# Applicability of Mathematics

Suppose that an alien civilization exists, in a planet somehow similar to our own (oxygen-based, plants, animals), in an evolutionary stage similar to ours (large cities, advanced communications, near-planet space travel etc.).

It's obvious that they must have developed Mathematics in one way or another. Probably in some domains their Mathematics may be more advanced and in some other domains it's the other way around.

The question is - regarding the common domains - this: are the Mathematical concepts and constructs (nearly) the same; are these Mathematics "common" regarding the same applications in real life?

In a way this question "touches" the "invented or discovered" question regarding Mathematics (already heavilly discussed), but what I am really interested in, is if there must be a "convergence" in the concepts, or by stating it in another way: will a potential interraction (of the two civilizations) be of huge significance (related to Mathematics) or just a matter of "translating" (mapping) one concept to another?

Note: I am not talking about simple concepts as numbers and operations, but of advanced mathematical constructs that are lying behind our nowadays civilization.

addition: Ok, I will rephrase the question clarifying some things, taking into account some already posted answers:

Our mathematics are generally based on axiomatic foundations. Gödel proved that mathematics is incomplete: whatever system of axioms we assume, there are statements that are true but that cannot be proved using only these axioms.

(For example, Goldbach's conjecture is one of the unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers. The conjecture has been shown to hold for all integers until to 4×10^18 but remains unproven. Some beleive that by starting with a different set of axions it could be proved.)

It is reasonable to accept that the alien civilization would have a different set of axiomatic foundations in the respective math domains. So then, there must be some "things" they have proved that we haven't and the other way around.

1. Do you agree with the conclusion?

2. Would then this be an inconsistency, in the then new world of mathematics (the two combined together)?

3. How then would mathematics evolve, as a response to such a confrontation?

• Presumably, they will have different Axiomatic foundations. Similarly, I imagine in 100 years, we Humans will have different Axiomatic Foundations. If one Axiomatic framework is better for proving things, or has an easier way to be implemented into other disciplines, we would adopt the other axiomatic system. I doubt we could combine the systems, I wouldn't expect two random axiomatic systems two be consistent ( or "combinable- as the syntax and meta-mathematics would be different) Commented Nov 5, 2023 at 22:36

I suspect your question might be closed on the grounds that it is inviting speculative options, but I will take the opportunity to put forward a contrasting alternative to Jo's excellent and thought-provoking answer.

Mathematics is a huge subject, and the boundaries of the subset we have explored have been pushed outwards in all kinds of directions over at least a couple of millennia. Some mathematical ideas are relevant to physics and various branches of engineering. My expectation, therefore, is that if the alien civilisation has developed similar capabilities to ours, they will have encountered and developed the necessary models of physics, electronics, mechanical and structural engineering, and so on, which means they must have explored the related parts of mathematics. Whether they articulate them in a similar way, is another matter. There are umpteen different ways of expressing the models of quantum theory, for example, some of which are superficially quite unlike others, so it is possible to adopt different approaches to address essentially the same question.

Whether the civilisation will have explored the same subset of pure mathematics as we have explored to date seems highly unlikely to me. The areas in which intellectual effort is focussed, and how much effort is expanded, is largely a social and cultural matter. Perhaps the alien civilisation will spend much more or less on funding for mathematical research. Perhaps their priorities directing research could be quite different from ours. Perhaps the development of mathematics on their planet might have been influenced by dominant individuals who had different interests from their counterparts on Earth.

• good job Marco! Commented Nov 4, 2023 at 14:38
• ok, so these "necessary models of" could be different or articulated in a different way - I agree - but the "related parts of mathematics" you say, should they be mathematically equivalent? Or to put it in another way, if you "merge" these mathematics, would there exist inconsistencies? Commented Nov 4, 2023 at 15:47

I assume that we agree that one can only speculate about mathematics on other planets. Because we only know one single case, mathematics on earth.

1. Nevertheless let’s assume that the hypothetical society also counts things and uses natural numbers. On earth this step has been invented several times in human societies.

Hence I speculate that at least in some societies some people generalize and introduce mathematics as a theory of abstract entities.

I consider the concept of prime numbers one of the basic concept after the invention of numbers. Until today questions about prime numbers trigger the development of new mathematical concepts, which are abstract, sophisticated and powerful: line bundles, scheaves, schemes, homology and cohomology etc.

I do not expect that we will find the same concepts in any society which develops mathematics: As an analogue I consider the fact that sometimes there have been invented different proofs for the same theorem. For more details and an example see https://math.stackexchange.com/questions/1009922/what-are-the-theorems-in-mathematics-which-can-be-proved-using-completely-differ Hence I expect that the hypothetical society will develop also quite different concepts and mathematical theorys than we did on earth. The concepts will not correspond bijectively, i.e. they do not carry over in both directions.

2. In one of the rockets determined to navigate through the interstellar space there has been included the “Pioneer plate” with some information about out solar system and its planets. The representation uses numbers in binary code from the mathematical side and information based on the hydrogen atom from physics. This fact testifies the hope that beeings from other planets can decode our basic concepts from mathematics and physics. See the Pioneer plate

3. Added in reply to a comment of the OP’s, asking

"Could then if the two math worlds face each other, we all realize that there are contradictions?”

Under the assumption that the two math worlds rely on the same 2-valued logic and that both require that mathematical statements have to be proved – as we do on earth – there cannot be contradictions between the two math worlds. Alternatives yes, but contradictions no.

Of course one can question whether the hypothetical math world employs a different logic, which disposes the law of contradiction. Such attempts have been made under the name “paraconsistent logic”. But I do not understand the use of such a logic calculus, see paraconsisten logic.

• ok, I follow your thinking. But can't we assume for ex. regarding flat space geomerty (Euclidean) that the same theorems (major understanding points or results) exist (independant of axiomatic foundation)? So we come up to the same conclusions. And going up in complexity, is there a point when we start making different conclusions regarding other domains of math? Could then if the two math worlds face each other, we all realize that there are contradictions? Commented Nov 4, 2023 at 15:24