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The beginning bit of the physics answer is the answer. Pi comes from logic, not nature. This is a bit like when you're trying to ask your kid what the area of
by lordnacho 5y ago
The beginning bit of the physics answer is the answer.
Pi comes from logic, not nature.
This is a bit like when you're trying to ask your kid what the area of the triangle is, and the kid tells you you've drawn the lines bent, or the corners aren't sharp. It's actually not terribly easy to explain to them that they're supposed to understand the idealized entity, and what exactly those ideals are, because you also can't actually draw a triangle with no width and then expect them to appreciate that the qualities you want to expose are somehow exposed when you're breaking your own rules.
- fooker 5y agoNow draw the same triangle on a curved surface. No matter how the idea of area comes from logic, the fundamental assumption for the commonly known formula to calculate area is that we are doing euclidean geometry. We do not yet know if there is some similar assumption hiding behind our ignorance of physics.
- HPsquared 5y agoEven in a non-Euclidean geometry, wouldn't π still emerge (or be approached in the limit) in "flat" areas?
- FreeFull 5y agoWith regular spherical and hyperbolic geometry, yes, it is flat-ish if you look at a small enough area. On the other hand, if you have a different distance metric, the ratio of a circle's diameter to circumference can be a different constant. For example, in taxicab geometry, it is 4
- DonHopkins 5y agoIn Uber Geometry, it can vary from 5 to 14 during surge pricing.
- HPsquared 5y agoIn London geometry, the distance depends where you are in the one-way system.
- milesvp 5y agoI know this is sort of a low quality joke post, but I don’t think I’d ever thought about a directed taxi-cab distance before. I’m not yet convinced it’s a useful model outside GIS, but I may spend considerable time today thinking about it. Thank you.
- HPsquared 5y agoHappy to be of service, in filth it will be found!
- KineticLensman 5y agoTIL that Taxicab circles are squares with sides oriented at a 45° angle to the coordinate axes.
- HPsquared 5y agoIt depends if you define a circle as "constant distance from centre point" or "minimum (circumference/area)".
- aj3 5y agoEuclidian geometry would be the same in any universe though, as it does not depend on objective physical reality.
- biofox 5y agoIt could be argued that the Euclidean plane is its own universe.
- mypalmike 5y agoThe Euclidean plane is a conceptual model also not based on physics.
- coldtea 5y agoIt is based on observing (and needing to measure) plots of land. The abstraction came later. Didn't Hume already prove we cannot invent anything not related to our prior experience (which ultimate goes to the universe's natural laws we experience).
- gmadsen 5y agohow is general relativity related to any prior experience of the 1920s?
- belugacat 5y agoTrains and clocks, mostly!
- gmadsen 5y agothat is special relatively , not general
- RichardCA 5y agoThe maths that underpin GR were developed before Einstein. Without the maths you have no ontology, no way to evaluate the truth of a scientific claim. https://en.wikipedia.org/wiki/Tullio_Levi-Civita https://en.wikipedia.org/wiki/Tullio_Levi-Civita
- JoeAltmaier 5y agoYes yes. But you can ask that question, and we can answer it, regardless of whether our space is curved or flat. PI is the same in a flat space in any universe where you ask the question, because we're doing a mental exercise, not measuring it locally.
- TomSwirly 5y ago> PI is the same in a flat space in any universe pi is the same in _any_ space, flat or not. Indeed, the space we are in is probably not flat, but pi is unchanged.
- philipswood 5y agoTrue but Pi (mathematical constant Pi) isn't necessarily the ratio between the circumference and the diameter in those spaces.
- anothernewdude 5y agoI figure it'd be different in any non-euclidean geometry.
- tremon 5y agoBut we can get (hypothesize) non-euclidean geometry right here in our Universe. So wouldn't that make the "euclidean" part of the answer orthogonal to the "universe" part?
- yk 5y agoNo, in non-euclidean geometry we have a notion of tangent space, where we measure angles and that looks the same as usual flat space. That is, what is meant with smaller and smaller circles in the physics answer. So the difference between euclidean and non euclidean geometry vanishes at very small scales and you can just define pi as the value in the limit.
