1: Why do we say that there can't be other logics/mathematics than those we have?
2: Logic and maths are independent of reality. Then, if we have invented a logic/math based on reality, would it be wrong or false?
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1Welcome to the forum. You’ll probably need to spell those questions out a bit more. I don’t think anyone says we can’t have other ‘versions’ of logic or math. In fact, we do have plenty of alternatives to classical logic and math - and more may be conceived of in the future. As for the second question, what do you mean ‘logic and maths are independent of reality’? On a broadly Platonist/Realist view, logic and math are part of reality – albeit an abstract part. Even the Nominalist can (and should) acknowledge that math give something like an ‘accurate description’ of reality.– MarkOxfordCommented Feb 6, 2018 at 20:17
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11: we do not say that, and we have plenty of different ones; 2: they are not independent of reality but transparently derived from applications, in less than six degrees of separation. See What are the differences between philosophies presupposing one Logic versus many logics? Inconsistent Mathematics, Constructive Mathematics, Predicative mathematics.– ConifoldCommented Feb 6, 2018 at 21:24
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1: here's Stephen Wolfram talking about a universe of different "mathematicses" (axiom systems) and our particular choice being just a historical artifact– ngnCommented Feb 7, 2018 at 1:30
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Why do you confuse logic and mathematics? There are distinct TYPES of logic: Aristotelian logic, mathematical logic, modal logic, etc.– LogikalCommented Apr 12, 2018 at 17:37
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Both of these have been asked, recently, and were already duplicates then.– user9166Commented Apr 12, 2018 at 19:38
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2 Answers
Who says that you can't have other logic or mathematics than we have?
This is simply not the case, for some alternative logic systems see here, and in mathematics new mathematical systems can be constructed by changing the axioms to produce difference formulations. Some of these can behave quite differently from conventional mathematics.
We use the systems that we use because they have proven useful and, to a large degree, behave in a similar way to the way the observable world seems to work.
Logic and mathematics are not independent of reality but have been abstracted from reality and are relying on tools of reality when being pursued and expressed in monologue, dialoge and general discourse.
Unfortunately this fact has not been recognized even by otherwise very able men. Einstein asks: "How is it possible that mathematics, which is a product of human thinking independent of all experience, fits reality in such an excellent way?" [A. Einstein: "Geometrie und Erfahrung", Festvortrag, Berlin (1921), reprinted in A. Einstein: "Mein Weltbild", C. Seelig (ed.), Ullstein, Frankfurt (1966) p. 119]
Without mental images from sensory impressions and experience thinking is impossible. Without reality (which includes the apparatus required for thinking as well as the objects of thinking – we never think of an abstractum "number 3" but always of three things or the written 3 or the spoken word or any materialization which could have supplied the abstraction) mathematics could not have evolved like a universe could not have evolved without energy and mass. Therefore real mathematics agrees with reality in the excellent way it does.
Einstein answers his question in a relativizing way: "In so far the theorems of mathematics concern reality they are not certain, and in so far as they are certain they do not concern reality." [loc cit]
He states a contraposition (R ==> ¬C) <==> (C ==> ¬R). Both statements are equivalent. Both statements are false. To contradict them a counterexample is sufficient. A theorem of mathematics is the law of commutation of addition of natural numbers a + b = b + a. It can be proven in every case in the reality of a wallet with two pockets.
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It's not really clear how this addresses the question. Commented Feb 11, 2018 at 22:40
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@possibleWorld: The question is: Why do we say that there can't be other logics/mathematics than those we have? The answer is: There is only one possible world and only one consistent abstraction of this world.– Hilbert7Commented Feb 12, 2018 at 16:13
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@Wilhelm: Is there only one consistent abstraction of this world? That seems a pretty incredible claim. It's certainly true that conventional logic and mathematics are powerful and useful tools for describing it but I'm entirely unconvinced they're the only way of doing so. Quite apart from anything else, there are a great many highly similar means of construction. Commented Apr 13, 2018 at 15:02
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@Jack Aidley: I agree that in details there can be big differences, but there are no contradictions of basic mathematics and logic possible - at least as far as these are experimentally verifyable. (Transfinite set theory does not belong to this realm.)– Hilbert7Commented Apr 13, 2018 at 15:30