This book is based on two premises: one cannot understand philosophy of mathematics without understanding mathematics and one cannot understand mathematics without doing mathematics. It draws readers into philosophy of mathematics by having them do mathematics. It offers 298 exercises, covering philosophically important material, presented in a philosophically informed way. The exercises give readers opportunities to recreate some mathematics that will illuminate important readings in philosophy of mathematics. Topics include primitive recursive arithmetic, Peano arithmetic, Gödel's theorems, interpretability, the hierarchy of (...) sets, Frege arithmetic, and intuitionist sentential logic. The book is intended for readers who understand basic properties of the natural and real numbers and have some background in formal logic. (shrink)

CarloCellucci. Rethinking Knowledge: The Heuristic View. Cham, Switzerland: Springer International Publishing, 2017. ISBN 978-3-319-53236-3, 978-3-319-53237-0. Pp. xx + 428††.

In a finitary closure space, irreducible sets behave like two-valued models, with membership playing the role of satisfaction. If f is a function on such a space and the membership of in an irreducible set is determined by the presence or absence of the inputs in that set, then f is a kind of truth function. The existence of some of these truth functions is enough to guarantee that every irreducible set is maximally consistent. The closure space is then said (...) to be expressive. This paper identifies the two-valued truth functional conditions that guarantee expressiveness. (shrink)

This essay shows that some recent work by George Weaver can be reformulated in an especially perspicuous way within the theory of closure systems. Closure theoretic generalizations of some theorems of Robert Goldblatt are presented. And, more generally, the relation between closure systems and the deducibility relations of Goldblatt is explored.

Say that a property is topological if and only if it is invariant under homeomorphism. Homeomorphism would be a successful criterion for the equivalence of logical systems only if every logically significant property of every logical system were topological. Alas, homeomorphisms are sometimes insensitive to distinctions that logicians value: properties such as functional completeness are not topological. So logics are not just devices for exploring closure topologies. One still wonders, though, how much of logic is topological. This essay examines some (...) logically significant properties that are topological (or are topological in some important class). In the process, we learn something about the conditions under which the meaning of a connective can be "given by the connective's role in inference.". (shrink)

The expressive truth functions of two-valued logic have all been identified. This paper begins the task of identifying the expressive truth functions of n-valued logic by characterizing the unary ones. These functions have distinctive algebraic, semantic, and closure-theoretic properties.

This essay defends the following two claims: (1) liraitation-of-size reasoning yields enough sets to meet the needs of most mathematicians; (2) set formation and mereological fusion share enough logical features to justify placing both in the genus composition (even when the components of a set are taken to be its members rather than its subsets).

Competent speakers of natural languages can borrow reference from one another. You can arrange for your utterances of ‘Kirksville’ to refer to the same thing as my utterances of ‘Kirksville’. We can then talk about the same thing when we discuss Kirksville. In cases like this, you borrow “ aboutness ” from me by borrowing reference. Now suppose I wish to initiate a line of reasoning applicable to any prime number. I might signal my intention by saying, “Let p be (...) any prime.” In this context, I will be using the term ‘p’ to reason about the primes. Although ‘p’ helps me secure the aboutness of my discourse, it may seem wrong to say that ‘p’ refers to anything. Be that as it may, this paper explores what mathematical discourse would be like if mathematicians were able to borrow freely from one another not just the reference of terms that clearly refer, but, more generally, the sort of aboutness present in a line of reasoning leading up to a universal generalization. The paper also gives reasons for believing that aboutness of this sort really is freely transferable. A key implication will be that the concept “set of natural numbers” suffers from no mathematically significant indeterminacy that can be coherently discussed. (shrink)

Arecent paper by George Boolos suggests that it is philosophically respectable to use monadic second order logic in one’s explication of the iterative concept of set. I shall here give a partial indication of the new range of theories of the iterative hierarchy which are thus madeavailable to philosophers of set theory.

Arecent paper by George Boolos suggests that it is philosophically respectable to use monadic second order logic in one’s explication of the iterative concept of set. I shall here give a partial indication of the new range of theories of the iterative hierarchy which are thus madeavailable to philosophers of set theory.

Is the Axiom of Foundation a truth or a falsehood about the universe of sets? Such questions have always struck this reviewer as unmathematical and a bit silly. Of course, that appraisal is more like a grunt of disapproval than an argument. We would welcome arguments. Justin Clarke-Doane offers a pretty good one that goes roughly as follows.Mathematicians making reasoned judgments about set-theoretic principles sometimes take positions that appear incompatible: one mathematician making a claim that would contradict the claim of (...) another if both claims were treated as assertions about a single universe of sets. If the claims are sincerely made and contradict each another, one party to the dispute must be expressing a false belief. It is characteristic of reliable methods that they significantly impede the formation of false beliefs. Reliable methods for fixing set-theoretic beliefs significantly impede the formation of false beliefs about sets. So, if we knew which party in a set-theoretic dispute has got it wrong, we would have evidence that that party’s methods are unreliable. But it is frequently unclear why the reasons offered by one side should prevail over those offered by the other. Unable to pinpoint the error, we could find ourselves less confident about the methods of each party. We might even wonder if there is much hope of establishing the reliability of set-theoretic methods in general. (shrink)

Hans Vaihinger tried to explain how mathematical theories can be useful without being true or even coherent, arguing that mathematicians employ a special kind of fictional or "as if" reasoning that reliably extracts truths from absurdities. Moritz Pasch insisted that Vaihinger was wrong about the incoherence of core mathematical theories, but right about the utility of fictional discourse in mathematics. This essay explores this area of agreement between Pasch and Vaihinger. Pasch's position raises questions about structuralist interpretations of mathematics.

