6 ms·

Whenever I encounter this sort of abstract math (at least “abstract” for me) I start wondering what’s even “real”. Like, what is some foundational truth of real

by brap 1y ago

Whenever I encounter this sort of abstract math (at least “abstract” for me) I start wondering what’s even “real”. Like, what is some foundational truth of reality vs. stuff we just made up and keep exploring. Are these knots real? Are prime numbers real? Multiplication? Addition? Are natural numbers really “natural”? For example, one thing that always seemed bizarre to me for as long as I can remember is Pi. If circles are natural and numbers are natural, then why does their relationship seem so unnatural and arbitrary? You could imagine some advanced alien civilization, maybe in a completely different universe, that isn’t even aware of these concepts. But does it make them any less real? Sorry for rambling off topic like a meth addict, just hoping someone can enlighten me.

slickytail 1y ago
In the words of Kronecker: "God created the integers, all else is the work of man."
- srean 1y ago
  Had I been god I would have created scaled turns and left the rest for humans.
fedeb95 1y ago
I sometimes think about the same things. As of now, my best bet is that math is one of the disciplines studying exactly these questions.
fjfaase 1y ago
What is real? There are strong indications that what we experience as reality is an ilusion generated by what is usually refered to as the subconscious. One could argue that knots are more real than numbers. It is hard to find two equal looking apples and talk about two apples, because it requires the abstraction that the apples are equal, while it is obvious that they are not. While, I guess, we all have had the experience of strugling with untying knots in strings.
- jrowen 1y ago
  It’s more than strong indications. What any individual life form perceives is a unique subset or projection of reality. To the extent that “one true reality” exists, we are each viewing part of it through a different window.
lqet 1y ago
Philosophical problems regarding the fundamental nature of reality aside, this short clip is relevant to your question: > https://www.youtube.com/watch?v=tCUK2zRTcOc https://www.youtube.com/watch?v=tCUK2zRTcOc Translated transcript: Physics is a "Real Science". It deals with reality. Math is a structural science. It deals with the structure of thinking. These structures do not have to exist. They can exist, but they don't have to. That's a fundamental difference. The translation of mathematical concepts to reality is highly critical, I would say. You cannot just translate it directly, because this leads to such strange questions like "what would happen if we take the law of gravitation by old Newton and let r^2 go to zero?". Well, you can't! Because Heisenberg is standing down there.
- twiceaday 1y ago
  Math is a purely logical tool. None of it "exists." That makes no sense. Some of it can be used to model reality. We call such math "physics." And I think physics is significantly closer to math than to reality. It's just a collection of math that models some measurements on some scales with some precision. We have no idea how close we are to actual reality. I do not understand the framing of "translating math concepts directly into reality." It's backwards. You must have first chosen some math to model reality. If you get "bad" numbers it has nothing to do with translating math to reality. It has to do with how you translated reality into math.
  - brap 1y ago
    I think maybe I didn’t really explain myself properly. I didn’t mean that math is real in the sense that atoms are real. Perhaps “true” would be a better word. We know these things are true to us, but are they universally true? If that’s even a thing? Hope that makes more sense.
    - IAmBroom 1y ago
      The age-old problem of a respondent using different definitions of words than the OP. Socrates made a whole career out of it.
    - 1y ago
yujzgzc 1y ago
Yes these knots are real and can be experienced with a simple piece of rope. The prime property of numbers is also very real, a number N is prime if and only if arranging N items on a rectangular, regular grid can only be done if one of the sides of the rectangle is 1. Multiplication and addition are even more simply realized. The infinity of natural numbers is not as real, if what we mean by that is that we can directly experience it. It's a useful abstraction but there is, according to that abstraction, an infinity of "natural" numbers that mankind will not be able to ever write down, either as a number or as a formula. So infinity will always escape our immediate perception and remain fundamentally an abstraction. Real numbers are some of the least real of the numbers we deal with in math. They turn out to be a very useful abstraction but we can only observe things that approximate them. A physical circle isn't exactly pi times its diameter up to infinity decimals, if only because there is a limit to the precision of our measurements. To me the relationship between pi and numbers is not so unnatural but I have to look at a broader set of abstractions to make more sense of it, adding exponentials and complex numbers - in my opinion the fact that e^i.pi = 1 is a profound relationship between pi and natural numbers. But abstractions get changed all the time. Math as an academic discipline hasn't been around for more than 10,000 years and in that course of time abstractions have changed. It's very likely that the concept of infinity wouldn't have made sense to anyone 5,000 years ago when numbers were primarily used for accounting. Even 500 years ago the concept of a number that is a square root of -1 wouldn't have made sense. Forget aliens from another planet, I'm pretty sure we wouldn't be able to comprehend 100th century math if somehow a textbook time-traveled to us.
- IIAOPSW 1y ago
  I see infinity all the time. Go look at a one point perspective drawing.
  - tocs3 1y ago
    I think that is a little like pi. There is a limit to what we can measure. In a real life drawing on paper the "one point" is not dimensionless. There is a limit to what we can draw.
jaffa2 1y ago
Theres always an xkcd : https://xkcd.com/435/ https://xkcd.com/435/
