Mathematical Finitism

As a philosophy of mathematics, strict finitism has been traditionally concerned with the notion of feasibility, defended mostly by appealing to the physicality of mathematical practice. This has led the strict finitists to influence and be influenced by the field of computational complexity theory, under the widely held belief that this branch of mathematics is concerned with the study of what is “feasible in practice”. In this paper, I survey these ideas and contend that, contrary to popular belief, complexity theory (...) is not what the ultrafinitists think it is, and that it does not provide a theoretical framework in which to formalize their ideas —at least not while defending the material grounds for feasibility. I conclude that the subject matter of complexity theory is not proving physical resource bounds in computation, but rather proving the absence of exploitable properties in a search space. (shrink)

We introduce two related concepts that together provide a physical interpretation of a familiar mathematical phenomenon. The first, the identity boundary, designates the smallest quantity of a physical substance that retains the defining properties of that substance: the minimum portion of water that is still water, the minimum unit of gold that is still gold. For any physical object-class, this boundary is always finite and strictly positive — there is always a lower limit below which division destroys rather than partitions. (...) The second, the shadow object, arises precisely when this boundary is reached. When a single indivisible unit must be shared among multiple claimants, and when fair temporal distribution operates faster than any available measurement resolution, each claimant perceives not the object itself but a shadow: the time-averaged presence of the object. -/- Crucially, shadow objects are not copies. Each shadow is a partial temporal view of the same underlying individual. As a direct consequence, any interaction with a shadow is an interaction with the real object, and all other shadows update accordingly — not through any transmitted signal, but because they were never descriptions of separate objects. We call this the Shadow Interaction Principle. -/- The framework provides a physical semantics for non-terminating decimal representations: the expression 0.333... describes the time-averaged presence of a single indivisible unit cycling among three claimants below measurement resolution. The mathematical equality 0.333... = 1/3 is not challenged; rather, we provide an account of what physical situation this representation faithfully describes. The shadow is not a defect in our mathematics. It is a precise description of how the world must look when something cannot be divided, but must still be shared. (shrink)

Mathematical pluralism is roughly the view that any coherent foundational theory is equally correct. This is rough because a careful formulation must avoid contradiction (how could a theory and its negation both be correct?), distinguish foundational theories from algebraic ones, differ from fictionalism, and accommodate the indefinite extensibility of coherence. Even after these repairs, however, pluralism faces an insuperable problem of articulation. I consider several attempts to circumvent this problem, and argue that each fails. Their failure is parallel to the (...) self-refutation problem for global relativism. I propose instead to analyze pluralism on the model of Carnap’s Principle of Tolerance (POT), interpreted as a non-factual policy proposal. Drawing on quasi-realism and the deflationism about truth that it presupposes, I show how such a policy can be truth-apt and embedded in complex sentences without inflating it into a factual thesis. The important point is that the considerations that support or refute it are practical, not evidential. Along the way, I isolate an underappreciated shadow of the problem of articulation, the problem of mathematical agreement. Even what follows from what in a fixed logic – indeed, even whether an explicitly given string is a proof in a logic – can depend on the foundation we use to check. So, why do we agree about mathematics to the extent that we do? I close by showing how pluralism leads to a limited non-factualist pluralism about physics. In particular, whether a discrete Ising system behaves deterministically from almost all starting states provably switches between ZF + V=L and ZFC + Large Cardinals (sufficient to imply Projective Determinacy). (shrink)

The Implicit Commitments Thesis (ICT) states that in accepting a formal theory S, one is implicitly committed to additional statements, such as S’s consistency. ICT aligns with the intuition that accepting S without acknowledging its consistency is epistemically problematic. However, there is ongoing debate over ICT’s validity in all cases. Dean (2015) argued that ICT should be rejected because it clashes with the idea of epistemic stability. Additionally, the epistemological basis for considering consistency statements as implicit commitments remains underexplored. This (...) paper addresses these issues by analyzing ICT’s epistemological and conceptual underpinnings. First, it examines putative counterexamples, such as finitism and first-orderism, and argues that they do not undermine ICT’s general applicability. Then, it introduces an intuitive epistemic norm that supports the implicit acceptance of consistency within ICT’s framework. (shrink)

Strict finitism posits a largest natural number. The view is usually thought to be objectionably arbitrary. After all, there seems to be no apparent reason as to why the natural numbers should ‘stop’ at a specific point and not a bit later on the natural line. Drawing on how arguments from arbitrariness are employed in mereology, I propose several ways of understanding this objection against strict finitism. No matter how it is understood, I argue that it is always found wanting.

