To: Max T.
From: Geoffrey Klempner
Subject: Dummett's idea of an anti-realist theory of meaning
Date: 30 September 10:32

Dear Max,

Thank you for your e-mail of 17 September with your third essay for the Metaphysics program, entitled, 'Michael Dummett's Idea of an Anti-Realist Theory of Meaning', and your notes in response to unit 11.

Essay

You give a reasonably good characterization of Dummett's account of the difference between a realist and an anti-realist theory of meaning. However, you make a slip in the first paragraph when you say, 'A or B is true if it is the case that A is true and B is true.' That is of course true; if A is true and B is true then a fortiori (from the stronger premiss) 'A or B' is true. But we are concerned with necessary and sufficient conditions, i.e. with statements of the form 'XYZ is true *if and only if*...' It is not the case that A or B is true *only if* A is true and B is true, because the truth of A alone or B alone is sufficient for the truth of A or B.

There is a special point to be made about statements of the form A or B. According to Dummett, or-statements (as well as if...then...-statements). The truth of a disjunction obtains *in virtue of* the truth of one or other (or both) of the disjuncts. In other words, A or B is *made* true either by the truth of A, or by the truth of B, or by the truth of A and B.

You raise an interesting question about Michael Dummett's notion of mathematical proof. Dummett himself places great emphasis on the relative clarity of issues of meaning and justification in mathematics. The mathematical case is supposed to carry over to factual discourse, using the analogy between mathematical proof and empirical verification. But just what counts as a mathematical proof?

It is not necessary to prove that 5+7=12. However, using Peano's axioms, this can be done. Such a proof (defining '5' as the successor of the successor of the successor of the successor of the successor of '0' etc.) does not make the proposition 5+7=12 any more certain. In fact, the contrary holds. I am far more likely to doubt the soundness of any such proof (I had to check the number of times I wrote 'successor' twice!) than I am to doubt whether 5+7=12.

Having attended Dummett's lectures on Intuitionist mathematics at Oxford, I can testify that he was very much aware of the problem of different *levels* of understanding of mathematical notions. Primitive peoples, or young children have some understanding of numbers, but less than the school student who, e.g., has learned to do long division and multiplication. And the school student understands less than the university student who has learned number theory, and so on. So what does it *really* take to understand a mathematical statement? And what conclusions should we draw from any uncertainty on that point for an account of understanding of factual discourse?

It could be said that there is a parallel to be drawn with factual discourse. Someone who has studied philosophy and knows about the problem of perception, or the theory of immaterialism, understands more in the statement, 'The book is on the table' than someone who does not have this knowledge. (Similar points can be made involving knowledge of economics, or physics, or medicine.) However, there is a sense in which a person who does not know these finer points of philosophy does have a full grasp of the meaning of the statement made. The book is on the table. Any further examination of the statement is surplus to the grasp of what that statement *says*.

However, I am not sure how far this thought can be taken, or whether it constitutes a sufficient defence of Dummett's position against the question that you have raised.

All the best,

Geoffrey