By grounding mathematics in what can be constructed, intuitionism has far-reaching consequences for the practice of math, and for determining which statements can be deemed true. The most radical departure from standard math is that the law of excluded middle, a vaunted principle since the time of Aristotle, doesn’t hold. The law of excluded middle says that either a proposition is true, or its negation is true—a clear set of alternatives that offers a powerful mode of inference. But in Brouwer’s framework, statements about numbers might be neither true nor false at a given time, because the number’s exact value hasn’t yet revealed itself.
There’s no difference from standard math when it comes to numbers such as 4, or ½, or pi, the ratio of a circle’s circumference to its diameter. Even though pi is irrational, with no finite decimal expansion, there’s an algorithm for generating its decimal expansion, making pi just as determinate as a number like ½. But consider another number x that’s in the ballpark of ½.
Say the value of x is 0.4999, where further digits unfurl in a choice sequence. Maybe the sequence of 9s will continue forever, in which case x converges to exactly ½. (This fact, that 0.4999… = 0.5, is true in standard math as well, since x differs from ½ by less than any finite difference.)
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But if at some future point in the sequence, a digit other than 9 crops up—if, say, the value of x becomes 4.999999999999997…—then no matter what happens after that, x is less than ½. But before that happens, when all we know is 0.4999, “we don’t know whether or not a digit other than 9 will ever show up,” explained Carl Posy, a philosopher of mathematics at the Hebrew University of Jerusalem and a leading expert on intuitionist math. “At the time we consider this x, we cannot say that x is less than ½, nor can we say that x equals ½.” The proposition “x is equal to ½” is not true, and neither is its negation. The law of the excluded middle doesn’t hold.
Moreover, the continuum can’t be cleanly divided into two parts consisting of all numbers less than ½ and all those greater than or equal to ½. “If you try to cut the continuum in half, this number x is going to stick to the knife, and it won’t be on the left or on the right,” said Posy. “The continuum is viscous; it’s sticky.”
Hilbert compared the removal of the law of excluded middle from math to “prohibiting the boxer the use of his fists,” because the principle underlies much mathematical deduction. Although Brouwer’s intuitionist framework compelled and fascinated the likes of Kurt Gödel and Hermann Weyl, standard math, with its real numbers, dominates because of ease of use.
Gisin first encountered intuitionist math at a meeting last May attended by Posy. When the two got to talking, Gisin quickly saw a connection between the unspooling decimal digits of numbers in this mathematical framework and the physical notion of time in the universe. Materializing digits seemed to naturally correspond to the sequence of moments defining the present, when the uncertain future becomes concrete reality. The lack of the law of excluded middle is akin to indeterministic propositions about the future.