea is that an analytic statement is a true statement which is so solely in virtue of the meanings of the words used. That is, it is a trivial truth in the sense that if you properly understand the concepts involved, if you know their meaning, then you cannot deny what is said. Quine s point is that the concept of analyticity is ill-founded. His argument is found in his Two Dogmas of Empiricism 13 and runs along the following lines: First, analytic statements have been defined as statements whose denials are self-contradictory. Any similar definition lacks explanatory power. These two notions are the two sides of a single dubious coin 14. In a similar vein, Quine proceeds to scrutinize a variety of attempts made to clarify the notion. He considers Kant s metaphorical notion of containment, Frege s work on the relation of meaning and reference. He traces the explanatory ground for meaning to that of synonymy, but, again, he ends up with an explanation owing to an understanding of some other, equally ill-founded, concept. The way through definition seems appealing, but adds nothing to our understanding. All such, accept the postulating kind, hinges on prior relationships of synonymy 15. So we re back at where we started. Further, Quine has a go at interchangeability salva veritate. He considers the predicate necessarily, but points out that such a predicate does indeed presuppose a satisfactory 10 p102, this is closely linked to the Quinean view, that stems from Duhem, and which is called the Duhem Thesis, that physical theories are accepted or rejected as a whole, and not statement by statement [Cambridge Dictionary of Philosophy]. 11 p [Quine], II.5, p [Quine], I. 14 [Quine], II.1, p [Quine], II.3, p27. 11

12 understanding of analyticity. Such an understanding can be obtained in an external language, but troubles appear here as well: Extensional agreement, what synonymy becomes in this context, is far from the type of cognitive synonymy that is needed for the elucidation of analyticity. In an intentional language, analyticity is presupposed. Quine s last attempt is about semantic rules. Obviously, a semantic rule, itself containing analyticity, is of no help. We know very well which instances of the language we might apply the predicate of analyticity to, but we know nothing more of what this says about the considered sentences. We might then consider, Quine proposes, differentiating among true statements; any analytic statement is of course among these. We might say that sentences true by virtue of specific semantic rules are to be labelled analytic. But even here, nothing is gained. For the reference to semantic rule is as dubious and unclear as is the notion of analyticity itself. Quine s conclusion is thus that any attempt to clarify what is meant in the use of analytic is condemned to fail. In his own words, for all its a priori reasonableness, a boundary between analytic and synthetic statements simply has not been drawn 16. Naturalism As we have seen, Quine claims that ontological questions are to be understood in, and only in, the framework of over-all science. But these are not the only philosophical questions that have dim evidential standards. As Quine only finds one legitimate ground for understanding statements about how it is, philosophical questions are intelligible just in so far as they have counterparts in this context. That is, questions on epistemology, for example, can only be properly understood in their scientific versions, as questions of how human beings, as understood by science, acquire information of the physical world, as understood by science. There is no other understanding of these questions. The traditional philosophical goal of epistemology has been to found science on some secure, extra-scientific cornerstone, but we must give up the Cartesian dream, Quine urges. In sum then, this abandonment of traditional philosophical goals, due to lack of intelligibility, Quine calls naturalism. This is the recognition that it is within science itself, and not in some prior philosophy, that reality is to be identified and described 17. Thus, science appears as an inquiry into reality, fallible and corrigible but not answerable to any supra-scientific tribunal, and not in need of any justification beyond observation and the hypothetico-deductive method 18. The abandonment of the goals of a first philosophy means not that all questions asked by philosophers are left behind. Those which are possible to ask inside the scientific framework are as good as any scientific such, but these must now be understood on scientific grounds, and answered within the theory. In this way the part of philosophy that endures Quine s treatment becomes continuous with science, as Maddy puts it. 16 [Quine], II.5, p37. Also cited in [Maddy], p Quine, cited on p Quine, cited on p

