The Benacerraf Problem of Mathematical Truth and Knowledge Before philosophical theorizing, people tend to believe that most of the claims generally accepted in mathematics—claims like “2+3=5” and “there are infinitely many prime numbers”—are true, and that people know many of them. Even after philosophical theorizing, most people remain committed to mathematical truth and mathematical knowledge. … Continue reading The Benacerraf Problem of Mathematical Truth and Knowledge →.

Benacerraf has presented two problems for the philosophy of mathematics. These are the problem of identification and the problem of representation. This paper aims to reconstruct the latter problem and to unpack its undermining bearing on the version of Ontic Structural Realism that frames scientific representations in terms of abstract structures. I argue that the dichotomy between mathematical structures and physical ones cannot be used to address the Benacerraf problem but strengthens it. I conclude (...) by arguing that versions of OSR that do not rely on mathematical frameworks for representational purposes need not be vulnerable to Benacerraf’s second problem. (shrink)

In "Mathematical Truth", Paul Benacerraf articulated an epistemological problem for mathematical realism. His formulation of the problem relied on a causal theory of knowledge which is now widely rejected. But it is generally agreed that Benacerraf was onto a genuine problem for mathematical realism nevertheless. Hartry Field describes it as the problem of explaining the reliability of our mathematical beliefs, realistically construed. In this paper, I argue that the Benacerraf Problem cannot be (...) made out. There simply is no intelligible problem that satisfies all of the constraints which have been placed on the Benacerraf Problem. The point generalizes to all arguments with the structure of the Benacerraf Problem aimed at realism about a domain meeting certain conditions. Such arguments include so-called "Evolutionary Debunking Arguments" aimed at moral realism. I conclude with some suggestions about the relationship between the Benacerraf Problem and the Gettier Problem. (shrink)

In “Mathematical Truth,” Paul Benacerraf presented an epistemological problem for mathematical realism.

Writers in the propositions literature consider the Benacerraf objection serious, often decisive. The objection figures heavily in dismissing standard theories of propositions of the past, notably set-theoretic theories. I argue that the situation is more complicated. After explicating the propositional Benacerraf problem, I focus on a classic set-theoretic theory of propositions, the possible worlds theory, and argue that methodological considerations influence the objection’s success.

Realists about modality offer an attractive semantics for modal discourse in terms of possible worlds, but standard accounts of the worlds—as properties, propositions, or causally-isolated concreta—invoke entities with which we can’t interact. If realism is true, how can we know anything about modal matters? Let's call this "the Benacerraf Problem." I suggest that C. I. Lewis has an intriguing answer to it. Given that we’re willing to disentangle some of Lewis’s insights from his phenomenalism, we can take the (...) following line. If the Benacerraf Problem is a genuine one, then it threatens all knowledge—not just modal knowledge. But then it leads to a general and implausible form of skepticism, not a limited and more plausible debunking argument. Hence, whatever we’re willing to say about skepticism we should say about the Benacerraf Problem. (shrink)

Set-theoretic pluralism is an increasingly influential position in the philosophy of set theory (Balaguer [1998], Linksy and Zalta [1995], Hamkins [2012]). There is considerable room for debate about how best to formulate set-theoretic pluralism, and even about whether the view is coherent. But there is widespread agreement as to what there is to recommend the view (given that it can be formulated coherently). Unlike set-theoretic universalism, set-theoretic pluralism affords an answer to Benacerraf’s epistemological challenge. The purpose of this paper (...) is to determine what Benacerraf’s challenge could be such that this view is warranted. I argue that it could not be any of the challenges with which it has been traditionally identified by its advocates, like of Benacerraf and Field. Not only are none of the challenges easier for the pluralist to meet. None satisfies a key constraint that has been placed on Benacerraf’s challenge. However, I argue that Benacerraf’s challenge could be the challenge to show that our set-theoretic beliefs are safe – i.e., to show that we could not have easily had false ones. Whether the pluralist is, in fact, better positioned to show that our set-theoretic beliefs are safe turns on a broadly empirical conjecture which is outstanding. If this conjecture proves to be false, then it is unclear what the epistemological argument for set-theoretic pluralism is supposed to be. (shrink)

Speaks defends the view that propositions are properties: for example, the proposition that grass is green is the property being such that grass is green. We argue that there is no reason to prefer Speaks's theory to analogous but competing theories that identify propositions with, say, 2-adic relations. This style of argument has recently been deployed by many, including Moore and King, against the view that propositions are n-tuples, and by Caplan and Tillman against King's view that propositions are facts (...) of a special sort. We offer our argument as an objection to the view that propositions are unsaturated relations. (shrink)

