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The difficulty in formulating my question lies in defining what I mean by "unique."

What I mean by "uniqueness": For example, the concepts that

  • 1 + 1 = 2
  • 5 + 2 = 7
  • 6 x 3 = 18
  • 6 - 9 = -3
  • etc.

only represent one unique class of concepts, movement about a number line of integers. Perhaps a second unique concept would be the extension of this number line to include all real numbers, complex numbers, etc. Other unique concepts be may the idea of limits, geometric perimeters, etc.

I suppose what I mean by "uniqueness" of a concept is a concept that cannot be derived or is an extension of other concepts. Instead, there must be some intellectual ingenuity that perhaps a human may be able to come up with but not a calculator.

I understand that my definition of "uniqueness" may be imperfect, but I do not believe an imperfect definition prevent meaningful discussion. We lack perfect definitions for what truth is, but can we still discuss truth in a meaningful way?

Considering this, are there infinitely many unique mathematical concepts? Is the landscape of mathematical knowledge in this sense infinite or is it finite? If humans of limitless intellect could live for any arbitrary amount of time, would we run out of ideas?

And if one says yes to my question because one could propose infinitely many axioms that are "unique," then how is this possible in a finite world? To conceptualize these "unique" axioms, one should have some sort of corresponding neurological structure. Aren't there finitely many neurological structures?

closed as unclear what you're asking by Not_Here, Eliran, Mark Andrews, Conifold, L.M. Student Nov 14 '18 at 21:46

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  • Clearly you mean to avoid 1 = 1, 2 = 2, 3 = 3, ... as an example of infinitely many concepts. But now you even want to call all arithmetic statements one concept. So you will need a very precise definition of "concept" before your question is meaningful. – user4894 Nov 9 '18 at 2:13
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    ps: "there must be some intellectual ingenuity that perhaps a human may be able to come up with but not a calculator." -- When we figure out which tasks those are -- if in fact there are any -- it will answer a question of lot of people are wondering about! – user4894 Nov 9 '18 at 2:14
  • Is 1+1=2 a "concept" ? – Mauro ALLEGRANZA Nov 9 '18 at 7:22
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    "I suppose what I mean by "uniqueness" of a concept is a concept that cannot be derived or is an extension of other concepts. Instead, there must be some intellectual ingenuity that perhaps a human may be able to come up with but not a calculator." This doesn't make any sense given what you've tried to give as an example. Maybe you have a really interesting question here but as it stands it is too vague to be a good suit for this site. We might be able to discuss some things that we can't define perfectly, but there is a limit to vagueness and I think your question crosses that line. – Not_Here Nov 9 '18 at 23:10
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    @MacroGuy I think that you should think more intensely about what exactly you mean by a mathematical concept, give that a well formed definition, and then it makes more sense to talk about what makes two concepts unique or dissimilar. Like, I can't tell if you think an identity statement is one concept and functions themselves are a different concept, or whatever, if I don't know what exactly you mean by a concept. Sure we could ask what makes the identity statements you listed different concepts from each other, but I don't think that makes sense until we actually know what a concept is first – Not_Here Nov 10 '18 at 19:53
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This is an interesting question, given you use the word infinite. It turns out this question cannot even be asked without a unique concept of "infinity," so we have to limit ourselves to systems of thought which admit a concept of infinity in the first place, such as set theory.

My own opinion would be the best place to look for an answer would be category theory. Category theory is one of the many attempts to make a single foundation for all of mathematics (which would then argue that there is exactly 1 unique concept). Of course whether or not it meets that high expectation is another question entirely.

The reason I recommend category theory is because its concept of a functor and the symmetries of morphisms is a powerful mathematical tool to describe something you might call "uniqueness." They let you "map" one category into another. For example, you can "embed" the integers into the real number line via an epimorphism.

This approach naturally leads to the idea of symmetries, which leads to group theory. Group theory will probably capture a large portion of what you are thinking of as "uniqueness," and its far more tangible than category theory is. For example, group theory can be used to show that you can embed the concept of "rotation" in 3 dimensions into a quaternion (a 4 dimensional structure with particular properties).

