I've this friend who asked me to ask this to some respectable mathematician or physicist. So, I decided to post this here.

Can there be universe with different laws of mathematics?

I tried to tell him that mathematics is how we define it.

He asked me if 2+2 could be 3. I do not have a clue what that means, so I think he wants to know if the Laws of Thought could be violated.

I said that even if it is, we cannot think of it so why bother?

The subsequent discussion was very cryptic for me but in conclusion he asked me to post it here.

So, please leave your comments. Thank You.

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TopNewestThis is always an interesting subject. Theoretical physicists and cosmologists imagine there might be "different universes", because "laws of physics might be different in each". The analogy with mathematics is that it has definitions and axioms, which can "be different", so we already have "different abstract universes of mathematics". But it doesn't stop us from discovering and understanding those different abstract universes of mathematics, which is currently easier for us to do now than it is for us to discover other "different universes". In fact, we're able to imagine such different universes and their different laws of physics through the power of mathematics.

See comments below about Jake Lai's question about the Axiom of Choice.

A related question, not synonymous with yours posted here, is whether or not we could someday run across an alien civilization that uses mathematics that contradicts ours. But if any two peoples, whether they're presently on Earth, or one of them is in the past or in the future or alien, at least agree to definitions and axioms of a given branch of mathematics, they are going to arrive at the same conclusions. Our mathematics may differ because it's possible that we're starting with different definitions and axioms---but nothing precludes us from working out consequences of such different definitions and axioms right here and now! Ready to try your hand at imagining some alien mathematics?

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So it's your contention that, if we can agree to a common set of axioms, then we and the aliens will necessarily always reach the same conclusions. This seems like the mathematical version of the premise to "Chance and Necessity" by Jacques Monod. If we and the aliens by chance have been working with the same set of axioms prior to "first contact", then by necessity none of our conclusions will contradict theirs. Would we not also have to have been working with the same set of rules of inference, or are these not in play?

I am also wondering if all the elements of the (short) list of "Laws of Thought" are challenged by quantum logic. I can see that being the case for the Laws of Non-Contradiction and the Excluded Middle, but the Law of Identity would still seem to be on solid footing. You would know better than I would, though, so I'm just curious what your thoughts are.

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"If we can agree to a common set of axioms, then we and the aliens will necessarily always reach the same conclusions" depends on both of us agreeing to a kind of a mathematics that delivers repeatable results. It's just one kind of formalism, one that follows, say, the classic three laws of logic as you've listed. But it's hardly the only formalism possible, and quantum logic hints at many more. In a way, as art imitates life, and often conversely, physical reality follows mathematical reality, and conversely as well. We're familiar with Newtonian physics, which is repeatable and deterministic, and yet there's the reality of quantum physics, which are not, but nevertheless there exists a kind of a logic (or madness) behind quantum physics. It's quite possible we could meet with advanced aliens that prefers thinking at the quantum level (because in many ways it's actually more efficient in delivering results), so that we'd get the perception that they must be engaged in very intelligent abstract thought, and yet such abstract thought appear to be very alien to our own "kind of mathematics".

But, as I keep saying here, the power and the beauty of mathematics is that there's nothing to PRECLUDE us from imagining and developing such "alien mathematics"!

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Depends. What does he mean by "3"? Or by "2"? For example 2 is defined as the successor of 1, which is why 1+1=2. If 3 is defined as the successor of 2, then we will have 2+1 = 3. But, if instead, 4 was defined as the successor of 2, and 3 is defined as the successor of 4, then yes we do have 2+2 = 3.

Alternatively, \( 2 + 2 \equiv 3 \pmod{1} \). So yes, there are universes (even on earth), where we allow for that arithmetic.

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Could there be an universe where the laws of thought do not hold?

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We're almost already doing that right now in the absolutely weird and often incomprehensible field of quantum mechanics. Quantum mechanics has already stirred vigorous re-examination of the foundations of logic, i.e., it's shaking up our notions about "laws of thought". If you want to see some otherworldly "laws of thought", check out quantum logic.

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Maybe there is another type of maths where you can divide by zero....

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Wheel theory!

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Someone reinvented the wheel!? :) I hadn't heard of Wheel Theory before, so thanks for mentioning it. It's a bit weird, but apparently it serves as a better model for the way a computer actually does arithmetic, with "apparently" being the operative word here. Of course, as per our discussion, it doesn't need to be useful, it just needs to be. :D

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Well, some people believe that other universes have different physics from ours, so maybe. Though, scientist do not really believe the different physics theory because there is no proof to support it.