There are two general questions which many views in the philosophy of mathematics can be seen as addressing: what are mathematical objects, and how do we have knowledge of them? Naturally, the answers given to these questions are linked, since whatever account we give of how we have knowledge of mathematical objects surely has to take into account what sorts of things we claim they are; conversely, whatever account we give of the nature of mathematical objects must be accompanied by (...) a corresponding account of how it is that we acquire knowledge of those objects. The connection between these problems results in what is often called "Benacerraf's Problem", which is a dilemma that many philosophical views about mathematical objects face. It will be my goal here to present a view, attributed to Richard Dedekind, which approaches the initial questions in a different way than many other philosophical views do, and in doing so, avoids the dilemma given by Benacerraf's problem. (shrink)

Paul Benacerraf’s argument that mathematical realism is apparently incompatible with mathematical knowledge has been widely thought to also show that a priori knowledge in general is problematic. Although many philosophers have rejected Benacerraf’s argument because it assumes a causal theory of knowledge, some maintain that Benacerraf nevertheless put his finger on a genuine problem, even though he didn’t state the problem in its most challenging form. After diagnosing what went wrong with Benacerraf’s argument, I (...) argue that a new, more challenging, version of Benacerraf’s problem can be constructed. The new version—what I call the Defeater Version—of Benacerraf’s problem makes use of a no-defeater condition on knowledge and justification. I conclude by arguing that the best way to avoid the problem is to construct a theory of how a priori judgments reliably track the facts. I also suggest four different kinds of theories worth pursuing. (shrink)

Benacerraf’s problem is justly famous. It’s had a major influence on the philosophy of mathematics right from its initial appearance, an influence that continues up through the present moment.

In a series of recent publications, Jeffrey King (The nature and structure of content, 2007; Proc Aristot Soc 109(3):257–277, 2009; Philos Stud, 2012) argues for a view on which propositions are facts. He also argues against views on which propositions are set-theoretical objects, in part because such views face Benacerraf problems. In this paper, we argue that, when it comes to Benacerraf problems, King’s view doesn’t fare any better than its set-theoretical rivals do. Finally, we argue that his (...) view faces a further Benacerraf problem, one that threatens to undercut his explanation of why propositions have truth-conditions. If correct, our arguments undercut King’s main motivation for accepting his view over its rivals. (shrink)

Hartry Field’s epistemological challenge to the mathematical platonist is often cast as an improvement on Paul Benacerraf’s original epistemological challenge. I disagree. While Field’s challenge is more difficult for the platonist to address than Benacerraf’s, I argue that this is because Field’s version is a special case of what I call the ‘sociological challenge’. The sociological challenge applies equally to platonists and fictionalists, and addressing it requires a serious examination of mathematical practice. I argue that the non-sociological part (...) of Field’s challenge amounts to a minor reformulation of Benacerraf’s original challenge. So, I contend, Field’s challenge is not an improvement on Benacerraf’s. What is new to Field’s challenge is as much a problem for the fictionalist as it is for the platonist. (shrink)

In this article, I discuss a trivialization worry for Hartry Field’s official formulation of the access problem for mathematical realists, which was pointed out by Øystein Linnebo. I argue that various attempted reformulations of the Benacerraf problem fail to block trivialization, but that access worriers can better defend themselves by sticking closer to Hartry Field’s initial informal characterization of the access problem in terms of general epistemic norms of coincidence avoidance.

Evolutionary debunking arguments purport to show that robust moral realism, the metaethical view that there are non-natural and mind-independent moral properties and facts that we can know about, is incompatible with evolutionary explanations of morality. One of the most prominent evolutionary debunking arguments is advanced by Sharon Street, who argues that if moral realism were true, then objective moral knowledge is unlikely because realist moral properties are evolutionary irrelevant and moral beliefs about those properties would not be selected for. However, (...) no evolutionary, causal explanation plays an essential role in reaching the argument’s epistemological conclusion. Street’s argument depends on the Benacerraf-Field challenge, which is the challenge to explain the reliability of our moral beliefs about causally inert moral properties. The Benacerraf-Field challenge relies on metaphysically necessary facts about realist moral properties, rather than on contingent Darwinian facts about the origin of our moral beliefs. Attempting to include an essential causal empirical premise yet avoiding recourse to the Benacerraf-Field problem yields an argument that is either self-defeating or of limited scope. Ultimately, evolutionary, causal explanations of our moral beliefs and their consequences do not present the strongest case against robust moral realism. Rather, the question is whether knowledge of casually-inert, mind-intendent properties is plausible at all. (shrink)

In this article, I discuss a trivialization worry for Hartry Field’s official formulation of the access problem for mathematical realists, which was pointed out by Øystein Linnebo (and has recently been made much of by Justin Clarke-Doane). I argue that various attempted reformulations of the Benacerraf problem fail to block trivialization, but that access worriers can better defend themselves by sticking closer to Hartry Field’s initial informal characterization of the access problem in terms of (something like) (...) general epistemic norms of coincidence avoidance. (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)