- adornKey 1y ago
  Nice line, but it isn't fully complete. After the Mathematicians there's Logic - and Philosophy - And in the end you complete the circle and go all back to Sociology again. One issue I sometimes witnessed myself was that Mathematicians sometimes form Groups that behave like pathological examples from Sociology. E.g. there was the Monty-Hall problem, where societies of mathematicians had a meltdown. Sadly I've seen this a few times when Sociology/Mass psychology simply trumped Math in Power.
CJefferson 1y ago
To me, the least real thing in maths is, ironically, the real numbers. As you dig through integers, fractions, square roots, solutions to polynomials, things a turing machine can output, you get to increasingly large classes of numbers which are still all countably infinite. At some point I realised I'd covered anything I could ever imagine caring about and was still in a countable set.
- JdeBP 1y ago
  The entirely opposite perspective is quite interesting: The "natural numbers" are the biggest mis-nomer in mathematics. They are the most un-Natural ones. The numbers that occur in Nature are almost always complex, and are neither integers nor rationals (nor even algebraics). When you approach reality through the lens of mathematics that concentrates the most upon these countable sets, you very often end up with infinite series in order to express physical reality, from Feynman sums to Taylor expansions.
  - rini17 1y ago
    But you can't really have chemistry without working with natural numbers of atoms, measured in moles. Recently they decided to explicitly fix a mole (Avogadro's constant) to be exactly 6.02214076×10^23 which is a natural number. Semiconductor manufacturing on nanometer scales deals with individual atoms and electrons too. Yes, modeling their behavior needs complex numbers, but their amounts are natural numbers.
  - srean 1y ago
    I agree. Had humanity made turning the more fundamental operation than counting that would have sped up our mathematical journey. The Naturals would have fallen off from it as an exercise of counting turns. The calculus of scaled rotation is so beautiful. The sacrificial lamb is the unique ordering relation.
  - kcexn 1y ago
    The natural numbers are 'natural' because they are definite quantities that can be used for counting. Taylor expansions about a point of a function requires that the function has a derivative defined at that point. The derivative itself is the point at which an infinite sequence (say, of incrementally closer approximations) converges. So derivatives and Taylor series are really more of an arbitrary precision approximation of a value rather than a concrete exact quantity. Arbitrary precision approximation just happens to be a very elegant way to model the physical world around us. For truly exact solutions, you still have to work with the naturals (and rationals, etc.)
Byamarro 1y ago
Math is about creating mental models. Sometimes we want to model something in real life and try to use math for this - this is physics. But even then, the model is not real, it's a model (not even a 1:1 one on top of that). It usually tries to capture some cherry picked traits of reality i.e. when will a planet be in 60 days ignoring all its "atoms"[1]. That's because we want to have some predictive power and we can't simulate whole reality. Wolfram calls these selective traits that can be calculated without calculating everything else "pockets of reducability". Do they exist? Imho no, planets don't fundamentally exist, they're mental constructs we've created for a group of particles so that our brains won't explode. If planets don't exist, so do their position etc. The things about models is that they're usually simplifications of the thing they model, with only the parts of it that interest us. Modeling is so natural for us that we often fail to realize that we're projecting. We're projecting content of our minds onto reality and then we start to ask questions out of confusion such as "does my mind concept exist". Your mind concept is a neutral pattern in your mind, that's it. [1] atoms are mental concepts as well ofc
- movpasd 1y ago
  I believe this is called epistemic pragmatism in philosophy: https://en.wikipedia.org/wiki/Pragmatism https://en.wikipedia.org/wiki/Pragmatism
JdeBP 1y ago
More usually, people imagine the reverse of the advanced alien civilizations: that the thing that we and they are most likely to have in common is the concept of obtaining the ratio between a circle's circumference and its diameter, whereas the things that they possibly aren't even aware of are going to be concepts like economics or poetry.
kurlberg 1y ago
Fun historical fact: knot theory got a big boost when lord Kelvin (yeah, that one) proposed understanding atoms by thinking of them as "knotted vortices in the ether".
rini17 1y ago
I see it like natural sciences strive to do replicable experiments in outside world, while math strives to do replicable experiments in mind. Not everything is transferable from one domain to the other but we keep finding many parallels between these two, which is surprising. But that's all we have, no foundational truths, no clear natural/unnatural divide here.
kannanvijayan 1y ago
I don't have an answer to your questions, but I think these thoughts are not uncommon for people who get into these topics. The relationship between the reals, including Pi, and the countables such as the naturals/integers/rationals is suggestive of some deeper truth. The ratio between the areas of a unit circle (or hypersphere in whatever dimension you choose) and a unit square (or hypercube in that dimension) in any system will always require infinite precision to describe. Make the areas between the circle and the square equal, and the infinite precision moves into the ratio between their lower order dimensional measures (circumfence, surface area, etc.). You can't describe a system that expresses the one, in terms of a system that expresses the other, without requiring infinite precision (and thus infinite information). Furthermore, it really seems like a bunch of the really fundamental reals (pi, e), have a pretty deep connection to algebras of rotations (both pi and e relate strongly to rotations) What that seems to suggest to me is that if the universe is discrete, then the discreteness must be biased towards one of these modes or the other - i.e. it is natively one and approximates the other. You can have a discrete universe where you have natural rotational relationships, or natural linear relationships, but not both at the same time.