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Is the axiom of choice correct?

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It's an axiom, which means it's up to you if you want to assume it to be true. You can equally assume it to be false, and thus define a different branch of mathematics. The classic example is the Parallel postulate, of which there are at least 3 variations, 1) given a line and through a given point not on it, only 1 line can be parallel to the first 2) ditto, but no lines can be parallel, 3) ditto, but many lines can be parallel, the 3 variations which have led to 1) Euclidean geometry, 2) Riemann geometry, and 3) Lobachevskian geometry. All of them "not incorrect".

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Good examples! I just wanted to bring up the often controversial axiom of choice to make the point about there being no absolute mathematics, and that different paradigms of mathematics already exist simultaneously, each with their own unique set of axioms.

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The ultimate goal of mathematics is to eliminate any need for intelligent thought.

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I STRONGLY and VOCIFEROUSLY disagree with that view!

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I second Michael on this one; see my comments to Bobbym None below.

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Maybe, but over here that is not what I mean by mathematics.

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@John Muradeli Your views please?

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NO!

Or, that's my initial response.

Then, YES!

Either way, I will put this on my "to do when you have room to breathe" - list. Thanks!

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@Arron Kau , @Calvin Lin , @bobbym none : I'd be obliged to have your opinion here.

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"The ultimate goal of mathematics is to eliminate any need for intelligent thought." Whitehead may have been prophetic in that view. The Discretists led by Zeilberger and us computational boys could not agree more. The formalists think that mathematics is just a game with rules invented by humans for humans.

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Mathematics has no "goal". It has axioms and rules of our choosing, (see Michael's comments above), can be used as a prosaic tool or an artist's paint brush, be a salvation for compulsive minds, and more, but it certainly has no overarching goal. Philosophy has goals but no rules, physics has both goals and (some) rules, and mathematics has rules but no goals. AI researchers and advocates may share Whitehead's view, but I doubt Gödel or von Neumann did. To this end, a quote from the latter: "The whole system [i.e., "pure" mathematics] seems to function without any direction, without any reference to usefulness, and without any desire to do things which are useful." The fact that so much of this "uselessness" eventually finds a use is besides the point; just because something ends up being useful doesn't mean that there was any initial objective to be so.

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Over here, we are picturing mathematics as an activity rather than a structure. I do not mean that mathematics is supposed to take over intelligence but think of it, the more mechanical the results the cooler it is. Would you rather prove a theorem that is much generalised or would you want it to have five cases? Which do you think would be more elegant?

It is not possible to define intelligence but let's accept equating it with raw processing power for a second. Which algorithm would a mathematician want to develop more? An O(n^2) algorithm or an O(n log(n)) algorithm?

Think for a second what if the quartic formula was as easy as the quadratic formula. Would you call the simplifier a good mathematician?

Isn't a mathematician with a general solution with a problem far better than a mathematician who can solve a few cases and gives you an algorithm that is time consuming and have to make you think how to optimise it? E.g, compare Fermat's Little theorem and Euler generalisation.

And oh, take the popular example of Wolfram|Alpha. Many people on Brilliant (not me) look down upon its users believing that it is of low intelligence to use it. But wouldn't you consider yourself a great mathematician, if you could program your own Wolfram|Alpha?

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Even though Alfred Whitehead spent much of his life in trying to develop the "definite axiomatization of mathematics" (and failed grandly!), I have my doubts that he actually said that quote. It doesn't quite sound like him.

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Though mathematics actually encourages more intelligent and creative thought, it is driven by the inspiration to eliminate the need of it from wherever possible.

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In trying to verify the real origin of the quote, "The ultimate goal of mathematics is to eliminate all need for intelligent thought", attributed to Alfred North Whitehead, I've accidentally found a nice PDF book (free!) that seems like an excellent reference book for problems typically found in Brilliant

Concrete Mathematics

I still can't find out if Alfred is actually the author of that quote. But the context of it has to do with "mechanization of mathematics", i.e. the project of putting it all into a computer in algorithmic form, so that the "drudgery of mathematical computation", including proofs, is eliminated, and we all just go ask Wolfram Alpha. I'm sure Stephen Wolfram would just like that.

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Only kidding, I found it in A=B

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