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Edit: I think the point of my question's been missed. I'm well aware pi is well defined and calculable. However my question has to do with the metaphysical, not the physical. Not how does pi work out to 3.14, but why, and how to describe the position that the why is for no reason in particular. Why C/D works out to 3.14 seems arbitrary, but arbitrary is the wrong way to describe it. Clearly I need to think about how to state my question more clearly- thanks for your answers.

Lately I've been stuck with the question of how to describe the nature of pi (and other mathematical constants), and how despite pi's definition of being C/D of a circle on the Cartesian plane, its value working out to be 3.1415... seems to be arbitrary. That is, obviously pi isn't arbitrary, because it is dependent on the properties of a circle, but it seems arbitrary that C/D turns out to be 3.1415... rather than, say, 4, or any other number.

Is there a better way to describe this observation?

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    To clarify the intent of your question, is the fact that 2 + 2 = 4 arbitrary? Why isn't it 5? It seems to me that this is the question you're asking. It has nothing to do with pi, which is the perfectly deterministic outcome of any number of closed-form expressions. mathworld.wolfram.com/PiFormulas.html
    – user4894
    Jun 16, 2017 at 19:41
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    Look at it from a different perspective. It had to be a number. If we chose randomly a real number, this number would have been transcendental. If it turned out to be an integer, I wouldn't be convinced that it is random. Now, I'm not saying that the value of pi is random, but if it was not a transcendental number, it would imply the existence of a relation we haven't yet discovered.
    – tst
    Jun 16, 2017 at 19:56
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    It might help to note that pi is tied up intimately with e, the other seemingly random transcendental constant that appears liberally throughout math, by the jarringly convenient fact that e ^ (pi * i) = 1.
    – user9166
    Jun 16, 2017 at 20:06
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    In our universe? But there are no circles and no exact measurements in the physical universe. Pi is a constant in abstract, formal Euclidean geometry. Just as 4 is a constant in number theory, even though you could not measure a distance of exactly 4 miles in the physical world.
    – user4894
    Jun 16, 2017 at 21:15
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    You might have to take this question and reframe it backwards: "What does it mean to you for a number to be 'arbitrary.'" Instead of focusing on the number (which I would argue most people don't choose to identify as "arbitrary"), we can focus on what you mean by the word. Words can mean different things to different people, and we might be able to identify a word which has the same meaning to you, but is more consistent in its meaning among other people.
    – Cort Ammon
    Jun 16, 2017 at 22:28

4 Answers 4

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Just to add my own exhortations to the above.

Mathematical constants are fixed, and are true in all possible worlds, unlike physical constants like Planck's. It seems like you're asking for a physical explanation, which is no better than asking the same for deductive logic.

Philosophers can ask questions about mathematics. e.g. the metaphysics or epistemology of mathematics. Even, why the golden ratio is aesthetically pleasing.

Not all questions have answers, beyond just because.

I wonder why you asked this question, and why 4/5=0.8 does not puzzle you.

In answer to the question as it is, I think by"arbitrary" you could mean unmemorable. Even-though you can derive it quite easily via e.g. geometry.

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It sort of sounds like you're getting stuck on the numerical expression, which is subjective because Pi is also 11.00100100001111110110... and 3.243F6A8885A308D313198A2E0... [See π in Different Bases]

This is a little bit outside of my field, but I'm going to go out on a limb and say Pi is not arbitrary, but is the ratio of the circumference of a circle to it's diameter, always.

When I googled "arbitrary" in relation to Pi, I came up with many results, but none suggesting that Pi itself is arbitrary. Possibly I am misunderstanding arbitrary in the context the context you are using it...

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    This is my field, and you are perfectly right. There is nothing arbitrary about pi. It does not even depend on the choice of unit. You can't decide what the ratio of circumference to diameter is; it is fixed and we call this fixed thing pi.
    – Olivier
    Jun 17, 2017 at 3:44
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despite pi's definition of being C/D of a circle on the Cartesian plane, its value working out to be 3.1415... seems to be arbitrary.

There is nothing arbitrary about the value of pi working out to be 3.1415... . Mathematicians do not meet up to fix the values of constants!

it seems arbitrary that C/D turns out to be 3.1415... rather than, say, 4, or any other number.

It could not be anything else than 3.1415... .

Is there a better way to describe this observation?

Your "observation" seems to be a misconception.

Other remark. Even though your question doesn't admit a very profound answer, at least you are questioning yourself. What are numbers? Why can we use numbers to represent the ratio of two planar "lengths"? These are important questions that have non-trivial answers.

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Flat space is only one among a number of different kinds of spaces.

We live on a sphere, so, in fact, none of the circles we actually see on the globe have pi as the ratio of their circumference to their diameter. Circles on a sphere do not have this as a fixed ratio.

But we have the mental image of a flat space, where pi is well defined by this relation. How arbitrary was that choice?

If it is to some degree arbitrary, why should any particular part of its structure be less so? For instance, the length of a diagonal in flat space is just as irrational and no more regular than pi.

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  • Circles on the sphere are still circles of euclidean space and have pi as the ratio of circumference to diameter.
    – Olivier
    Jun 17, 2017 at 23:31
  • @Olivier Not within the space itself. Consider a spherical planet. Take the Equator. The diameter of the equator is half the circumference of the planet as measured along the surface -- the shortest path on the surface between antipodal points goes through a pole. The circumference, divided by the diameter, is then 2 and not pi.
    – user9166
    Jun 18, 2017 at 0:58
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    I have added a link to a reference
    – user9166
    Jun 18, 2017 at 1:07
  • the best answer on this thread. i'm tired of answers without references when they state the obvious
    – user25714
    Jun 20, 2017 at 22:10

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