Gödel’s first incompleteness theorem is widely interpreted as a refutation of Hilbert’s program for a complete, consistent foundation of mathematics. This interpretation presupposes a reversible, Platonic setting in which perfect bijective encoding (Gödel numbering) is possible. I argue that physical reasoning—performed by humans, computers, and any finite causal process—operates under different constraints: information is irreversibly encoded, computation is finite‑state, and the arrow of time blocks the syntactic self‑reference required for diagonalization. I introduce a framework of Physical Formal Systems (PFS) that (...) replaces Gödel numbering with one‑way cryptographic hashing, rendering Gödel sentences unconstructible and restoring a physically grounded form of completeness by exhaustive finite search. This reconciles Hilbert’s vision with Gödel’s theorems: incompleteness governs the abstract “map” of infinite, reversible mathematics, while completeness describes the “territory” of real computation. (shrink)

I am grateful to Sean Carroll, Paul Horwich, and Sarah McGrath for their stimulating responses to Morality and Mathematics (M&M). Their arguments concern the reality of unapplied mathematics, the practical import of moral facts, and the deliberative and explanatory roles of evaluative theories. In what follows, I address their responses, as well as some broader issues.

Goldbach's conjecture is simply proved in Hilbert arithmetic. However, that proof is either invalid ("incomplete") or false ("contradictory") in the standard mathematics obeying Gödel's objections about the relation of arithmetic to set theory. The proof uses the "apophatic" (holistic) reformulation of the Kochen-Specker theorem and the fundamental randomness of primes in Hilbert arithmetic: both confirmed to be true in previous papers. A few other conjectures, about twin primes, k-twin primes, k-tuple primes (a part of the Hardy-Littlewood conjecture) including about infinite (...) tuples are deduced from Goldbach's conjecture as corollaries in Hilbert arithmetic. The hypothesis that the asymptotic behaviour predicted by Hardy and Littlewood is false in Hilbert arithmetic is reasoned. A few philosophical observations about "crossing Rubicon to reality" after interpreting the Gleason and Kochen-Specker theorems to the meta-space of human experience to be Hilbert space and considering the opposition of two-versus three-dimensionality meant by them where the former is associated with Modernity and the latter, with Postmodernity: thus, both defined rigorously and mathematically unlike today's vague humanitarian (cultural and relativistic) research about them and that historical development led to it. Another and quite simple proof of the theorem about the asymptotic limit of the number of primes until any natural number is suggested. The link between Goldbach's conjecture and Riemann's hypothesis is demonstrated to be inherent and almost obvious in Hilbert arithmetic. The conclusion explains why the proofs of centuries-old puzzles about primes are so "scandalously" brief and simple in Hilbert arithmetic. (shrink)

A classical reconstruction of Wright’s first-order logic of strict finitism is presented. Strict finitism is a constructive standpoint of mathematics that is more restrictive than intuitionism. Wright sketched the semantics of said logic in Wright (Realism, Meaning and Truth, chap 4, 2nd edition in 1993. Blackwell Publishers, Oxford, Cambridge, pp.107–75, 1982), in his strict finitistic metatheory. Yamada (J Philos Log. https://doi.org/10.1007/s10992-022-09698-w, 2023) proposed, as its classical reconstruction, a propositional logic of strict finitism under an auxiliary condition that makes the logic (...) correspond with intuitionistic propositional logic. In this paper, we extend the propositional logic to a first-order logic that does not assume the condition. We will provide a sound and complete pair of a Kripke-style semantics and a natural deduction system, and show that if the condition is imposed, then the logic exhibits natural extensions of Yamada (2023)’s results. (shrink)

According to Weyl, “‘inexhaustibility’ is essential to the infinite”. However, he distinguishes two kinds of inexhaustible, or merely potential, domains: those that are “extensionally determinate” and those that are not. This article clarifies Weyl's distinction and explains its enduring logical and philosophical significance. The distinction sheds lights on the contemporary debate about potentialism, which in turn affords a deeper understanding of Weyl.

Following the finitist’s rejection of the complete totality of the natural numbers, a finitist language allows only propositional connectives and bounded quantifiers in the formula-construction but not unbounded quantifiers. This is opposed to the currently standard framework, a first-order language. We conduct axiomatic studies on the notion of truth in the framework of finitist arithmetic in which at least smash function $\#$ is available. We propose finitist variants of Tarski ramified truth theories up to rank $\omega $, of Kripke–Feferman truth (...) theory and of Friedman–Sheard truth theory, and show that all of these have the same strength as the finitist arithmetic of one higher level along Grzegorczyk hierarchy. On the other hand, we also show that adding Burgess-style groundedness schema, adjusted to the finitist setting, makes Kripke–Feferman truth theory as strong as primitive recursive arithmetic. Meanwhile, we obtain some basic results on finitist theories of (full and hat) inductive definitions and on the second order axiom of hat inductive definitions for positive operators. (shrink)