- schiffern 1y ago
  >The ratio between the areas of a unit circle (or hypersphere in whatever dimension you choose) and a unit square (or hypercube in that dimension) in any system will always require infinite precision to describe. Easily fixed! I choose 1 dimension. :)
  - kannanvijayan 1y ago
    Hah, nice find :)
    - schiffern 1y ago
      Good show, and I appreciate your sentiment about the "messiness" of pi. There's a unit-converting calculator[0] that supports exact rational numbers and will carry undefined variables through algebraically. With a little hacking, you can redefine degrees in terms in an exact rational multiple of pi radians. Pi is effectively being defined as a new fundamental unit dimension, like distance. Trig functions can be overloaded to output an exact representation when it detects one of the exact trigonometric values[1] eg cos(60°) = 1/2. It will now give output values as "X + Y PI", or you can optionally collapse that to an inexact decimal with an eval[] function. That's the closest I got to containing the "messiness" of pi. Eventually I hit a wall because Frink doesn't support exact square roots, so most exact values would be decimals anyway. Still, I can dream! [0] https://frinklang.org/ https://frinklang.org/ [1] https://en.wikipedia.org/wiki/Exact_trigonometric_values https://en.wikipedia.org/wiki/Exact_trigonometric_values
jibal 1y ago
> If circles are natural and numbers are natural, then why does their relationship seem so unnatural and arbitrary? It is not in any way unnatural or arbitrary. However, there are no circles in nature. > You could imagine some advanced alien civilization, maybe in a completely different universe, that isn’t even aware of these concepts. I can't actually imagine that ... advancement in the physical world requires at least mastery of the most basic facts of arithmetic. > just hoping someone can enlighten me I suggest that you first need some basic grounding in math and philosophy.
amiga386 1y ago
I'm fairly confident that most mathematics are real, i.e. they have real world analogues. Pi is just an increasingly close look at the ratio between a circle's diameter and circumference. I'm willing to believe elecromagnetic fields are real - you can see the effects magnets (and electromagnets) have on ferrous material. You can really broadcast electromagnetic waves, induce currents in metals, all that. I'm willing to believe atoms, quarks, electrons, photons, etc. are real. Forces (electrical charge, weak and strong nuclear force, gravity) are real. What I'm not willing to believe is that quantum fields in general are real, that physical components are not real and don't literally move, they're just "interactions" with and "fluctuations" in the different quantum fields. I refuse to believe that matter doesn't exist and it's merely numbers or vectors arranged a grid. That's a step too far. That's surely just a mathematical abstraction. And yet, the numbers these abstractions produce match so well with physical observations. What's going on?
- BobbyTables2 1y ago
  What about the particles that randomly pop in and out of existence? If one thinks about it, electromagnetism is really bizarre. How can two electrons actually repel each other? Sure, they do, but it’s practically witchcraft. Magnetism is even more weird.
  - amiga386 1y ago
    > What about the particles that randomly pop in and out of existence? I like to imagine they're somehow just an observational error, otherwise the https://en.wikipedia.org/wiki/One-electron_universe https://en.wikipedia.org/wiki/One-electron_universe is real and we get a universe-sized '—All You Zombies—' > How can two electrons actually repel each other Indeed. I think it's something we can only intuit, I don't think we've really gotten to the bottom of it. Trying to push two electrons together feels like trying to push a car up a hill, or pressing on springs. The force you fight against is just there and you feel its resistance
- pelorat 1y ago
  Wait until you hear about the gluon, the mediator of the strong force, which is an excitation in the gluon field, and is also the only other particle that is massless and moves at C. However unlike the photon the excitation has a really short range because gluons interact with gluons and form flux tubes between quarks, the further you pull two quarks apart, the more energy you need to use, eventually the energy is so great that it spawns a new quark from the vacuum. Compared to EM it's just weird as hell and tbh I don't like it.
eprparadox 1y ago
there's a great episode of Mindscape where Max Tegmark takes this idea and runs with it: https://www.preposterousuniverse.com/podcast/2019/12/02/75-max-tegmark-on-reality-simulation-and-the-multiverse/ https://www.preposterousuniverse.com/podcast/2019/12/02/75-m...
kandel 1y ago
My pet philosophy is that math is real because the objects have persistent effects, like with the "if a tree falls in the forest..." riddle. Something that isn't real would be a story, because things do not have effects in it. If a function is one-to-one, it has a (right? left? keep forgetting which one)-inverse. But if Moshe the imaginary forgot the milk, his wife may or may not shout at him, whichever way the story teller decides to take the story... So a function being one-to-one is real, but Moshe the imaginary forgetting the milk isn't. I like this view when I'm being befuddled by a result, especially some ad absurdum argument. I tell myself: this thing is true, so if it wasn't we'd just need to look hard enough to find somewhere where two effects clash.
ctenb 1y ago
Math is about discovering universal truths: Given a set of axioms and following theorems, the theorems will apply in any scenario where the axioms are true. So that makes maths both invented (the axioms) and discovered (the theorems) and real in any situation where it applies.

