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## Weird math question

Here's an interesting question I came up with dealing with the so-called "universal" language of math. (I asked one brilliant educated person this, but it never hurts to get multiple opinions.)

Could you have an advanced mathematical system equal or superior to current human mathematics that did not include the concept of prime numbers or even recognize their existance?

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hmmm; it would seem that the number 13 is not going to be divisible by any other numbers except itself and one regardless of the number system or mathematical system employed. The mathematician may not recognize this, but I would like to see the arithmetic that does not.

3. No, for the reason TheHalcyonYear stated. It is a universal language because it is based upon universal truths. If we have six units of anything, say apples or oranges, then regardless of the mathematical system used for that number, those units can be split up into two sets of three or three sets of two, but a prime number of units cannot be split up into more than one set of whole units.

4. It would be a system without the integers, essentially, which would mean we really can't call it "mathematics". Anything with a notion of set or counting will allow definition of primes.

5. Originally Posted by Professor Tanhauser!
Could you have an advanced mathematical system equal or superior to current human mathematics that did not include the concept of prime numbers or even recognize their existance?
Although this might seem like a technical question that might be answered on the basis of accepted axioms, it is really asking for a value judgement. Since anyone answering it here is apt to be a human, I suspect that they would tend to favor "human mathematics" even if they tried to be impartial.

However, we can appeal to the greatest mathematician that the world has ever known, Gauss. The theory of prime numbers is a part of the area of mathematics called Number Theory, which he considered to be the Queen of Mathematics. In a non-human mathematics, Number Theory may not be comparable to a queen, maybe only to a duchess or a baron, but it would still be missing something substantial so it wouldn't be superior in all aspects.

It'd kinda be like which tastes better, moose gristle or bear sinew? both have their adherents.

6. Originally Posted by Professor Tanhauser!
Could you have an advanced mathematical system equal or superior to current human mathematics that did not include the concept of prime numbers or even recognize their existance?
What's the definition of "superior" in regard to mathematics? Is trigonometry superior to geometry? Or do you mean "more difficult to understand?" As far as I'm concerned, addition is quite superior enough.

7. From a non-math member's perspective, I enjoyed and could relate to hhEb09'1 answer, the core of which was human centric prejudice.
To me, math "represents" our reality, our world, our universe. But math is not how we experience the same, only how we "represent" it. So intelligence with a different "take" on the same universe we live in may have a different method of abstacting it.

Maybe their experience of a gorup of 13 of our objects is that "every thing, every group, every distance, every speed, everything in the universe" is equally divisable by two without a need for fractions of decimals.

I can't picture that right now, but I percieve the possibility.,

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Ok, I was too brief earlier, allow me to clarify:

The creators of this mathematic system really didn't see anything special about what humans call "Prime numbers". I mean, to them the whole bit about a number being indivisible by anything but 1 and itself is like "So?"

The attribute nothing of significance to it, and in some ways don't even acknowledge it. 13? How about 3.25x4=13? 3? .6x5. 2? 1/4x8 and so on.

To the people using this math, human obsession with prime numbers and formulas generating them and so on is just incomprehensible and they see nothing special about them at all.

One person said that advanced math had to make a big deal re prime numbers, I'm not a big enough math guru to follow his logic though.

As to "advanced" math, let me say that in this case advanced means a mathematic system that supports an advanced scientific and technological level of advancement.

So given this elaboration could there be a mathematical system that was advanced enough to support high technology and that really didn't consider prime numbers to be of any significance or even have a special name for them?

9. Originally Posted by Professor Tanhauser!
The creators of this mathematic system really didn't see anything special about what humans call "Prime numbers". I mean, to them the whole bit about a number being indivisible by anything but 1 and itself is like "So?"

The attribute nothing of significance to it, and in some ways don't even acknowledge it. 13? How about 3.25x4=13? 3? .6x5. 2? 1/4x8 and so on.

To the people using this math, human obsession with prime numbers and formulas generating them and so on is just incomprehensible and they see nothing special about them at all.
Actually, I'm human (though whether to believe that or not is your choice) and I have no particular obsession with prime numbers. It's a concept I understand, but my feeling is also, like "so"?

Originally Posted by Professor Tanhauser!
As to "advanced" math, let me say that in this case advanced means a mathematic system that supports an advanced scientific and technological level of advancement.

So given this elaboration could there be a mathematical system that was advanced enough to support high technology and that really didn't consider prime numbers to be of any significance or even have a special name for them?
In that case, I doubt they would not have a name for it, though it could be a compound like "numbers not divisible by other numbers." As long as they understand arithmetic, it seems like a property they will come up with. A more difficult issue might be "odd" and "even". I wonder if it's related to the fact that we have 2 of many things (eyes, ears, hands, feet).