The previous Part I of the paper discusses the option of the Gödel incompleteness statement (1931: whether “Satz VI” or “Satz X”) to be an axiom due to the pair of the axiom of induction in arithmetic and the axiom of infinity in set theory after interpreting them as logical negations to each other. The present Part II considers the previous Gödel’s paper (1930) (and more precisely, the negation of “Satz VII”, or “the completeness theorem”) as a necessary condition for (...) granting the Gödel incompleteness statement to be a theorem just as the statement itself, to be an axiom. Then, the “completeness paper” can be interpreted as relevant to Hilbert mathematics, according to which mathematics and reality as well as arithmetic and set theory are rather entangled or complementary rather than mathematics to obey reality able only to create models of the latter. According to that, both papers (1930; 1931) can be seen as advocating Russell’s logicism or the intensional propositional logic versus both extensional arithmetic and set theory. Reconstructing history of philosophy, Aristotle’s logic and doctrine can be opposed to those of Plato or the pre-Socratic schools as establishing ontology or intensionality versus extensionality. Husserl’s phenomenology can be analogically realized including and particularly as philosophy of mathematics. One can identify propositional logic and set theory by virtue of Gödel’s completeness theorem (1930: “Satz VII”) and even both and arithmetic in the sense of the “compactness theorem” (1930: “Satz X”) therefore opposing the latter to the “incompleteness paper” (1931). An approach identifying homomorphically propositional logic and set theory as the same structure of Boolean algebra, and arithmetic as the “half” of it in a rigorous construction involving information and its unit of a bit. Propositional logic and set theory are correspondingly identified as the shared zero-order logic of the class of all first-order logics and the class at issue correspondingly. Then, quantum mechanics does not need any quantum logics, but only the relation of propositional logic, set theory, arithmetic, and information: rather a change of the attitude into more mathematical, philosophical, and speculative than physical, empirical and experimental. Hilbert’s epsilon calculus can be situated in the same framework of the relation of propositional logic and the class of all mathematical theories. The horizon of Part III investigating Hilbert mathematics (i.e. according to the Pythagorean viewpoint about the world as mathematical) versus Gödel mathematics (i.e. the usual understanding of mathematics as all mathematical models of the world external to it) is outlined. (shrink)

This chapter addresses Georg Cantor’s transfinite numbers, with some emphasis upon the philosophical questions they raise. We recall that Cantor’s attempts to justify transfinites were directed at mathematicians and philosophers alike—both ancient and contemporary. How does he address objections to actually infinite quantities coming from the philosophical quarter—in particular those of Aristotle—as well as the mathematicians and philosophers of his own day? Does a workable mathematics of real and actually infinite quantities survive both sources of criticism? Not only are transfinites (...) consistent and applicable, he concludes, but real. Moreover, as it turns out, philosophy figures prominently in mathematical developments following Cantor—in particular, in the genesis of the three most prominent philosophies of mathematics of recent times: formalism, logicism, and intuitionism. This chapter makes the case, not only that Cantor’s claims led to this great divide in mathematical thought, but also that some of the prominent difficulties faced by these modern schools may be avoided via the philosophy of mathematics proposed and defended by Aristotle and Thomas Aquinas. (shrink)

In 'Forever Finite: The Case Against Infinity' (Rond Books, 2023), the author argues that, despite its cultural popularity, infinity is not a logical concept and consequently cannot be a property of anything that exists in the real world. This article summarizes the main points in 'Forever Finite', including its overview of what debunking infinity entails for conceptual thought in philosophy, mathematics, science, cosmology, and theology.

EXPANDED EDITION (eBook): -/- Infinity Is Not What It Seems...Infinity is commonly assumed to be a logical concept, reliable for conducting mathematics, describing the Universe, and understanding the divine. Most of us are educated to take for granted that there exist infinite sets of numbers, that lines contain an infinite number of points, that space is infinite in expanse, that time has an infinite succession of events, that possibilities are infinite in quantity, and over half of the world’s population believes (...) in a divine Creator infinite in knowledge, power, and benevolence. -/- According to this treatise, such assumptions are mistaken. In reality, to be is to be finite. The implications of this assessment are profound: the Universe and even God must necessarily be finite. -/- The author makes a compelling case against infinity, refuting its most prominent advocates. Any defense of the infinite will find it challenging to answer the arguments laid out in this book. But regardless of the reader’s position, FOREVER FINITE offers plenty of thought-provoking material for anyone interested in the subject of infinity from the perspectives of philosophy, mathematics, science, and theology. -/- This electronic edition of FOREVER FINITE is offered free online for all and is expanded with content that does not appear in the published edition due to print production page limitations. From the Introduction to the Conclusion, Chapters 1–27 are identical to the published print edition, including the page numbering for the chapters. This electronic edition adds an appendix and associated content—specifically, updates to the book’s back matter (68 new endnotes, 17 new bibliographic references, 3 new figure credits, 14 new glossary terms) and front matter (an updated table of contents and this preface note). (shrink)

