Jody Azzouni

A new approach to informal rigorous mathematical proof is offered. To this end, algorithmic devices are characterized and their central role in mathematical proof delineated. It is then shown how all the puzzling aspects of mathematical proof, including its peculiar capacity to convince its practitioners, are explained by algorithmic devices. Diagrammatic reasoning is also characterized in terms of algorithmic devices, and the algorithmic device view of mathematical proof is compared to alternative construals of informal proof to show its superiority.

Taking (a proposition) to be true is an epistemic theme appearing throughout George E. Smith’s work; this includes his marvelous new book with Raghav Seth, Brownian motion and molecular reality: A study in theory-mediated measurement (2020; hereafter Smith and Seth). They use this notion to categorize changes in scientific perspectives both towards the ontological claim that molecules exist and towards molecular-kinetic theory. They illustrate the shift in viewpoint occurring over the successive editions of Ostwald’s and Nernst’s respective textbooks. Nernst writes, in his 6th edition, that, “the theory begins to lose its hypothetical character”; later he writes, “… we may well acknowledge that the theory has lost its hypothetical character” (Smith and Seth, 3). Ostwald does an “about-face on atomism” (2): In earlier editions he characterizes the “atomic conception” as a “convenient mode of representation,” as having “enormous heuristic value” (2). In the 4th edition (1909) he writes (2)

Justin Clarke-Doane offers what purports to be a stand-alone argument, relying on Gödel’s second incompleteness theorem, that if we hold that PA + Con(PA) and PA + ~ Con(PA) are equally true of their intended subjects, then there is no objective fact as to whether PA is consistent. It is shown that the argument is fallacious, although illuminating: The fallaciousness of the argument arises from a 20th-century shift in our understanding of interpreted languages from the view—derived from our experience of language—of sentences as intrinsically interpreted to one which sharply distinguishes syntax and semantics, and treats uninterpreted syntactic forms as endowed with interpretations by models. It is shown that this syntax/semantic distinction, because it is “unintuitive” induces fallacies such as the one that Clarke-Doane’s argument exemplifies.

An extensive exploration of the special properties of ‘exist’ is here undertaken. Two of several results are: Denying that `exist’ has associated with it a set of necessary and sufficient conditions has seemed to a number of philosophers to imply metaphysical nihilism . This is because it has seemed that without such conditions the target domain of `existence’ is arbitrarily open. I show this is wrong. Second, my analysis sheds light on the puzzling question of what we are asking when we ask of something, `Does it exist?’ and mean that question in an ontically relevant way

Quine, in his 1980 forward to From a Logical Point of View wrote: ‘The time for revision is past. The book is dated, and its dates are 1953 and 1961’ (p. viii). Quine wrote modestly about himself, as he almost always did. His point—not stated loudly—was that the book was now an important historical document, and so its author had lost the right to tamper with it. The book I’m reviewing may seem different. Although Kripke’s Locke Lectures were given in 1973, the book has appeared in print forty years later. Kripke writes: ‘… the transcript has been available at the Oxford University library for many years … and it has had, since then, a modest life of its own, passed on among members of the profession, sometimes even being discussed and criticized in print’ (pp. xi-xii). Kripke understates the transcript’s impact. It influenced everyone in the profession who wrote on these topics, if only because it was read carefully and absorbed by so many philosophers working in this area. Thomasson, on a back-blurb, writes: ‘For decades getting a copy of these lectures has been a holy grail for philosophers working on fiction’. And in the Oxford advertising, included with my review copy of the book, I read: ‘At last in print—the most famous unpublished philosophy manuscript of recent decades’.

Tim Crane characterizes the cognitive role of singular thought via singular mental files: the application of such files to more than one object is senseless. As many do, he thus stresses the contrast between ‘singular’ and ‘general’. I give a counterexample, plurally-directed singular thought, and I offer alternative characterizations of singular thought—better described as ‘objects-directed thought’—initially in terms of the defeasibility of the descriptions associated with one's thinking of an object, and then more broadly in terms of whether descriptions of the object or description-independent epistemic routes to the object are primarily operative in an agent's thinking.

Ask a philosopher what a proof is, and you’re likely to get an answer hii empaszng one or another regimentationl of that notion in terms of a finite sequence of formalized statements, each of which is either an axiom or is derived from an axiom by certain inference rules. (Wecan call this the formal conception of proof) Ask a mathematician what a proof is, and you will rbbl poay get a different-looking answer. Instead of stressing a partic- l uar regimented notion of proof, the answer the mathematician will give ilikl..

