Brouwer's Conception of Truth

Abstract

In this paper it is argued that the understanding of Brouwer as replacing truth conditions with assertability or proof conditions, in particular as codified in the so-called Brouwer-Heyting-Kolmogorov Interpretation, is misleading and conflates a weak and a strong notion of truth that have to be kept apart to understand Brouwer properly: truth-as-anticipation and truth-in-content. These notions are explained, exegetical documentation provided and semi-formal recursive definitions are given. Consider the sequence γ = {cn}, where cn is defined to be equal to (−1/2)k1 if k1 is smaller than n and the sequence 0123456789 appears for the first time in the decimal expansion of π with the 0 at the kth 1 decimal position, and equal to (−1/2)n if there is no such k1. Then define r to be limn→∞ cn. Brouwer [1924a] claims that the proposition that r equals 0, is neither true nor false. It would only be true if we had a proof that there are no 0123456789-sequences in the decimal expansion of π, and it would only be false if we knew of such a sequence.1 Assuming that it is true or false in the absence of such knowledge amounts to assuming that the decimal expansion of π has extra-mental existence. When making such a claim, which notion of truth does Brouwer apply as his alternative to the rejected Platonic notion? The answer to this (purely exegetical) question is the goal of this paper. (Or perhaps more accurately: it is an attempt at a rational reconstruction.) Before I present what I think is the correct interpretation of Brouwer, I will discuss and reject three other interpretations that have been made or could be made of his writings. I do not do this with a merely negative aim, but because my interpretation combines elements from those three and is, therefore, best understood and motivated by comparison with them. The first interpretation is that what is true can only be so because of what has actually been constructed. The second is that not only actual but also all potential constructions can serve as truth makers. (These first two options are quite naive and are not serious contenders but they serve the purpose of stage setting.) And the third is that truth is equated with proof. This third We actually do today: there is a 0123456789-sequence beginning at decimal number 17,387,594,880 as well as at several other, later positions [Wells, 1986]. But as Brouwer [1951b] points out there will probably always be an ample supply of other examples that can be used instead. Hence, this is not really relevant, so we will just stick to Brouwer’s example.