Crispin Wright in his 1982 paper argues for strict finitism, a constructive standpoint that is more restrictive than intuitionism. In its appendix, he proposes models of strict finitistic arithmetic. They are tree-like structures, formed in his strict finitistic metatheory, of equations between numerals on which concrete arithmetical sentences are evaluated. As a first step towards classical formalisation of strict finitism, we propose their counterparts in the classical metatheory with one additional assumption, and then extract the propositional part of ‘strict finitistic (...) logic’ from it and investigate. We will provide a sound and complete pair of a Kripke-style semantics and a sequent calculus, and compare with other logics. The logic lacks the law of excluded middle and Modus Ponens and is weaker than classical logic, but stronger than any proper intermediate logics in terms of theoremhood. In fact, all the other well-known classical theorems are found to be theorems. Finally, we will make an observation that models of this semantics can be seen as nodes of an intuitionistic model. (shrink)

Strict finitism is a minority view in the philosophy of mathematics. In this paper, we develop a strict finite axiomatic system for geometric constructions in which only constructions that are executable by simple tools in a small number of steps are permitted. We aim to demonstrate that as far as the applications of synthetic geometry to real-world constructions are concerned, there are viable strict finite alternatives to classical geometry where by one can prove analogs to fundamental results in classical geometry. (...) We consider this as one of many early steps investigating the extent to which strict finite foundations can be developed for the application of mathematics to the real-world. (shrink)

This book is dedicated to V.A. Yankov’s seminal contributions to the theory of propositional logics. His papers, published in the 1960s, are highly cited even today. The Yankov characteristic formulas have become a very useful tool in propositional, modal and algebraic logic. The papers contributed to this book provide the new results on different generalizations and applications of characteristic formulas in propositional, modal and algebraic logics. In particular, an exposition of Yankov’s results and their applications in algebraic logic, the theory (...) of admissible rules and refutation systems is included in the book. In addition, the reader can find the studies on splitting and join-splitting in intermediate propositional logics that are based on Yankov-type formulas which are closely related to canonical formulas, and the study of properties of predicate extensions of non-classical propositional logics. The book also contains an exposition of Yankov’s revolutionary approach to constructive proof theory. The editors also include Yankov’s contributions to history and philosophy of mathematics and foundations of mathematics, as well as an examination of his original interpretation of history of Greek philosophy and mathematics. (shrink)

The Hilbert program was actually a specific approach for proving consistency, a kind of constructive model theory. Quantifiers were supposed to be replaced by ε-terms. εxA(x) was supposed to denote a witness to ∃xA(x), or something arbitrary if there is none. The Hilbertians claimed that in any proof in a number-theoretic system S, each ε-term can be replaced by a numeral, making each line provable and true. This implies that S must not only be consistent, but also 1-consistent. Here we (...) show that if the result is supposed to be provable within S, a statement about all Pi-0-2 statements that subsumes itself within its own scope must be provable, yielding a contradiction. The result resembles Gödel's but arises naturally out of the Hilbert program itself. (shrink)

This paper argues that, insofar as we doubt the bivalence of the Continuum Hypothesis or the truth of the Axiom of Choice, we should also doubt the consistency of third-order arithmetic, both the classical and intuitionistic versions. -/- Underlying this argument is the following philosophical view. Mathematical belief springs from certain intuitions, each of which can be either accepted or doubted in its entirety, but not half-accepted. Therefore, our beliefs about reality, bivalence, choice and consistency should all be aligned.

We investigate which part of Brouwer’s Intuitionistic Mathematics is finitistically justifiable or guaranteed in Hilbert’s Finitism, in the same way as similar investigations on Classical Mathematics (i.e., which part is equiconsistent with$\textbf {PRA}$or consistent provably in$\textbf {PRA}$) already done quite extensively in proof theory and reverse mathematics. While we already knew a contrast from the classical situation concerning the continuity principle, more contrasts turn out: we show that several principles are finitistically justifiable or guaranteed which are classically not. Among them (...) are:(i)fan theorem for decidable fans but arbitrary bars;(ii)continuity principle and the axiom of choice both for arbitrary formulae; and(iii)$\Sigma _2$induction and dependent choice. We also show that Markov’s principle MP does not change this situation; that neither does lesser limited principle of omniscience LLPO (except the choice along functions); but that limited principle of omniscience LPO makes the situation completely classical. (shrink)

The paper examines Yankov’s contribution to the history of mathematical logic and the foundations of mathematics. It concerns the public communication of Markov’s critical attitude towards Brouwer’s intuitionistic mathematics from the point of view of his constructive mathematics and the commentary on A.S. Esenin-Vol’pin program of ultra-intuitionistic foundations of mathematics.