The paper is devoted to analysis of P. Benacerraf’s argument against set-theoretic reductionist realism which is a fragment of a broader argument, know as the “identification problem”. The analyzed fragment of P. Benacerraf’s argument concerns the possibility of reducing of mathematical notions to set-theoretic notions. The paper presents a reconstruction of P. Benacerraf’s original argumentation, its analysis and also several possible objections proposed by P. Benacerraf himself about 30 years later after the original publication. Namely, (...) he claimed (1) that a set-theoretic definition of natural numbers in G. Frege’s fashion can serve as a proper and unique set-theoretic definition, (2) that his argument doesn’t undermine eliminative reductionsts’ position, (3) that even if there are no argument possible in favor of some particular set-theoretic definition of natural numbers one may take set-theoretic realism for granted. An analysis of the mentioned possible objections shows their dependence on a number of additional premises. The paper demonstrates that P. Benacerraf’s objections on his own argument against set-theoretic realism either have a pragmatic character themselves or essentially rely on additional arguments that are justified pragmatically or require additional argumentation. For example, his possible objections require that set theory is considered as the only true foundational theory in mathematics, and that it has several important pragmatic virtues, like convenience of use to formalize other mathematical theories. In some cases, P. Benacerraf’s objections on their own, or the indicated additional principles may well be called into question, which demonstrates the insufficiency of P. Benacerraf’s objections against his original argument. Without the mentioned pragmatic arguments P. Benacerraf’s objections become a kind of belief in mysticism. Accordingly, his doubts about his own argument against set-theoretical realism seem insufficient to reject the problem of identification and save the position of set-theoretical realism from collapse. (shrink)

One of the important challenges in the philosophy of mathematics is to account for the semantics of sentences that express mathematical propositions while simultaneously explaining our access to their contents. This is Benacerraf’s Dilemma. In this dissertation, I argue that cognitive science furnishes new tools by means of which we can make progress on this problem. The foundation of the solution, I argue, must be an ontologically realist, albeit non-platonist, conception of mathematical reality. The semantic portion of the (...) problem can be addressed by accepting a Chomskyan conception of natural languages and a matching internalist, mentalist and nativist view of semantics. A helpful perspective on the epistemic aspect of the puzzle can be gained by translating Kurt G ̈odel’s neo-Kantian conception of the nature of mathematics and its objects into modern, cognitive terms. (shrink)

The Benacerraf’s problem is a problem about how we can attain mathematical knowledge: mathematical entities are entities not located in space-time; we exist in spacetime; so, it does not seem that we could have a causal connection with mathematical entities in order to attain mathematical knowledge. In this paper, I propose a solution to the Benacerraf’s problem supported by the Quinean doctrines of naturalism, confirmational holism and postulation. I show that we have empirical knowledge of (...) centres of mass and of entities outside of our light cone and that these entities are inefficacious causality entities, at least,with us. At the end, I defend the existential knowledge of centres of mass and of entities outside of our light cone against the Eleatic principle of Cheyne that we only could attain existential knowledge of entities by a causal connection. (shrink)

In this article, I discuss Hawthorne'€™s contextualist solution to Benacerraf'€™s dilemma. He wants to find a satisfactory epistemology to go with realist ontology, namely with causally inaccessible mathematical and modal entities. I claim that he is unsuccessful. The contextualist theories of knowledge attributions were primarily developed as a response to the skeptical argument based on the deductive closure principle. Hawthorne uses the same strategy in his attempt to solve the epistemologist puzzle facing the proponents of mathematical and modal realism, (...) but this problem is of a different nature than the skeptical one. The contextualist theory of knowledge attributions cannot help us with the question about the nature of mathematical and modal reality and how they can be known. I further argue that Hawthorne'€™s account does not say anything about a priori status of mathematical and modal knowledge. Later, Hawthorne adds to his account an implausible claim that in some contexts a gettierized belief counts as knowledge. (shrink)

Does Wittgenstein have a coherent philosophy of mathematics? Here, I will be concerned with showing that the answer is positive. However, given that his life-long philosophical perspective on mathematics tends to be misleading, I focus on the specific problem posed by Paul Benacerraf in ‘Mathematical Truth’, that is: the puzzle about how to reconcile the metaphysics with the epistemology for mathematics. My aim is to show that there is an adequate anti-platonistic solution to that puzzle in the mature (...) writings of the Austrian philosopher. (shrink)

A range of current truth-value realist philosophies of mathematics allow one to reduce the Benacerraf Problem to a problem concerning mathematicians' ability to recognize which conceptions of pure mathematical structures are coherent – in a sense which can be cashed out in terms of logical possibility. In this paper I will clarify what it takes to solve this `residual' access problem and then present a framework for solving it.

