Michael Dummett

Wang's paradox

This paper bears on three different topics: observational predicates and phenomenal properties; vagueness; and strict finitism as a philosophy of mathematics. Of these three, only the last requires any preliminary comment. Constructivist philosophies of mathematics insist that the meanings of all terms, including logical constants, appearing in mathematical statements must be given in relation to constructions which we are capable of effecting, and of our capacity to recognise such constructions as providing proofs of those statements; and, further, that the principles of reasoning which, in assessing the cogency of such proofs, we acknowledge as valid must be justifiable in terms of the meanings of the logical constants and of other expressions as so given. The most powerful form of argument in favour of such a constructivist view is that which insists that there is no other means by which we can give meaning to mathematical expressions. We learn, and can only learn, their meanings by a training in their use; and that means a training in effecting mathematical constructions, and in recording them within the language of mathematics. There is no means by which we could derive from such a training a grasp of anything transcending it, such as a notion of truth and falsity for mathematical statements independent of our means of recognising their truth-values. Traditional constructivism has allowed that the mathematical constructions by reference to which the meanings of mathematical terms are to be given may be ones which we are capable of effecting only in principle. It makes no difference if they are too complex or, simply, too lengthy for any human being, or even the whole human race in collaboration, to effect in practice. Strict finitism rejects this concession to traditional views, and insists, rather, that the meanings of our terms must be given by reference to constructions which we can in practice carry out, and to criteria of correct proof on which we are in practice prepared to rely: and the strict finitist employs against the old-fashioned constructivist arguments of exactly the same form as the constructivist has been

The Philosophical Basis of Intuitionistic Logic

Publisher Summary This chapter discusses the philosophical basis of intuitionistic logic. The chapter explains that within mathematical reasoning, the canons of classical logic is in favor of those of intuitionistic logic. The chapter also emphasizes the standpoint of the intuitionists themselves that classical mathematics employs forms of reasoning that are not valid on any legitimate construal of mathematical statements. The chapter also discusses the most fundamental feature of intuitionistic mathematics and its underlying logic. However, it does not discuss intuitionistic mathematics with other respects (such as the theory of free choice sequences) in which it differs from classical mathematics. It is, therefore, possible to conduct the discussion wholly at the level of elementary number theory.

Thought and reality

Preface 1. Facts and Propositions 2. Semantics and Metaphysics 3. Truth and Meaning 4. Truth-Conditional Semantics 5. Justificationist Theories of Meaning 6. Tense and Time 7. Reality As It Is In Itself 8. God and the World

Common Sense and Physics

In The Central Questions of Philosophy (1973), chapter 5, section E, pp. 108–11,* Professor Ayer inquires into the compatibility of ‘the scientific view of the nature of physical objects and that which can be attributed to common sense’. We are presented with three alternative answers. One is that physical theory constitutes, in Ramsey’s terminology, a secondary system, the primary system being the world as conceived by common sense, or, rather, as we learn later (pp. 142–5), an attenuated version of it, stripped of dispositional and causal properties. The primary system embodies ‘the sum total of… purely factual propositions’ (p. 33); the function of the secondary system is ‘purely explanatory’, and the entities to which it refers, in so far as they cannot be identified with those figuring in the primary system, are simply conceptual tools serving to arrange the primary facts (pp. 109–10). This, then, is simply a version of instrumentalism: the actual facts, the hard facts, those that we really believe to obtain, are those of the primary system; the statements of scientific theory represent fictions, in which we do not really believe (as Ramsey confessed that he did not really believe in astronomy), but which we devise as a vivid means of encapsulating patterns and regularities detectable amongst the primary facts.

What Does the Appeal to Use Do for the Theory of Meaning

Consider the following style of argument. What would one say, e.g., ‘Either he is your brother or he isn’t,’ for? Well, it is tantamount to saying, ‘There must be a definite answer: there are no two ways about it.’ We say this when someone is shilly-shallying, behaving as if it were no more right to say the one thing than the other: so the utterance of that instance of the law of excluded middle is an expression of the conviction that the sentence, ‘He is your brother,’ has a definite sense. That, therefore, is the meaning of the sentence, ‘Either he is your brother or he isn’t’: that is its use in the language.

Frege and Kant on geometry

In his Grundlagen, Frege held that geometrical truths.are synthetic a priori, and that they rest on intuition. From this it has been concluded that he thought, like Kant, that space and time are a priori intuitions and that physical objects are mere appearances. It is plausible that Frege always believed geometrical truths to be synthetic a priori; the virtual disappearance of the word ‘intuition’ from his writings from after 1885 until 1924 suggests, on the other hand, that he became dissatisfied with the notion of intuition as he had employed it in Grundlagen. The belief that a priori intuition is a source of knowledge does not in itself entail idealism: that is a question about what it is that makes true the propositions known in this way. In Grundlagen, Frege expressly states that geometrical truths are objective in the sense of being independent of our intuition. This shows that, even at that period, Frege did not draw the idealist conclusion drawn by Kant.

Dummett on the time-continuum

Michael Dummett claims that the classical model of time as a continuum of instants has to be rejected. In his view, “it allows as possibilities what reason rules out, and leaves it to the contingent laws of physics to rule out what a good model of physical reality would not even be able to describe.” This paper argues otherwise.

I. Frege as a Realist

H. Sluga (Inquiry, Vol. 18 [1975], No. 4) has criticized me for representing Frege as a realist. He holds that, for Frege, abstract objects were not real: this rests on a mistranslation and a negle...

DOES QUANTIFICATION INVOLVE IDENTITY

How are ‘every’,’ some’ and their like related to ‘the same as’? Is there any ground for saying that an understanding of the former—of expressions of generality—presupposes an understanding of the latter— of expressions of identity? Let us begin with formalised languages, those framed by use of the apparatus of predicate logic. In a second-order language, identity is definable, so no problem arises. If, in a first-order language, identity is taken as primitive, it is at least un-problematic that an interpretation of the language must involve a relation of identity over the domain. But suppose that a language is framed within first-order logic without identity: is there then any need to assume that an interpretation of the language will involve or invoke a relation of identity over the domain?

Objectivity and reality in Lotze and Frege

Frege held that logical objects are objective but not wirklich, and that psychologism follows from the mistake of believing whatever is not wirklich to be subjective. It has been suggested that Freges use of the terms ‘objective∗ and ’wirklich’ is in line with that found in Lotzes Logic; from this it has been inferred that Freges doctrines have been misinterpreted as being ontological in character, but that they really belong to epistemology. In fact, Lotze held that something may be the same for all thinkers, and yet may exist only in thought, not independently of it. For Frege, by contrast, there is nothing intermediate between the content of a single consciousness and what exists independently of being thought at all. This crucial disagreement underlies the divergence between Freges realism and Lotzes idealism.