Alexander Paseau

Naturalism in Mathematics and the Authority of Philosophy

Naturalism in the philosophy of mathematics is the view that philosophy cannot legitimately gainsay mathematics. I distinguish between reinterpretation and reconstruction naturalism: the former states that philosophy cannot legitimately sanction a reinterpretation of mathematics (i.e. an interpretation different from the standard one); the latter that philosophy cannot legitimately change standard mathematics (as opposed to its interpretation). I begin by showing that neither form of naturalism is self-refuting. I then focus on reinterpretation naturalism, which comes in two forms, and examine the only available argument for it. I argue that this argument, the so-called Failure Argument, itself fails. My overall conclusion is that although there is no self-refutation argument against reinterpretation naturalism, there are as yet no good reasons to accept it. 1. Naturalism in mathematics2. The consistency of mathematical naturalism3. The failure argument4. Objections to the failure argument5. Philosophy as the default Naturalism in mathematics The consistency of mathematical naturalism The failure argument Objections to the failure argument Philosophy as the default

Knowledge of Mathematics without Proof

Mathematicians do not claim to know a proposition unless they think they possess a proof of it. For all their confidence in the truth of a proposition with weighty non-deductive support (for example, the Riemann hypothesis), they maintain that, strictly speaking, the proposition remains unknown until such time as someone has proved it. This article challenges this conception of knowledge, which is quasi-universal within mathematics. We present four arguments to the effect that non-deductive evidence can yield knowledge of a mathematical proposition. We also show that some of what mathematicians take to be deductive knowledge is in fact non-deductive. 1   Introduction 2   Why It Might Matter 3   Two Further Examples and Preliminaries 4   An Exclusive Epistemic Virtue of Proof? 5   Analyses of Knowledge 6   The Inductive Basis of (Some) Deduction 7   Physical to Mathematical Linkages 8   Conclusion 1   Introduction 2   Why It Might Matter 3   Two Further Examples and Preliminaries 4   An Exclusive Epistemic Virtue of Proof? 5   Analyses of Knowledge 6   The Inductive Basis of (Some) Deduction 7   Physical to Mathematical Linkages 8   Conclusion

Proofs of the Compactness Theorem

In this study, several proofs of the compactness theorem for propositional logic with countably many atomic sentences are compared. Thereby some steps are taken towards a systematic philosophical study of the compactness theorem. In addition, some related data and morals for the theory of mathematical explanation are presented.

Reducing Arithmetic to Set Theory

The revival of the philosophy of mathematics in the 60s following its post-1931 slump left us with two conflicting positions on arithmetic’s ontological relationship to set theory. W.V. Quine’s view, presented in Word and Object (1960), was that numbers are sets. The opposing view was advanced in another milestone of twentieth-century philosophy of mathematics, Paul Benacerraf’s “What Numbers Could Not Be” (1965): one of the things numbers could not be, it explained, was sets; the other thing numbers could not be, even more dramatically, was objects. Curiously, although Benacerraf’s article appeared in the heyday of Quine’s influence, it declined to engage the Quinean position squarely, even seemed to think it was not its business to do so. Despite that, in my experience, most philosophers believe that Benacerraf’s article put paid to the reductionist view that numbers are sets (though perhaps not the view that numbers are objects). My chapter will attempt to overturn this orthodoxy.

Fairness and aggregation

Sometimes, two unfair distributions cancel out in aggregate. Paradoxically, two distributions each of which is fair in isolation may give rise to aggregate unfairness. When assessing the fairness of distributions, it therefore matters whether we assess transactions piecemeal or focus only on the overall result. This piece illustrates these difficulties for two leading theories of fairness (proportionality and shortfall minimization) before offering a formal proof that no non-trivial theory guarantees aggregativity. This is not intended as a criticism of any particular theory, but as a datum that must be taken into account in constructing a theory of fairness.

The ‘stop after k girls or N children’ policy

1. Introduction In the article [1], a follow-up to my article [2], Christian and Trustrum cite empirical evidence that the probability of a family giving birth to a boy or to a girl may vary from family to family. Letting b i and g i = 1 − b i respectively denote the probability of the i th family giving birth to a boy or to a girl, the suggestion is that b i may be distinct from b j for distinct i and j . As they point out, this has implications for the expected society-wide number of boys and girls. Following my discussion of the ‘stop after k girls’ policy, Christian and Trustrum introduce the related policy ‘stop after k girls or N children’. They argue that the expected number of children under this latter policy is an increasing function of b i (when 0 k N ). The present article complements their discussion by examining this alternative stopping policy in more detail.

Motivating Reductionism about Sets

The paper raises some difficulties for the typical motivations behind set reductionism, the view that sets are reducible to entities identified independently of set theory.

The Open-Endedness of the Set Concept and the Semantics of Set Theory

Some philosophers have argued that the open-endedness of the set concept has revisionary consequences for the semantics and logic of set theory. I consider (several variants of) an argument for this claim, premissed on the view that quantification in mathematics cannot outrun our conceptual abilities. The argument urges a non-standard semantics for set theory that allegedly sanctions a non-classical logic. I show that the views about quantification the argument relies on turn out to sanction a classical semantics and logic after all. More generally, this article constitutes a case study in whether the need to account for conceptual progress can ever motivate a revision of semantics or logic. I end by expressing skepticism about the prospects of a so-called non-proof-based justification for this kind of revisionism about set theory.

Erratum to: A measure of inferential-role preservation

In line 3 of footnote 8 on page 4, ‘allow’ should be ‘disallow’. In line 8 of page 5, F1 should be 1 and F2 should be 2. Similarly for lines 1, 2, 3, 7, 8, 13 and 14 of page 6. The entry in row 20 column 6 of the table on page 5 should be 1 rather than 0. The entry ∃∀xFx in row 30 column 5 of the table on page 5 should be ∀xFx . In line 27 of page 13, ‘it’ should be ‘them’. Four lines from the end of section 12.3 on page 20, ‘premisses’ should be ‘premiss sets’, and {∃≥n is a natural number} should be {∃≥n : n is a natural number}. In line 4 of footnote 30 on page 20, Q0 = P1 should be Q0 = {P1}.

A measure of inferential-role preservation

The point of formalisation is to model various aspects of natural language. Perhaps the main use to which formalisation is put is to model and explain inferential relations between different sentences. Judged solely by this objective, a formalisation is successful in modelling the inferential network of natural language sentences to the extent that it mirrors this network. There is surprisingly little literature on the criteria of good formalisation, and even less on the question of what it is for a formalisation to mirror the inferential network of a natural language or some fragment of it. This paper takes some exploratory steps towards a quantitative account of the main ingredient in the goodness of a formalisation. We introduce and critically examine a mathematical model of how well a formalisation mirrors natural-language inferential relations.