Rigorous proof is supposed to guarantee that the premises invoked imply the conclusion reached, and the problem of rigor may be described as that of bringing together the perspectives of formal logic and mathematical practice on how this is to be achieved. This problem has recently raised a lot of discussion among philosophers of mathematics. We survey some possible solutions and argue that failure to understand its terms properly has led to misunderstandings in the literature.

Number theory abounds with conjectures asserting that every natural number has some arithmetic property. An example is Goldbach’s Conjecture, which states that every even number greater than 2 is the sum of two primes. Enumerative inductive evidence for such conjectures usually consists of small cases. In the absence of supporting reasons, mathematicians mistrust such evidence for arithmetical generalisations, more so than most other forms of non-deductive evidence. Some philosophers have also expressed scepticism about the value of enumerative inductive evidence in (...) arithmetic. But why? Perhaps the best argument is that known instances of an arithmetical conjecture are almost always small: they appear at the start of the natural number sequence. Evidence of this kind consequently suffers from size bias. My essay shows that this sort of scepticism comes in many different flavours, raises some challenges for them all, and explores their respective responses. (shrink)

Criteria of acceptability for mathematical proofs are field-dependent. In topology, though not in most other domains, it is sometimes acceptable to appeal to visual intuition to support inferential steps. In previous work :829–842, 2014; Lolli, Panza, Venturi From logic to practice, Springer, Berlin, 2015; Larvor Mathematical cultures, Springer, Berlin, 2016) my co-author and I aimed at spelling out how topological proofs work on their own terms, without appealing to formal proofs which might be associated with them. In this article, I (...) address two criticisms that have been raised in Tatton-Brown against our approach: that it leads to a form of relativism according to which validity is equated with social agreement and that it implies an antiformalizability thesis according to which it is not the case that all rigorous mathematical proofs can be formalized. I reject both criticisms and suggest that our previous case studies provide insight into the plausibility of two related but quite different theses. (shrink)

In this work I study the main tenets of the logicist philosophy of mathematics. I deal, basically, with two problems: (1) To what extent can one dispense with intuition in mathematics? (2) What is the appropriate logic for the purposes of logicism? By means of my considerations I try to determine the pros and cons of logicism. My standpoint favors the logicist line of thought. -/- .

The existence of fundamental moral disagreements is a central problem for moral realism and has often been contrasted with an alleged absence of disagreement in mathematics. However, mathematicians do in fact disagree on fundamental questions, for example on which set-theoretic axioms are true, and some philosophers have argued that this increases the plausibility of moral vis-à-vis mathematical realism. I argue that the analogy between mathematical and moral disagreement is not as straightforward as those arguments present it. In particular, I argue (...) that pluralist accounts of mathematics render fundamental mathematical disagreements compatible with mathematical realism in a way in which moral disagreements and moral realism are not. 11. (shrink)

'Reason, Causation and Compatibility with the Phenomena' strives to give answers to the philosophical problem of the interplay between realism, explanation and experience. This book is a compilation of essays that recollect significant conceptions of rival terms such as determinism and freedom, reason and appearance, power and knowledge. This title discusses the progress made in epistemology and natural philosophy, especially the steps that led from the ancient theory of atomism to the modern quantum theory, and from mathematization to analytic philosophy. (...) Moreover, it provides possible gateways from modern deadlocks of theory either through approaches to consciousness or through historical critique of intellectual authorities. This work will be of interest to those either researching or studying in colleges and universities, especially in the departments of philosophy, history of science, philosophy of science, philosophy of physics and quantum mechanics, history of ideas and culture. Greek and Latin Literature students and instructors may also find this book to be both a fascinating and valuable point of reference. (shrink)

In this paper a parallel is drawn between the problem of epistemic access to abstract objects in mathematics and the problem of epistemic access to idealized systems in the physical sciences. On this basis it is argued that some recent and more traditional approaches to solving these problems are problematic.

