Are space ships in game of life invented or discovered? I've come to believe that's ultimately exactly the same question.
We're talking about rich structures that emerge from simple rule sets. That's really all there is to mathematics.
We started from structures that provide helpful abstractions for things in real life, which might lead some to believe that mathematics points to some hidden reality behind things, but that's ultimately just spiritual thinking in rational sounding clothing.
It is sometimes said that it is surprising that nature follows laws that are mathematical, but that's the wrong way around. Mathematics just provides helpful abstractions that we can apply to whatever we like. Saying nature is mathematical is just saying that nature is consistent in the laws it follows.
I 100% agree with everything you said, I just wanted to make this point even more forcefully:
> Saying nature is mathematical is just saying that nature is consistent in the laws it follows.
It's more like "wherever nature isn't mathematical, we don't think about it as being mathematical", so saying "nature is mathematical" is very strongly tautological.
It's the same kind of situation as "why is everything linear?", or "why is everything an oscillator". The answer is more like "the things that aren't can't be easily described mathematically, and so we don't. What shakes out is mostly linear or quadratic (oscillators)".
Physics is just one (non-)choice of axioms. Math that works in the world is physics, it's all discovered. You can't invent math and expect physics to follow.
Other math is invented through the choice of axioms and then we discovery the theorems that follow from those.
This is a compelling way of looking at things for sure; to me it seems like you're saying math is just a language (like English) that we can use to describe the world. English is no more "right" than some other made up language, so same goes for math, and thats certainly seems true in some cases. However there are things that are intrinsic to mathematics that also seem intrinsic to existence, like prime numbers. Many things about prime numbers are 'discovered' as if they were elements of nature that need to be revealed through study. For example, the distribution of primes and the distribution of twin primes are things likely to have been discovered by an alien intelligence, assuming one exists. What would their math look like? Different for sure, but I think it's likely they still studied the primes, hypothetically.
I agree with you that what I'm saying is that maths is ultimately a kind of language that we use to describe the world. I wouldn't agree with calling it "just" language or "like English". It's different in that consistency is strictly enforced, such that simple rules can lead to rich emergent structures. That's what I was getting at with the game of life metaphor.
So, if you follow my argument, the fact that there are prime numbers is not 'intrinsic to mathematics', it's the other way around -- we use rules that lead to rich structures, because those provide useful, consistent abstractions to talk about things. Prime numbers are among those rich structures.
To respond to a sibling comment that seems somewhat related -- The fact that we can calculate some physical constants with high precision is not a counter argument to that. What we are doing is fitting a mathematical abstraction to data; once we have done that, we can find more physical truths by assuming consistency and performing logical inference. The fact that a constant can be calculated then just means that we have a consistent theory for the physical structures from which that constant arises.
Those "why" questions are exactly what math and science do not answer (:
Not saying that philosophy or religion is particularly good at answering them either, but that's more the domain you're getting into, I think.
Brief excerpt from this Feynman interview[1]
"Say pop, I noticed something: when I pull the wagon, the ball rolls to
the back ... and when I'm pulling along and I suddenly stop, the ball
rolls to the *front* of the wagon." And I said: "why is that?"
And he said: "That, nobody knows. But the general principle is that things
that are moving tend to keep on moving, and things that are standing still
tend to stand still... This tendency is called "inertia," but nobody knows
why it is true."
Is it intelligible that it might not have been? What would it mean for nature to be inconsistent? What would a universe look like that didn't follow laws?
We don't know, and mathematics is not the tool for that. The consistency of nature is considered by some to be one of the few things science has to take by faith. Same thing with "brain-in-a-jar" arguments, we cannot ever prove that we are not just being completely fooled by our senses that our universe exists.
I went to an in-person Q&A featuring a Fields medalist. The audience was a collection of undergraduate and high school math students, with a few professors in attendance.
One of the young students asked exactly this question to which everyone in the audience collectively groaned. The Fields medalist gave a short answer, something along the lines of "I don't know a single mathematician that thinks it's invented."
He was being polite, but you could tell he didn't think there was anything else interesting to say.
It's both. The axioms are invented, the corpus of theorems is discovered. As once the axioms are chosen the provable theorems are already fixed.
But the axioms are a choice, and we can pick different ones. The common choice of axioms is utilitarian, they lead to interesting math that helps us describe the universe.
Proving a theorem given a set of axioms is a search problem. Given a set of axioms you can apply rules of inference to generate the graph of all provable theorems. Proving a theorem is about finding a path from the axioms to the vertex which is your theorem.
But you can make the same case for axioms - that they are not invented but discovered through a process of search in the space of axioms.
I'm not sure I see why the axioms were not also discovered though? Choice between irreducible assumptions does not seem to make them any more 'invented'.
Are the theories beyond axiom fundamentally different if axioms are changed, though? And if not, aren't then axioms merely props or placeholders for invariants?
I think it's one of the those things where it doesn't matter what the answer is because it doesn't provide a useful lens for advancing your mathematical thinking.
Plato conceived of Truth as being the light through which things could be conceived. At the highest level was ideas (eg the idea of "Chair", not any particular chair), below that was mathematical objects, followed by physical objects and then shadows.
So Plato would say that math is closer to truth than a chair, but not than the idea of Chair.
When you define a set of axioms, you implicitly define all possible math inside this set. Mathematicians then "discover" useful or interesting math in this implicit pile.
In turn, introduction of a new (useful) axioms set can be called an "invention".
Different sets of axioms open the doors to different mathematical universes? I'm given to understand that's the general concensus on what Gödel's incompleteness theorems mean, practically speaking.
Not really. Goedel's incompleteness theorem says that no consistent set of axioms can prove all true statements. Moreover, once you take a few unprovable, but true, things and add them to you axioms, there will still be things that are true that you cannot prove.
