# Can there be a universe with different mathematics?

I do not know what exactly I mean by other universes, but I just have a feeling that mathematics is somehow inevitable.

For example the law of "excluded middle" (LEM). If there are aliens, can we assume they logically accept the law of "excluded middle"? If there are different universes, can they develop completely different mathematics?

edit: I know that there are formal systems that do not accept LEM, but what I really mean is that they really think LEM is a bad idea, as we think `p ^ !p` is a bad idea, they just intuitively do not think LEM is right, and the canonical model does not include LEM at all.

Anyway, this is just an example.

• It is law of excluded middle and it dates back to Aristole [see Wiki: "The earliest known formulation is Aristotle's principle of non-contradiction, first proposed in On Interpretation, where he says that of two contradictory propositions (i.e. where one proposition is the negation of the other) one must be true, and the other false"]. There are modern logician and mathematicians that do not accept it [again, see Wiki: "Many modern logic systems reject the law of excluded middle"]; so it is not necessary to wait for ET... Commented Feb 17, 2014 at 15:59
• +!: not a bad question. It shows how pervasive the idea of non-contradiction is that one has to go to another universe to think of them, which is in effect what we do when we interpret universe in the appropriate way... Commented Feb 17, 2014 at 18:37
• Yes and no. Mathematics and logic are not exactly empirical, so universes with different laws can be accomodated (perhaps not most naturally) by the same mathematics and logic, see Is Logic Empirical? and Is geometry mathematical or empirical? Commented Dec 20, 2016 at 1:46
• @MauroALLEGRANZA --- "...where he says that of two contradictory propositions (i.e. where one proposition is the negation of the other) one must be true, and the other false"]".--- This is not a statement about Reality but a rule for the dialectic (The Rule for Contradictory Pairs). It is not optional and it would apply regardless of the nature of Reality. Where we ibnore it the dialectic won't work. Garbage in, garbage out, as they say. . .
– user20253
Commented Feb 14, 2020 at 11:02
• Depends on whether or not there can be a different universe. An empiricist would argue, based on current science, no.
– J D
Commented Feb 14, 2020 at 16:17

Yes, we live in one. What was regarded as mathematics 2000 years ago is not what we regard as mathematics today. Gauss published the first acceptable proof of the Fundamental Theorem of Algebra; but Gauss's proof would not be acceptable from an undergrad today. Standards of rigor, as well as our understanding of the topology of the real line, have changed considerably since then.

Mathematics is a historically-contingent activity of humans. Not only could mathematics be different on a different planet or in another universe; which are of course unprovable one way or the other; but mathematics could and actually has been different at different eras on this planet.

Just consider the rise of computers, experimental mathematics, machine proof systems, and computatibility theory. It's likely that math in 100 years will be very different than math is now. Zermelo-Fraenkel set theory is less than 100 years old. What if on some other planet they never discovered it, but rather skipped to some other framework?

Now, you may be referring not to the mathematics as a historically and culturally contingent human activity; but rather as some sort of Platonic thing that is "out there" that we can discover. To which I'd ask: Where is your evidence that such a thing exists? And if it does, then which human mathematics is the one, true mathematics? The math of 1000 years ago? The math of today? Or the math of 1000 years from now?

I do realize that you're asking if it's possible that in some other universe, 2 + 2 is 3. I have no idea. I don't think the question is meaningful. I think I'm wearing my formalist hat today.

• If you presume that the Universe is "natural" then I doubt there can be much different math from one place to another. If however we are living in, say, a simulation (A) you might expect that the head gamer can do as he pleases with the rules. In another simulation (B) it could be set up so that if you physically add two apples to two apples you always end up with three apples. We in simulation A would ask, "Where did the missing apple go". In B it always happens like that so they would likely formulate their mathematics accordingly even if they notice the peculiarity of the missing apple. Commented Dec 19, 2016 at 20:29

The interesting issue is not if somewhere somebody can "think of" a contradictory mathematics.

In this world, there are alredy researches about inconsistent mathematics (see SEP Inconsistent Mathematics.

The relevant issue is : how they works ? what we can do with them ?

I think you are referring to simple math , like arithmetic. A universe where counting does not exist. You can't count objects. Numbers have no meaning. Space and time have no meaning and there is something else in its place. A place where logic does not exist. I don't have the answer, I am just trying to clarify your question. Why would some kind of mathematics be inevitable? There could be ways a universe could exist without math.

I think it is a very good question and I would like to hear some answers myself. Could there be a universe where the concept of order and logic and numbers and objects and space and time don't exist ? This would preclude math as we know it. Maybe there is something better than math in other universes where the concept of an object does not exist but other things exist that we cannot fathom in our universe because it is impossible to even imagine.

1. Are you asking if in another universe 2 + 2 = 5 is true but with the same meaning of the concepts behind “2”, “+”, “=” and “5” as for us? It can only be, if successful (correct) rational inference might yield truths which are not necessary truths. But if that’s so, is unanswerable.

