Introduction into Theory of Mathematical Phenomena

Revue internationale de philosophie, 2005

Gödel began his 1951 Gibbs Lecture by stating: "Research in the foundations of mathematics during the past few decades has produced some results which seem to me of interest, not only in themselves, but also with regard to their implications for the traditional philosophical problems about the nature of mathematics." (Gödel 1951) Gödel is referring here especially to his own incompleteness theorems (Gödel 1931). Gödel's first incompleteness theorem (as improved by Rosser (1936)) says that for any consistent formalized system F, which contains elementary arithmetic, there exists a sentence G F of the language of the system which is true but unprovable in that system. Gödel's second incompleteness theorem states that no consistent formal system can prove its own consistency. 1 These results are unquestionably among the most philosophically important logicomathematical discoveries ever made. However, there is also ample misunderstanding and confusion surrounding them. The aim of this paper is to review and evaluate various philosophical interpretations of Gödel's theorems and their consequences, as well as to clarify some confusions. I have argued for my own interpretation of Hilbert's program in detail in Raatikainen (2003a).

2020

Those incompleteness theorems mean the relation of (Peano) arithmetic and (ZFC) set theory, or philosophically, the relation of arithmetical finiteness and actual infinity. The same is managed in the framework of set theory by the axiom of choice (respectively, by the equivalent well-ordering "theorem'). One may discuss that incompleteness form the viewpoint of set theory by the axiom of choice rather than the usual viewpoint meant in the proof of theorems. The logical corollaries from that "nonstandard" viewpoint the relation of set theory and arithmetic are demonstrated.

2004

2007

Miscellanea Logica VI, UK FF, Praha 2006, pp. 45–64, 2006

When posing the old question ‘What are arithmetical truths about?’ (‘What is their epistemic status?’ or ‘How are they possible?’) we find ourselves standing in the shadow of Gödel, just as our predecessors stood in the shadow of Kant. Of course, this observation may be a bit misleading if only for the reason that Gödel’s famous incompleteness theorems are not of a philosophical nature, at least not in the first place. There are plenty of texts, however, explaining them as philosophically relevant, i.e. as having some philosophical implications. In this article I am not aiming to add a new interpretation to the old ones. Rather, I am proposing to see the incompleteness as a link in the chain of certain great (positive or negative) foundational results such as Frege’s calculization of logic, Russell’s paradox, G¨odel’s completeness theorem, Gentzen’s proof of consistency etc. The foundational line described in this way can then be critically examined as relatively successful with respect to some of its leading ideas and as unsuccessful with respect to others. What I have particularly in mind here is the idea of reducing arithmetic to logic, with its decisive influence on the rebirth and subsequent development of modern (mathematical) logic. Hence, the key issue of this article may be formulated as follows: ‘What do Gödel’s theorems tell us about the alleged analyticity or syntheticity of arithmetic?’.

The present first part about the eventual completeness of mathematics (called “Hilbert mathematics”) is concentrated on the Gödel incompleteness (1931) statement: weather it is an axiom rather than a theorem inferable from the axioms of (Peano) arithmetic, (ZFC) set theory, and propositional logic. The main argument consists in the contradiction of the axiom of induction in arithmetic and the axiom of infinity in set theory. Thus, the pair of arithmetic and set are similar to Euclidean and non-Euclidean geometries distinguishably only by the Fifth postulate: correspondingly, by the axiom of finiteness (induction) versus that of finiteness being idempotent negations to each other. The axiom of choice transforms any set in a well-ordering either necessarily finite according to the axiom of induction or also optionally infinite according to the axiom of infinity. The Gödel incompleteness statement relies on the contradiction of the axioma of induction and infinity.

Melisa Vivanco, 2023

Some of the most influential programs in the philosophy of mathematics started from the philosophical study of natural numbers. On the one hand, our arithmetic intuitions appear earlier and more direct than other mathematical (and non-mathematical) intuitions. On the other hand, while arithmetic admits one of the first axiomatizations with wide acceptance within the mathematical practice, the study of natural numbers sets a methodological precedent that will later seek to be replicated in other areas, in particular, in the study of the most complex numerical structures. This course will address the main issues in the philosophical discussion on arithmetic. Among these topics are the ideas of the various classical doctrines on the foundations of arithmetic, from the milestone of Gödel’s incompleteness theorems to recent doctrines on the semantics of numerical expressions and arithmetic sentences. The class will cover debates about metaphysics and the epistemology of numbers and arithmetic truths.

