How can we know that our mathematics is right at all without free will?

For example, someone do maths and logic as 1+1=3 from basic axioms because of determinism under normal means of symbol.

Or like doing all of mathematics of some comlicated integral and equations.

Because if it happens in brain, and if brain work deterministic, so whatever happens one who observes can't know whether all he thinks or all calculations or all logics he doing is right at all.

If because of determinism, someone doing logic as 1+1=2 and 1+1=3 and 1+1=4, all are thinking themselves right as to what deterministically for them to think that right.

Or to give more example one thinks √2 = 1.4142… , other √2 = 3.2651890002…, and someone else would be √2= 9190101.26774749201….., all thinks they are doing right and exactly as, if that is deterministically determined there.

I believe there are two main point of view by which it may taken into account as one of platonists in which mathematical objects there in platonic plane, and other non-platonists views.

So question is:

What methods can we use to know whether our mathematics is right without free will? And is free will requires to know about right or consistent logic and mathematics?

• Seriously, this makes no sense at all. I can't fathom how all of this follows logically. Yet it's not the first time I see this very strange argument, so I think it is a relevant question deserving a response. – armand Dec 29 '20 at 17:16
• There is no "programming to" in determinism, it's a weird assumption to go by, if anything it shows a failure to get out of a teleological way of thinking, like your quote about superdeterminism. People whose brain is too deficient to see that 1+1 != 3 even in the face of evidence are long dead. I fully disagree that we need indeterminism to do tests, it's a non sequitur. If anything, "if I do one thing this happens, and if I do the the other thing that happens" is an intellectual process that requires determinism, the idea that the same experimental conditions will produce the same result. – armand Dec 30 '20 at 4:37
• @armand You are putting too much on words and motivations, "programmed to" vs "determined", teleology, what intellectual processes require, etc., makes no difference. That superdeterminism forecloses the possibility of uncovering modeling errors is a simple inference from its definition, Wikipedia quotes Zeilinger on it. It is true that we can not tell the difference between superdeterminism and indeterminism empirically, but it is also true that models we would come up with through testing under superdeterminism need not have any relation to the reality they model. – Conifold Dec 30 '20 at 6:27
• @armand I think you missed the point. "They do because they work" is a non-sequitur under superdeterminism, that's exactly the problem. The citation is for the details, but you can easily see yourself that the argument is valid.The usual inference from successful testing to likely match with reality breaks down without indeterminism in testing design. All you can do is dispute the premises or interpretation, but I do not see anything like that so far. – Conifold Dec 30 '20 at 9:38
• It does not follow at all. I don't see he argument is valid at all, so what i am left with is just reassertion of the same claim with no proof, "it becomes obvious once you assume it is", "just look better", or a citation from a guy whose physics skills I respect but gives no argument either. I did my homework and checked for a demonstration, but peanuts. That's pretty underwhelming. "The usual inference from successful testing to likely match with reality breaks down without indeterminism" -> why on earth? that makes no sense. Where is the demonstration ? – armand Dec 30 '20 at 14:43

Mathematics give the ability to make predictions. For example, I can demonstrate that if I make a triangle of sides 3, 4 and 5, the angle between 3 and 4 will be a straight angle. I can use this construction to build my house, the walls will be vertical and parallel, the roof straight, and the whole structure sound. If I am mistaken, my house collapses, I prepare too much or not enough materials. If I can't count the days I don't store enough food and wood for winter, etc... And each time I try, the same trick will always work. You can't build a circle whose circumference is not pi times its diameter.

Even square root of 2, you can verify it is roughly 1.4142 by making a straight triangle with two sides of length 1. Someone who would believe it's worth 3.2651 can check their mistake easily. Someone who would still believe it's closer to 3 than 1 after this simple test is crazy, their belief just don't match reality.

At no point is a decision ever involved. Either your house is sound or it collapses, either the diagonal of your square is 1.4142 or it is not. We don't decide the result, we observe it. Therefore free will or not is absolutely irrelevant.

So, we can check wether mathematics and logic are valid by observing that it works, it involves no decision whatsoever on our part, and therefore it has nothing to do with the question of free will.