In an earlier paper I suggested that we can solve the Benacerraf Problem – the problem of explaining how mathematical knowledge is possible on the assumption that the objects of mathematics are abstract and immaterial – by positing efficient causal relations between those abstract objects and our brains. The burden of the paper was to remove the appearance that relations between abstracta and concreta, far from being actual, are inconceivable. This alleged inconceivability has been derived from some (...) putative conditions on efficient causation: physical contact, energy transfer, reciprocal effects on the cause, and the idea that the cause must exist in time if its effects do. I argued that none of these conditions are a priori conditions on efficient causation, and thus do not pose an obstacle to the conceivability or metaphysical possibility of efficient causal relations between platonic objects and brains. In this essay I reply to three published challenges to this proposed solution to the Benacerraf Problem. (shrink)

What is mathematics about? And if it is about some sort of mathematical reality, how can we have access to it? This is the problem raised by Plato, which still today is the subject of lively philosophical disputes. This book traces the history of the problem, from its origins to its contemporary treatment. It discusses the answers given by Aristotle, Proclus and Kant, through Frege's and Russell's versions of logicism, Hilbert's formalism, Gödel's platonism, up to the the current (...) debate on Benacerraf's dilemma and the indispensability argument. Through the considerations of themes in the philosophy of language, ontology, and the philosophy of science, the book aims at offering an historically-informed introduction to the philosophy of mathematics, approached through the lenses of its most fundamental problem. (shrink)

Modal epistemology has been dominated by a focus on establishing an account either of how we have modal knowledge or how we have justified beliefs about modality. One component of this focus has been that necessity and possibility are basic access points for modal reasoning. For example, knowing that P is necessary plays a role in deducing that P is essential, and knowing that both P and ¬P are possible plays a role in knowing that P is accidental. Chalmers (2002) (...) and Williamson (2007) provide two good examples of contrasting views in modal epistemology that focus on providing an account of modal knowledge where necessity and possibility are basic access points for modal knowledge, and Yablo (1993) provides a good account of how we have justified beliefs about modality. In contrast to this tradition I argue for and outline a modal epistemology based on objectual understanding and essence, rather than knowledge or justification and necessity and possibility. The account employs a non-modal conception of essence and takes objectual understanding of essence, rather than knowledge of essence to be basic in modal reasoning. I begin by articulating Kvanvig’s (2003) account of objectual understanding, on which objectual understanding of Φ is not equivalent to propositional knowledge of Φ. I then argue that an epistemology of essence that uses property variation-in-imagination is better construed as a model that delivers objectual understanding of essence rather than knowledge of essence. I argue that this is so, since the latter and not the former runs into a version of the Meno paradox. I show how this account can be applied to two issues in modal epistemology: the Benacerraf problem for modality, and the architecture of modal knowledge. (shrink)

Suggests that the recent emphasis on Benacerraf's access problem locates the peculiarity of mathematical knowledge in the wrong place. Instead we should focus on the sense in which mathematical concepts are or might be "armchair concepts" – concepts about which non-trivial knowledge is obtainable a priori.

Various arguments have been put forward to show that Zeno-like paradoxes are still with us. A particularly interesting one involves a cube composed of colored slabs that geometrically decrease in thickness. We first point out that this argument has already been nullified by Paul Benacerraf. Then we show that nevertheless a further problem remains, one that withstands Benacerraf s critique. We explain that the new problem is isomorphic to two other Zeno-like predicaments: a problem described (...) by Alper and Bridger in 1998 and a modified version of the problem that Benardete introduced in 1964. Finally, we present a solution to the three isomorphic problems. (shrink)

This paper is a comparison of two structural theories of propositions: the theory proposed by Kazimierz Ajdukiewicz in the 1960s and the theory developed by Jeffrey King at the beginning of the 21st century. The first section of the paper is an overview of these theories. The second part is a detailed discussion of significant similarities shared by them. In this section, I also identify and analyze ways in which these theories differ and attempt to determine if these differences are (...) substantial or apparent. The last part is an attempt to determine if the discussed theories are capable of coping with the Benacerraf Problem. (shrink)

Those inclined to believe in the existence of propositions as traditionally conceived might seek to reduce them to some other type of entity. However, parsimonious propositionalists of this type are confronted with a choice of competing candidates – for example, sets of possible worlds, and various neo-Russellian and neo-Fregean constructions. It is argued that this choice is an arbitrary one, and that it closely resembles the type of problematic choice that, as Benacerraf pointed out, bedevils the attempt to reduce (...) numbers to sets – should the number 2 be identified with the set Ø or with the set Ø, Ø? An “argument from arbitrary identification” is formulated with the conclusion that propositions (and perhaps numbers) cannot be reduced away. Various responses to this argument are considered, but ultimately rejected. The paper concludes that the argument is sound: propositions, at least, are sui generis entities. (shrink)