I argue for the Wittgensteinian thesis that mathematical statements are expressions of norms, rather than descriptions of the world. An expression of a norm is a statement like a promise or a New Year's resolution, which says that someone is committed or entitled to a certain line of action. A expression of a norm is not a mere description of a regularity of human behavior, nor is it merely a descriptive statement which happens to entail a norms. The view can (...) be thought of as a sort of logicism for the logical expressivist---a person who believes that the purpose of logical language is to make explicit commitments and entitlements that are implicit in ordinary practice. The thesis that mathematical statements are expression of norms is a kind of logicism, not because it says that mathematics can be reduced to logic, but because it says that mathematical statements play the same role as logical statements. ;I contrast my position with two sets of views, an empiricist view, which says that mathematical knowledge is acquired and justified through experience, and a cluster of nativist and apriorist views, which say that mathematical knowledge is either hardwired into the human brain, or justified a priori, or both. To develop the empiricist view, I look at the work of Kitcher and Mill, arguing that although their ideas can withstand the criticisms brought against empiricism by Frege and others, they cannot reply to a version of the critique brought by Wittgenstein in the Remarks on the Foundations of Mathematics. To develop the nativist and apriorist views, I look at the work of contemporary developmental psychologists, like Gelman and Gallistel and Karen Wynn, as well as the work of philosophers who advocate the existence of a mathematical intuition, such as Kant, Husserl, and Parsons. After clarifying the definitions of "innate" and "a priori," I argue that the mechanisms proposed by the nativists cannot bring knowledge, and the existence of the mechanisms proposed by the apriorists is not supported by the arguments they give. (shrink)

A crucial part of the contemporary interest in logicism in the philosophy of mathematics resides in its idea that arithmetical knowledge may be based on logical knowledge. Here an implementation of this idea is considered that holds that knowledge of arithmetical principles may be based on two things: (i) knowledge of logical principles and (ii) knowledge that the arithmetical principles are representable in the logical principles. The notions of representation considered here are related to theory-based and structure-based notions of representation (...) from contemporary mathematical logic. It is argued that the theory-based versions of such logicism are either too liberal (the plethora problem) or are committed to intuitively incorrect closure conditions (the consistency problem). Structure-based versions must on the other hand respond to a charge of begging the question (the circularity problem) or explain how one may have a knowledge of structure in advance of a knowledge of axioms (the signature problem). This discussion is significant because it gives us a better idea of what a notion of representation must look like if it is to aid in realizing some of the traditional epistemic aims of logicism in the philosophy of mathematics. (shrink)

Contemporary philosophy’s three main naturalisms are methodological, ontological and epistemological. Methodological naturalism states that the only authoritative standards are those of science. Ontological and epistemological naturalism respectively state that all entities and all valid methods of inquiry are in some sense natural. In philosophy of mathematics of the past few decades methodological naturalism has received the lion’s share of the attention, so we concentrate on this. Ontological and epistemological naturalism in the philosophy of mathematics are discussed more briefly in section (...) 6. (shrink)

Mathematical fictionalism (or as I'll call it, fictionalism) is best thought of as a reaction to mathematical platonism. Platonism is the view that (a) there exist abstract mathematical objects (i.e., nonspatiotemporal mathematical objects), and (b) our mathematical sentences and theories provide true descriptions of such objects. So, for instance, on the platonist view, the sentence ‘3 is prime’ provides a straightforward description of a certain object—namely, the number 3—in much the same way that the sentence ‘Mars is red’ provides a (...) description of Mars. But whereas Mars is a physical object, the number 3 is (according to platonism) an abstract object. And abstract objects, platonists tell us, are wholly nonphysical, nonmental, nonspatial, nontemporal, and noncausal. Thus, on this view, the number 3 exists independently of us and our thinking, but it does not exist in space or time, it is not a physical or mental object, and it does not enter into causal relations with other objects. This view has been endorsed by Plato, Frege (1884, 1893-1903, 1919), Gödel (1964), and in some of their writings, Russell (1912) and Quine (1948, 1951), not to mention numerous more recent philosophers of mathematics, e.g., Putnam (1971), Parsons (1971), Steiner (1975), Resnik (1997), Shapiro (1997), Hale (1987), Wright (1983), Katz (1998), Zalta (1988), and Colyvan (2001). (shrink)

In his "Discussion" (1984), Elliott Sober offers some criticisms of the view about species--pluralistic realism--advocated in my 1984. Sober's comments divide into three parts. He attempts to show that species are not sets; he responds to my critique of David Hull's thesis that species are individuals; and he offers some arguments for the claim that species are "chunks of the genealogical nexus." I consider each of these objections in turn, arguing that each of them fails. I attempt to use Sober's (...) insightful critique to explain and defend pluralistic realism more fully. (shrink)

