Some consequences of defining mathematical objects constructively and mathematical truth effectively (original) (raw)
We show how removing faith-based beliefs in current philosophies of classical and constructive mathematics admits formal, evidence-based, definitions of constructive mathematics; of a constructively well-defined logic of a formal mathematical language; and of a constructively well-defined model of such a language. We argue that, from an evidence-based perspective, classical approaches which follow Hilbert's formal definitions of quantification can be labelled `theistic'; whilst constructive approaches based on Brouwer's philosophy of Intuitionism can be labelled `atheistic'. We then adopt what may be labelled a finitary, evidence-based, `agnostic' perspective and argue that Brouwerian atheism is merely a restricted perspective within the finitary agnostic perspective, whilst Hilbertian theism contradicts the finitary agnostic perspective. We then consider the argument that Tarski's classic definitions permit an intelligence---whether human or mechanistic---to admit finitary, evidence-based, definitions of the satisfaction and truth of the atomic formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two, hitherto unsuspected and essentially different, ways. We show that the two definitions correspond to two distinctly different---not necessarily evidence-based but complementary---assignments of satisfaction and truth to the compound formulas of PA over N. We further show that the PA axioms are true over N, and that the PA rules of inference preserve truth over N, under both the complementary interpretations; and conclude some unsuspected constructive consequences of such complementarity for the foundations of mathematics, logic, philosophy, and the physical sciences.
A Natural Interpretation of Classical Proofs
2006
of accuracy. On the other hand, the constructivist claims that to know that a particular sequence of rational numbers converges is to know how to approximate the corresponding real number to an arbitrarily given degree of accuracy. Hence the constructivist rejects the law of excluded middle. Using the law of excluded middle one can also argue that every sequence of rational numbers that does not converge must diverge. However, one can not from the knowledge that a particular sequence of rational numbers does not converge construct the corresponding witness. The example just given illustrates the direct nature of constructive existence, as opposed to the indirect nature of classical existence, and shows why the law of excluded middle is not accepted as a law of constructive logic. Finally, I would like to thank my supervisor, Per Martin-Löf, for posing the problem of investigating how the double-negation interpretation operates on derivations and not only on formulas as well as for his continued guidance of my work. Without him, this thesis would never have come into existence. Jens Brage Reynolds (1972), and Plotkin (1975) for the foundations of CPS translations and Reynolds (1993) for the early history of continuations. To my knowledge there is no interpretation of classical logic in constructive logic that makes full use of the syntactic-semantic method of constructive type theory. With this thesis I hope to fill this gap. The subject of interpretations of classical logic in constructive logic began with the double negation interpretation of classical logic in minimal logic due to Kolmogorov (1925). The double negation interpretation was then followed by the interpretation of Peano arithmetic in Heyting arithmetic due to Gödel (1933) and the interpretation of classical logic in intuitionistic logic due to Kuroda (1951). Yet it was not until Griffin (1990) showed how to extend the formulae-as-types correspondence to classical logic that significant growth took place. His solution was to include operations on the flow of control, similar to call/cc of Scheme, into the notion of computation given by a simply typed call-by-value λ-calculus. After that Parigot (1992) introduced his λµ-calculus to realize classical proofs as programs. The λµ-calculus extended the simply typed λ-calculus with operators that can be used to model operations on the flow of control. The development then took the form of CPS translations of different λµ-calculi into different λ-calculi. See Ong (1996) and Ong and Stewart (1997) for call-by-value respectively call-byname CPS translations of Parigot's λµ-calculus into the simply typed λ-calculus. See Selinger (2001, p. 24) for an informal description of the semantics of the λµcalculus.
A Theory of Mathematical Correctness and Mathematical Truth
Pacific Philosophical Quarterly, 2001
A theory of objective mathematical correctness is developed. The theory is consistent with both mathematical realism and mathematical antirealism, and versions of realism and anti-realism are developed that dovetail with the theory of correctness. It is argued that these are the best versions of realism and anti-realism and that the theory of correctness behind them is true. Along the way, it is shown that, contrary to the traditional wisdom, the question of whether undecidable sentences like the continuum hypothesis have objectively determinate truth values is independent of the question of whether mathematical realism is true.