by brap 1y ago

slickytail 1y ago

In the words of Kronecker: "God created the integers, all else is the work of man."

srean 1y ago
Had I been god I would have created scaled turns and left the rest for humans.

fedeb95 1y ago

I sometimes think about the same things. As of now, my best bet is that math is one of the disciplines studying exactly these questions.

fjfaase 1y ago

What is real? There are strong indications that what we experience as reality is an ilusion generated by what is usually refered to as the subconscious. One could argue that knots are more real than numbers. It is hard to find two equal looking apples and talk about two apples, because it requires the abstraction that the apples are equal, while it is obvious that they are not. While, I guess, we all have had the experience of strugling with untying knots in strings.

jrowen 1y ago
It’s more than strong indications. What any individual life form perceives is a unique subset or projection of reality. To the extent that “one true reality” exists, we are each viewing part of it through a different window.

lqet 1y ago

Philosophical problems regarding the fundamental nature of reality aside, this short clip is relevant to your question: > https://www.youtube.com/watch?v=tCUK2zRTcOc https://www.youtube.com/watch?v=tCUK2zRTcOc Translated transcript: Physics is a "Real Science". It deals with reality. Math is a structural science. It deals with the structure of thinking. These structures do not have to exist. They can exist, but they don't have to. That's a fundamental difference. The translation of mathematical concepts to reality is highly critical, I would say. You cannot just translate it directly, because this leads to such strange questions like "what would happen if we take the law of gravitation by old Newton and let r^2 go to zero?". Well, you can't! Because Heisenberg is standing down there.