The paper provides an interpretation of Wittgenstein’s claim that a mathematical proof must be surveyable. It will be argued that this claim specifies a precondition for the applicability of the word ‘proof’. Accordingly, the latter is applicable to a proof-pattern only if we can come to agree by mere observation whether or not the pattern possesses the relevant structural features. The claim is problematic. It does not imply any questionable finitist doctrine. But it cannot be said to articulate a feature (...) of our actual usage of the word ‘proof’. The claim can be dissociated, however, from two tenable doctrines of Wittgenstein, namely that proofs can be used as paradigms for corresponding proof concepts and that a proof is not an experiment. (shrink)

The analytic/synthetic distinction has played a central role in discussions of a priori knowledge throughout the twentieth century. One of the primary reasons for the prominence of this distinction is the widespread influence of the tradition of logical empiricism which endorsed the following principles: All analytic propositions are knowable a prioriand All propositions knowable a priori are analytic.Hence, proponents of the a priori often argue in support of the contention that the propositions of a particular discipline, say mathematics or logic, (...) are knowable a priori by arguing that it consists solely of analytic propositions. On the other hand, detractors of the a priori often reject such knowledge on the grounds that the analytic/synthetic distinction is not cogent. My primary goal in this paper is to challenge the prevalent acceptance of. (shrink)

The topic of this paper is our knowledge of the natural numbers, and in particular, our knowledge of the basic axioms for the natural numbers, namely the Peano axioms. The thesis defended in this paper is that knowledge of these axioms may be gained by recourse to judgements of probability. While considerations of probability have come to the forefront in recent epistemology, it seems safe to say that the thesis defended here is heterodox from the vantage point of traditional philosophy (...) of mathematics. So this paper focuses on providing a preliminary defense of this thesis, in that it focuses on responding to several objections. Some of these objections are from the classical literature, such as Frege's concern about indiscernibility and circularity, while other are more recent, such as Baker's concern about the unreliability of small samplings in the setting of arithmetic. Another family of objections suggests that we simply do not have access to probability assignments in the setting of arithmetic, either due to issues related to the~$\omega$-rule or to the non-computability and non-continuity of probability assignments. Articulating these objections and the responses to them involves developing some non-trivial results on probability assignments, such as a forcing argument to establish the existence of continuous probability assignments that may be computably approximated. In the concluding section, two problems for future work are discussed: developing the source of arithmetical confirmation and responding to the probabilistic liar. (shrink)

Several scholars have argued that Wittgenstein held the view that the notion of number is presupposed by the notion of one-one correlation, and that therefore Hume's principle is not a sound basis for a definition of number. I offer a new interpretation of the relevant fragments on philosophy of mathematics from Wittgenstein's Nachlass, showing that if different uses of ‘presupposition’ are understood in terms of de re and de dicto knowledge, Wittgenstein's argument against the Frege-Russell definition of number turns out (...) to be valid on its own terms, even though it depends on two epistemological principles logicist philosophers of mathematics may find too ‘constructivist’. (shrink)

Platonism is the view that there exist such things as abstract objects — where an abstract object is an object that does not exist in space or time and which is therefore entirely non-physical and nonmental. Platonism in this sense is a contemporary view. It is obviously related to the views of Plato in important ways, but it is not entirely clear that Plato endorsed this view, as it is defined here. In order to remain neutral on this question, the (...) term ‘platonism’ is spelled with a lower-case ‘p’. (See entry on Plato.) The most important figure in the development of modern platonism is Gottlob Frege (1884, 1892, 1893-1903, 1919). The view has also been endorsed by many others, including Kurt Gödel (1964), Bertrand Russell (1912), and W.V.O. Quine (1948, 1951). (shrink)

The present paper discusses a proposal which says,roughly and with several qualifications, that thecollection of mathematical truths is identical withthe set of theorems of ZFC. It is argued that thisproposal is not as easily dismissed as outright falseor philosophically incoherent as one might think. Some morals of this are drawn for the concept ofmathematical knowledge.