The Problem of Mathematical Truth
In current mathematical practice, mathematical knowledge (if it is achieved at all) is achieved by proving theorems on the basis of definitions and axioms. The problem is to understand how what is achieved thereby constitutes knowledge; more specifically, it is to develop a unified account of mathematical truth and mathematical knowledge, one that reveals their inner connection. What stands in our way, according to a very familiar argument of Benacerraf's, is that in mathematics there seems no way to combine a Tarskian semantics, according to which truth involves ineliminable reference to objects (either by way of singular terms or by way of quantifiers), with an adequate epistemology: either mathematical knowledge is by way of proof, in which case mathematical objects are irrelevant to mathematical knowledge and then we have no account of mathematical truth, or mathematical knowledge is not by way of proof because mathematical objects are constitutive of mathematical truth, but then we have no resources for understanding mathematical knowledge. I then trace the difficulties, in a series of stages, all the way down to our most basic conception of logic as formal and merely explicative: if mathematics is a practice of reasoning from concepts by logic alone then it ought, according to Kant, to be analytic, that is, merely explicative, not knowledge properly speaking at all. This, I submit, is the really hard problem of mathematical truth. Four responses are outlined, but only one holds out promise of resolving our difficulties, namely, that of Peirce and Frege. According to them, logic is a science, and hence experimental and fallible, symbolic language is contentful despite involving no reference to any objects, and proof is a constructive and hence fruitful process. Adequately developed, these ideas will enable us finally to resolve the problem of mathematical truth.
2010
I shall argue that a resolution of the PvNP problem requires building an iff bridge between the domain of provability and that of computability. The former concerns how a human intelligence decides the truth of number-theoretic relations, and is formalised by the first-order Peano Arithmetic PA following Dededekind's axiomatisation of Peano's Postulates. The latter concerns how a human intelligence computes the values of number-theoretic functions, and is formalised by the operations of a Turing Machine following Turing's analysis of computable functions. I shall show that such a bridge requires objective definitions of both an `algorithmic' interpretation of PA, and an `instantiational' interpretation of PA. I shall show that both interpretations are implicit in the definition of the subjectively defined `standard' interpretation of PA. However the existence of, and distinction between, the two objectively definable interpretations---and the fact that the former is sound whilst the latter is not---is obscured by the extraneous presumption under the `standard' interpretation of PA that Aristotle's particularisation must hold over the structure N of the natural numbers. I shall argue that recognising the falseness of this belief awaits a paradigm shift in our perception of the application of Tarski's analysis (of the concept of truth in the languages of the deductive sciences) to the `standard' interpretation of PA. I shall then show that an arithmetical formula [F] is PA-provable if, and only if, [F] interprets as true under an algorithmic interpretation of PA. I shall finally show how it then follows from Goedel's construction of a formally `undecidable' arithmetical proposition that there is a Halting-type PA formula which---by Tarski's definitions---is algorithmically verifiable as true, but not algorithmically computable as true, under a sound interpretation of PA.
In the first part of this investigation we highlight two, seemingly irreconcilable, beliefs that suggest an impending crisis in the teaching, research, and practice of mathematics: (1) the belief, with increasing, essentially faith-based, conviction and authority amongst academics that a first-order Set Theory such as ZFC can be treated as the lingua franca of mathematics, since its theorems-even when unfalsifiable-can be treated as 'knowledge' because they are finite proof sequences which are entailed finitarily by self-evidently Justified True Beliefs; and (2) the gradually emerging, but powerfully rooted in economic imperatives, belief that only those Justified True Beliefs can be treated as 'knowledge' which are, further, categorically communicable as Factually Grounded Beliefs-hence as comprehensible pre-formal 'mathematical truths'-by any intelligence that lays claims to a mathematical lingua franca. We argue that this seeming irreconcilability merely reflects a widening chasm between an increasing underestimation of the value to society of 'semantic arithmetical truth', and an increasing over-estimation of the value to society of 'syntactic set-theoretical provability'; a chasm which must be narrowed and bridged explicitly to avoid lending an illusory legitimacy-by-omission to the perilous concept of 'alternative facts'. We thus proffer a Complementarity Thesis which seeks to recognize that mathematical 'provability' and mathematical 'truth' need to be interdependent and complementary, 'evidence-based', assignments-by-convention to the formulas of a formal mathematical language; where (a) the goal of mathematical 'proofs' may be viewed as seeking to ensure that any mathematical language intended