Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

π in Other Universes

azeemba.com

Post reply on HN
Print view View original

81–90 of 113 posts

Re: π in Other Universes

#81

Post by gumby »

tsimionescu wrote: post #58
chongli wrote: post #44

Earlier quoted context omitted.

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...

Most mathematics originates from trying to solve physical or engineering problems. Typically physicists have been on the forefront of mathematical research - this has only really changed significantly in the last few decades. Also, mathematics as practiced is internally consistent. It is incomplete, though. That is how it stays afloat of Godel's result. Basically Godel's results showed that no matter how much we stri…

> Most mathematics originates from trying to solve physical or engineering problems.

Has this been true since the early 20th century? I have no feel for what constitutes "most" in the vast corpus of pure mathematics, so am not challenging your claim but rather am curious.

Re: π in Other Universes

#82

Post by [deleted] »

gumby wrote: post #81
tsimionescu wrote: post #58

Earlier quoted context omitted.

Most mathematics originates from trying to solve physical or engineering problems. Typically physicists have been on the forefront of mathematical research - this has only really changed significantly in the last few decades. Also, mathematics as practiced is internally consistent. It is incomplete, though. That is how it stays afloat of Godel's result. Basically Godel's results showed that no matter how much we stri…

> Most mathematics originates from trying to solve physical or engineering problems. Has this been true since the early 20th century? I have no feel for what constitutes "most" in the vast corpus of pure mathematics, so am not challenging your claim but rather am curious.

[deleted]

Re: π in Other Universes

#83

Post by tsimionescu »

gumby wrote: post #81
tsimionescu wrote: post #58

Earlier quoted context omitted.

Most mathematics originates from trying to solve physical or engineering problems. Typically physicists have been on the forefront of mathematical research - this has only really changed significantly in the last few decades. Also, mathematics as practiced is internally consistent. It is incomplete, though. That is how it stays afloat of Godel's result. Basically Godel's results showed that no matter how much we stri…

> Most mathematics originates from trying to solve physical or engineering problems. Has this been true since the early 20th century? I have no feel for what constitutes "most" in the vast corpus of pure mathematics, so am not challenging your claim but rather am curious.

You're right, that might actually be wrong.

However, the claim I was actually thinking of, which is right I think, is that the maths used in the physical revolutions of the turn of the century (SR, QM, GR, and probably QFT, QED, and QCD as well) was invented by physicists or by mathematicians working with physicists for the express purpose of developing this theories, not the other way around.

Also, the basis of mathematics and the first few thousand years were indeed motivated by these kinds of concerns.

Re: π in Other Universes

#84

Post by defrost »

tsimionescu wrote: post #83
gumby wrote: post #81

Earlier quoted context omitted.

> Most mathematics originates from trying to solve physical or engineering problems. Has this been true since the early 20th century? I have no feel for what constitutes "most" in the vast corpus of pure mathematics, so am not challenging your claim but rather am curious.

You're right, that might actually be wrong. However, the claim I was actually thinking of, which is right I think, is that the maths used in the physical revolutions of the turn of the century (SR, QM, GR, and probably QFT, QED, and QCD as well) was invented by physicists or by mathematicians working with physicists for the express purpose of developing this theories, not the other way around. Also, the basis of math…

I wouldn't agree - consider the hyperbolic transforms used to describe space time "bending" wrt relativity:

https://en.wikipedia.org/wiki/History_of_Lorentz_transformat...

    In mathematics, transformations equivalent to what was later known as Lorentz transformations in various dimensions were discussed in the 19th century in relation to the theory of quadratic forms, hyperbolic geometry, Möbius geometry, and sphere geometry, which is connected to the fact that the group of motions in hyperbolic space, the Möbius group or projective special linear group, and the Laguerre group are isomorphic to the Lorentz group.
Mathematicians were following up on "what happens when you discard one of Eucilids Axioms" and discovering there was an entire world of consistent hyperbolic geometry and more.