twiceaday 1y ago
Math is a purely logical tool. None of it "exists." That makes no sense. Some of it can be used to model reality. We call such math "physics." And I think physics is significantly closer to math than to reality. It's just a collection of math that models some measurements on some scales with some precision. We have no idea how close we are to actual reality. I do not understand the framing of "translating math concepts directly into reality." It's backwards. You must have first chosen some math to model reality. If you get "bad" numbers it has nothing to do with translating math to reality. It has to do with how you translated reality into math.
- brap 1y ago
  I think maybe I didn’t really explain myself properly. I didn’t mean that math is real in the sense that atoms are real. Perhaps “true” would be a better word. We know these things are true to us, but are they universally true? If that’s even a thing? Hope that makes more sense.
  - IAmBroom 1y ago
    The age-old problem of a respondent using different definitions of words than the OP. Socrates made a whole career out of it.
  - 1y ago

yujzgzc 1y ago

Yes these knots are real and can be experienced with a simple piece of rope. The prime property of numbers is also very real, a number N is prime if and only if arranging N items on a rectangular, regular grid can only be done if one of the sides of the rectangle is 1. Multiplication and addition are even more simply realized. The infinity of natural numbers is not as real, if what we mean by that is that we can directly experience it. It's a useful abstraction but there is, according to that abstraction, an infinity of "natural" numbers that mankind will not be able to ever write down, either as a number or as a formula. So infinity will always escape our immediate perception and remain fundamentally an abstraction. Real numbers are some of the least real of the numbers we deal with in math. They turn out to be a very useful abstraction but we can only observe things that approximate them. A physical circle isn't exactly pi times its diameter up to infinity decimals, if only because there is a limit to the precision of our measurements. To me the relationship between pi and numbers is not so unnatural but I have to look at a broader set of abstractions to make more sense of it, adding exponentials and complex numbers - in my opinion the fact that e^i.pi = 1 is a profound relationship between pi and natural numbers. But abstractions get changed all the time. Math as an academic discipline hasn't been around for more than 10,000 years and in that course of time abstractions have changed. It's very likely that the concept of infinity wouldn't have made sense to anyone 5,000 years ago when numbers were primarily used for accounting. Even 500 years ago the concept of a number that is a square root of -1 wouldn't have made sense. Forget aliens from another planet, I'm pretty sure we wouldn't be able to comprehend 100th century math if somehow a textbook time-traveled to us.

IIAOPSW 1y ago
I see infinity all the time. Go look at a one point perspective drawing.
- tocs3 1y ago
  I think that is a little like pi. There is a limit to what we can measure. In a real life drawing on paper the "one point" is not dimensionless. There is a limit to what we can draw.

jaffa2 1y ago

Theres always an xkcd : https://xkcd.com/435/ https://xkcd.com/435/

adornKey 1y ago
Nice line, but it isn't fully complete. After the Mathematicians there's Logic - and Philosophy - And in the end you complete the circle and go all back to Sociology again. One issue I sometimes witnessed myself was that Mathematicians sometimes form Groups that behave like pathological examples from Sociology. E.g. there was the Monty-Hall problem, where societies of mathematicians had a meltdown. Sadly I've seen this a few times when Sociology/Mass psychology simply trumped Math in Power.

CJefferson 1y ago

To me, the least real thing in maths is, ironically, the real numbers. As you dig through integers, fractions, square roots, solutions to polynomials, things a turing machine can output, you get to increasingly large classes of numbers which are still all countably infinite. At some point I realised I'd covered anything I could ever imagine caring about and was still in a countable set.

JdeBP 1y ago
The entirely opposite perspective is quite interesting: The "natural numbers" are the biggest mis-nomer in mathematics. They are the most un-Natural ones. The numbers that occur in Nature are almost always complex, and are neither integers nor rationals (nor even algebraics). When you approach reality through the lens of mathematics that concentrates the most upon these countable sets, you very often end up with infinite series in order to express physical reality, from Feynman sums to Taylor expansions.
- rini17 1y ago
  But you can't really have chemistry without working with natural numbers of atoms, measured in moles. Recently they decided to explicitly fix a mole (Avogadro's constant) to be exactly 6.02214076×10^23 which is a natural number. Semiconductor manufacturing on nanometer scales deals with individual atoms and electrons too. Yes, modeling their behavior needs complex numbers, but their amounts are natural numbers.
- srean 1y ago
  I agree. Had humanity made turning the more fundamental operation than counting that would have sped up our mathematical journey. The Naturals would have fallen off from it as an exercise of counting turns. The calculus of scaled rotation is so beautiful. The sacrificial lamb is the unique ordering relation.
- kcexn 1y ago
  The natural numbers are 'natural' because they are definite quantities that can be used for counting. Taylor expansions about a point of a function requires that the function has a derivative defined at that point. The derivative itself is the point at which an infinite sequence (say, of incrementally closer approximations) converges. So derivatives and Taylor series are really more of an arbitrary precision approximation of a value rather than a concrete exact quantity. Arbitrary precision approximation just happens to be a very elegant way to model the physical world around us. For truly exact solutions, you still have to work with the naturals (and rationals, etc.)

