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π in Other Universes

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Re: π in Other Universes

#41

Post by gumby »

lachlan_gray wrote: post #32
IanCal wrote: post #2

> Mathematics can be seen as a logic game. You start with a set of assumptions and you come up with all the logical conclusions you can from that. Then, if someone else finds a situation that fits those assumptions, they can benefit from the pre-discovered logical conclusions. This means that if some conclusions require fewer assumptions, then those conclusions are more generally applicable This is a really, really n…

It blows my mind to think of mathematics/logic almost like a huge cellular automaton. “axioms” don’t necessarily correspond to “truth”, to me they’re arbitrary constraints that can give rise to complexity. And sometimes the resulting systems can be useful

The whole of the puzzles of cosmology actually all might be obvious if we had a few different fundamental theorems. But because we hit on some that almost work, and then build upon them a hole edifice of mathematics that is internally consistent and almost fits the universe we keep beating on it, not realizing that backing up a little and then driving forward again at a slightly different angle might yield a simpler, and even more consistent and explanatory system.

Re: π in Other Universes

#42

Post by msds »

One thing this doesn't touch on is that there are multiple meaningful definitions of pi-like constants for the p-norm unit circle that don't necessarily agree with each other in p != 2. Defining pi as the area of the unit circle gives an entirely different set of values that satisfying some wonderful properties - in particular, that definition of pi turns out to be the periodicity constant for a (arguably) natural set of trigonometric functions for the p-circle. Furthermore, pi(p) = 2 Beta(1/p,1/p)/p...

However, this (circumference/arc-length based) definition of pi does have a fascinating property for conjugate p,q: pi(p) = pi(q)

"Squigonometry: The Study of Imperfect Circles" is a very fun reference for this sort of stuff.

Re: π in Other Universes

#43

Post by moring »

gumby wrote: post #41
lachlan_gray wrote: post #32

Earlier quoted context omitted.

It blows my mind to think of mathematics/logic almost like a huge cellular automaton. “axioms” don’t necessarily correspond to “truth”, to me they’re arbitrary constraints that can give rise to complexity. And sometimes the resulting systems can be useful

The whole of the puzzles of cosmology actually all might be obvious if we had a few different fundamental theorems. But because we hit on some that almost work, and then build upon them a hole edifice of mathematics that is internally consistent and almost fits the universe we keep beating on it, not realizing that backing up a little and then driving forward again at a slightly different angle might yield a simpler,…

This has happened, and is happening all the time. Many groundbraking theories in physics can be framed this way. The problem is that "slightly different angle" is a huge space, so scientists throw a lot of theories at it and see what sticks.

Re: π in Other Universes

#44

Post by chongli »

gumby wrote: post #41
lachlan_gray wrote: post #32

Earlier quoted context omitted.

It blows my mind to think of mathematics/logic almost like a huge cellular automaton. “axioms” don’t necessarily correspond to “truth”, to me they’re arbitrary constraints that can give rise to complexity. And sometimes the resulting systems can be useful

The whole of the puzzles of cosmology actually all might be obvious if we had a few different fundamental theorems. But because we hit on some that almost work, and then build upon them a hole edifice of mathematics that is internally consistent and almost fits the universe we keep beating on it, not realizing that backing up a little and then driving forward again at a slightly different angle might yield a simpler,…

Most mathematics has no application to science whatsoever. It's a huge parts bin which scientists delve into when they build their models. And then much of the work is in trying to shoehorn the mathematics into being tractable.

Mathematics is also not provably internally consistent. This was famously shown by Gödel [1].

[1] https://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theor...

Re: π in Other Universes

#46

Post by moring »

nlitened wrote: post #37
tzs wrote: post #16

Note that even if another universe has a different π when it comes to geometry they are still going to also have an important constant that has the same value as our π. E.g., the zeros of the function defined by the series x - x^3/3! + x^5/5! - x^7/7! + ... are nπ where n is an integer and π is our π. Another place our pi will come up is in the exponential function. It's periodic with period 2πi.

> Another place our pi will come up is in the exponential function. It's periodic with period 2πi. Isn't it the opposite? As I understand, we (European civilization humans) historically _define_ our complex exponential function to have a period of 2πi to match the period of our previously defined sin and cos functions. We could have defined it to have another period — for example, if we define "360° angle" to be equa…

Where in the definition of the complex exponential function is π used? IIRC its: the exponential function is defined to be its own derivative (note 1), i is the square root of -1, and exp(ix) is observed to have a period of 2π. There isn't any arbitrary choices in there that could be said to be defined in such a way that π results.

note 1: exp(x) can alternatively be defined by the exponential series, but that series does contain arbitrary numbers that could be said to be selected in such a way that π results.

Re: π in Other Universes

#47

Post by Miiko »

TheOtherHobbes wrote: post #6

All of these assume your background metric is Euclidean. If your background 2D metric is a projection of a warped 3D space, you can make π as big as you want by tugging on the centre of the circle.

It's not the background metric but the space geometry is assumed Euclidean - in non-Euclidean geometry the ratio of of the circumference of any circle to the diameter of that is not a constant, it depends on such diameter (so you simply cannot define 'pi' in that case)

Re: π in Other Universes

#48

Post by sepen77 »

lfnoise wrote: post #18

This person is not a sailor. Sailing orthogonal to the wind, a "beam reach", is the fastest point of sail due to the lift of the sail.

I didn't know anything about sailing, but your one comment made me search up point of sail and now you've opened my eyes to something that was a mystery to me for all my life -- how sailboats can "course made good" against the wind. Thank you.. this stuff is amazing, and sailing is an incredible science!

Re: π in Other Universes

#49

Post by thriftwy »

lachlan_gray wrote: post #32
IanCal wrote: post #2

> Mathematics can be seen as a logic game. You start with a set of assumptions and you come up with all the logical conclusions you can from that. Then, if someone else finds a situation that fits those assumptions, they can benefit from the pre-discovered logical conclusions. This means that if some conclusions require fewer assumptions, then those conclusions are more generally applicable This is a really, really n…

It blows my mind to think of mathematics/logic almost like a huge cellular automaton. “axioms” don’t necessarily correspond to “truth”, to me they’re arbitrary constraints that can give rise to complexity. And sometimes the resulting systems can be useful

Axioms are not wrong if you can derive some math from them.

They may not correspond to anything in our world, and then we usually discover something that does.

Re: π in Other Universes

#50

Post by WiSaGaN »

This largely depends on how one defines pi. I believe that the concept of R^n (Euclidean space) exists even in entirely different physical spaces. This is because Euclidean space represents a universally recognized idea of simple space in terms of curvature. For instance, in any world, the concept of '0' represents simplicity. In this context, pi will always remain constant.
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