Charles Chihara's new book develops and defends a structural view of the nature of mathematics, and uses it to explain a number of striking features of mathematics that have puzzled philosophers for centuries. The view is used to show that, in order to understand how mathematical systems are applied in science and everyday life, it is not necessary to assume that its theorems either presuppose mathematical objects or are even true. Chihara builds upon his previous work, in which (...) he presented a new system of mathematics, the constructibility theory, which did not make reference to, or presuppose, mathematical objects. Now he develops the project further by analysing mathematical systems currently used by scientists to show how such systems are compatible with this nominalistic outlook. He advances several new ways of undermining the heavily discussed indispensability argument for the existence of mathematical objects made famous by Willard Quine and Hilary Putnam. And Chihara presents a rationale for the nominalistic outlook that is quite different from those generally put forward, which he maintains have led to serious misunderstandings.A Structural Account of Mathematics will be required reading for anyone working in this field. (shrink)

Charles Chihara gives a thorough critical exposition of modal realism, the philosophical doctrine that there exist many possible worlds of which the actual world--the universe in which we live--is just one. The striking success of possible-worlds semantics in modal logic has made this ontological doctrine attractive. Modal realists maintain that philosophers must accept the existence of possible worlds if they wish to have the benefit of using possible-worlds semantics to assess modal arguments and explain modal principles. Chihara challenges (...) this claim, and argues instead for modality without worlds; he offers a new account of the role of interpretations or structures of the formal languages of logic. (shrink)

Charles Chihara gives a thorough critical exposition of modal realism, the philosophical doctrine that there exist many possible worlds of which the actual world--the universe in which we live--is just one. The striking success of possible-worlds semantics in modal logic has made thisontological doctrine attractive. Modal realists maintain that philosophers must accept the existence of possible worlds if they wish to have the benefit of using possible-worlds semantics to assess modal arguments and explain modal principles. Chihara challenges this (...) claim, and argues instead formodality without worlds; he offers a new account of the role of interpretations or structures of the formal languages of logic. (shrink)

This book is concerned with `the problem of existence in mathematics'. It develops a mathematical system in which there are no existence assertions but only assertions of the constructibility of certain sorts of things. It explores the philosophical implications of such an approach through an examination of the writings of Field, Burgess, Maddy, Kitcher, and others.

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)

A powerful challenge to some highly influential theories, this book offers a thorough critical exposition of modal realism, the philosophical doctrine that many possible worlds exist of which our own universe is just one. Chihara challenges this claim and offers a new argument for modality without worlds.

This paper explores some lines of argument in wittgenstein's post-Tractatus writings in order to indicate the relations between wittgenstein's philosophical psychology, On the one hand, And his philosophy of language, His epistemology, And his doctrines about the nature of philosophical analysis on the other. The authors maintain that the later writings of wittgenstein express a coherent doctrine in which an operationalistic analysis of confirmation and language supports a philosophical psychology of a type the authors call "logical behaviorism." they also maintain (...) that there are good grounds for rejecting the philosophical theory implicit in wittgenstein's later works. In particular, They first argue that wittgenstein's position leads to some implausible conclusions concerning the nature of language and psychology; second, They maintain that the arguments wittgenstein provides are inconclusive; and third, They sketch an alternative position which they believe avoids many of the difficulties implicit in wittgenstein's philosophy. (shrink)

“Probably Peirce’s best-known works are the first two articles in a series of six that originally were collectively entitled Illustrations of the Logic of Science and published in Popular Science Monthly from November 1877 through August 1878. The first is entitled ‘The Fixation of Belief’ and the second is entitled ‘How to Make Our Ideas Clear.’ In the first of these papers Peirce defended, in a manner consistent with not accepting naive realism, the superiority of the scientific method over other (...) methods of overcoming doubt and ‘fixing belief.’” — Robert Burch, “Charles Sanders Peirce,” entry in the Stanford Encyclopedia of Philosophy (2021 revision) -/- “Pragmatist epistemologies often explore how we can carry out inquiries in a self-controlled and fruitful way. (Where much analytic epistemology centres around the concept of knowledge, considered as an idealised end-point of human thought, pragmatist epistemology centres around the concept of inquiry, considered as the process of knowledge-seeking and how we can improve it.) So pragmatists often provide rich accounts of the capacities or virtues that we must possess in order to inquire well, and the rules or guiding principles that we should adopt. A canonical account is Peirce’s classic early paper ‘The Fixation of Belief‘. Here Peirce states that inquiry is a struggle to replace doubt with “settled belief“, and that the only method of inquiry that can make sense of the fact that at least some of us are disturbed by inconsistent beliefs, and will subsequently reflect upon which methods of fixing belief are correct is the Method of Science, which draws on the Pragmatic Maxim described above. This contrasts with three other methods of fixing belief: i) refusing to consider evidence contrary to one’s favored beliefs (the Method of Tenacity), ii) accepting an institution’s dictates (the Method of Authority), iii) developing the most rationally coherent or elegant-seeming belief-set (the A Priori Method).“ — Catherine Legg, “Pragmatism,” entry in the Stanford Encyclopedia of Philosophy (2021 revision). (shrink)