- mjburgess 5y agoI assume you're diving things into "from logic" or "from nature". However that's a highly misleading (if not false) dichotomy. "Logic" refers to the most basic set of logical operations (AND, OR, etc.) and their use. Pi does not come from this. It comes from Mathematics. If we define World to be "a set of true claims" then we can define worlds without any mathematical claims in; however the process of defining the world likely requires logic. (And so Mathematics != Logic). In any case, as the article shows, the definition of Pi is relative to a distance metric. "Pi = Circ/Diam" which is ill-defined. One can construct a World (as defined above) which admits only claims relative to a given distance metric ("A Taxi-Cab World"). At this world, "Pi" has a different value. It isn't also clear that the value of "Pi" here is given purely by mathematics. As its value is relative to a distance metric, and we choose that metric for empirical reasons ("it applies to our world").
- pdpi 5y agoThe question "Is pi the same in all universes?" is poorly defined. You're talking about pi as "the circle constant for whatever universe I'm in", whereas the GP is talking about pi as "the circle constant for a very specific geometry (that we once accepted as representative of the universe we're in)"
- lordnacho 5y agoNone of the other metrics come from nature either, that's the point. "Logic" is more of a catchall for "not from nature" than a specific axiomatic system.
- mypalmike 5y agoI'm no mathematician but it's my understanding that axiomatic set theory is considered foundational for both what you call mathematics and also for discrete mathematics (aka logic).
- foldr 5y agoAxiomatic set theory can't very well be considered foundational for logic, as its axioms are formulas of first-order predicate logic.
- makerofthings 5y agoThat’s exactly what my son does and I hadn’t really thought about why until this post. Really interesting comment. I might try asking him to imagine the shapes instead of looking at my diagrams, I doubt he would imagine my terrible art skills.
- lordnacho 5y agoYeah it's pretty limited what you can describe in words. You have to sort of "get it" with the diagrams what is being said.
- throwaaskjdfh 5y ago> Pi comes from logic, not nature. What if logic is different in different universes? Logic is universal... but is it multiversal?
- mekoka 5y agoCan you conceive of a universe where the abstract quantity 1 added to another abstract quantity 1 does not equal the abstract quantity 2? How would that work? Edit: added the word abstract to make it clear that the context of the question is the realm of logic, and not the boundaries of nature (physics).
- V-2 5y agoI (for one) can't conceive it, but I also cannot really conceive what's happening in a black hole, or at the quantum level. And the weirdness of those phenomenons is nothing in comparison to a "brand new" different reality.
- mekoka 5y agoBlack hole, quantum. Physics i.e nature, not logic.
- less_less 5y agoThere could be a universe where "thingness" doesn't work like in ours -- e.g. where things aren't portable, or where the natural world doesn't divide easily into things. Maybe quantity wouldn't mean anything in such a universe. More concretely, there are lots of ways that the mathematics of a different universe could be so different from ours that π is at best a theoretical concept. There could be a universe where spacetime is discrete, à la Conway's Game of Life. We might even live in such a universe, but the discretization in ours is too small to probe. There could be a universe with a different distance metric, such as a L1 ("taxicab") or L∞ (max difference), so that "circles" look like squares. Or spacetime, or even "thingness" could be modulo some number p -- if p=2, then 1+1 = 0. Space could still be infinite, though -- it could have more dimensions, or it could be based on (e.g.) a complete field of characteristic p. Or the metric could be p-adic, where two numbers are close if their difference is divisible by many powers of p -- kind of like the US highway system, where highways 80 and 880 are nearby. These metrics have rules like the triangle equality: the longest two sides of a triangle are the same length.
- pacman128 5y agoThis brings up the thing that really bothered me about Carl Sagan's novel _Contact_. A book that I really liked otherwise. Warning, this might be considered a spoiler: A pattern found deep in the computed binary digits of pi are used to "prove" that the universe was created by an intelligence.
- neurobashing 5y agothat's a common trope, though. Baxter did it the Xeelee Sequence: the monads selected our particular universe "seed" bc it was amicable to life, and the proto-Xeelee and others tweaked universal constants (via unexplained methods) in the first few millionths of a second after the Big Bang that allowed for, well, us. And there's too many to list of the whole "Ancients/Precursors" who basically made intelligent life or whatever.
- coldtea 5y agoWhat part of this is "really bothering"? We can make a simulation today, and have it obey such natural laws (of course programmed behavior) + logic (ditto), so that we could encode whatever we want inside a physical or geometrical constant as it appears inside the simulation.
