Some consequences of defining mathematical objects constructively and mathematical truth effectively (original) (raw)
Related papers
2002
Standard expositions of classical first order theory-rooted primarily in the works of Cantor, Gödel, Tarski, and Turing-argue that the truth of some propositions of significant formal mathematical languages, under any interpretation, is nonalgorithmic, and essentially unverifiable constructively. This, implicitly, admits nonconstructive elements into the domain of the interpretation. In this essay, we consider some arguments for, and consequences of, an interpretation of classical foundational concepts in which we define mathematical objects constructively, and mathematical truth effectively. 1 The earlier, arXived, title of this essay was "Some consequences of a recursive number-theoretic relation that is not the standard interpretation of any of its formal expressions".
Some Sober Conceptions of Mathematical Truth
Activity and Sign, 2005
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:
A principle, according to which any scientific theory can be mathematized, is investigated. That theory is presupposed to be a consistent text, which can be exhaustedly represented by a certain mathematical structure constructively. In thus used, the term " theory " includes all hypotheses as yet unconfirmed as already rejected. The investigation of the sketch of a possible proof of the principle demonstrates that it should be accepted rather a metamathematical axiom about the relation of mathematics and reality. Its investigation needs philosophical means. Husserl's phenomenology is what is used, and then the conception of " bracketing reality " is modelled to generalize Peano arithmetic in its relation to set theory in the foundation of mathematics. The obtained model is equivalent to the generalization of Peano arithmetic by means of replacing the axiom of induction with that of transfinite induction. Accepting or rejecting the principle, two kinds of mathematics appear differing from each other by its relation to reality. Accepting the principle, mathematics has to include reality within itself in a kind of Pythagoreanism. These two kinds are called in paper correspondingly Hilbert mathematics and Gödel mathematics. The sketch of the proof of the principle demonstrates that the generalization of Peano arithmetic as above can be interpreted as a model of Hilbert mathematics into Gödel mathematics therefore showing that the former is not less consistent than the latter, and the principle is an independent axiom. An information interpretation of Hilbert mathematics is involved. It is a kind of ontology of information. Thus the problem which of the two mathematics is more relevant to our being (rather than reality for reality is external only to Gödel mathematics) is discussed. An information interpretation of the Schrödinger equation is involved to illustrate the above problem.
On the Semantics of Mathematical Statements
Manuscrito Revista Internacional De Filosofia, 1996
Husserl developed-independently of Frege-a semantics of sense and reference. There are, however, some important differences, specially with respect to the references of statements. According to Husserl, an assertive sentence refers to a state of affairs, which was its basis what he called a situation of affairs. Situations of affairs could also be considered as an alternative referent for statements on their own right, although for Husserl they were simply a sort of referential basis. Both Husserlian states of affairs and situations of affairs are extensional. Tarskian semantics can be rendered as a sort of state of affairs semantics. However, to assess adequately the existence of dual theorems in mathematics and, more generally, seemingly unrelated interderivable statements like the Axiom of Choice and its many equivalents, states of affairs (and truth-values) are not enough. We need a sort of refinement of the notion of a situation of affairs, namely what we have called elsewhere an abstract situation of affairs. We are going to introduce abstract situations of affairs as equivalence classes of states of affairs denoted by closed sentences of a given language which are true in the same models. We first sketch the procedure for a first-order many-sorted language and then for a second-order manysorted language.
Proof-theoretic Semantics for Classical Mathematics
Synthese, 2006
The picture of mathematics as being about constructing objects of various sorts and proving the constructed objects equal or unequal is an attractive one, going back at least to Euclid. On this picture, what counts as a mathematical object is specified once and for all by effective rules of construction. In the last century, this picture arose in a richer form with Brouwer's intuitionism. In his hands (for example, in his proof of the Bar Theorem), proofs themselves became constructed mathematical objects, the objects of mathematical study, and with Heyting's development of intuitionistic logic, this conception of proof became quite explicit. Today it finds its most elegant expression in the Curry-Howard theory of types, in which a proposition may be regarded, at least in principle, as simply a type of object, namely the type of its proofs. When we speak of 'proof-theoretic semantics' for mathematics, it is of course this point of view that we have in mind. On this view, objects are given or constructed as objects of a ginen type. That an object is of this or that type is thus not a matter for discovery or proof. One consequence of this view is that equality of types must be a decidable relation. For, if an object is constructed as an object of type A and A and B are equal, then the object is of type B, too, and this must be determinable. One pleasant feature of the type theoretic point of view is that the laws of logic are no longer 'empty': The laws governing the type ∀x : A.F (x) = Π x:A F (x) simply express our general notion of a function, and the laws governing ∃x : A.
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.