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I have been thinking about the axiomatization of geometry, and I don't know one thing. Imagine you are defining triangles:

Definition [Triangle]: A triangle is a polygon with 3 sides.

In this case, is it required to define $3$ (and, by extension, $\mathbb{N}$)?

Edit: Maybe I should explain myself more clearly. What I am doing is the following: I have the definition of triangle, now what do I have to define previously so that the definition can be completely understood? In this case, it is clear that I must define what a polygon is and what is a side/vertex of a polygon. I'm wondering whether I have to define "3", but I am not complaining specifically with number 3, it just happens that it is the number I needed to use in this case.

J. C.
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  • the bigger question seems to be: how do you want do define a "polygon"? and "number of sides"? – Simon Nov 25 '16 at 23:26
  • Maybe it's better to speak about number of vertices instead of sides and I don't think the definition of polygon matters much for the question. Although, I think of a polygon including its interior. – J. C. Nov 26 '16 at 21:07
  • That is not the point I was asking. What kind of axioms are you doing? Naively, one might define A polygon as a subset of $\mathbb{R}^2$ with certain properties. In that case, you already defined real numbers, so natural numbers should be no problem. I thought you might have an alternative definition. An actual "axiomatic geometry", without any numbers at all. I would be interested to see that. Even though your axioms might not be fully done yet, one needs more context in order to understand your original question. – Simon Nov 26 '16 at 21:12
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    Also, you can replace "has 3 sides" with "there are sides x, y and z, such that $x\neq y$, $y\neq z$, $y\neq x$. Every side of the polygon is one of x,y or z.". That is a lot more to write, but there is no number in there. – Simon Nov 26 '16 at 21:14
  • I am not quite working on axioms, but with definitions. I guess my definition of polygon would be one like Hilbert's, but including the interior. As for your second comment, it isn't a good solution because then I can't talk about $n$-gons, for general $n$. – J. C. Nov 26 '16 at 22:49
  • And that definition is what? In any case, I don't understand your question. If you are not interested in fundamental axioms, what is the problem with "3"? Its a perfectly fine number. – Simon Nov 26 '16 at 22:53
  • Maybe I should explain myself more clearly. What I am doing is the following: I have the definition of triangle, now what do I have to define previously so that the definition can be completely understood? In this case, it is clear that I must define what a polygon is and what is a side/vertice of a polygon. I'm wondering whether I have to define "3", but I am not complaining specifically with number 3, it just happens that it is the number I needed to use in this case. – J. C. Nov 27 '16 at 13:54
  • You could define what a natural number $n$ is without defining the set of all natural numbers $\Bbb N$. That is, you could come up with a definition that allows you to use any natural number $n$, yet without defining $\Bbb N$. You would need to find a way to define the property of being "finite". See https://en.wikipedia.org/wiki/Finite_number, also https://en.wikipedia.org/wiki/Finite_set, also https://en.wikipedia.org/wiki/Natural_number – Mirko Nov 29 '16 at 18:55
  • I understand what you're saying, but what I want is to use particular numbers, not to define what a natural number is. For example, to know what a triangle is, you need to know what 3 is, so that you can know what is to have 3 sides. – J. C. Nov 29 '16 at 19:11

2 Answers2

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In order to write down definitions and axioms (without which those definitions don't really make sense), you need a logic - that is, a formal language in which you can write these statements.

The standard example is first-order logic, although there are others. And Euclidean geometry is easily formulated in first-order logic.

Now, the natural numbers are implicit in first-order logic! Specifically, defining what a sentence is, and what a proof is, requires us to already understand what a natural number is. And there's really no way around this, and nothing special about first-order logic in this regard (indeed, other logics tend to require more mathematical background - e.g. second-order logic basically requires all of set theory!).

So in order to have a context in which your definitions and axioms can be expressed, and theorems can be proved, you need to already have the natural numbers "in the background".

To clarify: it's not that we have to define "$3$" in order to express "A triangle is a polygon with $3$ sides," it's that we already need to understand $\mathbb{N}$ in order to be able to formulate the language within which this definition is being expressed.

This point of view can be seen in more detailed fashion at the answers to this question.