13 We already noted the holistic character of science. Its only principles are observation and the hypothetico-deductive method, and scientific claims are meaningful only in relation to other scientific claims and true only insofar as they explain our observational experiences. The naturalistic philosopher is thus no different than the scientist. He begins his reasoning within the inherited world as a going concern 19. Without any extra-scientific ground to hold on to, he must reconstruct science from within, while staying afloat in it 20. This is the picture of Neurath s Boat 21. Meaning according to Quine So given this naturalistic view of an unavoidable scientific context, or framework, what has Quine to say on the notion of meaning? This is important in view of the naïve questions raised in the introduction. Just like Carnap, Quine holds that meaning for a statement arises in relation to a given linguistic framework. This framework is to supply evidential rules which explain how the statement is to interact with the other statements of the framework. These rules should account for what inferences are to be drawn from a given statement, or what is to be deduced from a certain perceptual impression. For a sentence, this interplay with other sentences of the framework, and with observation, is all there is to its meaning. That is how Quine should be understood. Spelling this out, the meaning of a statement, or a question for that matter, is its relation to all other internal statements in terms of what is to be inferred from it, and what it can be inferred from. This is exactly the façon de parler earlier mentioned in connection to ontology. This lack of meaning in its more metaphysical sense is easily understood from Quine s attack on analyticity. A clear interdependence is identified between the two concepts, and as analyticity is judged as an empty notion, so must also any concept of meaning be considered outside the framework. Thus meaning outside science lacks explanatory ground, for the very same reasons as does analyticity. Quine on mathematics The consequence of Quine s philosophy for mathematics is immediate: Mathematics plays a major role in the over-all scientific theory, it is claimed to be indispensable for science 22, and as we adopt the framework of over-all science, Quine argues, then we must accept the ontology of that framework, and we must accept, as part of its ontology, what mathemat- 19 Quine, cited on p Quine, cited on p Neurath compared the scientific project to a ship on the sea that was continuously reconstructed, but in such a way that it could stay afloat. 22 H. Putnam, cited on p

14 ics claims there is 23. This is how a realism concerning mathematical objects develops out of Quinean thought. His view also answers to the question of meaning and meaningfulness of mathematics raised above. As all other scientific statements are to be understood, the mathematical assertion of the existence of numbers means nothing more than that in scientific practice we talk and behave as if there were numbers. That is all the meaning that can be assigned to statements of mathematics. On the other hand, this is all there is to meaning of any statement. So, a mathematical claim is just as meaningful any other scientific claim. But even more so. Quine explains the applicability of mathematics and its epistemologically secure character by saying that mathematical statements are central in the web of scientific knowledge, and in this sense mathematical claims are even more meaningful than the average scientific claim. Now, Quinean thought does not leave mathematics as it is, but retains only a minor part. If mathematical claims gain their intelligibility and mathematical existential claims their justification solely through their role in science, not much of pure mathematics could be spared. Or this is at least Quine s conclusion 24. We have seen how Quine rejects the concept of analyticity, since there can be no clarifying explanation of its meaning, or of what is an analytic truth, and what is not. The Carnapian solution to epistemological questions on mathematics would be to say that mathematical claims are analytical claims, and so they are secure, for they are trivially true. Quine rejects this solution, because, as noted above, analyticity has not yet been explained in a satisfying way. 23 This is, together with Quine s ontological commitment, commonly known as the Indispensability Argument for mathematical realism. 24 p105 14

