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Mathematicians constantly create new axiom systems, and there are different concepts of information. My view is that mathematics gets where I am because mathematicians invented notations to write down the concepts that are discovered on the plane of axioms.

A rock could neither develop an axiom nor initiate a mathematical science of any kind, in fact, no simple inorganic structure could. This is obvious, I know, but the problem is that we have no idea what the minimum biological structure is necessary for mathematical inferences to occur.

What I mean is that our mathematics was developed by us, beings endowed with vision that can understand a specific band of the electromagnetic field, that can translate a specific band of the spectrum of mechanical waves (sound), that interprets tactile phenomena of a unique and singular form and that enjoys a cognitive structure that I probably don't think exists anywhere in the universe other than here.

I believe they could do something like finism, which is one way to do that is taking the axiom of infinity and putting the following axiom: a set has no bijection with a part of its own

That mathematics is a universal language as first imagined Pythagoras of Samos? Would intelligent species from other planets develop the same axioms as we do? And even if they started from different axioms, could they get where we are?

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    $\begingroup$ Is there a question here? $\endgroup$ Commented Jun 28, 2021 at 12:24
  • $\begingroup$ @JeffZeitlin Yep-i-yes! "Would intelligent species from other planets develop the same axioms as we do?" Still, that doesn't mean it's a question we can give an answer too as it is stated :/ ... $\endgroup$ Commented Jun 28, 2021 at 13:05
  • $\begingroup$ @Behemooth Hi and welcome to WB:SE! As your question stands, it's really hard to give any answer as you don't give any details what your aliens are or where are they living. In other words, it's sweeping too broad on a high-concept. You'd need to add a lot (and I mean it, a lot!) more data to find the consequences on maths evolution. On top of species intel, maybe could you focus on a specific part of maths, like a part of geometry or algebra to make it more specific? $\endgroup$ Commented Jun 28, 2021 at 13:10
  • $\begingroup$ Also, check the tour and help center if you haven't yet :). $\endgroup$ Commented Jun 28, 2021 at 13:13
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    $\begingroup$ I would say that all intelligent species will most certainly develop the same number theory as we did -- after all, arithmetic needs no input from the senses and no intuition of Euclidean space: it is entirely abstract. Anyway, I'm voting to close in the expectation that the question will be edited to reflect an actual worldbuilding problem. (Ah, and speaking about mathematics absolutely requires proper terminology; what you wanted to say was that a set cannot have a bijection with a proper subset -- as it appears in the question the statement cannot be satisfied by any set.) $\endgroup$
    – AlexP
    Commented Jun 28, 2021 at 13:23

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Our entire system of mathematics is founded on the concept of unity, and the basic premise that one plus one equals two. Take fractions, for instance. 'one', or 'unity', is subdivided into OTHER 'unities' which are no longer unities, but sub-unities? Why are they not just broken down into more 'unities', such that '1 broken up = 2', or '1 becomes 5', or '1 = 58' in some operation '=' called 'breaking' or 'becomes'? 'Unity' is indeed a very abstract concept that other minds might not develop the same way the human mind has.

Our concept of 'unity' is based entirely on 'intuition' and 'supposition'. A system of math based on an entirely different supposition would be entirely different. We take it for granted, but it was not always so. For instance, the entire concept of counting would be different. More like Roman Numerals, where every 'quantity' would have a different symbol, unrelated to the previous symbol, and there would be no sequential progression from one to the other. In point of fact, some posit that Roman Numerals evolved the symbol 'V' to represent 'a hand's worth' (five fingers) and 'X' 'both hands worth' (ten fingers). Would a person born with six fingers on the hand instead of five see 'V' and 'X' as different amounts than someone with five fingers? If the idea was to use a 'one-to-one' correlation between fingers and the objects being 'counted', one can easily imagine two people, one with five fingers and one with six fingers on their hands, could be in dispute over what 'one hand's worth' would be, and how to represent it in symbols.

Had we never developed the Arabic number system, our concept of mathematics would be very different indeed. We now take it for granted, but it is NOT that old a concept, as mathematics goes.

European Expansion The first mention of Arabic numbers in the West is found in the "Codex Vigilanus," a historical account of Hispania published in 976. Pope Sylvester II began to spread knowledge of Arabic numerals throughout Europe beginning in the 980s. As a student, Sylvester studied a form of mathematics and requested that Italian and Algerian scholars translate some of the earlier mathematical texts into common European languages. This was accomplished more fully in 1202 with a book by Leonardo of Pisa called "Liber Abaci."

The turning point in our mathematical system, and what made it the way it is, was the invention of the concept of 'zero'. Making the absence of anything an actual number, and assigning it a value, changed everything. That came as a natural progression from 1+1=2 to 1-1= well, what?

Another turning point, which developed only because our minds were able to handle extreme abstract concepts, was algebra. The concept or representing the 'unknown' by a symbol. There have been studies that relate how the development of algebraic concepts is dependent on the unique properties of a subset of human minds and human cognition. The fact that there are many, many humans that just can not comprehend algebra at all indicates exactly how much it is dependent on constructs and cognitive abilities that have evolved in human minds.

Why did the development of algebra lag behind geometry for so many centuries? Why do today’s pupils have difficulties with even the simplest word problems? What prevented generations of mathematicians from accepting the idea of the irrational and the negative numbers? What are the roots of the difficulties experienced by students confronted with the concept of complex number for the first time?

It is neither by chance, nor by mere carelessness, that my list of questions is a mixture of psychological and historical puzzles. As different as they seem at first glance, these two sets of problems may in fact have much in common. Indeed, there are good reasons to expect that, when scrutinized, the phylogeny and ontogeny of mathematics will reveal more than marginal similarities. At least, this is what follows from the constructivist view according to which learning consists in the reconstruction of knowledge.

Consider also out mathematical symbol "=". Exactly what does it mean? For instance, it has an entirely different meaning in computer science ('becomes') than in algebra ('has the same value as') and in expressions ('replace with' or 'substitute') and in topology ('transforms into'). A mind that is based on patterns and wholistic processing instead of sequential processes and logical step-wise processing would have a very different concept of 'equivalence', with a very different symbolic representation.

It can not be absolutely expected that every intelligent mind would make progressions in mathematical thought in the same way that we have.

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  • $\begingroup$ "Our entire system of mathematics is founded on the concept of unity, and the basic premise that one plus one equals two": Do you have any reference backing up this assertion? Pretty please! My imagination is much too limited to imagine a way of how this could possibly be true, and I would very much like to expand its horizons. $\endgroup$
    – AlexP
    Commented Jun 28, 2021 at 16:13
  • $\begingroup$ @AlexP "The nine Peano axioms contain three types of statements. The first axiom asserts the existence of at least one member of the set of natural numbers. The next four are general statements about equality; in modern treatments these are often not taken as part of the Peano axioms, but rather as axioms of the "underlying logic"." en.wikipedia.org/wiki/Peano_axioms Unity and equality are axioms that all math is based on. $\endgroup$ Commented Jun 29, 2021 at 3:15

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