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Originally Posted by Professor Tanhauser!
The creators of this mathematic system really didn't see anything special about what humans call "Prime numbers". I mean, to them the whole bit about a number being indivisible by anything but 1 and itself is like "So?"

The attribute nothing of significance to it, and in some ways don't even acknowledge it. 13? How about 3.25x4=13? 3? .6x5. 2? 1/4x8 and so on.

To the people using this math, human obsession with prime numbers and formulas generating them and so on is just incomprehensible and they see nothing special about them at all.
In other words, these are people who can work out, if they were bothered, that the moon is there, but see no reason to look at it or find out anything about it. The tides happen, etc, but they think this is a matter of no importance.

Prime numbers turn up in an important fashion in so many bits of of mathematics, theory of commutative rings, complex analysis (Riemann hypothesis), cryptography, etc, that calling them an incomprehensible human obsession seems to be not just narrow-minded but foolish. They are the building blocks not just of the number system, but of the algebraic structures we do mathematics with, and turn up to harass us even in analysis.

11. One could wonder if it would be conceivable to develop mathematics starting with reals without going by the integers first.

It's hard for me to think of the kind of world someone could live in for that developments to happen, it couldn't be a someone doing the development for starters.

One thing's rather sure, it wouldn't be an advance mathematics, as primes or primelike set elements show up all over the place once you have rings.

12. Originally Posted by HenrikOlsen
One could wonder if it would be conceivable to develop mathematics starting with reals without going by the integers first.

It's hard for me to think of the kind of world someone could live in for that developments to happen, it couldn't be a someone doing the development for starters.

One thing's rather sure, it wouldn't be an advance mathematics, as primes or primelike set elements show up all over the place once you have rings.
I guess that's what the good professor is really asking. Not that the result would be superior or even equal to our own mathematics, but that it's possible to get things done without it at all. Jens seems to get by fine without the refinements, so obviously it's possible, right?