15 Maddy s philosophy In view of Maddys overriding concern to investigate the possibility of settling undecided issues in set theory, it is easy to understand a temptation for a naturalistic approach similar to that of Quine on natural science. She reads the spirit of naturalism as synonymous for philosophical modesty towards methodological considerations. Maddy takes her position to be a variation of Quine s naturalism, but not a philosophy of mathematics, rather a position on the proper relations between the philosophy of mathematics and the practice of mathematics 25. We need to know what the philosophical result of such methodological interests is. Maddy starts out with Quinean naturalism; actual scientific methods are fundamental. Not far gone into investigations of the physical world, two facts become obvious. Mathematics is central to our scientific study of the world. And, the methods of mathematics differ markedly from those of natural science 26. The first observation underlies arguments of indispensability for mathematical realism. Maddy takes history and actual fact as a guide to take a step away from Quine; she offers to mathematics a similar, self-supporting, framework of its own. This is clearly in opposition to Quine, but in accordance with the naturalistic spirit of his philosophy. Following Quine s parallel, Maddy thus takes mathematics to be not answerable to any extra-mathematical tribunal and not in need of any justification beyond proof and the axiomatic method 27. So far the methodological concerns are central. But what are the philosophical effects? In a Quinean fashion, Maddy asks herself what mathematical method has to say on ontology. If we are to investigate philosophical questions on mathematics, then it would be interesting to know, as in the case of natural science, which ones might be taken as continuous with mathematics. Knowing that, we would know which of the questions can be asked inside the mathematical framework, and thus which can be asked with some real hope of finding answers 28. This was how Quine succeeded in saving at least some philosophical questions. In short, she finds not much. Firstly, Maddy notes that common sense is more forthcoming on physical than mathematical ontology 29. Moving to the level of mathematical theories, we find existential claims in abundance, but not the slightest hint of the nature of this existence. Maddy does find some hope for ontological guidance at the level of mathematical discussion, the level at which, for instance, discussions of justifications of CH and similar, takes place. But this raises the question of whether these discussions are continuous with mathematics. The outcome is daunting, and she ends the section, The methods of natural sciences [ ] also tell us that ordinary physical objects and many unobservables exist, [ ]; but the methods of mathematics [ ] tells us no more than that certain mathematical objects exist p p p p p p192 15

16 The outcome of this investigation, then, is nought. Maddy writes, our mathematical naturalist relinquishes her last hope of ontological guidance from the practice of mathematics 31. Moving to other parts of traditional philosophy, the picture is the same. She writes, it seems no traditional epistemological questions about mathematics and only the barest ontological questions about mathematics can be naturalized as mathematical questions. So, we are to infer that ontological questions are not mathematical questions. The result is that naturalism is not a metaphysical rival to realism 32. Thus the shift for a proper mathematical framework is mainly of methodological nature. A scientific study of mathematics in a naturalistic spirit evaluates mathematical progress on its own terms, but what are the consequences for the philosophical questions on mathematics? All such are considered first philosophy, and are hence, in Carnaps sense, pseudo-questions. Just one philosophical question remains, and that is What exists?. Its intra-mathematical counterpart has a trivial answer: Numbers, sets, functions, etc. All things that appears directly after an in mathematical literature. And the matter ends here. 31 p p205 16

17 Criticism: Quine The Quinean account has many virtues, indeed, a whole bunch of insights is woven into the fabric, but they are not always as apparent and explicitly presented as one might please. Besides, there are a number of questions to be asked. At first sight, Quine is not very successful in responding to the naïve questions of the philosophy. Although mathematicians throughout the ages have reported on eternal virtues of the field, Quine rejects most of these. Mathematics is in so far as it is intelligible at all, that is, is applied in science no more than a scientific theory amongst others, and is, therefore, as fallible and corrigible as the surrounding, supporting fabric of theory, that is over-all science. Thus, mathematics provides no knowledge of a more secure kind than does science in general. This is most certainly opposed to the general naïve philosophical view. Further, mathematics has no meaning in this far-reaching sense; an existential claim of mathematics is a way of speech, no more, no less. Pure mathematics is not meaningful at all. As a part of the actual scientific framework, mathematics is in no way the only possible field of this sort. Quine holds that we cannot consider our science to be the only possible science 33, and unfortunately mathematics goes with it. Before plunging into specific worries about the Quinean view, a necessary remark must be made. The naturalistic view in no way does away with philosophy according to itself, but changes, so to say, the labelling of the issues involved. Where the classical philosopher discusses meaning in a first-philosophy sense, in the sense of meaning in itself, the natural philosopher hears just nothing. What is meaning to the naturalist is merely how the issue is related to all other issues of knowledge. What is meaning for a term is just the way we use that term in ordinary speech. This means that if the classical philosopher were to describe the naturalistic view, then he would say that the naturalist denies any meaning, or existence for that matter, in itself, and that the question is now reduced to that of how the parlance goes. The naturalist, on the other hand, would say that the classical philosopher is mistaken in his thoughts of meaning, as he considers there to be a meaning of meaning which goes beyond the usage of the word. Uttering there is something more to meaning, would result in nothing more than what goes well with naturalism. Because, were there no other meaning to there s something more to meaning than the usage of the terms and the grammatical construction, then the sentence would indeed fail to say anything about something non-existing. The naturalist claims thus that we simply fail to talk about meaning or existence, because as we say anything on these matters, then what we say is meaningless accept for the relation of the spoken word to the rest of human language usage. Comparing with another famous example, if one could say that the king of France is bald fails in making sense because of lack of referent, then the naturalist would say (according to the classical philosopher) that the king of France is bald fails to make sense (be meaningful), for there is no such thing as reference (or meaning), while the naturalist himself would say that the king of France is bald makes perfect sense, and that the classical philosopher talks about invented (pseudo-) worries in 33 p181 17