to formally represent our pre-formal conceptual metaphors and their inter-relatedness is unambiguous, and free from contradiction; whilst (b) the goal of mathematical 'truths' must be viewed as seeking to 'validate' that any such representation does, indeed, uniquely identify and adequately represent such metaphors and their inter-relatedness. Our thesis is that, by universally ignoring the need to introduce goal (b) in our curriculum, the teaching of, and research in, mathematics at the postgraduate level cannnot justify its value to society beyond the mere intellectual stimulation of individual minds. In the second part we appeal to some recent-and hitherto unsuspected-unarguably constructive Tarskian interpretations, of the first-order Peano Arithmetic PA, which establish PA as both finitarily consistent, and categorical. Since we also show that the second order subsystem ACA 0 of Peano Arithmetic and PA cannot both be assumed provably consistent, we conclude that there can be no mathematical, or meta-mathematical, proof of consistency for Set Theory. Hence the theorems of any Set Theory are not sufficient for distinguishing between (i) what we can believe to be a 'mathematical truth'; (ii) what we can evidence as a 'mathematical truth'; and (iii) what we ought not to believe is a 'mathematical truth'; whilst the theorems of the first-order Peano Arithmetic PA are sufficient for distinguishing between (i), (ii) and (iii). We conclude from this that the holy grail of mathematics ought to be 'arithmetical truth', not 'set-theoretical proof'.
TOWARD A MINIMALIST FOUNDATION FOR CONSTRUCTIVE MATHEMATICS
The two main views in modern constructive mathematics, usually associated with constructive type theory and topos theory, are compatible with the classical view, but they are incompatible with each other, in a sense explained by some specific results which we briefly review. So it is desirable to design a common core which is compatible with all the theories in which mathematics has been developed, like Zermelo–Fraenkel set theory, topos theory, Martin-Löf's type theory, etc. and can be understood as it is by any mathematician, whatever foundation is adopted. A requirement with increasing importance is that of developing mathematics in such a way that it can be formalized on a computer. This theoretically means that the foundation should obey the proofs-as-programs paradigm. We claim that to satisfy both requirements it is necessary to use a minimal type theory mTT, which is obtained from Martin-Löf's type theory by relaxing the identification of propositions with sets. This ground type theory mTT is intensional and is needed for formalization. A 'toolbox' of extensional concepts, needed to do mathematics , is built on it. The common core is obtained at this level, by subtraction. The underlying conceptual novelty is that one should give up to the expectation of an all-embracing foundation, also in the concrete sense that one needs a formal system living at two different levels of abstraction. After abandoning the classical view and all its limits, a prospective constructivist is faced with the problem of choosing among a variety of views. They generally share the choice for intuitionistic logic, but they differ in mathematical principles. The principal distinction is between two views. One maintains that the meaning of mathematics lies in its computational content, and thus keeps its formalization in a computer language in mind. It is usually associated with type theory; the axiom of choice then turns out to be valid and (hence) the powerset axiom is not accepted [29]. The other favours the mathematical structure beyond its particular presentations. It is usually expressed through category theory and often identified with topos theory. Extensionality is an essential feature, the powerset axiom is mostly accepted and hence the axiom of choice is not accepted as valid [23]. Both views are reasonable, well motivated and apparently cannot be dispensed with. It is however a matter of fact that they are incompatible, in the sense that by accepting both one is forced back to the classical view. The same tension is revealed also as a technological challenge: while implementation of mathematics in a computer is intrinsically intensional (just think of the fact that
Publications de l'Institut Math?matique (Belgrade)
The concept of definition is usually not covered in mathematical logic textbooks. The definability of classes of structures is dealt with in model theory but the definability of concepts within a given structure is not. Our aim is to deal with these kind of definitions. We also address some of the implications for teaching and learning mathematics.
Oxford Handbooks Online
This chapter discusses four questions concerning the nature and role of the concept of truth in mathematics. First, the question as to whether the concept of truth is needed in a philosophical account of mathematics is answered affirmatively: a formalist approach to the language of mathematics is inadequate. Next, following Frege, a classical conception of mathematical truth is defended, involving the existence of mathematical objects. The third question concerns the relation between the existence of mathematical objects and the objectivity of mathematical truth. A traditional platonist seeks to explain the latter in terms of the former, while Frege reverses this order of explanation. Finally, the question regarding the extent to which mathematical statements have objective truth-values is discussed.