This means that you're not going to find two mutually exclusive sets of axioms that, between the two of them, can prove any and all true things. This suggests to me that there's not a Library of Babel of mathematical universes, it's more or less one big universe, with a lot of black holes.
Not sure why this was downvoted. If I said something inaccurate or misunderstood Goedel's incompleteness theorem, then I'd like to know. It's something that's fascinated me for decades.
Gödel tells us that every formal system contains contradictions and unprovable statements (save for very simple systems). I’ve never heard a summary remotely akin to yours, so I’m unsure if it’s equivalent.
I was introduced to incompleteness in the context of Turing Machines and the halting problem in a CS course. I think that if you are introduced to it as part of a foundations of mathematics course then the focus is on what it implies about mathematical reasoning. As ever if you can find the right bit of Wikipedia then there's a fascinating explanation clearly written by someone who really knows about it that I don't understand but hints at worlds of thought that I will spend time exploring when I don't have to get this fucking demo to work or deal with my boss. So maybe never...
It definitely does not say that. Only that we cannot make a proof of consistency of them. It is possible, and believed, that the Zermelo–Fraenkel axioms we use for set theory for example are consistent. But we cannot prove that because of Godel.
Can these universes be mutual subsets? I'm grappling with the incompleteness theorems as they pertain to statements that are not contained in the set of possibilities which they describe, as in Gödel Escher Bach when discussing ideas foundational to Hofstadter's strange loop concept. The self-descriptive nature of information or lack thereof, essentially the existence of recursion without exclusion of the seed itself is stretching my mind. I think I need to start G.E.D. from the beginning (i.e. re-read the 40 pages I made it through).
Keep reading it, page 1 really isn't the start. Each thread starts in a different place and we are introduced to some of them in the middle of their story.
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We're talking about rich structures that emerge from simple rule sets. That's really all there is to mathematics.
We started from structures that provide helpful abstractions for things in real life, which might lead some to believe that mathematics points to some hidden reality behind things, but that's ultimately just spiritual thinking in rational sounding clothing.
It is sometimes said that it is surprising that nature follows laws that are mathematical, but that's the wrong way around. Mathematics just provides helpful abstractions that we can apply to whatever we like. Saying nature is mathematical is just saying that nature is consistent in the laws it follows.
> Saying nature is mathematical is just saying that nature is consistent in the laws it follows.
It's more like "wherever nature isn't mathematical, we don't think about it as being mathematical", so saying "nature is mathematical" is very strongly tautological.
It's the same kind of situation as "why is everything linear?", or "why is everything an oscillator". The answer is more like "the things that aren't can't be easily described mathematically, and so we don't. What shakes out is mostly linear or quadratic (oscillators)".
Other math is invented through the choice of axioms and then we discovery the theorems that follow from those.
So, if you follow my argument, the fact that there are prime numbers is not 'intrinsic to mathematics', it's the other way around -- we use rules that lead to rich structures, because those provide useful, consistent abstractions to talk about things. Prime numbers are among those rich structures.
To respond to a sibling comment that seems somewhat related -- The fact that we can calculate some physical constants with high precision is not a counter argument to that. What we are doing is fitting a mathematical abstraction to data; once we have done that, we can find more physical truths by assuming consistency and performing logical inference. The fact that a constant can be calculated then just means that we have a consistent theory for the physical structures from which that constant arises.
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Just because an approximation is really good in some cases doesn't mean it's more than an approximation, does it?
Not saying that philosophy or religion is particularly good at answering them either, but that's more the domain you're getting into, I think.
Brief excerpt from this Feynman interview[1]
[1] https://www.youtube.com/watch?v=NjUSO4u2di0Deleted Comment
Ultimately it's the only plausible explanation as to why mathematics applies to the physical world.
One of the young students asked exactly this question to which everyone in the audience collectively groaned. The Fields medalist gave a short answer, something along the lines of "I don't know a single mathematician that thinks it's invented."
He was being polite, but you could tell he didn't think there was anything else interesting to say.
But the axioms are a choice, and we can pick different ones. The common choice of axioms is utilitarian, they lead to interesting math that helps us describe the universe.
Choosing to study molecular biology doesn’t mean cells are a human invention.
But you can make the same case for axioms - that they are not invented but discovered through a process of search in the space of axioms.
I've read that Godel's result and diagonalization procedure shows that they exist (not invented).
If humans didn’t exist would the idea of a chair still exist on some plane, or is it only in our minds?
I think most today would say it’s in our heads—-and so is mathematics—-yet it’s still thought-provoking.
Notice that even the universe follows that structure.
I didn't mean "a chair" i meant the [universal idea](https://en.wikipedia.org/wiki/Universal_(metaphysics)) of chair.
Plato conceived of Truth as being the light through which things could be conceived. At the highest level was ideas (eg the idea of "Chair", not any particular chair), below that was mathematical objects, followed by physical objects and then shadows.
So Plato would say that math is closer to truth than a chair, but not than the idea of Chair.
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In turn, introduction of a new (useful) axioms set can be called an "invention".
After watching it, I fall on the side of invention.
This means that you're not going to find two mutually exclusive sets of axioms that, between the two of them, can prove any and all true things. This suggests to me that there's not a Library of Babel of mathematical universes, it's more or less one big universe, with a lot of black holes.
https://en.wikipedia.org/wiki/Foundations_of_mathematics#Phi...
It definitely does not say that. Only that we cannot make a proof of consistency of them. It is possible, and believed, that the Zermelo–Fraenkel axioms we use for set theory for example are consistent. But we cannot prove that because of Godel.
Roger Penrose – Is Mathematics Invented or Discovered? [video] - https://news.ycombinator.com/item?id=22896671 - April 2020 (311 comments)