Why anti-realists and classical mathematicians cannot get along (Q5950501)

The author concludes: ``From Heyting semantics -- the natural outcome of Dummett's arguments for anti-realism -- we see that LEM [law of excluded middle] and optimism/immodesty are intimately bound together. So if the Dummettian anti-realist maintains Church's thesis, or its formal statement (CT) [see page 60], or some other semblance of modesty, she cannot have any truck with LEM or any metaphysical thesis that underwrites LEM. The anti-realist who endorses modesty has a principal reason to reject (the truth of some consequences of) LEM outright, not just demur from its assertion or its status as a logical truth. [\dots]. If one is determined to accept classical mathematics and still maintain a sense of modesty, then he must thereby reject the thesis that Heyting semantics provides the meaning of the logical operators of this classical mathematics. Consequently, he must reject Dummett's arguments leading to anti-realism, either out of hand or by finding some fault in the reasoning'' [p. 62].