• While I agree with you and agree that the question is confusing, I believe it was aiming at axioms, and here we might say judgements of the will enter into the picture. For example, Kroneker's objections to Cantor or judgements about Euclid's fifth postulate, the concept of the infinitesimals in Newtons' calculus, or the real mapping controversies raised by Richardsons' coastal paradox. I'm no expert, but my understanding is there are open controversies in "choosing" axioms, which are referred to judgment and may even have "real" applied consequences. So, free will at some "meta" level? – Nelson Alexander Dec 29 '20 at 19:04
• @NelsonAlexander: But determining the "correct" axioms is not at all based on free will, if the objective is to describe the physical world. From the above examples, without infinitesimals you cannot solve the equations of motion, so they are accepted by necessity, not by free will. – user000001 Dec 29 '20 at 19:18
• @user. I'd have to think more about it. But the fit between math and real-world prediction is never absolute and final. We "disprove" the universality of Newton by tossing out Euclid's fifth postulate and "choosing" Reimann's geometry. But it is not Eddington's experiment that "proves" Reimann's geometry is "correct." It only proves it was the right tool to "choose" for that job. Once axioms are fixed, "free choice" is formally limited, as in chess. But free will is still exercised both in the moves and in the choice of axioms or systems. – Nelson Alexander Dec 29 '20 at 19:57
• @ttnphns: ok. So you're just going to assume you're right ? That's a way to go. I am just gonna assume I am right too, then (^_^)/ – armand Dec 29 '20 at 22:47
• @Nelson Alexander: what disproved Newton is the discovery by Maxwell that light speed is constant and it's confirmation by Michelson and Morley (maybe some other discovery prior, I am no historian). We observed the hole in his theory, which determined us to find a new one, which led to the conclusion by Einstein that space was not euclidean and a system superseding Newton that has worked so far. Unless you too assume free will in any decision (but then why bother arguing ?) I don't see the necessity of free will anywhere in this picture. – armand Dec 29 '20 at 22:54

We just have to be thankful for our good fortune that we evolved to think in the partially-rational way we do, where evidence can persuade us of the truth of a proposition. And we have to be thankful to be born in a society with a well-developed system of mathematics available for everyone to learn. These factors, which lead you to accept mathematical truths, are the result of the circumstances of your birth, not "free will."

It's conceivable that an "intelligent" creature might believe that 1+1=3 or that sqrt(2) = 97 and even that they would would not be persuaded differently by any contrary evidence. But a creature that thinks like this is going to contradict itself and have difficulty using this mathematics to achieve practical goals. A creature like this has some serious internal "bugs" in its method of reasoning that extend far beyond mathematics. As a result, creatures that think like this would have been selected against, and died out. The human method of allowing evidence to persuade us in the way that it does simply happened to be more effective at propagating our genes.

There are two languages: common language, which is messy, and mathematics, which is much more precise, and contains basic physics. Indeed, all common languages are more or less isomorphic (same shape and preoccupations). Mathematics is the part of common language which, given precise axioms, is the simplest and irreducibly deduced from those simplest notions (in physics, thus nature). “Physics” is a compendium of how nature looks, for sure, or how it works, de facto. How nature looks, as deduced from experiments, has varied in the last 100 million years… and that description is getting increasingly precise, as demonstrated by our ever greater power in making nature do as we wish.

But how nature works inside brains has become ever more powerful and precise ever since there are brains, and they have grown. Neurology is an emergent part of nature. Thus it is factual, being natural, and we also call its basic architecture mathematics, when we describe it. For example, basic category theory looks like the simplest abstraction of basic neurology restricted to the simplest axons…

Thus elucidated, counting becomes a matter of neural networks. 1 + 1 = 2 can be directly envisioned as a semantic description of a (very useful) neural network which has appeared in advanced species. That makes “2” a description of some neuronal architecture. There is no free will there. “2” is just the label for a particular type of neural network found in nature.

As a result of being the product of emerging neuronal networks, there is no more free will in “2” than in the Iron nucleus (Fe 56). And so on it goes: “pi” is the length of the circumference of a circle of radius 1. No free will there, either.

Nor is there for multiplication of real numbers. Even better: one gets in complex numbers by trying to build a multiplication in the plane which generalizes the multiplication of real numbers. There is a way to do this (multiplying distances to the origin, adding angles from the real axis): it enables us to get square roots of negative numbers… some numbers which multiplied by themselves, have a negative square. Not much freedom there. But then something spectacular happens: this gives the best description of light (including momentum, energy and polarization)... And as such becomes the basic language of Quantum Physics.

How could that all be? Does that mean that our brain and how we build networks there, is not free from Quantum Physics? Indeed. Let’s inverse the question: how could the brain be free of Quantum Physics, considering, well, that Physics, Nature in Greek, is Quantum? Would that not be considering that brains are not natural?

If somehow there is no free will in the nature of the neural networks (and thus mathematics) we build, where could free will be? Well, in which kind of networks we decide to build, then? The networks themselves, at their simplest, are mathematics, and thus mathematics is digital… So is language. Being digital, and finite (in its mode of construction) make languages and mathematics, limited and pre-ordained. But Quantum Physics itself is based on a continuum, and that brings the freedom… of the butterfly effect. Free will is a subtle thing.

The famous mathematician Richard Dedekind said numbers were the work of God, and the rest of mathematics the work of man. It is probably wiser to acknowledge that we, or at least our mathematics, are the work of physics… self-describing...