A response is given here to Benacerraf's (1965) non-uniqueness (or multiple-reductions) objection to mathematical platonism. It is argued that non-uniqueness is simply not a problem for platonism; more specifically, it is argued that platonists can simply embrace non-uniqueness—i.e., that one can endorse the thesis that our mathematical theories truly describe collections of abstract mathematical objects while rejecting the thesis that such theories truly describe unique collections of such objects. I also argue that part of the motivation for this (...) stance is that it dovetails with the correct response to Benacerraf's other objection to platonism, i.e., his (1973) epistemological objection. (shrink)

Reductionist realist accounts of certain entities, such as the natural numbers and propositions, have been taken to be fatally undermined by what we may call the problem of arbitrary identification. The problem is that there are multiple and equally adequate reductions of the natural numbers to sets (see Benacerraf, 1965), as well as of propositions to unstructured or structured entities (see, e.g., Bealer, 1998; King, Soames, & Speaks, 2014; Melia, 1992). This paper sets out to solve the (...) problem by canvassing what we may call the arbitrary reference strategy. The main claims of such strategy are 2. First, we do not know which objects are the referents of proposition and numerical terms since their reference is fixed arbitrarily. Second, our ignorance of which object is picked out as the referent does not entail that no object is referred to by the relevant expression. Different articulations of the strategy are assessed, and a new one is defended. (shrink)

Various arguments have been put forward to show that Zeno-like paradoxes are still with us. A particularly interesting one involves a cube composed of colored slabs that geometrically decrease in thickness. We first point out that this argument has already been nullified by Paul Benacerraf. Then we show that nevertheless a further problem remains, one that withstands Benacerraf’s critique. We explain that the new problem is isomorphic to two other Zeno-like predicaments: a problem described by (...) Alper and Bridger in 1998 and a modified version of the problem that Benardete introduced in 1964. Finally, we present a solution to the three isomorphic problems. (shrink)

I critically examine an evolutionary debunking argument against moral realism. The key premise of the argument is that there is no adequate explanation of our moral reliability. I search for the strongest version of the argument; this involves exploring how ‘adequate explanation’ could be understood such that the key premise comes out true. Finally, I give a reductio: in the sense in which there is no adequate explanation of our moral reliability, there is equally no adequate explanation of our inductive (...) reliability. Thus, the argument that would debunk our moral views would also, absurdly, debunk all inductive reasoning. (shrink)

The conflict between Platonic realism and Constructivism marks a watershed in philosophy of mathematics. Among other things, the controversy over the Axiom of Choice is typical of the conflict. Platonists accept the Axiom of Choice, which allows a set consisting of the members resulting from infinitely many arbitrary choices, while Constructivists reject the Axiom of Choice and confine themselves to sets consisting of effectively specifiable members. Indeed there are seemingly unpleasant consequences of the Axiom of Choice. The non-constructive nature of (...) the Axiom of Choice leads to the existence of non-Lebesgue measurable sets, which in turn yields the Banach-Tarski Paradox. But the Banach-Tarski Paradox is so called in the sense that it is a counter-intuitive theorem. To corroborate my view that mathematical truths are of non-constructive nature, I shall draw upon Gödel’s Incompleteness Theorems. This also shows the limitations inherent in formal methods. Indeed the Löwenheim-Skolem Theorem and the Skolem Paradox seem to pose a threat to Platonists. In this light, Quine/Putnam’s arguments come to take on a clear meaning. According to the model-theoretic arguments, the Axiom of Choice depends for its truth-value upon the model in which it is placed. In my view, however, this is another limitation inherent in formal methods, not a defect for Platonists. To see this, we shall examine how mathematical models have been developed in the actual practice of mathematics. I argue that most mathematicians accept the Axiom of Choice because the existence of non-Lebesgue measurable sets and the Well-Ordering of reals open the possibility of more fruitful mathematics. Finally, after responding to Benacerraf’s challenge to Platonism, I conclude that in mathematics, as distinct from natural sciences, there is a close connection between essence and existence. Actual mathematical theories are the parts of the maximally logically consistent theory that describes mathematical reality. (shrink)

In this paper, I will attempt to develop and defend a common form of intuitive resistance to the companions in guilt argument. I will argue that one can reasonably believe there are promising solutions to the access problem for mathematical realism that don’t translate to moral realism. In particular, I will suggest that the structuralist project of accounting for mathematical knowledge in terms of some form of logical knowledge offers significant hope of success while no analogous approach offers such (...) hope for moral realism. (shrink)