Ontogeny doesn't have to recapitulate phylogeny. I imagine that even today there's a vast class of people who, when faced with packing 23 objects in a box just use
Code:
```+ + + + +
+ + + +
+ + + + +
+ + + +
+ + + + +```
and be done with it, not worrying that 23 can't be factored.

13. If you can have a non-commutative math I don't see why a non-prime math couldn't be constructed somehow.

14. I would certainly expect that such a thing would be possible. All it would require is some other path to all the proofs that involve primes, and some other way to do encryption, and so forth. There might be some fundamental mathematical constructs that we haven't thought of yet, that connect with other ways to make proofs and so forth, that avoid the need for primes. One might imagine that if it were possible, someone like Gauss would have thought of it, but it's hard to anticipate what cannot be since it's just reasoning from incredulity.

15. Originally Posted by Ken G
All it would require is some other path to all the proofs that involve primes,
Or, you can just ignore most theorems that limit themselves to integers, which some consider artificial anyway.
and some other way to do encryption, and so forth.
The Data Encryption Standard (DES) was adopted a couple years before the RSA algorithm was described. It's been used since, although it was superceded this millennium by AES, the Advanced Encryption Standard.

16. Originally Posted by hhEb09'1
Or, you can just ignore most theorems that limit themselves to integers, which some consider artificial anyway.
All numbers are artificial. If we limited outselves to not having a concept of integers, we would lose some mathematical power, the power to count. It is certainly true that one can imagine an intelligence functioning in an environment where counting is not useful, but you couldn't have digital computers and so forth. I am interpreting the OP as asking, could you have technology like we have now but no concept of prime numbers-- could mathematics that functions like our own simply have taken another route that sidesteps that issue. If we take a more general type of OP interpretation, we can expand it to mathematics itself-- could an advanced intelligence function and achieve with no concept of mathematics itself?

17. Originally Posted by Ken G
All numbers are artificial. If we limited outselves to not having a concept of integers, we would lose some mathematical power, the power to count. It is certainly true that one can imagine an intelligence functioning in an environment where counting is not useful, but you couldn't have digital computers and so forth. I am interpreting the OP as asking, could you have technology like we have now but no concept of prime numbers-- could mathematics that functions like our own simply have taken another route that sidesteps that issue.
I didn't mean that integers are artificial, what I was referring to were the theorems about integers. There are theorems whose scope is limited to integers--some people consider that limitation too artificial.

I'm not trying to get rid of the integers.
If we take a more general type of OP interpretation, we can expand it to mathematics itself-- could an advanced intelligence function and achieve with no concept of mathematics itself?
A whole 'nother can of worms

18. Originally Posted by hhEb09'1
I'm not trying to get rid of the integers.
But if you would keep integers and counting, then you would need theorems about them. The theorems might be achievable without recognizing prime numbers, however.
A whole 'nother can of worms
You can say that again. But please don't-- I hate "nother"!

19. Originally Posted by Professor Tanhauser!
Ok, I was too brief earlier, allow me to clarify:

The creators of this mathematic system really didn't see anything special about what humans call "Prime numbers". I mean, to them the whole bit about a number being indivisible by anything but 1 and itself is like "So?"
Really? The multiplicative basis of the counting numbers is nothing special? Don't think because you wouldn't have studied it means it is of no value! As far as I know, primes were known, in some form or other, by the ancients. The precise definition has been solidified and generalized since, and so much mathematics (useful as well as beautiful) comes from or is dependent on them (or is proven with their help) it's hard to think it a coincidence.

It is certainly not the case that someone came up with some random idea of numbers divisible only by themselves or 1 and ran with it for no reason.

20. Originally Posted by HenrikOlsen
One could wonder if it would be conceivable to develop mathematics starting with reals without going by the integers first.

It's hard for me to think of the kind of world someone could live in for that developments to happen, it couldn't be a someone doing the development for starters.

One thing's rather sure, it wouldn't be an advance mathematics, as primes or primelike set elements show up all over the place once you have rings.
Certainly, when talking about lengths of lines, and you can add two of them together...oh wait, no you can't, because you have no "two"! You might, I guess, see the counting numbers and ignore them. Though after a while, someone would surely notice and ultimately come up with primes!

21. Originally Posted by loglo
If you can have a non-commutative math I don't see why a non-prime math couldn't be constructed somehow.
well, we have noncommutative rings as part of mathematics--not quite the same as saying math is noncommutative.

Sure, the real numbers have no primes unless you single out the integers as well.

22. Originally Posted by tdvance
As far as I know, primes were known, in some form or other, by the ancients. The precise definition has been solidified and generalized since, and so much mathematics (useful as well as beautiful) comes from or is dependent on them (or is proven with their help) it's hard to think it a coincidence.
But another way to address the question is, let's say there is a different kind of mathematical structure or entity, from which every theorem we have on rings over the integers can be derived. What if the ancients had happened on that structure instead, or what if intelligence had evolved in a different environment more conducive to grasping that different mathematical structure. If that were possible, and it's hard to know it isn't just because we haven't thought of such a structure, call it the aggregate property of quantors, then that other intelligence might now be asking the opposite question-- they might be asking, is it possible to understand their mathematics without quantors? Maybe their mathematics has all our theorems and more, so they see rings over the integers as some trivial questor application. Perhaps instead of counting blocks, their kids drop blocks in water and measure their volume, relative to a universal standard block volume. They'd say "I have 3.04 +/- .02 block quantors, let's play". They'd still have integers, the concept that 3.04 is close to 3 would have to emerge, so they'd have all theorems about integers that we prove with prime factorizations, but maybe their approach finds another way to prove them that is more intuitive to the structures they're used to thinking about. And I don't just mean they'd have the real numbers, we have that too, they'd have to have some way to manipulate or think about the reals and the integers that we have not thought of, embedding them in some different kind of structure the way we eventually embedded them in the complex plane. I can only speak hypothetically, because if I knew what those structures were, I'd prove them myself!

23. Originally Posted by Professor Tanhauser!
Ok, I was too brief earlier, allow me to clarify:

The creators of this mathematic system really didn't see anything special about what humans call "Prime numbers". I mean, to them the whole bit about a number being indivisible by anything but 1 and itself is like "So?"