- pacman128 5y agoWe can't make a simulation that changes the result of infinite series. They were computing pi, not measuring it. This wasn't changing the curvature of space, it was changing math itself.
- coldtea 5y ago>We can't make a simulation that changes the result of infinite series. Huh? It's trivial to do so. You just give the simulation the appropriate algebra that applies to all their measurements. Not more difficult than setting 12=66 to programming languages that allow it (there are some, it can also be done in others like Python through some trickery [1]), and thus changing all the subsequent calculations done there. [1] https://www.reddit.com/r/Python/comments/2441cv/can_you_change_the_value_of_1/ https://www.reddit.com/r/Python/comments/2441cv/can_you_chan...
- coldtea 5y ago>Pi comes from logic, not nature. Logic itself is a human construct that comes from observing nature - God's surely didn't hand it over to us. It is also not necessarily a singular thing (same way there are different logics as well as different geometries, and so on). A universe with different natural laws might also obey a different logic. In fact, even in this universe, even basic logic propositions are said to be uncertain/untrue under specific scales or physical environments (quantum micro scales, inside a singularity, and so on). >It's actually not terribly easy to explain to them that they're supposed to understand the idealized entity There's no singular "idealized entity". For example there are geometries where triangle angles add up to > 180 degrees.
- mcherm 5y ago> Logic itself is a human construct that comes from observing nature - God's surely didn't hand it over to us. I'm not sure that is true. If we estimate Pi by tossing needles onto a grid of lines[1] then we are obtaining the value by observing nature. But, while that's a fun exercise, it isn't how mathematicians actually determine the value of Pi. They use formulas[2] based on elementary numbers, such as this: Pi = Sqrt( 6 * Sum[k=1..infinity](1/k^2) ). Maybe you believe that the numbers 1, 2, 3... are different in different universes. Personally, I can't fathom that. But if you accept that the basic counting numbers are fundamental to logic and reason instead of being derived from nature and somehow varying from one universe to another, then simply square those numbers, take the reciprocals, sum the series, multiply by 6 and take the square root -- that's Pi. There ARE various geometries possible. But those consist of different definitions of what terms like "straight line" mean. The geometries all have the same value for the fundamental circle constant, Pi. Whether we choose to talk about Pi or instead about Tau (where Tau = 2Pi) is a social convention. But what the value is (of both constants) is a fundamental property of mathematical reasoning independent of convention. [1] https://www.sciencefriday.com/articles/estimate-pi-by-dropping-sticks/ https://www.sciencefriday.com/articles/estimate-pi-by-droppi... [2] https://mathworld.wolfram.com/PiFormulas.html https://mathworld.wolfram.com/PiFormulas.html
- coldtea 5y ago>But, while that's a fun exercise, it isn't how mathematicians actually determine the value of Pi. They use formulas[2] based on elementary numbers, such as this: Pi = Sqrt( 6 Sum[k=1..infinity](1/k^2) ).* Yes, but this didn't come from God (or the Platonic realm of Ideas) into mathematicians heads. Humans arrived at that by first observing natural laws / behavior (e.g. 1 apple + 1 apple = 2 apples, or observing that somebody can't be A and not A, and so on) and progressing from those into abstractions. Nature comes first, not abstraction. Nor there is some plane where mathematical abstraction or logic exists outside of the universe. It exists only (a) explicitly on people's heads (which, unaraguably, are part of nature), (b) as implicitly observed on natural behavior by said people. >Maybe you believe that the numbers 1, 2, 3... are different in different universes. Personally, I can't fathom that. No, I believe that numbers don't exist. What would "1" even be, as an entity in our very universe? There is a rock or a planet or a tiger, but not an 1. Countable things do (but not necessarily on another universe, where e.g. there might be just a single entity). Also, we already have boolean algebra in this universe, which has different properties compared to regular algebra. In that sense a logic circuit is already a kind of universe where the algebra that governs it is different. One could well imagine a universe that only observes circuit board like behavior.
- d0mine 5y agoWe live in a world that is well described by Euclidien geometry for many applications. It doesn't mean it is the only possible useful geometry. If we define Pi as a ratio C/D, it can be different. Look at the example in the article.