We are reliable about logic in the sense that we by-and-large believe logical truths and disbelieve logical falsehoods. Given that logic is an objective subject matter, it is difficult to provide a satisfying explanation of our reliability. This generates a significant epistemological challenge, analogous to the well-known Benacerraf-Field problem for mathematical Platonism. One initially plausible way to answer the challenge is to appeal to evolution by natural selection. The central idea is that being able to correctly deductively reason (...) conferred a heritable survival advantage upon our ancestors. However, there are several arguments that purport to show that evolutionary accounts cannot even in principle explain how it is that we are reliable about logic. In this paper, I address these arguments. I show that there is no general reason to think that evolutionary accounts are incapable of explaining our reliability about logic. (shrink)

In this article, I present a novel account of a priori warrant, which I then use to examine the relationship between a priori and a posteriori warrant in mathematics. According to this account of a priori warrant, the reason that a posteriori warrant is subordinate to a priori warrant in mathematics is because processes that produce a priori warrant are reliable independent of the contexts in which they are used, whereas this is not true for processes that produce a posteriori (...) warrant. Following this, I show how this difference explains why a priori warranting processes, such as proof (and mathematical intuition) can, and a posteriori warranting processes cannot, definitively establish mathematical results. I then show why, in the event of a conflict between a priori and a posteriori warranting processes, the former undermine, and cannot be undermined by, the latter. Finally, I explain the connection between apriority and mathematical necessity, but do so in a way that allows for the existence of contingent a priori knowledge. (shrink)

The paper begins with an argument against eliminativism with respect to the propositional attitudes. There follows an argument that concepts are sui generis ante rem entities. A nonreductionist view of concepts and propositions is then sketched. This provides the background for a theory of concept possession, which forms the bulk of the paper. The central idea is that concept possession is to be analyzed in terms of a certain kind of pattern of reliability in one’s intuitions regarding the behavior of (...) the concept. The challenge is to find an analysis that is at once noncircular and fully general. Environmentalism, anti-individualism, holism, analyticity, etc. provide additional hurdles. The paper closes with a discussion of the theory’s implications for the Wittgenstein-Kripke puzzle about rule-following and the Benacerraf problem concerning mathematical knowledge. (shrink)

The philosophy of mathematics of the later Wittgenstein is normally not taken very seriously. According to a popular objection, it cannot account for mathematical necessity. Other critics have dismissed Wittgenstein's approach on the grounds that his anti-platonism is unable to explain mathematical objectivity. This latter objection would be endorsed by somebody who agreed with Paul Benacerraf that any anti-platonistic view fails to describe mathematical truth. This paper focuses on the problem proposed by Benacerraf of reconciling the semantics (...) with the epistemology for mathematics. It is claimed that there is a way of solving Benacerrafs problem along the lines suggested by Wittgenstein's later remarks on mathematics. This will require demonstrating that a satisfactory conception of mathematical objectivity can be extracted from his mature philosophy. (shrink)

I attempt to accommodate the phenomenon of vagueness with classical logic and bivalence. I hold that for any vague predicate there is a sharp cut-off between the things that satisfy it and the things that don’t; I claim that this is due to the greater naturalness of one of the candidate meanings of that predicate. I extend the view to give an account of arbitrary reference and a solution to Benacerraf problems. I end by exploring the idea that it (...) is ontically indeterminate what the most natural meanings are, and hence ontically indeterminate where the sharp cut-off in a sorites series is. (shrink)

Stance-independent nonnaturalist moral realism is subject to two related epistemological objections. First, there is the metaethical descendant of the Benacerraf problem. Second, there are evolutionary debunking arguments. Standard attempts to solve these epistemological problems have not appealed to any particular moral epistemology. The focus on these epistemologically neutral responses leaves many interesting theoretical stones unturned. Exploring the ability of particular theories in moral epistemology to handle these difficult epistemological objections can help illuminate strengths or weaknesses within these theories (...) themselves, as well as opening up potentially unexplored avenues for responding to deeply entrenched concerns about our epistemic access to the moral properties. In this paper, I assess the prospects of an empiricist and perceptualist model of moral knowledge for responding to epistemological arguments against non-skeptical moral realism. I argue that such a view has powerful responses to these objections that are not open to other moral epistemologists. The upshot is that if some version of the view is correct, then the realist has less to fear from Benacerraf and evolutionary debunking–style epistemological objections. Insofar as one is already a committed realist, then, this provides some indirect support for moral perceptualism. (shrink)

This paper is a comparison of two structural theories of propositions: the theory proposed by Kazimierz Ajdukiewicz in the 1960s and the theory developed by Jeffrey King at the beginning of the 21st century. The first section of the paper is an overview of these theories. The second part is a detailed discussion of significant similarities shared by them. In this section, I also identify and analyze ways in which these theories differ and attempt to determine if these differences are (...) substantial or apparent. The last part is an attempt to determine if the discussed theories are capable of coping with the Benacerraf Problem. (shrink)