18 discussing reference and meaning as such. To avoid a situation where debate is impossible, simply as one party claims that the other fails to say what he wants to say, I now take position, for practical reasons, in the classical camp. Thus, I will allow the idea of a possible real existence, meaning, etc, and say that the naturalist denies this faculty. 34 Mathematics in the web Mathematical claims seem to be of epistemological excellence. Nothing seems to be more secure than claims of mathematics. Why this is so, Quine explains by saying that their position is central in the fabric of the over-all scientific theory. So, Quine offers a reason, why we do refrain from questioning these theses as long as we have anything else left to put in doubt first. But, of course, we could now ask why mathematical claims are so central. And as far as Maddy s account goes, we find no answers. That mathematics is of high fidelity, we knew in advance, but what, in mathematics, makes it so secure? This could well be asked even if we considered merely a degree of security as in the case of Quine. However, the picture Quine gives goes well with the over-all view, and it is easy to understand the unpleasant implications for the Quinean project of accepting mathematics as a quite different faculty of knowledge. If mathematical claims are true in a much stronger sense than scientific claims, then this must be explained, presumably in terms of concepts not far from the rejected concept of analyticity. For, if a claim is true in a stronger sense than by virtue of observational data, what might that truth depend on other than meaning? Of course, there is always the possibility of psychologism. We could say that mathematical security is due to our mental apparatus rather than an external meaning in this sense. But, then we would still be interested in what makes mathematical states of mind more central to the field of knowledge than scientific states of mind. If there were states of minds, or theorizing patterns, of greater generality, thus applicable to a wider range of scientific situations, then this generality, and the reasons for it explained in mental properties would be interesting indeed in revealing features of scientific progress. Thus the conclusion is still the same. Quine says that mathematics is indispensable for science, and therefore central in the web, but he does not explain why. If mathematics is to be questionable, it seems natural to ask for a fair description of how this might actually happen. What would it look like to reject a central claim of mathematics? I would be surprised if there were a single example of a change of mathematical theory because of incompatibility with any other body of claims of the over-all science. Meaning Quine argues that outer questions are pseudo-questions, but despite his relativizing manners, he does not seem to notice that the claim that outside issues are meaningless is itself an utterance in the context, and therefore relative to this context. The point here is that if 34 Compare for the introductory note in Quine s On What There Is, [Quine], I. 18

