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Can it be said that a number is a category is a set?

There is such a variety of ideas on numbers, categories and sets that probably anything one says about them will be controversial, but I was wondering what would prevent us from saying a number is a category and a category is a set.

  • If you are using the common math meanings, a number is the simplest instance of a category, but a category is always more than a set. Even the category that is a number is usually an ordered set. There are lots of categories that are quite a lot more complex than the category of having a given number of elements. And there is a lot more to a category than the set that underlies it -- it contains relations among all the elements of that set, and those relations have been selected to reflect a given purpose. – user9166 Jan 27 at 19:30
  • @jobermark - Thanks. So a number may be considered a (simple) category, Can a number not be considered a simple set? – PeterJ Jan 28 at 9:27
  • It can. Traditionally 0 = {}, 1 = {0}, 2 = {0,1} after Kronecker. It is convenient to have your canonical notion of a number be as a set so that counting is a natural mapping process. And this also makes less-than be the subset operation. Less formalistic views of the world consider the numbers to be Ur-elements, individisible, and thus not sets. – user9166 Jan 28 at 18:18
  • The four primary branches of symbolic logic all have their own pet canonical representation of the natural numbers. The category of categories that are ordered sets, Kronecker's piles of braces, and you can also consider the numbers as models. Zero would be the free model, that adds no constraints to an existing model. One is the model {exists a}, Two {exists a, exists b, a != b}, Then {a !=b != c != a} and so on. Proof theory takes succession, since the essence of a proof is a succession of steps, as more basic than any other concept, so the model of the integers is unanalyzed. – user9166 Jan 28 at 18:51
  • At this point, this is becoming an answer, and is basically all in Mauro's links. – user9166 Jan 28 at 18:58
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For category theory, see Natural number object.

For set theory, see Set-theoretic definition of natural numbers.

And see Category of sets for a link between the two theories.

See also the post Categorical foundations without set theory and Philosophical Significance of Category Theory.

From a mathematical point of view, the proposed construction of the natural numbers using set theory is a way to define mathematically a structure (i.e. a mathematical object) satisying the axioms of numbers, i.e. "performing" in the same way that numbers do.

Is this the way to catch the "real essence" of numbers ?

From the point of view of mathematical practice, the question is quite negligeable and useless.

For a philosophical point of view, the issue is "on the table" since Pythagoreanism and Plato and is still open.

  • Thanks. Regrettably I cannot follow most of the linked articles and am out of my depth. As a non-mathematician I'm still struggling to grasp the difference between a category and a set. My naive idea of a category is a 'category of thought'. – PeterJ Jan 28 at 9:37
  • The reason for naming "categories" the objects of Category Theory is obscure to me. The link with Aristotelian and Kantian categories seems to me highly unlikely. – Mauro ALLEGRANZA Jan 28 at 14:35
  • Hmm. Interesting comment. I don't have enough maths to hold an opinion but perhaps my naive idea of a category is more like that of earlier philosophers. Anyway. my question has been well-answered so thanks. . – PeterJ Jan 29 at 10:43
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This is not a simple affirmative.

First, we have to understand exactly what you mean by "set". There are many Set Theories, with different objectives.

Second, categories are not necessarily sets. This is a very complicated topic; someone could argue that it makes no sense to talk about categories that are not sets, but the language of categories is in principle independent of a set theory (but uses some intuitive notion of "collection").

Here I can make a point. There is a difference between the language that we use to talk about math and math. For a realist, math has some reality that does not depend on the language we use to express that reality. So, according to this view set theory and category could be seen as just "ways" to express that reality. When you say a "number" is a "category" then we have to be clear if we are talking about numbers as "real things" in some sense that coincide with categories as "real things" in that same sense, or if you are saying just that a number is a linguistic entity that is referred to by the language that we use for categories.

For a Platonist, there is a whole world where math exists and from that world we just experience its shadow. For an Aristotelian, math exists in the things, in some sense, there is no other world necessarily. So, if the ontological status of math is some kind of realism, then you could have in the own reality of math a problem to say that "a number is a category", you could linguistically say something like that, but not in reality. If the approach to the question pressuposes some kind of nominalism, and you think that math doesn't really exist, is just a linguistic entity, then you could say something like that.

  • Thanks. These things always turn out to nuanced and difficult. . – PeterJ Feb 17 at 15:58

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