This book publishes the previously unpublished Wittgenstein-Skinner Archive held in Trinity College Cambridge Wren Library. The principal Editor is Arthur Gibson, joined by the Editor Niamh O'Mahony in the editing project. The manuscripts were transcribed by Arthur Gibson, checked and edited by Niamh O'Mahony and Arthur Gibson, with additional assistance from Kelsey Gibson. The Chapters that reproduce the Archive, including the Preface, and Part I (chapters 1 and 2) are authored by Arthur Gibson. Arthur Gibson and Niamh O'Mahony jointly edited (...) the Appendices and variously authored parts of the Appendices' introductions and notes. The Foreword was contributed by Professor Brian McGuinness. (shrink)

Ordinarily, the order in which some objects are attached to a scale does not affect the total weight measured by the scale. This principle is shown to fail in certain cases involving infinitely many objects. In these cases, we can produce any desired reading of the scale merely by changing the order in which a fixed collection of objects are attached to the scale. This puzzling phenomenon brings out the metaphysical significance of a theorem about infinite series that is well (...) known by mathematicians but has so far eluded philosophical scrutiny. (shrink)

Strict finitism is the position that only those natural numbers exist that we can represent in practice. Michael Dummett, in a paper called Wang's Paradox, famously tried to show that strict finitism is an incoherent position. By using the Sorites paradox, he claimed that certain predicates the strict finitist is committed to are incoherent. More recently, Ofra Magidor objected to Dummett's claims, arguing that Dummett fails to show the incoherence of strict finitism. In this paper, I shall investigate whether Magidor (...) is successful in preventing Dummett from proving the incoherence of strict finitism. Though not all the counterarguments Magidor presents are successful, she does in the end manage to corner Dummett. There remains an opportunity for Dummett to insist on the incoherence of strict finitism, but this is a very small opening. The final conclusion of this paper is that Dummett cannot logically prove the incoherence of strict finitism, even though a limited chance for success remains. (shrink)

This article is about the ontological dispute between finitists, who claim that only finitely many numbers exist, and infinitists, who claim that infinitely many numbers exist. Van Bendegem set out to solve the 'general problem' for finitism: how can one recast finite fragments of classical mathematics in finitist terms? To solve this problem Van Bendegem comes up with a new brand of finitism, namely so-called 'apophatic finitism'. In this article it will be argued that apophatic finitism is unable to represent (...) the negative ontological commitments of infinitism or, in other words, that which does not exist according to infinitism. However, there is a brand of infinitism, so-called 'apophatic infinitism', that is able to represent both the positive and the negative ontological commitments of apophatic finitism. Unfortunately, apophatic finitism cannot adopt that way without losing the ability to represent the positive ontological commitments of infinitism. (shrink)

Strict finitism is the position that only those natural numbers exist that we can represent in practice. Michael Dummett, in a paper called Wang’s Paradox, famously tried to show that strict finitism is an incoherent position. By using the Sorites paradox, he claimed that certain predicates the strict finitist is committed to are incoherent. More recently, Ofra Magidor objected to Dummett’s claims, arguing that Dummett fails to show the incoherence of strict finitism. In this paper, I shall investigate whether Magidor (...) is successful in preventing Dummett from proving the incoherence of strict finitism. Though not all the counterarguments Magidor presents are successful, she does in the end manages to corner Dummett. There remains a small opportunity for Dummett to insist on the incoherence of strict finitism, but this is a very small opening. The final conclusion of this paper is that Dummett cannot logically prove the incoherence of strict finitism, even though not all hope is lost. (shrink)

On pense généralement que l'impossibilité, l'incomplétdulité, la paracohérence, l'indécidabilité, le hasard, la calcul, le paradoxe, l'incertitude et les limites de la raison sont des questions scientifiques physiques ou mathématiques disparates ayant peu ou rien dans terrain d'entente. Je suggère qu'ils sont en grande partie des problèmes philosophiques standard (c.-à-d., jeux de langue) qui ont été la plupart du temps résolus par Wittgenstein plus de 80 ans. Je fournis un bref résumé de quelques-unes des principales conclusions de deux des plus éminents (...) étudiants du comportement des temps modernes, Ludwig Wittgenstein et John Searle, sur la structure logique de l'intentionnalité (esprit, langue, comportement), en prenant comme point de départ La découverte fondamentale de Wittgenstein, à savoir que tous les problèmes véritablement « philosophiques » sont les mêmes, les confusions sur la façon d'utiliser la langue dans un contexte particulier, et donc toutes les solutions sont les mêmes— en regardant comment la langue peut être utilisée dans le contexte en cause afin que sa vérité (Conditions de satisfaction ou COS) sont claires. Le problème fondamental est que l'on peut dire n'importe quoi, mais on ne peut pas signifier (état clair COS pour) toute déclaration arbitraire et le sens n'est possible que dans un contexte très spécifique. Je dissé que quelques écrits de quelques-uns des principaux commentateurs sur ces questions d'un point de vue wittgensteinien dans le cadre de la perspective moderne des deux systèmes de pensée (popularisé comme «penser vite, penser lentement»), en utilisant une nouvelle table de intentionnalité et la nomenclature de nouveaux systèmes doubles. Je montre qu'il s'agit d'un puissant heuristique pour décrire la vraie nature de ces questions scientifiques, physiques ou mathématiques putatives qui sont vraiment mieux abordés comme des problèmes philosophiques standard de la façon dont la langue doit être utilisée (jeux de langue dans Wittgenstein terminologie). (shrink)