Regardless, if you use category theory in its dare-I-say purest form, all categories are classes of things. A class of things can be even larger than infinity. It can be larger than an uncountable infinity. In fact, it can be so large that the only word we can use to describe it is "large." (that's the mathematical term).

And of particular interest if you're looking at how to capture the idea of neurological structures related to mathematics, the Lie groups cover the smooth symmetries over a differentiable manifold. I mention them because most-if-not-all of modern physics assumes that the world is well described by a differentiable manifold. If you use the scientific term "neuron," you are likely invoking assumptions from physics.

And since embedding "concepts" in neurological structures rapidly leads itself to the use of symmetry arguments (there should be some symmetry between the conservation of energy and the neurons which encode the conservation of energy in our brain), its worth looking at the C series of Lie groups. They are particularly useful for capturing symmetries in Hamiltonian systems -- which is basically all of physics as we know it. (they also happen to be a countably infinite series of groups)

  • A category may well contain a proper class of objects and/or morphisms; but are you saying there could be infinitely many categories? Can you point to, say, more than a hundred of them, without going to n-categories? I'm trying to understand what you mean here. Likewise surely group theory is one "concept." You can't point to the proper class of all groups to satisfy OP's quest. – user4894 Nov 9 '18 at 15:31
  • The size of the category of categories appears to depend on what formal approach one takes, but from what I have seen, all of them assume that there is at least an infinite nuimber of categories, and some even start from the cateogry of categories being large. As for group theory, the reason I point to it is because the OP specifically invoked neurology. If you're using a scientifically bounded concept of "concepts," there's a really good chance the resulting concept of concepts can be described using Lie groups due to the connections between the word "concept" and "symmetry." – Cort Ammon Nov 9 '18 at 16:23
  • Why would we need a definition of infinity? It's pretty simple. If we have a list of unique mathematical concepts, does it end or not? – David Thornley Nov 9 '18 at 19:04
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Mathematical concepts in that sense are not completely mathematical. There is nothing about integers that make them significant in mathematics. There are branches of mathematics that don't necessarily use them (such as abstract algebra). We find that integers are part of some very useful and interesting branches of math, and they're intuitively obvious from basic observation of the world around us.

Therefore, a mathematical concept is part math and part psychology. It's something that an intelligent entity considers important or interesting.

Clearly, then, the number of mathematical concepts that exist in the observable universe is finite, since they require a certain amount of thought, and the amount of thought possible in the observable universe so far is finite (although very large). The number that will ever exist is also finite, since we can't expect intelligence to last after the last stars burn out in trillions of years.

So, what if we had an infinite universe to work with (such as the long-discredited Steady State theory as an alternative to the Big Bang)?

Individual brains are finite, so no one brain can hold an infinite number of concepts. Suppose we had infinitely many finite brains. A mathematical concept has to be explainable, and the explanation can't exceed a finite length. The number of concepts at any given time has to be finite.

Suppose we were to construct an infinitely expandable brain or other conscious and intelligent device, and expand it forever. At any given time, it has a finite number of concepts. Given any number, there is a time at which it would have more mathematical concepts than that, so we've finally potentially found infinite mathematical concepts.

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I'm going to say this all boils down to whether you're talking about practical mathematics or theoretical mathematics.

I mean, if we're just talking theory, I'd have a hard time believing that the number of concepts is finite.

  • Kevinality: the property of a number if, when you add all its digits together, is equal to a factor of 21.
  • Kevinesqueness: the property of two numbers, where sum of all the factors of their product is a prime number.
  • Kevinosity: the property of a vector, where its absolute magnitude's square is cleanly divisible by the cube of a factor of 7.

... sure, none of those concepts are useful at all (I think?) But I'm pretty sure that there's no limit to the silliness I can come up with along those lines (especially if I start feeding them into new definitions - "Kevinic: A set of numbers where at least half have Kevinality.")

But practical/applied mathematics? That's almost certainly finite. Because unless a concept is meaningfully applicable to some real-world phenomenon, it's not really a useful concept. And unless you believe the universe cannot be expressed in a finite amount of equations/laws/etc, the number of practical/meaningful mathematical concepts has to be finite as well.

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