3

Okay I was wrong. I can’t delete the post but I’ve accepted it’s a dumb question and moved on :)

  • 1
    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. http://mathworld.wolfram.com/PiFormulas.html – user4894 Jun 16 '17 at 19:41
  • 1
    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 '17 at 19:56
  • 1
    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. –  Jun 16 '17 at 20:06
  • @user4894, I see your point, but I guess my question has more to do with how pi is a constant in our universe. Yes, we have formulas that yield us this constant, but it's value is fundamentally defined by the ratio of a the circumference of a circle and its diameter. It seems in my mind that there's no particular reason this ratio/constant turned out to be 3.14- it just is. My question is: is 'arbitrary' the right way to describe this? –  Jun 16 '17 at 20:10
  • 1
    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 '17 at 21:15
  • 3
    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 '17 at 22:28
  • Here's an easy proof that 2.8 < pi < 4. Just look at the inscribed and circumscribed squares. https://betterexplained.com/articles/prehistoric-calculus-discovering-pi/ – user4894 Jun 16 '17 at 22:45
  • Duke Zhou's answer is a very useful answer to the question. In base pi units pi = 1, so of course the length or make up of its decimal expansion is irrelevant to whether or not its 'arbitrary'. I think other people are correct that your use of 'arbitrary' is somewhat dubious (possibly just unclear) but if you are stuck on the fact that pi is a nonrepeating infinite decimal when written in base 10, you are getting caught up on [the signifier and not the signified](https://en.wikipedia.org/wiki/Signified_and_signifier). Pi is anything but arbitrary, it is ubiquitous across all of math. – Not_Here Jun 17 '17 at 00:13
  • 1
    i get the impression the OP is essentially asking, "Why does the ratio of a circle's circumference to its diameter equal 3.1415... rather than any other value?" – Alexis Jun 17 '17 at 01:38
  • (i.e. base-ten 3.1415...) – Alexis Jun 17 '17 at 01:49
  • 1
    Or even: "Why does the ratio of a circle's circumference to its diameter have the specific value it does, regardless of the base used to represent that value?" – Alexis Jun 17 '17 at 01:59
  • 1
    @Alexis I get a similar impression, but I'm hoping that we can help coax out a more powerful question. As is, I would argue the question is 'mu'. It is a question that should be unasked because any answer will lead to suffering. Questions in the form of "Why is this very specific thing exactly as it is, as opposed to something else" have a terrible tendency of leading people down undesirable paths. ("If only I hadn't kissed my wife goodbye before she left, she might have gotten to the traintracks slightly earlier and not been hit by the train.") – Cort Ammon Jun 17 '17 at 23:53
  • how could it be different? if it's a mathematical constant, can we really ask the question you pose? –  Jun 20 '17 at 22:13
  • we **can** ask questions like [this](https://philosophy.stackexchange.com/questions/38239/what-are-some-arguments-for-the-golden-ratio-making-things-more-aesthetically-pl/38240#38240) because it's not mathematics. unless you're asking about the status of mathematical truth, which has a tonne of answers here already i'm sure –  Jun 20 '17 at 22:15
  • (To beat a dead horse) The issue with your thinking is that you are stuck on the numerals (again the signs instead of what the signs signify). The ratio between the circumference and the diameter of a circle is exactly what it is, the ratio between the circumference and the diameter of a circle. You are basically saying "it feels arbitrary that A = A" because you are getting too caught up on what sign you picked to represent A. Pi is an infinite decimal in base 10, in base pi it is the numeral 1. It doesn't matter what sign you give it, you are still talking about C/D. – Not_Here Jun 20 '17 at 22:19
  • god did it imvho –  Jun 21 '17 at 00:57
  • If I have understood you correctly, I believe where you say *arbitrary*, you may mean *coincidental*. That is to say, the number has no special signifigance, excepting that it coincidentally happens to be the fundamental ratio of circumference to diameter. –  Jun 23 '17 at 13:32
  • I think there was a US State Legislature which declared pi to be 3.00. you could move there. Your car tires will turn into hexagons though. It will be a bumpy ride! – Scott Rowe Sep 19 '22 at 19:17
  • 1
    After five years you deleted your question? Someone might have learned something from it. Why isn't there an edit lock after a couple of days? – user4894 Sep 20 '22 at 20:59
  • @ScottRowe Not necessarily. Stan Wagon's square wheeled bicycle. https://www.macalester.edu/mscs/multimedia/squarewheeledbike/squarewheelbike/ – user4894 Sep 20 '22 at 23:16
  • I wish you would put the question back, I actually found it compelling and important. I was just making a silly comment. – Scott Rowe Sep 21 '22 at 02:32
  • 1
    It's a good question. And Duke Zhou comes closer to a good answer than others. But not quite good enough for mathematical purposes – Rushi Sep 21 '22 at 02:53
  • @Rusi-packing-up but definitely close enough for government work. – Scott Rowe Sep 22 '22 at 00:29

5 Answers5

8

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...

DukeZhou
  • 463
  • 1
  • 3
  • 12
  • 5
    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 '17 at 03:44
3

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.

Olivier
  • 270
  • 1
  • 10
2

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.

Glorfindel
  • 279
  • 1
  • 4
  • 11
  • Circles on the sphere are still circles of euclidean space and have pi as the ratio of circumference to diameter. – Olivier Jun 17 '17 at 23:31
  • 1
    @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. –  Jun 18 '17 at 00:58
  • 1
    I have added a link to a reference –  Jun 18 '17 at 01:07
  • the best answer on this thread. i'm tired of answers without references when they state the obvious –  Jun 20 '17 at 22:10
1

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.

0

Pi is not arbitrary. It is half the ratio of the circumference to the radius. Or variations on this theme. This its definition. And is true in whatever world you want to try to work out what the value of pi is.

Mozibur Ullah
  • 1
  • 14
  • 88
  • 234