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Is there a conflict between Brouwer's intuitionism and the claim that the free will must not exist because nature is deterministic ? It seems to me that the act of mental construction of mathematical truths must be determined by initial conditions so one can claim that these mental constructs are somehow determined by the laws of physics and initial data. Furthermore , one can imagine that it could be entirely possible that intelligent lifeforms that can dream up mathematical constructions did not in fact exist. Does it mean that math didn't exist by then ? The universe with no intelligent human beings was governed by some laws that are mathematical. Does this lead to a contradiction ?

  • Brouwer might have jokingly turned the argument around and said that since intuitionism is true physical determinism must be false. In seriousness, I do not think it works either way. It is enough for Brouwer's "lawless sequences" to be merely possible, regardless of how this world actually is. Of course, Brouwer himself believed in actual indeterminism of the "creative subject", but that is philosophy, not mathematics. Besides, physical measurements are always of finite precision, so disagreements about the infinite are empirically moot. – Conifold Jun 20 '17 at 23:37
  • is mathematics a natural science? – user20153 Jun 21 '17 at 22:12
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Determinism has nothing to do with the laws of physics.

The "laws" of physics are man-made. The behavior of the universe did not dictate our physics. Trial and error did. We chose what to describe and how to describe it. We were often wrong, and we probably still are. If none of modern physics is incorrect, we will shortly have no use for physicists.

All that Brouwer is saying with Inutitionism, at root, is that mathematics is also man-made, and we should look at it with the natural suspicion, and the natural courage, that belongs to all things we have made. It is just another science, and properly as much of a science as physics is. We know that current physics contradicts older physics. So to assume that mathematics is going to somehow remain permanently without internal contradictions is an odd approach to reality.

So these two issues are wholly disconnected. Whether the universe we are studying is subject to perfect rules is not related to whether we know them, because we don't. Nor can having really good approximations to observed phenomena tell us one way or another whether the underlying rules are perfect.

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