Some time later:

    In physics, Lorentz transformations became known at the beginning of the 20th century, when it was discovered that they exhibit the symmetry of Maxwell's equations. Subsequently, they became fundamental to all of physics, because they formed the basis of special relativity in which they exhibit the symmetry of Minkowski spacetime, making the speed of light invariant between different inertial frames.
If you read mathematics histories it's a common complaint that it's nigh on impossible to discover something new and esoteric that doesn't soon end up with a military application; the ongoing search for interesting but useless mathematics is akin to the search for the fountain of youth.

It is the case (IIRC) that quaterions arose directly from Hamilton's search for a better way to describe mechanical motions in three dimension spaces - ie created to be useful from the outset.

Re: π in Other Universes

#85

Post by gumby »

tsimionescu wrote: post #83
gumby wrote: post #81

Earlier quoted context omitted.

> Most mathematics originates from trying to solve physical or engineering problems. Has this been true since the early 20th century? I have no feel for what constitutes "most" in the vast corpus of pure mathematics, so am not challenging your claim but rather am curious.

You're right, that might actually be wrong. However, the claim I was actually thinking of, which is right I think, is that the maths used in the physical revolutions of the turn of the century (SR, QM, GR, and probably QFT, QED, and QCD as well) was invented by physicists or by mathematicians working with physicists for the express purpose of developing this theories, not the other way around. Also, the basis of math…

Yes, I agree to some extent with the restricted claim, which only (slowly) started to break down in the 17th century in the west.

A lot of Indian mathematics was rather abstract going back to Vedic times, but since they didn't develop the concept of proof, it sadly had little impact on other mathematics practice (except as inspiration to Persian and Arab scholars) other than the the famous cases of zero and positional notation. The mathematical documents I've seen from that practice have been in the form of essays.

I know little of Chinese or Mesoamerican mathematics and wonder where they were on this axis. It seems pretty likely that maths started in support of astronomy/planting predictions in the cultures I know of so likely also for East Asia and the Americas, but whither thence did it go?

Re: π in Other Universes

#87

Post by mnw21cam »

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.

A beam reach isn't necessarily the fastest point of sail. It depends on the boat, the efficiency (lift/drag ratio) of the sail, and the efficiency of the centreboard/keel (again, lift/drag ratio), but a reach of some kind is likely to be the fastest - it just won't be exactly perpendicular to the true wind direction. It'll also vary with the wind speed, wave height, weight distribution, etc.

Re: π in Other Universes

#88

Post by gus_massa »

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.

I think it's better to say that π is the same number everywhere 3.14... , but in other universe you don't use π in the formula of the length of a circle.

* Manhattan (L_1): C = 8 R

...

* Euclidean (L_2): C = 2π R

...

* Maximal Distance (L_infinity): C = 8 R

Re: π in Other Universes

#89

Post by zoomablemind »

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…

Fact that pi is irrational may point at something fundamental missing in our knowledge system.

It appears that we cannot precisely measure circle length/area in units of radius and vice versa. Basically, the unity as such does not exist in our knowledge, nor can we truly comprehend infinity.

Perhaps, unity and infinity are just our abstractions for something else.

Re: π in Other Universes

#90

Post by jerf »

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…

Mathematics is humanity's longest running, largest-scoped, most complicated game.

It also happens to be useful, and you can dive into a lot of philosophy about that which is all very interesting. The utility itself is a large thing on its own. But I think of that utility as something separate from the game itself. The game is just a game. You can do whatever you want with it. If you want to convert your cookbook to hexadecimal just for fun, you can. The fact that it is (broadly speaking) useless, that it will produce no new knowledge, and if anything negative utility in general, doesn't mean you can't do it.

That's the game.

You can also try to play the game to prove the Twin Prime conjecture. That's a much harder level.

This game is scalable to all ages and skill levels, has the best level variety, and can be done with anything from just your personal noggin, to a pencil & paper, to the largest computing cluster in the world. Technically all other games you play are a subset of this game; that may not always be a useful way to think of it, but it is technically true. And while there are a few rules, generally, nobody can tell you how to play it. You want to color pretty pictures? The game has lots of ways of doing that. You want to smash atoms together? The game can help with that. You want to simply count to the highest number you possibly can? Go for it. It's a very popular play with the younger players, but anyone can do it.

Post reply on HN