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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?

Mauro ALLEGRANZA
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Hare Krishna
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  • "someone do maths and logic as 1+1=3 from basic axioms because of determinism under normal means of symbol." what does it mean ??? – Mauro ALLEGRANZA Dec 29 '20 at 16:58
  • "one thinks √2 = 1.4142… , other √2 = 3.2651890002…" Where? In what school ? – Mauro ALLEGRANZA Dec 29 '20 at 17:00
  • We "work" everyday **with** mathematics and we are confident that it is right: because God created it? maybe. Because our brain is "hardwired" with it? maybe. Because the "community" of mathematicians agree on it? maybe. – Mauro ALLEGRANZA Dec 29 '20 at 17:02
  • "one thinks √2 = 1.4142… , other √2 = 3.2651890002…" : Nope. Mathematics can be thought of as a system; it is a puzzle which, so to speak, "fits" together. Therefore unless you change *rules* or what symbols √, 1,2,3,4,5,6,7,8,9,0 mean, √2 is always 1.4142. – Ajax Dec 29 '20 at 17:12
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    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
  • @armand Why not? If everyone is programmed to be wrong then there is no way to break out of it even by accident. I do not think free will is necessary, but one does need indeterminism to make direct sense of counterfactuals involved in setting up testing: if I do one thing this happens, and if I do the the other thing that happens. The breakdown of falsifiability is a well-known consequence of [superdeterminism](https://en.wikipedia.org/wiki/Superdeterminism):"*nature could determine what our questions are, and that could guide our questions such that we arrive at a false picture of nature*". – Conifold Dec 30 '20 at 04:13
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    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 04:37
  • "non sequitur" might be too strong, but it's definitely not obvious enough that I can accept it at face value. Is there a complete development of this idea ? Someone mentioned Kant the other day. – armand Dec 30 '20 at 05:03
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    @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 06:27
  • They need not, but they obviously do simply because they work. The proof of the pudding is eating the pudding. I dont care who said it, what they said still does not make any sense. Beware to not read too much into citations in layman terms from quantum physicists... – armand Dec 30 '20 at 08:37
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    @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 09:38
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    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
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    got it. "apples fall each time we look, but, if determinism is true, our understanding of gravity *might be* flawed because there *could be* some prior correlation making it so that apple happen to fall only when we look" ? this is nonsense. And what if we are a brain in a vat and what if the world was entirely created yesterday ? As long as such a correlation is not demonstrated, this is highly unparcimonious. I can understand why it is a major problem for QM, because it breaks an assumption of rigorous probability calculations. But in our everyday life, apples do reliably fall. – armand Dec 30 '20 at 16:51
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    @armand How do you expect such a correlation to be demonstrated if it is determined that it will not be? It is ironic that you judge things as nonsense based on common sense stereotypes when your assumption of superdeterminism is at variance with them. Forget about "sense" and track the logic explicitly, and not just extravagant examples. The likely generality of experimental conclusions is based on the assumption that experimental conditions are representative, that randomization washes out spurious correlations. Determinism's confounders are far less contrived than evil demons or vats. – Conifold Dec 31 '20 at 05:57
  • That's your problem. It is you who is in solipsism/omphalos theory territory. Have fun. – armand Jan 04 '21 at 04:12

3 Answers3

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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.

armand
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    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
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    @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
  • `At no point is a decision ever involved. We don't decide the result, we observe it.` It is miraculous how one can conclude so wrongly after an otherwise sound bulk of an answer. We _understand_ (= nonreflectively interpret) an observation, don't we?- here is where decision is lurking, and freedom of it, too. No, I do not ready-observe that the circumference is 2pir. I get to agree with it newly every time I come to recall it, and only because me being currently in the path to be precise and in consent with the "rules", (cont.) – ttnphns Dec 29 '20 at 22:07
  • (cont.) else the length might easily be not 2pir for me, really. Sure, the latter would be invalid from praxis pow, but validity notion is orthogonal to the problem of decision/freedom. – ttnphns Dec 29 '20 at 22:07
  • ...To reword it: It is because I'm freely wishing to build me a right house I feel obliged to treat the possible relevant rules earnestly with caution, and only in this context the 2pir law can be observed as true. – ttnphns Dec 29 '20 at 22:30
  • @ttnphns: sure, if you presuppose any decision is made by free will, then it's not difficult to conclude we have free will. That's just begging the question. – armand Dec 29 '20 at 22:36
  • But that is what I've been saying in conversation with you all along. (And not only I sole.) Free will should be assumed, not proven. https://philosophy.stackexchange.com/a/65137/28067 – ttnphns Dec 29 '20 at 22:43
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    @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
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    @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
  • @armand, pardon for returning yet again (I'll leave now). Not _any_ decision is free, but a so called "fundamental" one, also called "project", which open us world-where-we-are-in. There is a hierarchy of decisions, "derivative" ones being less free. https://philosophy.stackexchange.com/a/45304/28067. Good night, cheers. – ttnphns Dec 29 '20 at 23:07
  • Sure, if it's fallacy party time let's add special pleading to begging the question. – armand Dec 29 '20 at 23:17
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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.

causative
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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...

Patrice Ayme
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