One of the richest and most salient applications of a non-classical logic is the matter of how mathematics operates within its province. Historically, this is most evident in the case of intuitionism, insofar as the intuitionistic standpoints with respect to deduction and mathematical practice are tightly bound together. Yet even in the case of Robert Meyer's relevant arithmetic R#, that a robust and compelling theory of arithmetic can be erected on relevant foundations speaks to the maturity of relevant logics. Accordingly, as connexive logic matures as a field, the topography of mathematics against a connexive backdrop becomes more and more compelling. The contraclassicality of connexive logics entails that the development of connexive mathematics will be more complex---and, arguably, more interesting---than intuitionistic or relevant accounts. For example, although formally undecidable sentences in classical Peano arithmetic remain independent of its intuitionistic and relevant counterparts, there exist undecidable sentences of classical arithmetic that will become decidable modulo any reasonable connexive arithmetic. In, e.g., Peano arithmetic, the Gödel sentence G is undecidable. Classically, this entails that the sentence ~(G->~G) is likewise undecidable. Of course, in a connexive logic L and connexive arithmetic L#, L# will prove ~(G->~G), witnessing that some classically undecidable statements in number theory become decidable connexively. Although this example is extremely simple, it demonstrates that there are many subtle questions that uniquely arise in a connexive mathematics. In this paper, I wish to make a few comments on how mathematics---in particular, arithmetic---must behave if formulated connexively. We will first consider some relevant historical and philosophical topics, such as Łukasiewicz' number-theoretic argument against Aristotle's Thesis, before taking a foray into the formalization of modest subsystems of arithmetic in Richard Angell's PA1 and PA2, observing some of the pathologies that will greet arithmetic in these settings.

This paper investigates the relation between Carnap and Quine's views on analyticity on the one hand, and their views on philosophical analysis or explication on the other. I argue that the stance each takes on what constitutes a successful explication largely dictates the view they take on analyticity. I show that although acknowledged by neither party (in fact Quine frequently expressed his agreement with Carnap on this subject) their views on explication are substantially different. I argue that this difference not only explains their differences on the question of analyticity, but points to a Quinean way to answer a challenge that Quine posed to Carnap. The answer to this challenge leads to a Quinean view of analyticity such that arithmetical truths are analytic, according to Quine's own remarks, and set theory is at least defensibly analytic.

The conflict between Platonic realism and Constructivism marks a watershed in philosophy of mathematics. Among other things, the controversy over the Axiom of Choice is typical of the conflict. Platonists accept the Axiom of Choice, which allows a set consisting of the members resulting from infinitely many arbitrary choices, while Constructivists reject the Axiom of Choice and confine themselves to sets consisting of effectively specifiable members. Indeed there are seemingly unpleasant consequences of the Axiom of Choice. The non-constructive nature of the Axiom of Choice leads to the existence of non-Lebesgue measurable sets, which in turn yields the Banach-Tarski Paradox. But the Banach-Tarski Paradox is so called in the sense that it is a counter-intuitive theorem. To corroborate my view that mathematical truths are of non-constructive nature, I shall draw upon Gödel's Incompleteness Theorems. This also shows the limitations inherent in formal methods. Indeed the Löwenheim-Skolem Theorem and the Skolem Paradox seem to pose a threat to υlatonists. In this light, Quine/υutnam's arguments come to take on a clear meaning. According to the model-theoretic arguments, the Axiom of Choice depends for its truth-value upon the model in which it is placed. In my view, however, this is another limitation inherent in formal methods, not a defect for Platonists. To see this, we shall examine how mathematical models have been developed in the actual practice of mathematics. I argue that most mathematicians accept the Axiom of Choice because the existence of non-Lebesgue measurable sets and the Well-Ordering of reals open the possibility of more fruitful mathematics. Finally, after responding to ψenacerraf's challenge to Platonism, I conclude that in mathematics, as distinct from natural sciences, there is a close connection between essence and existence. Actual mathematical theories are the parts of the maximally logically consistent theory that describes mathematical reality. 1 I will use the word-Constructivism‖ in a broader sense than ψrower's ωonstructivism. In ψrower's Constructivism mathematical entities are constructible in our mind. But I will use the word-Constructivism‖ in a narrower sense than Gödel's axiom of constructibility. Gödel's Axiom of Constructibility is a much stronger assumption than Constructivism as I call it. Choice that we couldn't otherwise. Platonists accept the Axiom of Choice, which allows a set consisting of the members resulting from infinitely many arbitrary choices, while Constructivists reject the Axiom of Choice and confine themselves to sets consisting of effectively specifiable members (Chapter 1). Lebesgue's theory of measure will set the stage for discussing the Banach-Tarski Paradox and the existence of measurable cardinals in later chapters. Also, since Lebesgue is one of the French Constructivists, it is interesting to see the non-constructive nature of Lebesgue measure creates an irreconcilable tension with Lebesgue's skeptical attitude toward the Axiom of Choice (Chapter 2). The Hausdorff Paradox is the prototype of the Banach-Tarski Paradox. Informally, the Hausdorff Paradox states that a sphere is decomposed into finite number of pieces and reassembled by rigid motions to form two copies of almost the same size as the original. Here-almost‖ means-except on a countable subset.‖ ψanach and Tarski made improvement on the Hausdorff Paradox by eliminating the need to exclude a countable subset from a sphere. Informally, the Banach-Tarski Paradox states that a sphere is decomposed into finite number of pieces and reassembled by rigid motions to form two copies of exactly the same size as the original. The Banach-Tarski Paradox deepened the skepticism about the Axiom of Choice. But the Banach-Tarski Paradox is so called in the sense that it is a counter-intuitive theorem, as distinct from a logical contradiction or a fallacious reasoning. I argue that we should accept the Banach-Tarski Paradox as a Platonic truth and rejects epistemology based on a mathematical intuition (Chapter 3). Next, from a slightly different perspective, I corroborate my view that mathematical truths are of non-constructive nature. Once we got the undecidability of Peano Arithmetic (PA), Gödel's First Incompleteness Theorem is immediate. The set of true sentences in PA is not recursively enumerable. But the set of theorems (provable sentences) in PA is recursively enumerable. So it is easy to see that there is a sentence that is true but unprovable. This implies that there are some arithmetical truths we cannot get access to in an effective way. We also have to note Gödel's Incompleteness Theorems show that there are limitations inherent in formal methods (Chapter 4).