The attribute nothing of significance to it, and in some ways don't even acknowledge it. 13? How about 3.25x4=13? 3? .6x5. 2? 1/4x8 and so on.

To the people using this math, human obsession with prime numbers and formulas generating them and so on is just incomprehensible and they see nothing special about them at all.

One person said that advanced math had to make a big deal re prime numbers, I'm not a big enough math guru to follow his logic though.

As to "advanced" math, let me say that in this case advanced means a mathematic system that supports an advanced scientific and technological level of advancement.

So given this elaboration could there be a mathematical system that was advanced enough to support high technology and that really didn't consider prime numbers to be of any significance or even have a special name for them?
It would never happen in practice and couldn't even be done after the fact without at least already realizing they are there. The first thing used to generate any mathematical system would be integers and then sets of integers, so the prime numbers are realized right away with the comparison of those sets. Even if we were to use something like 3.25x4=13, we are still using the decimal point to separate the positions between integers and fractions, so a knowledge of integers is still present. In other words, to realize the significance of integers is to realize the significance of primes. There is no way around it. It is universal.

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Without integers, you'd have no power series, no polygons, no polynomials, no vector spaces of integral dimension, no Euclidean space, no geometry, no quantum mechanics, no music, no digital computers, no networks or maps,....

There'd be very little left. Integers are everywhere naturally in the world and universe around us.

25. Originally Posted by Ivan Viehoff
Without integers, you'd have no power series, no polygons, no polynomials, no vector spaces of integral dimension, no Euclidean space, no geometry, no quantum mechanics, no music, no digital computers, no networks or maps,....

There'd be very little left. Integers are everywhere naturally in the world and universe around us.
Yahbut the reals contain the integers. We wouldn't have to get rid of the integers, just get rid of the theorems concerned with primality.

I'll have to think how extensive this would be...

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Originally Posted by hhEb09'1
Yahbut the reals contain the integers.
There is a subset of the reals which can be put in one to one correspondence with the integers. But according to the usual set-theoretical constructions of the reals and the integers, they are not exactly the same. And if you've ever programmed a computer the old-fashioned way to calculate things, you would know that 1 and 1.0 are not treated in the same way, and you typically get different answers depending upon which you use.

If those seem like quibbles, here's a genuine, practical difference. Since the generic real number cannot be specified with 100% accuracy, generally speaking we have a range of uncertainty about it. But with an integer or rational number there is no such uncertainty.

There are many things for which only that subset of the reals called the integers will do - they naturally self-identify themselves as the method of counting discrete objects, etc. And since discrete objects are part of mathematics, be they dimensions, polynomials or whatever, then their behaviour depends upon the properties of the integers. Doing geometry, quantum mechanics, etc, we soon realise that what we are doing relates to the properties of finite groups. Groups with a prime number of members have important and distinctive properties. That kind of stuff infects all sorts of areas of non-discrete mathematics.

You just can't escape the integers.

27. Originally Posted by Ivan Viehoff
If those seem like quibbles, here's a genuine, practical difference. Since the generic real number cannot be specified with 100% accuracy, generally speaking we have a range of uncertainty about it. But with an integer or rational number there is no such uncertainty.
I'll accept those as quibbles, and point out that that genuine difference is also not insurmountable. Maybe the generic real number can't, but rational numbers can, and that's all we really need there.
You just can't escape the integers.
The OP is not suggesting avoiding the integers though, just the concept of prime.

But would "the set of all orders of finite groups that have an element whose order (period) is the order of the group" be considered equivalent to a definition of prime? It's clearly equivalent, but does it satisfy the OP?

28. Originally Posted by Ivan Viehoff
Doing geometry, quantum mechanics, etc, we soon realise that what we are doing relates to the properties of finite groups. Groups with a prime number of members have important and distinctive properties.
Certainly some other intelligence would want access to all these outcomes. But their concept of a finite group might be a seemingly unremarkable application of some other mathematical structure, that was the point I was getting at. One of the deepest questions underpinning metaphysics is whether the universe hands us our physics, and what we are doing in our brains is just an inevitable part of this connection, or whether we create our own physics based very much on what we are doing in our brains. So one way to approach the OP question is the more general question of whether it is possible to gain as powerful of an understanding of nature, via some other mechanism or approach to intelligence, using a completely different physics-- one that may use very different mathematical structures, or even nothing at all that our intelligence would recognize as mathematics or logic. It's very hard to know if that is possible or not, given the limitations of our imagination.

And maybe imagination isn't even the key limitation-- as I once put in my sig, it is customary to think that our imagination limits the models we can create, but I see the greater problem as being that our models limit our imagination. There are certain ways we have chosen to look at the world, based on the way our genes regulate our brains and the mental constructs we pass down from generation to generation, and it's hard to know what other possibilities are out there.

29. Originally Posted by hhEb09'1
I'll accept those as quibbles, and point out that that genuine difference is also not insurmountable. Maybe the generic real number can't, but rational numbers can, and that's all we really need there.The OP is not suggesting avoiding the integers though, just the concept of prime.
Rational numbers are ratios of integers. Having those without having a concept of primes seems even more unlikely than having integers and not having primes. The minimal representation of a rational is the one where there is no common factor between the numerator and denominator. Any use of rationals requires awareness that there are multiple representations and that this minimal representation exists. This requires factorization, and the awareness that there are certain numbers that can not be factored.

30. Originally Posted by cjameshuff
Rational numbers are ratios of integers. Having those without having a concept of primes seems even more unlikely than having integers and not having primes. The minimal representation of a rational is the one where there is no common factor between the numerator and denominator. Any use of rationals requires awareness that there are multiple representations and that this minimal representation exists. This requires factorization, and the awareness that there are certain numbers that can not be factored.
How important is that? I mean, 22/7 and 335/113 are both reasonable representations of the same number, right? Why not?

I think that's what the OP is asking, whether it is truly important or not.

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