Strict Finitism, Feasibility, and the Sorites

Dean, Walter

doi:10.1017/s1755020318000163

Cited by 15 publications

(14 citation statements)

References 75 publications

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“…Proof. By induction on + 1 , we show that F is a function N 1 → N 2 and that it satisfies the conditions (5). We then use induction on + 2 to prove that F is onto.…”

Section: -270])mentioning

confidence: 99%

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Philosophical Uses of Categoricity Arguments

Maddy¹,

Väänánen²

2022

Preprint

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Mathematicians and philosophers have appealed to categoricity arguments in a surprisingly varied range of contexts. One familiar example calls on second-order categoricity in an attempt to show that the Continuum Hypothesis, despite its formal independence, has a determinate truth value, but this doesn't exhaust the uses of categoricity even in set theory, not to mention its appearance in various roles in discussions of arithmetic. Here we compare and contrast a sampling of these deployments to get a sense of when these arguments tend to succeed and when they tend to fail. Our story begins with two historical landmarks, Dedekind and Zermelo, on arithmetic and set theory, respectively, and ends with two leading contemporary writers, Charles Parsons and the co-authors Tim Button and Sean Walsh, again on arithmetic and set theory, respectively. In between, we pause over the well-known contribution of Georg Kreisel. In each case, 1 we ask: what does the author set out to accomplish, philosophically?; what do they actually do (or what can be done), mathematically?; and does what's done (or can be done) accomplish what they set out to do? We find this focus on context illuminating: these authors have qualitatively different philosophical goals, and what works for one might not work for another.

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“…Proof. By induction on + 1 , we show that F is a function N 1 → N 2 and that it satisfies the conditions (5). We then use induction on + 2 to prove that F is onto.…”

Section: -270])mentioning

confidence: 99%

“…He then introduces the key idea of a 'φ-chain' ( §37): 5 Definition 1. If φ is a function from S to S, and K ⊆ S, then K is a φ-chain iff ∀x ∈ K(φ(x) ∈ K).…”

mentioning

confidence: 99%

Philosophical Uses of Categoricity Arguments

Maddy¹,

Väänánen²

2022

Preprint

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“…the characterization of logical omniscience at its relation to a priori knowledge and rationality -see Cherniak 1986 andDean 2016a, §4.7), philosophy of language (e.g. the semantics of vague predicates -see Dean 2018), philosophy of mind and cognitive science (e.g. the performance/competence distinction -see Pylyshyn 1984 andPantsar 2019), social choice theory (e.g.…”

Section: Basic Notions In Complexity Theorymentioning

confidence: 99%

“…507-508), (Lavine 1994, p. 248), (Troelstra 2011, p. 153), (Feferman et al 2000, p. 410), (Gaifman 2004, p. 16). See also (Dean 2018) for more on the historical and technical connections between attempts to analyze feasibility, the sorites paradox, and natural language vagueness.…”

Section: Intractability and Np-completenessmentioning

confidence: 99%

Computational Complexity Theory and the Philosophy of Mathematics†

Dean

2019

Philosophia Mathematica

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Computational complexity theory is a subfield of computer science originating in computability theory and the study of algorithms for solving practical mathematical problems. Amongst its aims is classifying problems by their degree of difficulty — i.e., how hard they are to solve computationally. This paper highlights the significance of complexity theory relative to questions traditionally asked by philosophers of mathematics while also attempting to isolate some new ones — e.g., about the notion of feasibility in mathematics, the $\mathbf{P} \neq \mathbf{NP}$ problem and why it has proven hard to resolve, and the role of non-classical modes of computation and proof.

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“…(Defending the Axioms, pp. [27][28][29]. One clear moral for mathematics in application is that we are not in fact uncovering the underlying mathematical structures realized in the world.…”

Section: 3mentioning

confidence: 99%

On mathematical realism and applicability of hyperreals

Bottazzi¹,

Kanovei²,

Katz³

et al. 2019

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We argue that Robinson's hyperreals have just as much claim to applicability as the garden variety reals.In a recent text, Easwaran and Towsner (ET) analyze the applicability of mathematical techniques in the sciences, and introduce a distinction between techniques that are applicable and those that are merely instrumental. Unfortunately the authors have not shown that their distinction is a clear and fruitful one, as the examples they provide are superficial and unconvincing. Moreover, their analysis is vitiated by a reliance on a naive version of object realism which has long been abandoned by most philosophical realists in favor of truth-value realism. ET's argument against the applicability of hyperreals based on automorphisms of hyperreal models involves massaging the evidence and is similarly unconvincing. The purpose of the ET text is to argue that Robinson's infinitesimal analysis is merely instrumental rather than applicable. Yet in spite of Robinson's techniques being applied in physics, probability, and economics (see e.g., [70, Chapter IX], [1], [76], [60]), ET don't bother to provide a meaningful analysis of even a single case in which these techniques are used. Instead, ET produce page after page of speculations mainly imitating Connesian chimera-type arguments 'from first principles' against Robinson. In an earlier paper Easwaran endorsed real applicability of the σ-additivity of measures, whereas the ET text rejects real applicability of the axiom of choice, voicing a preference for ZF. Since it is consistent with ZF that the Lebesgue measure is not σ-additive, Easwaran is thereby walking back his earlier endorsement. We note a related inaccuracy in the textbook Measure Theory by Paul Halmos. ET's arguments are unacceptable to mathematicians because they ignore a large body of applications of infinitesimals in science, and massage the evidence of some crucial mathematical details to conform with their philosophical conclusions.

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Strict Finitism, Feasibility, and the Sorites

Cited by 15 publications

References 75 publications

Philosophical Uses of Categoricity Arguments

Philosophical Uses of Categoricity Arguments

Computational Complexity Theory and the Philosophy of Mathematics†

On mathematical realism and applicability of hyperreals

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