The subject of this paper is the philosophical problem of accounting for the relationship between mathematics and non-mathematical reality. The first section, devoted to the importance of the problem, suggests that many of the reasons for engaging in philosophy at all make an account of the relationship between mathematics and reality a priority, not only in philosophy of mathematics and philosophy of science, but also in general epistemology/metaphysics. This is followed by a (rather brief) survey of the major, traditional philosophies (...) of mathematics indicating how each is prepared to deal with the present problem. It is shown that (the standard formulations of) some views seem to deny outright that there is a relationship between mathematics and any non-mathematical reality; such philosophies are clearly unacceptable. Other views leave the relationship rather mysterious and, thus, are incomplete at best. The final, more speculative section provides the direction of a positive account. A structuralist philosophy of mathematics is outlined and it is proposed that mathematics applies to reality though the discovery of mathematical structures underlying the non-mathematical universe. (shrink)

According to a classical view in the philosophy of language, the reference of a term is determined by a property of the term which supervenes on the history of its use. A contrasting view is that a term's reference is determined by how it is properly interpreted, in accordance with certain constraints or conditions of adequacy on interpretations. Causal theories of reference of the sort associated with Hilary Putnam, Saul Kripke and Michael Devitt are versions of the first view, while (...) defenders of determination by interpretation theories include Donald Davidson, Daniel Dennett and John Haugeland. I use a variant of Putnam's Twin Earth thought experiment to argue against the first view generally, and causal theories of reference in particular, then go on to argue that properly-formulated version of the principle of charity can account for the intuitions that seem to support causal theories. Finally, I apply my version of interpretationism to the problem of reference to abstract objects and compare it with some of Wittgenstein's and Quine's views about language. (shrink)

Though many working mathematicians embrace a rough and ready form of Platonism, that venerable position has suffered a checkered philosophical career. Indeed the three schools of thought with which most of us began our official philosophizing about mathematics—Intuitionism, Formalism, and Logicism—all stand in fundamental disagreement with Platonism. Nevertheless, various versions of Platonistic thinking survive in contemporary philosophical circles. The aim of this paper is to describe these views, and, as my title suggests, to trace their roots.I'll begin with some preliminary (...) remarks about the big three schools. This seems a reasonable approach to the issues both because most observers are familiar, at least in a general way, with the tenets of Intuitionism, Formalism, and Logicism, and because it is in reaction to these that contemporary Platonism has taken shape. (shrink)

Complete inferential rigour is achieved by breaking down arguments into steps that are as small as possible: inferential ‘atoms’. For example, a mathematical or philosophical argument may be made completely inferentially rigorous by decomposing its inferential steps into the type of step found in a natural deduction system. It is commonly thought that atomization, paradigmatically in mathematics but also more generally, is pro tanto epistemically valuable. The paper considers some plausible candidates for the epistemic value arising from atomization and finds (...) that none of them fits the bill. In particular, atomized arguments do not ground the correctness of their unatomized counterparts; they are typically not credence-preserving; and they need not reveal the source of inferential disagreement. The moral this suggests is that complete rigour is not even a defeasible epistemic ideal. (shrink)

The constructive mathematics developed by Bishop in Foundations of Constructive Analysis succeeded in gaining the attention of mathematicians, but discussions of its underlying philosophy are still rare in the literature. Commentators seem to conclude, from Bishop’s rejection of choice sequences and his severe criticism of Brouwerian intuitionism, that he is not an intuitionist–broadly understood as someone who maintains that mathematics is a mental creation, mathematics is meaningful and eludes formalization, mathematical objects are mind-dependent constructions given in intuition, and mathematical truths (...) are experienceable. This paper develops and defends an intuitionistic interpretation of Bishop’s philosophical views. (shrink)

Mark Jay Steiner, a brilliant and influential philosopher of mathematics, whose interests and accomplishments extended beyond that field as well, passed away on.

Kripke observes that the decimal numerals have the buck-stopping property: when a number is given in decimal notation, there is no further question of what number it is. What makes them special in this way? According to Kripke, it is because of structural revelation: each decimal numeral represents the structure of the corresponding number. Though insightful, I argue, this account has some counterintuitive consequences. Then I sketch an alternative account of the buck-stopping property in terms of how we specify the (...) positions of numbers in the progression. (shrink)