A feature of agnostic views—views that officially express ignorance about the existence of something —is that they are widely perceived to be epistemically more cautious than views that are committed to the entities in question. This is often seen as giving agnostics a debating advantage: all things being equal, fence-sitters have smaller argumentative burdens. Otávio Bueno argues in this way for what he calls “agnostic nominalism,” the view that we don’t know whether ontologically-independent Platonic objects exist. I show that agnostic nominalism, so called, can be sustained only in ways that don’t give agnostic nominalists debating advantages: the position must either be deduced from antecedently-held broader sceptical assumptions or it requires manufacturing potential referents for one’s terms that artificially generate grounds for scepticism. Neither maneuver leads to an agnostic nominalism with a debating advantage over its nominalist opponents

A new approach to truth is offered which dispenses with the truth predicate, and replaces it with a special kind of quantifier which simultaneously binds variables in sentential and nominal positions. The resulting theory of truth for a (first-order) language is shown to be able to handle blind truth ascriptions, and is shown to be compatible with a characterization of the semantic and syntactic principles governing that language. Comparisons with other approaches to truth are drawn. An axiomatization of AU-quantifiers and a model theory for them is given, and an appendix contains a completeness proof

Mistaken reasons for thinking diagrammatic proofs aren't rigorous are explored. The main result is that a confusion between the contents of a proof procedure (what's expressed by the referential elements in a proof procedure) and the unarticulated mathematical aspects of a proof procedure (how that proof procedure is enabled) gives the impression that diagrammatic proofs are less rigorous than language proofs. An additional (and independent) factor is treating the impossibility of naturally generalizing a diagrammatic proof procedure as an indication of lack of rigor

Kant discovered a philosophical problem with mathematical proof. Despite being a priori , its methodology involves more than analytic truth. But what else is involved? This problem is widely taken to have been solved by Frege’s extension of logic beyond its restricted (and largely Aristotelian) form. Nevertheless, a successor problem remains: both traditional and contemporary (classical) mathematical proofs, although conforming to the norms of contemporary (classical) logic, never were, and still aren’t, executed by mathematicians in a way that transparently reveals why these proofs—written in the vernacular to this very day—succeed in conforming to those norms.

This monograph presents Azzouni’s new approach to the rule-following paradox. His solution leaves intact an isolated individual’s capacity to follow rules, and it simultaneously avoids replacing the truth conditions for meaning-talk with mere assertability conditions for that talk. Kripke’s influential version of Wittgenstein’s rule-following paradox—and Wittgenstein’s views more generally—on the contrary, make rule-following practices and assertions about those practices subject to community norms without which they lose their cogency. Azzouni summarizes and develops Kripke’s original version of Wittgenstein’s rule-following paradox to make salient the linchpin assumptions of the paradox. By doing so, Azzouni reveals how compelling Kripke’s earlier work on the paradox was. Objections raised over the years by Fodor, Forbes Ginsborg, Goldfarb, Tait, Wright, and many others, are all shown to fail. No straight solution can be made to work. Azzouni illustrates this in detail by showing that a popular family of straight solutions due to Lewis and refined by Williams, “reference magnetism,” fail as well. And yet an overlooked sceptical solution is still available in logical space. Azzouni describes a series of “disposition-meaning” private languages that he shows can be successfully used by a population of speakers to communicate with one another despite their ideolectical character. The same sorts of languages enable solitary “Robinson Crusoes” to survive and flourish in their island habitats. These languages—sufficiently refined—have the same properties normal human languages have; and this is the key to solving the rule-following paradox without sacrificing the individual’s authority over her self-imposed rules or her ability to follow those rules. Azzouni concludes this unusual monograph by uncovering a striking resemblance between the rule-following paradox and Hume’s problem of induction: he shows the rule-following paradox to be a corollary of Hume’s problem that arises when the problem of induction is applied to an individual’s own abilities to follow rules.

Kripke’s version of Wittgenstein’s rule-following paradox has been influential. My concern is with how it—and Wittgenstein’s views more generally—have been perceived as undercutting the individualistic picture of mathematical practice: the view that individuals— Robinson Crusoes —can, entirely independently of a community, engage in cogent mathematics, and indeed have “private languages.” What has been denied is that phrases like “correctly counting” can be applied to such individuals because these normative notions can only be applied cogently in a context involving community standards. I attempt to show that this shocking corollary doesn’t follow even if Kripke’s Wittgensteinian objections to dispositional approaches to rule-following are largely right. My reason for claiming this is that there is another solution to the rule-following paradox, one that doesn’t favor community standards over individual ones. Furthermore, it doesn’t replace truth conditions with assertability conditions; and this latter maneuver is essential to Kripke’s sceptical solution favoring the community over the individual.