According to most accounts of trust, you can only trust other people (or groups of people). To trust is to think that another has goodwill, or something to that effect. I sketch a different form of trust: the unquestioning attitude. What it is to trust, in this sense, is to settle one’s mind about something, to stop questioning it. To trust is to rely on a resource while suspending deliberation over its reliability. Trust lowers the barrier of monitoring, challenging, checking, (...) and questioning. Trust sets up open pipelines between yourself and parts of the external world. Trust permits external resources to have a similar relationship to one as one’s internal cognitive faculties. This creates efficiency, but at the price of exquisite vulnerability. We must trust in this way because we are cognitively limited beings in a cognitively overwhelming world. Crucially, we can hold the unquestioning attitude towards objects and technologies. When I trust my climbing rope, I stop worrying about its reliability. When I trust my online calendaring system, I simply go to the events indicated, without question. But, one might worry, how could one ever hold such a normatively loaded attitude as trust towards mere objects? How could it ever make sense to feel betrayed by an object? Trust is our engine for expanding and outsourcing our agency — for binding external processes into our practical selves. Thus, we can be betrayed by our smartphones in the same way that we can be betrayed by our memory. When we trust, we try to make something a part of our agency, and we are betrayed when our part lets us down. To unquestioningly trust something is to let it in—to attempt to bring it inside one’s practical functioning. This suggests a new form of gullibility: agential gullibility, which occurs when agents too hastily and carelessly integrate external resources into their own agency. (shrink)

Value capture occurs when an agent’s values are rich and subtle; they enter a social environment that presents simplified — typically quantified — versions of those values; and those simplified articulations come to dominate their practical reasoning. Examples include becoming motivated by FitBit’s step counts, Twitter Likes and Re-tweets, citation rates, ranked lists of best schools, and Grade Point Averages. We are vulnerable to value capture because of the competitive advantage that such crisp and clear expressions of value have in (...) our private reasoning and our public justification. There is, however, a price. In value capture, we take a central component of our autonomy — our ongoing deliberation over the exact articulation of our values — and we outsource it. And the metrics to which we outsource usually engineered for the interests of some external force, like a large-scale institution’s interest in cross-contextual comprehensibility and quick aggregability. That outsourcing cuts off one of the key benefits to personal deliberation. In value capture, we no longer adjust our values and their articulations in light of own rich experience of the world. Our values should often be carefully tailored to our particular selves or our small-scale communities, but in value capture, we buy our values off the rack. In some cases – like decreasing CO2 emissions – the costs of non-tailored values are outweighed by the benefit of precise collective coordination. In other cases, like in our aesthetic lives, they are not. This suggests that we should want different values suited to different scales. We should want value federalism. Some values are perhaps best pursued at the largest-scale level, others at smaller scales. The problem occurs when we exhibit an excess preference for the largest-scale values – when we consistently let the universal metrics swamp our quieter interests. (shrink)

Nominalism is the view that abstract mathematical objects like numbers, functions, and sets do not exist. The chapter articulates and defends a variety of nominalism, based on a reading of mathematical statements in terms of possible linguistic constructions. The chapter responds directly to a recent study of nominalism by Gideon Rosen and John Burgess, and develops a reply to the Quine-Putnam indispensability argument for the existence of mathematical objects.

In the final chapter of their book A Subject With No Object, John Burgess and Gideon Rosen raise the question of the value of the nominalistic reconstructions of mathematics that have been put forward in recent years, asking specifically what this body of work is good for. The authors conclude that these reconstructions are all inferior to current versions of mathematics (or science) and make no advances in science. This paper investigates the reasoning that led to such a negative appraisal, (...) and it produces a rebuttal to this reasoning. I am grateful to the following mathematicians who were kind enough to provide me with their thoughts about nonstandard analysis: Martin Davis, Laura Chihara, Ted Chihara, Steve Galovich, Bonnie Gold, and especially Roger Simons, whose comments about an earlier version of this paper were very helpful. Thanks also go to two referees for their useful suggestions and criticisms of an earlier version of this paper. (shrink)

The present paper will argue that, for too long, many nominalists have concentrated their researches on the question of whether one could make sense of applications of mathematics (especially in science) without presupposing the existence of mathematical objects. This was, no doubt, due to the enormous influence of Quine's "Indispensability Argument", which challenged the nominalist to come up with an explanation of how science could be done without referring to, or quantifying over, mathematical objects. I shall admonish nominalists to enlarge (...) the target of their investigations to include the many uses mathematicians make of concepts such as structures and models to advance pure mathematics. I shall illustrate my reasons for admonishing nominalists to strike out in these new directions by using Hartry Field's nominalistic view of mathematics as a model of a philosophy of mathematics that was developed in just the sort of way I argue one should guard against. I shall support my reasons by providing grounds for rejecting both Field's fictionalism and also his deflationist account of mathematical knowledge—doctrines that were formed largely in response to the Indispensability Argument. I shall then give a refutation of Mark Balaguer's argument for his thesis that fictionalism is "the best version of anti-realistic anti-platonism". (shrink)

Recent conversation has blurred two very different social epistemic phenomena: echo chambers and epistemic bubbles. Members of epistemic bubbles merely lack exposure to relevant information and arguments. Members of echo chambers, on the other hand, have been brought to systematically distrust all outside sources. In epistemic bubbles, other voices are not heard; in echo chambers, other voices are actively undermined. It is crucial to keep these phenomena distinct. First, echo chambers can explain the post-truth phenomena in a way that epistemic (...) bubbles cannot. Second, each type of structure requires a distinct intervention. Mere exposure to evidence can shatter an epistemic bubble, but may actually reinforce an echo chamber. Finally, echo chambers are much harder to escape. Once in their grip, an agent may act with epistemic virtue, but social context will pervert those actions. Escape from an echo chamber may require a radical rebooting of one's belief system. (shrink)