This article bears on four topics: observational predicates and phenomenal properties, vagueness, strict finitism as a philosophy of mathematics, and the analysis of feasible computability. It is argued that reactions to strict finitism point towards a semantics for vague predicates in the form of nonstandard models of weak arithmetical theories of the sort originally introduced to characterize the notion of feasibility as understood in computational complexity theory. The approach described eschews the use of nonclassical logic and related devices like degrees (...) of truth or supervaluation. Like epistemic approaches to vagueness, it may thus be smoothly integrated with the use of classical model theory as widely employed in natural language semantics. But unlike epistemicism, the described approach fails to imply either the existence of sharp boundaries or the failure of tolerance for soritical predicates. Applications ofmeasurement theory(in the sense of Krantz, Luce, Suppes, & Tversky (1971)) to vagueness in the nonstandard setting are also explored. (shrink)

This article introduces finitist set theory (FST) and shows how it can be applied in modeling finite nested structures. Mereology is a straightforward foundation for transitive chains of part-whole relations between individuals but is incapable of modeling antitransitive chains. Traditional set theories are capable of modeling transitive and antitransitive chains of relations, but due to their function as foundations of mathematics they come with features that make them unnecessarily difficult in modeling finite structures. FST has been designed to function as (...) a practical tool in modeling transitive and antitransitive chains of relations without suffering from difficulties of traditional set theories, and a major portion of the functionality of discrete mereology can be incorporated in FST. This makes FST a viable collection theory in ontological modeling. (shrink)

“Finitism” (Tait 1981) presents an argument that finitist number theory is primitive recursive arithmetic (PRA). The argument is based on taking seriously the “finite” in “finitism”. But the question remained: what did Hilbert (and Bernays) mean in the early 1920’s through the early 1930’s by “finitism” and in particular, did they restrict finitist number theory to PRA. In his dissertation (Zach 2003), Richard Zach pointed out that Hilbert endorsed results as finitist that require more than PRA for their proofs. Tait (...) 2002 and tait2005 argue that it is not clear that Hilbertwas aware that these results go beyond PRA. But that view is challenged in more recent times in Sieg/Ravaglia 2005 and by the editors of (the invaluable!) David Hilbert’s Lectures on the Foundations of Arithmetic and Logic 1917-1933 (Hilbert 2013). I will survey the old ground and then discuss the new challenge, which claims that, from the early 1920’s on, Hilbert accepted as finitist an enumeration function of the primitive recursive functions (which of course is not primitive recursive). The grounds for this are a reading of a passage in §7 of Grundlagen der Mathematik I and an argument for the consistency of PRA which goes back to 1922-1923 and is elaborated again in §7 of Grundlagen der Mathematik I. I will argue that their reading of the passage in question is a misreading and that the argument for the consistency of PRA uses, not an enumeration function for the primitive recursive functions, but rather mathematical induction on a Π02 predicate (i.e. of the form ∀x∃yϕ(x, y)), which was explicitly rejected by Hilbert as finitist - e.g. notably in Hilbert 1926. (shrink)

Gentzen's three consistency proofs for elementary number theory have a common aim that originates from Hilbert's Program, namely, the aim to justify the application of classical reasoning to quantified propositions in elementary number theory. In addition to this common aim, Gentzen gave a “finitist” interpretation to every number-theoretic proposition with his 1935 and 1936 consistency proofs. In the present paper, we investigate the relationship of this interpretation with intuitionism in terms of the debate between the Hilbert School and the Brouwer (...) School over the significance of consistency proofs. First, we argue that the interpretation had the role of responding to a Brouwer-style objection against the significance of consistency proofs. Second, we propose a way of understanding Gentzen's response to this objection from an intuitionist perspective. (shrink)