When ordinary people - mathematicians among them - take something to follow from something else, they are exposing the backbone of our self-ascribed ability to reason. Jody Azzouni investigates the connection between that ordinary notion of consequence and the formal analogues invented by logicians. One claim of the book is that, despite our apparent intuitive grasp of consequence, we do not introspect rules by which we reason, nor do we grasp the scope and range of the domain, as it were, of our reasoning. This point is illustrated with a close analysis of a paradigmatic case of ordinary reasoning: mathematical proof.

When ordinary people--mathematicians among them--take something to follow from something else, they are exposing the backbone of our self-ascribed ability to reason. Jody Azzouni investigates the connection between that ordinary notion of consequence and the formal analogues invented by logicians. One claim of the book is that, despite our apparent intuitive grasp of consequence, we do not introspect rules by which we reason, nor do we grasp the scope and range of the domain, as it were, of our reasoning. This point is illustrated with a close analysis of a paradigmatic case of ordinary reasoning: mathematical proof.

One major motivation for nominalism, at least according to Hartry Field, is the desirability of intrinsic explanations: explanations that don’t invoke objects that are causally irrelevant to the phenomena being explained. There is something right about the search for such explanations. But that search must be carefully implemented. Nothing is gained if, to avoid a certain class of objects, one only introduces other objects and relations that are just as nominalistically questionable. We will argue that this is the case for two alleged nominalist views: Field’s fictionalism, and Frank Arntzenius and Cian Dorr’s geometricalism. Central to our competing approach to nominalism is a distinction between terms that refer to objects and ones that instead code empirical phenomena while being referentially empty. We next contrast our approach to nominalism, which uses this term-grained distinction between coding and referring, with approaches that instead attempt to make a sentence-grained distinction between mathematical and non-mathematical content. We show the latter approach fails to be responsive to objections raised by van Fraassen. In the end, only one last approach to nominalism is left standing. 1 Introduction2 Troubles for Mathematical Fictionalism2.1 A distinction2.2 Field’s programme2.3 Parts of nominalistically acceptable entities are nominalistically acceptable2.4 Structural assumptions: How far can these be pushed?2.5 Cases of indispensable theories where the mathematical posits are known not to be real2.6 How does science distinguish between the indispensable structural presuppositions taken to be real and those that aren’t?2.7 Do physicists distinguish between mathematical posits and non-mathematical posits?2.8 Mathematical coding versus genuine metaphysics. Case study: Fields2.9 The upshot3 Troubles for Topologism4 Contrasts: Kitcher, Maddy, Sober, and van Fraassen4.1 Sober’s approach4.2 Maddy’s approach4.3 Kitcher’s approach4.4 Where do we go from here?4.5 Van Fraassen’s objection to Maddy’s approach to ontological commitment5 Conclusion.

A normative constraint on theories about objects which we take to be real is explored: such theories are required to track the properties of the objects which they are theories of. Epistemic views in which observation (and generalizations of it) play a central role, and holist views which see epistemic virtues as applicable only to whole theories, are contrasted in the light of this constraint. It's argued that global-style epistemic virtues can't meet the constraint, although (certain) epistemic views within which observation (and, more generally, thick epistemic access) are centralized do meet it.

  It is argued that the blind ascriptive role for the word true, its use, that is, in conjunction with descriptions of classes of sentences or with proper names of sentences (but not quote-names), is one which applies indiscriminately to sentences regardless of whether these are in languages we speak, can understand, or can translate into sentences that we do speak (and understand). Formal analogues of the ordinary word true as they arise in Tarskis seminal work, and in others, cannot replicate this essential role of the ordinary word true

It is argued that the blind ascriptive role for the word "true", its use, that is, in conjunction with descriptions of classes of sentences or with proper names of sentences, is one which applies indiscriminately to sentences regardless of whether these are in languages we speak, can understand, or can translate into sentences that we do speak. Formal analogues of the ordinary word "true" as they arise in Tarski's seminal work, and in others, cannot replicate this essential role of the ordinary word "true".