Byamarro 1y ago

Math is about creating mental models. Sometimes we want to model something in real life and try to use math for this - this is physics. But even then, the model is not real, it's a model (not even a 1:1 one on top of that). It usually tries to capture some cherry picked traits of reality i.e. when will a planet be in 60 days ignoring all its "atoms"[1]. That's because we want to have some predictive power and we can't simulate whole reality. Wolfram calls these selective traits that can be calculated without calculating everything else "pockets of reducability". Do they exist? Imho no, planets don't fundamentally exist, they're mental constructs we've created for a group of particles so that our brains won't explode. If planets don't exist, so do their position etc. The things about models is that they're usually simplifications of the thing they model, with only the parts of it that interest us. Modeling is so natural for us that we often fail to realize that we're projecting. We're projecting content of our minds onto reality and then we start to ask questions out of confusion such as "does my mind concept exist". Your mind concept is a neutral pattern in your mind, that's it. [1] atoms are mental concepts as well ofc

movpasd 1y ago
I believe this is called epistemic pragmatism in philosophy: https://en.wikipedia.org/wiki/Pragmatism https://en.wikipedia.org/wiki/Pragmatism

JdeBP 1y ago

More usually, people imagine the reverse of the advanced alien civilizations: that the thing that we and they are most likely to have in common is the concept of obtaining the ratio between a circle's circumference and its diameter, whereas the things that they possibly aren't even aware of are going to be concepts like economics or poetry.

kurlberg 1y ago

Fun historical fact: knot theory got a big boost when lord Kelvin (yeah, that one) proposed understanding atoms by thinking of them as "knotted vortices in the ether".

rini17 1y ago

I see it like natural sciences strive to do replicable experiments in outside world, while math strives to do replicable experiments in mind. Not everything is transferable from one domain to the other but we keep finding many parallels between these two, which is surprising. But that's all we have, no foundational truths, no clear natural/unnatural divide here.

kannanvijayan 1y ago

I don't have an answer to your questions, but I think these thoughts are not uncommon for people who get into these topics. The relationship between the reals, including Pi, and the countables such as the naturals/integers/rationals is suggestive of some deeper truth. The ratio between the areas of a unit circle (or hypersphere in whatever dimension you choose) and a unit square (or hypercube in that dimension) in any system will always require infinite precision to describe. Make the areas between the circle and the square equal, and the infinite precision moves into the ratio between their lower order dimensional measures (circumfence, surface area, etc.). You can't describe a system that expresses the one, in terms of a system that expresses the other, without requiring infinite precision (and thus infinite information). Furthermore, it really seems like a bunch of the really fundamental reals (pi, e), have a pretty deep connection to algebras of rotations (both pi and e relate strongly to rotations) What that seems to suggest to me is that if the universe is discrete, then the discreteness must be biased towards one of these modes or the other - i.e. it is natively one and approximates the other. You can have a discrete universe where you have natural rotational relationships, or natural linear relationships, but not both at the same time.

schiffern 1y ago
>The ratio between the areas of a unit circle (or hypersphere in whatever dimension you choose) and a unit square (or hypercube in that dimension) in any system will always require infinite precision to describe. Easily fixed! I choose 1 dimension. :)
- kannanvijayan 1y ago
  Hah, nice find :)
  - schiffern 1y ago
    Good show, and I appreciate your sentiment about the "messiness" of pi. There's a unit-converting calculator[0] that supports exact rational numbers and will carry undefined variables through algebraically. With a little hacking, you can redefine degrees in terms in an exact rational multiple of pi radians. Pi is effectively being defined as a new fundamental unit dimension, like distance. Trig functions can be overloaded to output an exact representation when it detects one of the exact trigonometric values[1] eg cos(60°) = 1/2. It will now give output values as "X + Y PI", or you can optionally collapse that to an inexact decimal with an eval[] function. That's the closest I got to containing the "messiness" of pi. Eventually I hit a wall because Frink doesn't support exact square roots, so most exact values would be decimals anyway. Still, I can dream! [0] https://frinklang.org/ https://frinklang.org/ [1] https://en.wikipedia.org/wiki/Exact_trigonometric_values https://en.wikipedia.org/wiki/Exact_trigonometric_values