In this paper, I will argue from a historical perspective that Gentzen’s 1935 consistency proof of 1st order Peano Arithmetic PA principally aimed to give a finitist interpretation of implication and this aspect of the 1935 proof emerged as the attempt to cope with the non-finiteness in BHK-interpretation of implication. My argument consists of two parts. First, I will explain that the fundamental idea of the 1935 proof is to show the soundness of PA on some finitist interpretation and Gentzen (...) had attempted to use BHK-interpretation as such an interpretation. It is claimed that he eventually thought that there is a ‘circularity’ which is due to the non-finiteness of BHK-interpretation of implication, so it cannot be used as a finitist interpretation. Secondly, I will argue that the principal aim of the 1935 proof is to give his own finitist interpretation of implication which avoids such a ‘circularity’. Finally, this paper’s conclusion and an open question are presented. (shrink)

Halbach has argued that Tarski biconditionals are not ontologically conservative over classical logic, but his argument is undermined by the fact that he cannot include a theory of arithmetic, which functions as a theory of syntax. This article is an improvement on Halbach's argument. By adding the Tarski biconditionals to inclusive negative free logic and the universal closure of minimal arithmetic, which is by itself an ontologically neutral combination, one can prove that at least one thing exists. The result can (...) then be strengthened to the conclusion that infinitely many things exist. Those things are not just all Gödel codes of sentences but rather all natural numbers. Against this background inclusive negative free logic collapses into noninclusive free logic, which collapses into classical logic. The consequences for ontological deflationism with respect to truth are discussed. (shrink)

Peano arithmetic cannot serve as the ground of mathematics for it is inconsistent to infinity, and infinity is necessary for its foundation. Though Peano arithmetic cannot be complemented by any axiom of infinity, there exists at least one (logical) axiomatics consistent to infinity. That is nothing else than a new reading at issue and comparative interpretation of Gödel’s papers (1930; 1931) meant here. Peano arithmetic admits anyway generalizations consistent to infinity and thus to some addable axiom(s) of infinity. The most (...) utilized example of those generalizations is the complex Hilbert space. Any generalization of Peano arithmetic consistent to infinity, e.g. the complex Hilbert space, can serve as a foundation for mathematics to found itself and by itself. (shrink)

An observation and a thesis: The observation is that, whatever the connection between Kant’s philosophy and Hilbert’s conception of finitism, Kant’s account of geometric reasoning shares an essential idea with the account of finitist number theory in “Finitism”, namely the idea of constructions f from ‘arbitrary’ or ‘generic’ objects of various types. The thesis is that, contrary to a substantial part of contemporary literature on the subject, when Kant referred to number and arithmetic, he was not referring to the natural (...) or whole numbers and their arithmetic, but rather to the real numbers and their arithmetic. (shrink)

At the most general level, the concept of finitism is typically characterized by saying that finitistic mathematics is that part of mathematics which does not appeal to completed infinite totalities and is endowed with some epistemological property that makes it secure or privileged. This paper argues that this characterization can in fact be sharpened in various ways, giving rise to different conceptions of finitism. The paper investigates these conceptions and shows that they sanction different portions of mathematics as finitistic.

It can be argued that only the equational theories of some sub-elementary function algebras are finitistic or intuitive according to a certain interpretation of Hilbert's conception of intuition. The purpose of this paper is to investigate the relation of those restricted forms of equational reasoning to classical quantifier logic in arithmetic. The conclusion reached is that Edward Nelson's ‘predicative arithmetic’ program, which makes essential use of classical quantifier logic, cannot be justified finitistically and thus requires a different philosophical foundation, possibly (...) as a restricted form of logicism. (shrink)

Detlefsen (1986) reads Hilbert's program as a sophisticated defense of instrumentalism, but Feferman (1998) has it that Hilbert's program leaves significant ontological questions unanswered. One such question is of the reference of individual number terms. Hilbert's use of admittedly "meaningless" signs for numbers and formulae appears to impair his ability to establish the reference of mathematical terms and the content of mathematical propositions (Weyl (1949); Kitcher (1976)). The paper traces the history and context of Hilbert's reasoning about signs, which illuminates (...) Hilbert's account of mathematical objectivity, axiomatics, idealization, and consistency. (shrink)

Quantum computer is considered as a generalization of Turing machine. The bits are substituted by qubits. In turn, a "qubit" is the generalization of "bit" referring to infinite sets or series. It extends the consept of calculation from finite processes and algorithms to infinite ones, impossible as to any Turing machines (such as our computers). However, the concept of quantum computer mets all paradoxes of infinity such as Gödel's incompletness theorems (1931), etc. A philosophical reflection on how quantum computer might (...) implement the idea of "infinite calculation" is the main subject. (shrink)

Greg Frost-Arnold’s book is a highly elegant edition and commentary of Carnap’s notes, claiming just as much as he is warranted on the basis of the manuscript and other relevant texts, and formulating his scholarly assumptions very carefully. Along the way he tries to unify the three historiographical strategies: narrative, argumentative and micro-historical.