2. Are you asking about the application of mathematics? That in another universe nothing in our mathematical arsenal could be successfully applied? That intelligent species there apply a kind of mathematics completely different? This is a much weaker claim than 1. But still basic arithmetic is so fundamental, how can there be a world with intelligent life where there is nothing to count? We can count any distinct “things” in the broadest sense. We can count apples, words and even ‘objects’ from mathematics and logic itself (including numbers: there are four numbers in the set {x : 2 < x < 7}). So it might be that that basic arithmetic is so fundamental that any kind of reality must metaphysically necessary adhere to it – though this is not true for other branches of mathematics like geometry. That our geometry gives us no insight in how space must be structured, is well accepted.

“Perhaps in another life we may be able to obtain insight into the nature of space which is now unattainable. Until then we must not place geometry in the same class with arithmetic, which is purely a priori, but with mechanics.” – C. F. Gauss

But again, this is ultimately unanswerable, too.

So, I've been thinking about this. I wanted to ask the question but makes sense there already is one. So let me have my take at an answer. Don't know if ur question is exactly this, but it makes sense that math in different universe with different rules would have 100% completely different math rules Humans would create to do math. So with same rules as our universe would humanity have created some different rules for maths to calculate on?

I think with basic mathematics (arithmetic and such) things won't be much different cause these aren't rules we created by ourselves but the way the universe works and we build our own maths from that using proofs and principles. So when humans did their calculations and we depended on our previous postulates or theories that made sense. So one mis-step into different ways or looking at things in some other universe may have a large effect into a huge difference from our advanced maths. They may find patterns where we couldn't and we may have found patterns where they didn't.

And that's why I believe math is always changing through out our history, current way of doing things even though it makes sense isn't the only way and it's always open for innovation and change, and that's why it always changes though time and u won't find the exact same maths we do now in 500 years or maybe even 100 years who knows. Maths is just so interesting

• Welcome to SE. Your answer has some interesting points to make, but it would be much better if you had explained yourself more fully. Each of your paragraphs considers a different interpretation of the question and gives an answer for that case. that. But you don't make that clear. Nor do you explain the reasons for your answer in each case. It would also help if you started by considering the possibility (of not accepting the Law of Excluded Middle, which is in the question. Commented Feb 15, 2023 at 18:45

If you go with the physical notion of multiverses then the mathematics will be the same. This is because at bottom the same theory holds with differently chosen universal constants. If you go with the Spinozoan notion of mutiverse then we have two possiblities: worlds where different mathematics holds, or rather, whefe the same fundamental mathematics holds but where different mathematics are prominent. This is based on his posit that two essential modes are thought and extension. However, there are worlds where mathematics is not possible. This is because there are an infinity of modes which are essentially indescribable because they don't fit into the modes of either extension or thought - the two modes that makes things intelligible to us.

I think it might be possible, assume that you have the capability to create a universe and create individuals with free wills, and you inserted a rule in the universe saying: "Every time some individuals finish counting something, you always remove one object physically from the universe".

After eons, the individuals in your universe began to develop consciousness and thus started counting objects, say from our perception that there are 5 bananas, but when they finish counting, one of the bananas just disappears. So they are left with 4 bananas.

Assuming counting is also the foundation of arithmetic, they might develop mathematical rule such as 1 + 2 = 2.

Because that is always true according to their observations.

But of course, this universe might not possibly exists, but that doesn't nullify the possibility that a universe with different mathematics exists unless you only want to talk about a universe that can possibly exist.

Mathematics is models reality, so, if reality changes, mathematics must change.

According to Kant, Mathematics raises as a priori knowledge, not directly from experience (there are no numbers in nature), but indirectly (we create the concept of object, unity, multiplicity, number, etc. in order to address natural phenomena). That is, we can't know spheres per se, we just internally create the concept of sphere in order to know, for example, apples. All logic and mathematics are just part of metaphysics (we can't use the scientific method to prove logic and mathematical facts, because any method and any part of science already depend on logic and mathematics; both are a priori for science).

So, all mathematical objects, like lines, spheres, numbers, etc. are just abstractions of real facts. Evidently, there are mathematical ideas that don't correspond trivially to real objects, like spheres might be understood as an abstraction of apples: they are just inferential mathematical results.

Of course, if the laws of nature change, mathematics would be different. For example, if there would be no objects (e.g. the alternative universe would be something like a foam), numbers wouldn't be necessary anymore. But the entities that live inside would develop new concepts to describe the foam-universe, for example, vector-like elements, with which they would perform other types of mathematical operations. And they would never know what a number is.

• Mathematics doesn't necessarily model reality, it models abstract truths. Our choice of what math to explore is inspired by reality (from counting cattle herds to launching satellites) but it is not a theoretical necessity. Commented Oct 23, 2023 at 13:09
• @WouterLievens Never said it is necessary for math to model reality. All your statements are contingent. Math can model metaphysical facts (what you call abstract truths) or physical facts. But mathematics is developed a priori to experience (see Kant's Transcendental Aesthetic), so its structure depends on nature. Commented Oct 23, 2023 at 17:44