This paper addresses John Burgess's answer to the ‘Benacerraf Problem’: How could we come justifiably to believe anything implying that there are numbers, given that it does not make sense to ascribe location or causal powers to numbers? Burgess responds that we should look at how mathematicians come to accept: There are prime numbers greater than 1010 That, according to Burgess, is how one can come justifiably to believe something implying that there are numbers. This paper investigates what (...) lies behind Burgess's answer and ends up as a rebuttal to Burgess's reasoning. (shrink)

This paper addresses John Burgess's answer to the ‘Benacerraf Problem’: How could we come justifiably to believe anything implying that there are numbers, given that it does not make sense to ascribe location or causal powers to numbers? Burgess responds that we should look at how mathematicians come to accept: There are prime numbers greater than 1010That, according to Burgess, is how one can come justifiably to believe something implying that there are numbers. This paper investigates what lies (...) behind Burgess's answer and ends up as a rebuttal to Burgess's reasoning. (shrink)

Neo-Russellian theories of structured propositions face challenges to do with both representation and structure which are sometimes called the problem of unity and the Benacerraf problem. In §i, I set out the problems and Jeffrey King's solution, which I take to be the best of its type, as well as an unfortunate consequence for that solution. In §§ii–iii, I diagnose what is going wrong with this line of thought. If I am right, it follows that the (...) class='Hi'>Benacerraf problem cannot be used to motivate the view that propositions are irreducible elements of our ontology. (shrink)

Field’s challenge to platonists is the challenge to explain the reliable match between mathematical truth and belief. The challenge grounds an objection claiming that platonists cannot provide such an explanation. This objection is often taken to be both neutral with respect to controversial epistemological assumptions, and a comparatively forceful objection against platonists. I argue that these two characteristics are in tension: no construal of the objection in the current literature realises both, and there are strong reasons to think that no (...) version of Field’s epistemological objection which has both Neutrality and Force can be construed. (shrink)

The philosophy of mathematics of the later Wittgenstein is normally not taken very seriously. According to a popular objection, it cannot account for mathematical necessity. Other critics have dismissed Wittgenstein's approach on the grounds that his anti-platonism is unable to explain mathematical objectivity. This latter objection would be endorsed by somebody who agreed with Paul Benacerraf that any anti-platonistic view fails to describe mathematical truth. This paper focuses on the problem proposed by Benacerraf of reconciling the semantics (...) with the epistemology for mathematics. It is claimed that there is a way of solving Benacerrafs problem along the lines suggested by Wittgenstein's later remarks on mathematics. This will require demonstrating that a satisfactory conception of mathematical objectivity can be extracted from his mature philosophy. (shrink)