Community
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Noah Schweber
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  • Out of curiosity, why the downvote? – Noah Schweber Nov 30 '16 at 20:20
  • I strongly disagree with what you are saying. – Rene Schipperus Nov 30 '16 at 20:22
  • @ReneSchipperus OK, do you want to tell me how you disagree with it? – Noah Schweber Nov 30 '16 at 20:22
  • Numbers are not implicit in logic. – Rene Schipperus Nov 30 '16 at 20:23
  • @ReneSchipperus I presume you meant to say "dense linear orders". My point is not that every first-order theory interprets $(\mathbb{N}, +, \times)$ - this is of course false. My point is that in order to form first-order logic in the first place, you need the natural numbers. How would you define first-order logic (or your favorite logic - whatever you're using to express the definitions and axioms of geometry in) without reference to natural numbers (or their moral equivalents, like finite strings of symbols)? – Noah Schweber Nov 30 '16 at 20:24
  • I dont see that you need numbers to define a formula. – Rene Schipperus Nov 30 '16 at 20:25
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    @ReneSchipperus A formula is a finite string of symbols with certain properties. In order to make sense of "finite string of symbols," we need to understand what "finite" means, and this takes us back to $\mathbb{N}$. Similarly for proofs. Or, how are you going to define the collection of formulas? To be entirely honest, until you tell me a little about how you propose to found logic without recourse to $\mathbb{N}$ in some guise, I'm not going to take your objection too seriously. One-sentence comments just telling me that I'm wrong aren't very convincing. – Noah Schweber Nov 30 '16 at 20:27
  • Are you saying there is a chicken and egg problem, you need a theory to define logic and you need logic to express a theory ? – Rene Schipperus Nov 30 '16 at 20:29
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    @ReneSchipperus Yes, essentially - similarly in spirit to the story What the Tortoise Said to Achilles. Mathematics requires something to get off the ground. – Noah Schweber Nov 30 '16 at 20:33
  • @ReneSchipperus I referred to the answers there, since they say more on the topic in more detail, and also include answers by other people. Now, do you want to explain why I'm wrong, or just keep insisting that I am? Frankly, I think this is very rude. – Noah Schweber Nov 30 '16 at 20:36
  • Well you could just as well say that the english language is implicit in first order logic. Is the german language also implicit in logic ? – Rene Schipperus Nov 30 '16 at 20:37
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    @ReneSchipperus OK, it's pretty clear to me that you don't want to discuss this in a serious way, so I'm going to bow out now. – Noah Schweber Nov 30 '16 at 20:38
  • Ok, well, dont go away mad, you wanted to know why I downvoted and I told you. I dont engage in arguments, and that is what I saw coming. Best wishes. – Rene Schipperus Nov 30 '16 at 20:42
  • So, should I assume 3 and natural numbers in general as undefined concepts? – J. C. Nov 30 '16 at 21:12
  • @JoãoC. Yes, that is what I am arguing. – Noah Schweber Nov 30 '16 at 21:16
  • @JoãoC. For instance, we can write down "Every triangle has three sides" in first-order logic (with an appropriate language, and some abbreviations) as: "$\forall x(T(x)\implies \exists a\exists b\exists c(a\not=b, b\not=c, a\not=c, Side(a, x), Side(b, x), Side(c, x), \mbox{ and }\forall d(Side(d, x)\implies d=a\mbox{ or }d=b\mbox{ or }d=c)))$". – Noah Schweber Nov 30 '16 at 21:20
  • Note that this nowhere mentions the number "$3$"; however, "$3$-ness" is built into the statement when I write "$\exists a\exists b\exists c$". The natural numbers are implicit (I would argue) in how we read and understand these formulas. – Noah Schweber Nov 30 '16 at 21:21
  • @Noah Ok, good point. However, how can you define general n-gons that way? – J. C. Nov 30 '16 at 21:25
  • @JoãoC. (I presume you don't mean for $n$ fixed - it's easy to define, say, a 172-gon in the same way, just way way longer.) That's actually substantially more complicated. An $n$-gon is a finite set of points with some properties. Saying "is finite" can't be done directly in first-order without a broader context - say, working within set theory (OK fine that's overkill but it's a natural overkill). The limitations of first-order logic in this regard are captured by the compactness theorem. But this is getting a bit off-track. – Noah Schweber Nov 30 '16 at 21:26
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Without attempting to rule on the question I would like to express my opinion that there are two first order logics. There is the naive logic we define using the english language an all that it entails (though I think numbers dont play a big role). There is a second formalized FOL defined within a theory such as ZFC or PA. Clearly, to prove theorems about FOL we need assumptions such as the axiom of choice and those are theorem about this formalized FOL.

Rene Schipperus
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  • I mostly agree with this answer, except for the paranthetical comment and the last sentence. Re: the paranthetical, I think the natural numbers (or a naive idea of them, at least) are implicit in the naive logic. I would be interested in seeing the contrary claim elaborated. Re: the last comment, the axiom of choice isn't really needed for first-order logic. At best, choice shows up in the compactness theorem for arbitrary languages, but (a) we don't need full choice for that, and (b) for foundational purposes, finite languages are enough, and choice isn't needed in any way for those. – Noah Schweber Nov 30 '16 at 21:05
  • (Of course, the comment about choice is really a nitpick, but I think it's one worth stating to forestall possible confusion down the road.) – Noah Schweber Nov 30 '16 at 21:06
  • @ Glad that you agree with the general idea, it avoids a chicken and egg problem. What I meant in the last sentence is what you said in you comment. Just that at some point we want to prove things about FOL and this has to be done in an axiomatic context. – Rene Schipperus Nov 30 '16 at 21:12
  • I don't really think it does avoid a chicken and egg problem, it just phrases it differently. The point is that somewhere we have to have something that's "in the background". The precise nature of that something - innate access to the "true" natural numbers if one is a Platonist (I am not), or a sufficiently-precise-but-not-formal background logic, or something else - is flexible, but something is needed to get off the ground. I think the only way out of the chicken-egg problem is to take something for granted. (I do suspect we disagree on the amount we have to take for granted.) – Noah Schweber Nov 30 '16 at 21:15