Hans Vaihinger tried to explain how mathematical theories can be useful without being true or even coherent, arguing that mathematicians employ a special kind of fictional or “as if” reasoning that reliably extracts truths from absurdities. Moritz Pasch insisted that Vaihinger was wrong about the incoherence of core mathematical theories, but right about the utility of fictional discourse in mathematics. This essay explores this area of agreement between Pasch and Vaihinger. Pasch’s position raises questions about structuralist interpretations of mathematics.

That mathematics makes for poor literature is a conclusion as uninteresting as it is inevitable—inevitable because were mathematical prose to score high on a scale of literary value, this result would do more to discredit the scale than glorify the prose. It may, however, help us better understand our cultural landscape if, without attempting a literary appraisal of mathematics or a mathematical appraisal of literature, we search for some community of interest between the formal sciences and the literary arts. This (...) search for common ground will also be an opportunity to learn from some voices seldom heard in conversations about mathematics. Even at the risk of giving our discussion an antiquated flavor, we will... (shrink)

The expressive truth functions of two-valued logic have all been characterized, as have the expressive unary truth functions of finitely-many-valued logic. This paper introduces some techniques for identifying expressive functions in three-valued logics.

Philosophers call it “contagion” when pretense influences belief, behavior, perception, or emotion. This pejorative terminology is justified in some cases: fantasy and imagination can exercise a pathological influence. This essay, however, reviews some logical techniques that allow pretense to govern belief in a rational and beneficial way. Philosophers might want similar techniques in their tool-kits when they explore interactions between belief and pretense.

We mathematical animals should be grateful that mathematics is instrumentally useful. We should not, however, forget its other contributions to human happiness. Bertrand Russell and John Dewey offer timely reminders that provide insight into the role of non-mathematicians in the evaluation of mathematics.

This note examines the mereological component of Geoffrey Hellman's most recent version of modal structuralism. There are plausible forms of agnosticism that benefit only a little from Hellman's mereological turn.

This book is about universals. "Mathematics," we learn, "is the theory of universals". Natural numbers turn out to be universals, as do real numbers, complex numbers, and sets. It is natural, therefore, that the reader demand some evidence that universals abound sufficiently to supply models of canonical mathematical theories. The author devotes nearly a third of his book to one potential source of such evidence: the Truthmaker axiom. In the case of some propositions P, Truthmaker entails that if P is (...) true, then there are things whose joint existence strictly implies P's truth. These things are truthmakers for P. Might Truthmaker, then, transform evidence for the truth of propositions into evidence for the existence of truthmaking universals? And might we thus be guaranteed a rich supply of numbers and sets? The author thinks not: "universals must not be construed as truthmakers". Something is a universal only if it can be instantiated by individuals x and y such that x's being an instance does not strictly imply y's being an instance. Since this characteristic of universals is incompatible with natural assumptions about truthmakers, the author's disheartening conclusion follows: "universals should not be expected to play any distinctive role in truthmaking". So Truthmaker is not a wellspring of universals. Why should we, then, believe that universals abound? One must understand, first, that the author embraces David Armstrong's a posteriori realism: "Everything there is is physical.... Hence universals, too, are physical. That is to say, the universals which exist are all real physical properties and relations among physical things". Universals are reasonably posited, then, only if their existence is entailed by our best accounts of the physical universe: "In cutting universals away from truthmakers, I am making their existence a matter, not for a priori proof from logic alone, but for total science". Sentences not interpretable in optimal physical theory will, on this view, not be modeled by universals and, hence, will be either bad mathematics or no part of mathematics at all. (shrink)

Despite the recent increase of interest in the work of Merleau-Ponty there is still a persistent tendency to overlook the uniqueness of the philosophical position he advances in Phenomenology of Perception. In this article I present a reading of Merleau-Ponty’s account of the phenomenological reduction that explains how it is original. I do this by contrasting his presentation of the reduction with that of the early Husserl, highlighting how his emphasis on the phenomenology of the ‘perceived world’ leads him to (...) reject Husserl’s conception of phenomenology as a ‘philosophical science,’ and the Kantian language in which the this account is framed. I go on to critically discuss the interpretations of the reduction advanced by Stephen Priest and Joel Smith as examples of readings that fail to fully grasp Merleau-Ponty’s account of the ‘natural attitude’ as resting on the inherent objectivizing structure that is built into perception itself. The way that these authors misinterpret Merleau-Ponty helps to make maximally clear the profound philosophical significance that he places on the phenomenology of the ‘perceived world’. (shrink)