Some of the most interesting recent work in philosophy of language and metaphysics is focused on questions about propositions, the abstract, truth-bearing contents of sentences and beliefs. The aim of this guide is to give instructors and students a road map for some significant work on propositions since the mid-1990s. This work falls roughly into two areas: challenges to the existence of propositions and theories about the nature and structure of propositions. The former includes both a widely discussed puzzle about (...) propositional designators as well as direct and indirect arguments against the existence of propositions. The latter is dominated by what is currently the central debate about the metaphysics of propositions, i.e. whether they are structured, composite entities or unstructured ontological simples. This issue has eclipsed older debates about whether propositions can be identified with sets of possible worlds or other kinds of sentence intensions. Author Recommends 1. Soames, Scott. 'Direct Reference, Propositional Attitudes, and Semantic Content.' Philosophical Topics 15 (1987): 47–87. Reprinted in Propositions and Attitudes . Eds. N. Salmon and S. Soames. Oxford: Oxford University Press, 1988. 197–239. Essential groundwork for more recent work on propositions. Soames gives a careful and exacting presentation of the case against identifying propositions with sets of possible worlds or other truth-supporting circumstances. Also contains a detailed statement of the Russellian conception of propositions on which propositions are ordered sets of objects, properties and relations. 2. King, Jeffrey. 'Designating Propositions.' The Philosophical Review 111 (2002): 341–71. Sometimes substituting a definite description for a corresponding 'that'-clause can lead to bizarre changes in truth-conditions: compare 'Bill fears that Hillary will be president' with 'Bill fears the proposition that Hillary will be president'. This puzzle about propositional designators threatens the relational analysis of propositional attitude reports, the view that 'believes' expresses a relation to the proposition designated by its 'that'-clause, and thereby poses an indirect threat to the existence of propositions. King's solution posits an ambiguity in verbs like 'fear' that embed both 'that'-clauses and definite descriptions. 3. Jubien, Michael. 'Propositions and the Objects of Thought.' Philosophical Studies 104 (2001): 47–62. A direct attack on the existence of propositions. Jubien deploys an analogue of the problem that Paul Benacerraf raised for set-theoretical reductions of numbers against metaphysical reductions of propositions. Just as numbers can be reduced to sets in many different ways, any reduction of propositions brings with it equally good variants, thus making any such reduction arbitrary and unmotivated. The only alternative is to treat propositions as abstract metaphysical primitives. As Jubien argues, however, abstract primitive entities are incapable of doing what propositions must do, i.e. represent objects and states of affairs on their own, without the input of thinking subjects. The upshot is the propositions cannot be reduced and they cannot be primitive, and so they must not exist. 4. Hanks, Peter. 'How Wittgenstein Defeated Russell's Multiple Relation Theory of Judgment.' Synthese 154 (2007): 121–46. Scepticism about propositions has recently led some philosophers, Jubien included, to resuscitate Russell's multiple relation theory of judgment, the idea that judgment is a many-place relation to objects, properties and relations. This paper explains why Russell himself abandoned that theory, and why the theory is still refuted by an objection due to Wittgenstein. 5. Hofweber, Thomas. 'Inexpressible Properties and Propositions.' Oxford Studies in Metaphysics . 2 vols. Ed. D. Zimmerman. Oxford: Oxford University Press, 2006. 155–206. An indirect attack on the existence of propositions. Hofweber argues that sentences like 'Bill believes something that Hillary asserted' do not commit us to the existence of propositions. His view is that propositional quantification is an instance of what he calls 'internal' or 'inferential role' quantification, a kind of quantification that carries no ontological implications. 6. Schiffer, Stephen. The Things We Mean . Oxford: Oxford University Press, 2003. esp. chs 1–2. Schiffer defends his theory of pleonastic propositions, on which propositions are unstructured, have no parts, and are very finely grained. 7. Bealer, George. 'Propositions.' Mind 107 (1998): 1–32. Bealer defends his algebraic theory of propositions, which, like Schiffer's pleonastic account, treats propositions as unstructured metaphysical simples. 8. King, Jeffrey. The Nature of and Structure of Content . Oxford: Oxford University Press, 2007. The best developed current theory of the structure in structured propositions. King identifies propositions with certain kinds of facts in which objects, properties and relations are bound together by amalgams of syntactic and semantic relations. 9. Hanks, Peter. 'Recent Work on Propositions.' Philosophy Compass 4 (2009): 1–18. A survey of work on propositions since the mid-1990s that complements this teaching and learning guide. Contains responses to Jubien's and Hofweber's arguments against propositions and critical discussions of Schiffer's pleonastic propositions and King's theory of propositional structure. Online Resources 1. http://plato.stanford.edu/entries/propositions/ Propositions (Matthew McGrath) 2. http://plato.stanford.edu/entries/propositions-structured/ Structured Propositions (Jeffrey King) 3. http://plato.stanford.edu/entries/propositions-singular/ Singular Propositions (Greg Fitch) Sample Partial Syllabus The following partial syllabus can be used as a unit on recent work on propositions in graduate level courses in philosophy of language or metaphysics. Week 1: A Substitution Puzzle About Propositional Designators King, Jeffrey. 'Designating Propositions'. Moltmann, Friederike. 'Propositional Attitudes Without Propositions.' Synthese 135 (2003): 77–118. Week 2: The Benacerraf Problem and Propositional Representation Benacerraf, Paul. 'What Numbers Could Not Be.' Philosophical Review 74 (1965): 47–73. Jubien, Michael. 'Propositions and the Objects of Thought.' Week 3: Propositional Quantification Hofweber, Thomas. 'Inexpressible Properties and Propositions'. Hofweber, Thomas. 'A Puzzle about Ontology.' Noûs 39 (2005): 256–83. Week 4: Schiffer on Pleonastic Propositions Schiffer, Stephen. 'Language-Created Language-Independent Entities.' Philosophical Topics 24 (1996): 149–67. Schiffer, Stephen. The Things We Mean , chs 1–2. Week 5: King on Structured Propositions King, Jeffrey. 'Structured Propositions and Complex Predicates.' Noûs , 29 (1995): 516–35. King, Jeffrey. The Nature and Structure of Content , chs 1–3. Focus Questions 1. Why does identifying propositions with sentence intensions, e.g. sets of possible worlds, 'require the attitudes to have a particular sort of closure under logical consequence, which they clearly don't have' (Mark Richard)? 2. How does the difference between (a) and (b) pose a threat to the existence of propositions? (a) Bill fears that Hillary will be president. (b) Bill fears the proposition that Hillary will be president. 3. What is the Benacerraf problem for metaphysical reductions of propositions? 4. Why must a proposition represent 'on its own cuff' (Michael Jubien)? Why is this a problem for the view that propositions are primitive abstract entities? 5. What does it mean to say that propositions are structured ? Give two different accounts of what propositional structure might be. (shrink)