Social Science Research Network, 2020

The paper introduces and utilizes a few new concepts: "nonstandard Peano arithmetic", "complementary Peano arithmetic", "Hilbert arithmetic". They identify the foundations of both mathematics and physics demonstrating the equivalence of the newly introduced Hilbert arithmetic and the separable complex Hilbert space of quantum mechanics in turn underlying physics and all the world. That new both mathematical and physical ground can be recognized as information complemented and generalized by quantum information. A few fundamental mathematical problems of the present such as Fermat's last theorem, four-color theorem as well as its new-formulated generalization as "four-letter theorem", Poincaré's conjecture, "P vs NP" are considered over again, from and within the newfounding conceptual reference frame of information, as illustrations. Simple or crucially simplifying solutions and proofs are demonstrated. The link between the consistent completeness of the system mathematics-physics on the ground of information and all the great mathematical problems of the present (rather than the enumerated ones) is suggested.

2010

Benacerraf's Dilemma (BD), as formulated by Paul Benacerraf in "Mathematical Truth," is about the apparent impossibility of reconciling a "standard" (i.e., classical Platonic) semantics of mathematics with a "reasonable" (i.e., causal, spatiotemporal) epistemology of cognizing true statements. In this paper I spell out a new solution to BD. I call this new solution a positive Kantian phenomenological solution for three reasons: (1) It accepts Benacerraf's preliminary philosophical assumptions about the nature of semantics and knowledge, as well as all the basic steps of BD, and then shows how we can, consistently with those very assumptions and premises, still reject the skeptical conclusion of BD and also adequately explain mathematical knowledge. (2) The standard semantics of mathematically necessary truth that I offer is based on Kant's philosophy of arithmetic, as interpreted by Charles Parsons and by me. The reasonable epistemology of mathematical knowledge that I offer is based on the phenomenology of logical and mathematical self-evidence developed by early Husserl in Logical Investigations and by early Wittgenstein in Tractatus Logico-Philosophicus.

It is not sufficient to supply an instance of Tarski’s schema, ⌈“p” is true if and only if p⌉ for a certain statement in order to get a definition of truth for this statement and thus fix a truth-condition for it. A definition of the truth of a statement x of a language L is a bi-conditional whose two members are two statements of a meta-language L’. Tarski’s schema simply suggests that a definition of truth for a certain segment x of a language L consists in a statement of the form: ⌈v(x) is true if and only if τ(x)⌉, where ⌈v(x)⌉ is the name of x in L’ and τ(x) is a function τ: S → S’ (S and S’ being the sets of the statements respectively of L end L’) which associates to x the statement of L’ expressed by the same sentence as that which expresses x in L. In order to get a definition of truth for x and thus fix a truth-condition for it, one has thus to specify the function τ. A conception of truth for a certain class X of mathematical statements is a general condition imposed on the truth-conditions for the statements of this class. It is advanced when the nature of the function τ is specified for the statements belonging to X. It is sober when there is no need to appeal to a controversial ontology in order to describe the conditions under which the statement τ(x) is assertible. Four sober conceptions of truth are presented and discussed.