Causal theory of knowledge has been used by some theoreticians who, dealing with the philosophy of mathematics, touched the subject of mathematical knowledge. Some of them discuss the necessity of the causal condition for justification, which creates the grounds for renewing the old conflict between empiricists and rationalists. Emphasizing the condition of causality as necessary for justifiability, causal theory has provided stimulus for the contemporary empiricists to venture on the so far unquestioned cognitive foundations of mathematics. However, in what sense (...) can we speak about the justifiability as causality when it comes to mathematical knowledge? Can we justify the mathematical statement 2+2=4 by means of causality-perceptibility? Do we need perceptibility, a visual image of apples and marbles we add up in our first mathematics lesson, in order to know that statement? Does the same go with the statement that any two points can be joined by a straight line, and that the sum of angles in any triangle is the angle the measure of which equals the sum of the measures of the two right angles? Even though the major part of this article offers a review of the attitudes of various authors who endeavor to discuss the application of the causal theory to the problem of understanding of mathematical knowledge, it advocates the idea that knowledge expressed in a mathematical statement is acquired and transmitted by evidence. Evidence is by itself a guarantee of truth. It convinces us of the accuracy of mathematical truths. Proving, we find out. The greatest part of what we know is based on evidence of the previous knowledge . Due to the previous knowledge, we can say that knowledge in mathematics is in itself different from the knowledge in empirical sciences, i.e. it does not require causal relation that is considered necessary condition of knowing in those sciences.Uzročna teorija znanja iskorištena je od strane nekih teoretičara koji su se, baveći se filozofijom matematike, dotakli teme matematičkog znanja. Neki od njih govore o nužnosti uzročnog uvjeta za opravdanje, čime se stvaraju uvjeti za obnavljanje starog sukoba empirista i racionalista. Uzročna teorija je, isticanjem uvjeta uzročnosti kao nužnog za opravdanost, dala poticaj suvremenim empiristima da krenu čak i na do tada nedodirljive spoznajne temelje matematike. Međutim, u kom smislu možemo govoriti o opravdanosti kao uzročnosti kad je riječ o matematičkom znanju? Da li uzročnošću-čulnošću možemo opravdati matematički iskaz 2+2=4? Je li nam za znanje tog iskaza neophodna čulnost, vizualna predodžba jabuka ili klikera koje zbrajamo na našim prvim satovima matematike u školi? Stoje li stvari isto i s iskazom da se kroz bilo koje dvije točke može povući točno jedan pravac, ili da je zbroj kutova bilo kojeg trokuta kut čija je mjera jednaka zbroju mjera dvaju pravih kutova? Iako je u ovom radu, uglavnom, dan pregled stavova autora koji pokušavaju govoriti o primjeni uzročne teorije u shvaćanju matematičkog znanja, ideja koja se podržava u njemu jest da je znanje, iskazano matematičkim tvrđenjem, dobiveno i preneseno dokazom. Dokaz, sam po sebi, jamac je istine. On nas uvjerava u točnost matematičkih tvrdnji. Dokazujući spoznajemo. Najveći dio onoga što znamo zasnovano je dokazom na prethodnim znanjima . Zbog prethodnog možemo reći da je znanje u matematici po svojoj prirodi drugačije od znanja u empirijskim znanostima, to jest da ne iziskuje uzročnu vezu koja se smatra nužnim uvjetom za znanje u tim znanostima. (shrink)

We argue that mathematical knowledge is context dependent. Our main argument is that on pain of distorting mathematical practice, one must analyse the notion of having available a proof, which supplies justification in mathematics, in a context dependent way.

Penelope Maddy has defended a modified version of mathematical platonism that involves the perception of some sets. Frederick Suppe has developed a conclusive reasons account of empirical knowledge that, when applied to the sets of interest to Maddy, yields that we have knowledge of these sets. Thus, Benacerraf's challenge to the platonist to account for mathematical knowledge has been met, at least in part. Moreover, it is argued that the modalities involved in Suppe's conclusive reasons account of knowledge can be (...) handled without recourse to either laws of nature or possible worlds, and that this approach is preferable. (shrink)

I develop a novel argument against the claim that ordinals are sets. In contrast to Benacerraf’s antireductionist argument, I make no use of covert epistemic assumptions. Instead, my argument uses considerations of ontological dependence. I draw on the datum that sets depend immediately and asymmetrically on their elements and argue that this datum is incompatible with reductionism, given plausible assumptions about the dependence profile of ordinals. In addition, I show that a structurally similar argument can be made against the claim (...) that cardinals are sets. (shrink)