Is it a contradiction, if
- mathematical objects are for me "mental constructions" -- like in intuitionism
but
- I accept classical mathematics and the Law oft excluded middle
?
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3@jobermark What does LEM have to do with Russell's paradox? The derivation of a contradiction from naïve comprehension is intuitionistically valid.– J.P.Commented Jul 27, 2015 at 19:58
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1@J.P. The paradox involves three steps. You have to say "1) it is unambiguous what is meant for x not to be in x (free self reference). 2) construct the set of all x with x not in x (unbounded universal quantification). 3) either x is in x or x is not in x (LEM) and you get a contradiction." So removing any of the steps solves the problem. Set theory cripples unbounded universal quantification, intuitionism undercuts LEM, and ramification or sequencing of types limit free self-reference. Each is a solution to this problem.– user9166Commented Jul 27, 2015 at 21:00
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1In intuitionism it is simply not true that either x has to be a member of itself or it has to not be a member of itself. It can be undeterminable whether x is a member of itself. "tertium datur".– user9166Commented Jul 27, 2015 at 21:07
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2@jobermark It's just false (and therefore misleading) to say that the paradox involves your step (3) (LEM). Let r be the Russell set. Suppose that r∈r, then ¬(r∈r), which is a contradiction, so by intuitionistically valid reductio, ¬(r∈r), and we discharge the assumption. Then, r∈r, and we have a contradiction, independent of any assumptions. At no stage is LEM used, all is used is the intuitionistically valid principle that if you can derive a contradiction from p, then you can infer ¬ p. See this question.– J.P.Commented Jul 29, 2015 at 7:11
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1@jobermark Whatever you're suggesting is not standard intuitionism, and goes well beyond just rejecting LEM, which is what was at issue here.– J.P.Commented Jul 29, 2015 at 17:21
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1 Answer
It depends on what you mean by "mental construction":
If you mean mental construction, i.e. just in mind, then there is no contradiction.
If you mean mental construction, i.e. constructible step by step by a finite algorithm, then it is a contradiction.
Added.
1) mental construction: Mathematical objects are ideas, they do not exist as physical objects. Sophisticated concepts like Hilbert space or Riemann manifold have been created by man and fixed by definitions. Hence they are mental constructions. This opinion is not shared by all philosophers of mathematics. Instead, some keep alive the discussion whether mathematical objects have been invented or have been detected.
2) mental construction: Mathematicians from the school of intuitionism reject the existence of mathematical objects like the set of all natural numbers. They accept "potential" infinity like constructing one number after the other - without an end. But they reject "actual" infinity, which equals the cardinalty of the set of natural numbers taken as a whole. Instead, they request that any object must be constructible. Therefore they do not accept the method of indirect proof. It builds on the law of the excluded middle.
answered Jul 26, 2015 at 16:46
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The second point is not quite honest. Intuitionism is not constructivism, it allows for the idea of the set of all integers, and proposes ideas that arise from intuition: for instance the idea of a real number as "an infinite pre-determined free-flowing stream of digits", which make no sense from the point of view of constructivism. Constructivism captures the intention behind continuity with an approximation. Intuitionism notes that the definition is not very useful in that it makes all functions continuous, and chooses to focus its attention elsewhere. The two are not the same thing.– user9166Commented Jul 27, 2015 at 16:58
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I should have said it allows for the idea of collections which contain all integers, not a unique such entity, and with no single best one among them. But still, it is not constructivism, because constructively, there is not an ambiguous collection of such things, there is no such thing.– user9166Commented Jul 29, 2015 at 15:07