jibal 1y ago

> If circles are natural and numbers are natural, then why does their relationship seem so unnatural and arbitrary? It is not in any way unnatural or arbitrary. However, there are no circles in nature. > You could imagine some advanced alien civilization, maybe in a completely different universe, that isn’t even aware of these concepts. I can't actually imagine that ... advancement in the physical world requires at least mastery of the most basic facts of arithmetic. > just hoping someone can enlighten me I suggest that you first need some basic grounding in math and philosophy.

amiga386 1y ago

I'm fairly confident that most mathematics are real, i.e. they have real world analogues. Pi is just an increasingly close look at the ratio between a circle's diameter and circumference. I'm willing to believe elecromagnetic fields are real - you can see the effects magnets (and electromagnets) have on ferrous material. You can really broadcast electromagnetic waves, induce currents in metals, all that. I'm willing to believe atoms, quarks, electrons, photons, etc. are real. Forces (electrical charge, weak and strong nuclear force, gravity) are real. What I'm not willing to believe is that quantum fields in general are real, that physical components are not real and don't literally move, they're just "interactions" with and "fluctuations" in the different quantum fields. I refuse to believe that matter doesn't exist and it's merely numbers or vectors arranged a grid. That's a step too far. That's surely just a mathematical abstraction. And yet, the numbers these abstractions produce match so well with physical observations. What's going on?

BobbyTables2 1y ago
What about the particles that randomly pop in and out of existence? If one thinks about it, electromagnetism is really bizarre. How can two electrons actually repel each other? Sure, they do, but it’s practically witchcraft. Magnetism is even more weird.
- amiga386 1y ago
  > What about the particles that randomly pop in and out of existence? I like to imagine they're somehow just an observational error, otherwise the https://en.wikipedia.org/wiki/One-electron_universe https://en.wikipedia.org/wiki/One-electron_universe is real and we get a universe-sized '—All You Zombies—' > How can two electrons actually repel each other Indeed. I think it's something we can only intuit, I don't think we've really gotten to the bottom of it. Trying to push two electrons together feels like trying to push a car up a hill, or pressing on springs. The force you fight against is just there and you feel its resistance
pelorat 1y ago
Wait until you hear about the gluon, the mediator of the strong force, which is an excitation in the gluon field, and is also the only other particle that is massless and moves at C. However unlike the photon the excitation has a really short range because gluons interact with gluons and form flux tubes between quarks, the further you pull two quarks apart, the more energy you need to use, eventually the energy is so great that it spawns a new quark from the vacuum. Compared to EM it's just weird as hell and tbh I don't like it.

eprparadox 1y ago

there's a great episode of Mindscape where Max Tegmark takes this idea and runs with it: https://www.preposterousuniverse.com/podcast/2019/12/02/75-max-tegmark-on-reality-simulation-and-the-multiverse/ https://www.preposterousuniverse.com/podcast/2019/12/02/75-m...

kandel 1y ago

My pet philosophy is that math is real because the objects have persistent effects, like with the "if a tree falls in the forest..." riddle. Something that isn't real would be a story, because things do not have effects in it. If a function is one-to-one, it has a (right? left? keep forgetting which one)-inverse. But if Moshe the imaginary forgot the milk, his wife may or may not shout at him, whichever way the story teller decides to take the story... So a function being one-to-one is real, but Moshe the imaginary forgetting the milk isn't. I like this view when I'm being befuddled by a result, especially some ad absurdum argument. I tell myself: this thing is true, so if it wasn't we'd just need to look hard enough to find somewhere where two effects clash.

ctenb 1y ago

Math is about discovering universal truths: Given a set of axioms and following theorems, the theorems will apply in any scenario where the axioms are true. So that makes maths both invented (the axioms) and discovered (the theorems) and real in any situation where it applies.