Are there at least some mathematical truths that would have been different had the laws of physics been different? Probably most mathematical truths would not change, but are there some that would? Or do all mathematical truths come prior to physics?
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It's the same answer as to your previous question, Are truth values of all mathematical statements immutable?, depends on what counts as "mathematical statement". Laws of physics have mathematical concepts in them (derivatives, integrals, etc.), so they are "mathematical" and obviously contingent on themselves. Mathematical abstractions, on the other hand, are divorced from any physical contingency by definition.– ConifoldDec 16, 2020 at 22:03
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3 Answers
On the view that the existence of mathematical objects and/or the truth of mathematical statements are contingent on us, humans, if the laws of physics had been different, due to the fine-tuning argument, we would almost certainly not exist along with all other forms of life.
Therefore, any mathematical statement that is regarded by mathematicians as true would be false, or at least fail to have any truth value at all.
This seems to me the only way mathematics can depend on physics. The orthodox views, however, is that mathematical statements are necessary and hold in any logically possible world, and physics is essentially concerned with contingent truths about this particular world, so there is no dependence of the sort you are asking about (besides the one I mentioned earlier).
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That mathematical statements are contingent on us does not correspond to intuitionism, even roughly. Intuitionistic truths derive from faculties of an ideal reasoning subject abstracted from space, time, physics or our existence. What you are talking about is postmodernistic social constructivism, not intuitionism.– ConifoldDec 16, 2020 at 23:38
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According to Wittgenstein, mathematical and logical statements are mere tautologies, it is to say, they establish a proposition A is equivalent to a proposition B, entirely based on the definition of A. In other terms, B is nothing but another way to say A, and no new knowledge is created (Kantians would say it is an analytical statement, not a synthetic one).
So for example, to say "sum of forces = mass * acceleration" is just another way to say "acceleration = sum of forces / mass". The rules of mathematics just tell us that it's ok to juggle the symbols in this way, and that if we take the former to be true, we have to admit the latter is too.
Physicists have understood that, by formalizing models of reality in mathematical statements, they can establish equivalence between a fact they measured and another they want to deduce. By plugging measured mass and acceleration in their formula, they can deduce the applied force, and also inversely deduce the acceleration applied to an object of known mass when applying a known force.
Yet, "laws" of physics are still mere models, and not reality. The Newton law of gravity has been a "law" only so far until we discovered it didn't work in some cases. When the reality does not fit the "law", it is obviously the model that is broken, not reality.
Thus, had the laws of physics be different, physicists (assuming they are not different enough to result in no physicist existing, of course) would have devised different models, fitting this other reality, but the mathematics they would have used to formalize and establish equivalence between formulas would still be the same.
Certainly mathematics is based, in a way, on "real world" physics. If, through some mystical means, gathering 4 sets of 4 eggs together always resulted in 17 eggs then the rules of multiplication would likely be different.
The fundamentals of math are based on reality. But then so is physics. Which came first, counting chickens or counting eggs?
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What the mathematician studies might (the key word is "might") be inspired by real life things, but the statements of mathematics are necessary. The physicist "studies real life" and its fundamental contingent truths. So, what you said, namely: The fundamentals of math are based on reality. But then so is physics. is false.– AlexDec 16, 2020 at 23:42
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@Aleksandr - So if "mathematics" declared that 1+1=3 that would be meaningful? Wouldn't people very quickly invent "mathbmatics" and "mathcmatics", trying again and again until something was found that matched reality? Dec 16, 2020 at 23:46
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There is no "declaring" a mathematical statement being a theorem in mathematics. So, there is no declaring "1+1=3".– AlexDec 16, 2020 at 23:49
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Exact same answer. There is no "defining" a mathematical statement to be a theorem in mathematics. To believe otherwise is to ignore how mathematics is actually practiced. I will grant that some mathematicians believe that the symbol "2" is just a shorthand for the expression "1+1". Will not work with "3", because the resulting statement is meaningless.– AlexDec 17, 2020 at 0:55