19 Quine provides a linguistic framework, which gives utterances in the framework meaning, then he has not yet provided grounds for the rejection of any other possible meaning to occur. His argument is a positive one, inner statements are intelligible, but his conclusions are negative, outer statements are not intelligible. If any kind of meaning occurred outside the framework, then this framework would not recognize it as meaning. For if meaning is always relative to a framework, as Carnap argues, then meaning A would not qualify as a meaning in framework B, that is, a meaning B. So, if Quine is to be understood in this relative sense, that a sentence that gains meaning from a context, cannot gain this meaning outside the context, then his remark is indeed trivial. And if he is to be understood to say that meaning is impossible whatsoever outside the context, then he has no supporting argument to offer. In this sense, Quine s argument for meaninglessness outside the scientific context, is itself meaningful inside the context, but then it follows that the argument can only work in a positive manner, inside, and not in a negative manner, outside, that is, it can justify the existence of meaning inside the context, but is useless in rejecting any possible meaning outside this context. If utterances are to be meaningful just in relation, then that is meaningful just in relation. The sentence p p is supposedly included in the Quinean fabric, but is it meaningful? He could say, no, it is trivial and empty. Or he could say yes, in relation to other sentences of science it is meaningful, but then, if we were to reject it, what could science possibly look like? Some sentences seems fundamental to the very project of science, to the very possibility of a Quinean fabric of belief. Are these of equivalent dignity as ordinary statements of science? Further, how are we to judge in the case of sentences of the following form: CH follows from the existence of certain large cardinals. This is a result of set theory, and it is proven with mathematical rigor and may therefore be considered true beyond any reasonable doubt. It seems it could not end up false however we were to interpret mathematics, science or logic. It seems meaningful as it claims something nontrivial. It seems to be about mathematics, but still within mathematics, as a logical utterance, that is, not a scientific claim. But how is it to be understood? I claim that Quine could not do justice to these kinds of example. And while not accounting properly for the logical part of knowledge, chance is considerable that he misses out on closely related mathematics. Ontology Metaphysics is of tradition a central concern of western philosophy. And therefore it seems natural that ontological questions have carried over to the philosophy of mathematics. The Quinean project, as well as Maddy s prolongation, is heavily concerned with what there is. Establishing her proper mathematical realm, Maddy asks for ontological guidance, but finds none. Her conclusion is that, as mathematics offers little guidance on the nature of existence of its objects, then questions of existence cannot be naturalized as proper mathematical questions. But this whole quest for ontological guidance rests, of course, on the assumption that mathematics actually claims something to exist, and that seems not to be evident. Surely, a first acquaintance with mathematics would provide 19

20 ground for believing that mathematics talks about existence, but a closer look will reveal that any statement of existence is preceded by a specification of the required universe of objects which is needed for the development of the theory in question. Thus, a statement of existence of a certain object is merely a statement of there being, among the already existing objects, an object of the intended kind. But, scrutinizing naturalism, we see that what now differentiates between our accounts is no more than a matter of words. For, if mathematics does talk about existence, then, in the naturalistic interpretation, following Maddy, this existence is of no more metaphysical connotation than the very usage of the mathematical existential claims, which is all that may be said of existence at all. Then, in naturalistic parlance, mathematics does talk about existence in the only way possible. That is, we may understand mathematics to make existential claims and then adopt the naturalistic view and thus make the existential claims of mathematics less metaphysically loaded, or we may say that mathematics does not talk of existence, referring to the higher order existence of first philosophy, in order to arrive at the same conclusion. So, in that case, and again, if the same point is really possible to make from the stance of classical philosophy, why this new labelling of the concepts? If this is really the case, then I cannot see why the naturalistic account becomes anything more than an observation on the limits of human knowledge. For interpreted in this way, the only arguments for the naturalistic definitions of the terms, rather than the classical, which are not arguments for limits of human knowledge, would be arguments in favour of the stance that there is no real existence, that there is no real meaning, etc. And those have not been provided. Cartesian doubt is not the way to begin Quine traces the failure of the epistemological project to Hume. He did not succeed in his attempts to found knowledge of bodies through identification with impressions and the reconstruction of statements as about impressions. His statements gained no increment of certainty by being construed as about impressions 35. The conclusion is that we cannot hope to found science on anything more secure, and we are back at Neurat s boat. Now, Quine notes that the very impulse to found science on a secure foundation is driven by a scepticism that is itself internal to science: the very notion of sensory illusion depends on a rudimentary science of common sense physical objects 36. As far as a scientific-free situation is imaginable, I cannot see why the possibility of the doubt that what I spontaneously think of something, need not be the case, is so tightly linked to science. If science means any human theorizing, if science is what thinking is, then of course this seems right, but trivially so. If the very idea of common sense physical objects is to be the root of scepticism, I cannot see how this would be contaminated by statements like I feel as one person, but there might be no way to specify me in a scientific way. It seems possible that, as soon as perception arises, there is a possibility of questioning that impression. That is Descartes insight, indeed. But this doubt applies equally well to any other theory 35 Quine cited on p p179 20