Context: Strict finitism is usually not taken seriously as a possible view on what mathematics is and how it functions. This is due mainly to unfamiliarity with the topic. Problem: First, it is necessary to present a “decent” history of strict finitism (which is now lacking) and, secondly, to show that common counterarguments against strict finitism can be properly addressed and refuted. Method: For the historical part, the historical material is situated in a broader context, and for the argumentative part, (...) an evaluation of arguments and counterarguments is presented. Results: The main result is that strict finitism is indeed a viable option, next to other constructive approaches, in (the foundations of) mathematics. Implications: Arguing for strict finitism is more complex than is usually thought. For future research, strict finitist mathematics itself needs to be written out in more detail to increase its credibility. In as far as strict finitism is a viable option, it will change our views on such “classics” as the platonist-constructivist discussion, the discovery-construction debate and the mysterious applicability problem (why is mathematics so successful in its applications?). Constructivist content: Strict finitism starts from the idea that counting is an act of labeling, hence the mathematician is an active subject right from the start. It differs from other constructivist views in that the finite limitations of the human subject are taken into account. (shrink)

Call an argument a ‘happy sorites’ if it is a sorites argument with true premises and a false conclusion. It is a striking fact that although most philosophers working on the sorites paradox find it at prima facie highly compelling that the premises of the sorites paradox are true and its conclusion false, few (if any) of the standard theories on the issue ultimately allow for happy sorites arguments. There is one philosophical view, however, that appears to allow for at (...) least some happy sorites arguments: strict finitism in the philosophy of mathematics. My aim in this paper is to explore to what extent this appearance is accurate. As we shall see, this question is far from trivial. In particular, I will discuss two arguments that threaten to show that strict finitism cannot consistently accept happy sorites arguments, but I will argue that (given reasonable assumptions on strict finitistic logic) these arguments can ultimately be avoided, and the view can indeed allow for happy sorites arguments. (shrink)

This chapter contains sections titled: * Brief History * How We Talk * Science and Infinity * Religion and Infinity * Concluding Remarks * Notes * References * Further Reading.

This article develops a critical investigation of the epistemological core of Hilbert's foundational project, the so-called the finitary attitude. The investigation proceeds by distinguishing different senses of 'number' and 'finitude' that have been used in the philosophical arguments. The usual notion of modern pure mathematics, i.e. the sense of number which is implicit in the notion of an arbitrary finite sequence and iteration is one sense of number and finitude. Another sense, of older origin, is connected with practices of counting (...) concrete things, and a third sense is linked up with the immediate intuitive experience of multitudes of concrete things. Hilbert's fìnitism is examined with respect to these differences, and it will be shown that there is a tendency to conflate the different senses of number and fìnitude, a tendency which has been a source of problems in the discussion of the foundations of mathematics and in the philosophy of logic and language. (shrink)

Context: Strict finitism is usually not taken seriously as a possible view on what mathematics is and how it functions. This is due mainly to unfamiliarity with the topic. Problem: First, it is necessary to present a “decent” history of strict finitism and, secondly, to show that common counterarguments against strict finitism can be properly addressed and refuted. Method: For the historical part, the historical material is situated in a broader context, and for the argumentative part, an evaluation of arguments (...) and counterarguments is presented. Results: The main result is that strict finitism is indeed a viable option, next to other constructive approaches, in mathematics. Implications: Arguing for strict finitism is more complex than is usually thought. For future research, strict finitist mathematics itself needs to be written out in more detail to increase its credibility. In as far as strict finitism is a viable option, it will change our views on such “classics” as the platonist-constructivist discussion, the discovery-construction debate and the mysterious applicability problem . Constructivist content: Strict finitism starts from the idea that counting is an act of labeling, hence the mathematician is an active subject right from the start. It differs from other constructivist views in that the finite limitations of the human subject are taken into account. (shrink)

This book intends to show that radical naturalism (or physicalism), nominalism and strict finitism account for the applications of classical mathematics in current scientific theories. The applied mathematical theories developed in the book include the basics of calculus, metric space theory, complex analysis, Lebesgue integration, Hilbert spaces, and semi-Riemann geometry (sufficient for the applications in classical quantum mechanics and general relativity). The fact that so much applied mathematics can be developed within such a weak, strictly finitistic system, is surprising in (...) itself. It also shows that the applications of those classical theories to the finite physical world can be translated into the applications of strict finitism, which demonstrates the applicability of those classical theories without assuming the literal truth of those theories or the reality of infinity. Both professional researchers and students of philosophy of mathematics will benefit greatly from reading this book. (shrink)