Is Mathematics Invented or Discovered? (2020) [video]
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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.
Just because an approximation is really good in some cases doesn't mean it's more than an approximation, does it?
That the universe chose to stick with a set of axioms that are minimal entropy and maximally predictive about the non-existent future means that some approximations are so arbitrarily close to the truth that they become indistinguishable.
> 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.
It's clearly a much deeper question, given even Wigner wrote a paper about it. He called it a miracle.
Look, ultimately, mathematics is abstract and the physical world is material. The fact that an abstract condition is able to affect material conditions is literally a miracle that can only be explained metaphysically.
One might say the universe has no obligation to be intelligible to us, and perhaps that true. But we have an obligation to intelligible beliefs. In being able to ask the question of why the world is law-based, we're rationally constrained in the answer we can accept. You can see it as an application of the anthropic principle, but I think it's more basic than that. If there is sense to be made of anything (in a logical sense that is prior to the universe), then the universe must also make sense.
Yet it appears like all physical effects are following such laws and rules. I.e, there appears to be something that breathes fire into our equations.
Wigner clearly delineates the problem on his famous paper about the applicability of mathematics. It's still a topic in philosophy to this day. I'm sure you appreciate that if it was a simple solution it wouldn't be a big deal.
>It's still a topic in philosophy to this day. I'm sure you appreciate that if it was a simple solution it wouldn't be a big deal.
The more philosophy I read, the less I appreciate this. It's clear to me that philosophy as an institution is perniciously dominated by the fashion of the day which undermines the idea of philosophical consensus (or lack thereof) as oriented towards truth.
I can imagine a world (maybe not ours) where there are no atoms, just ideas, in God's mind, and the ideas are interacting and are formed by something completely different than atoms. Like our dreams. Are there atoms in dreams? What about a dream of a person that turns into a bird and flies through the emptiness of space and disappears into a mist that starts dreaming of a person.......
So the question still remains and it is still open to discussion even if you insist that any universe must follow a rule-based system. Moreover, that would imply the existence of these rules in a kind of platonic sense. How would they give rise to a universe? This is not as simple as you make it to be, at all.
I think you're mistaken about philosophy; maybe early 20th century philosophy. Right now it seems like there's lots of interesting debates going on. But I'm just an observer.
I agree there are a lot of interesting debates going on. I just don't think philosophical consensus has much evidentiary value. Philosophy is the process of turning intuitions into concrete positions on philosophical subject matter. But no one's intuitions are better suited to me than my own. There's no substitute for just doing the work of understanding an idea and weighing the credence for oneself.
And I don't think it's that far off from our universe: we are likely something akin to a thought in a Mind. Anyways good chat - I think there's a big gap between our understanding of what is possible and what is, and how it can be given it is what we observe.
Ultimately it's the only plausible explanation as to why mathematics applies to the physical world.
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."
Placing an arbitrary barrier on the questions that can be asked is called dogma.
Example: Noether's Theorems explain why there are conservation laws. There are conservation laws because the universe observes certain symmetries.
In your example, you just moved the question of "why are there conservation laws" to "why are there symmetries?" The answer is still that we don't know. Though as you point out, understanding the relationship between the two is useful!
> Placing an arbitrary barrier on the questions that can be asked is called dogma.
I did not mean to place a barrier. It is fine to ask the question. But it is also fine (and honest) to answer it with "we don't know."
Edit: of course, please do not interpret "we don't know" as "we shouldn't try to find out" (:
There was a "why X" question, and then an equivalence was observed, saying "X = Y" (which is useful, to be sure), but at the end of the process, there is still only one "why" question — "why X" and "why Y" are now shown to be equivalent, but the question remains unanswered.
Maybe we disagree on what "why" even means, though (:
Why do apples fall from trees? Gravity. Why does gravity exist? Because matter bends space. Why does matter bend space? We don't know yet.
Ultimately, any X is either caused or uncaused. Not "X=Y", but either "Y=>X" or "X exists" - the latter being uncaused.
This is pretty basic philosophy.
1.
The original question was about the natural world, not math, and I'd say that the connection between the two is not guaranteed by anything. So any conclusion in math does not dictate reality. We simply have no "givens" in reality to work from. Though certainly mathematical models are useful.
2.
Going back to your Noether example: are you saying that conservation laws are caused by symmetries?
If the theorem says "X => Y", then does that mean Y is "caused by" X?
I don't think so. It just means that if you observe X, you can be sure Y is there too.
Suppose later we find out additional information: Y => X
Now, we have "X <=> Y", and certainly it would be unfair to say one of these "causes" the other, no?
That would fall in your "uncaused" category, I believe.
3.
The scientific method does not prove things to be true, ever. It only disproves wrong theories, by providing counterexamples.
So, if you have a theory about "why" something is a certain way, you will never fully confirm it with the scientific method. You will only discover when one of your proposals was wrong, never that it was right.
It may be the case that a lot of people try very hard to prove it wrong, and fail. And it may be the case that it is useful at predicting the future and other novelties. But it could still be wrong, and you would never know — maybe the counterexample will be found tomorrow.
----
I really do think the bedrock answer to "why do apples fall from trees" is "we don't know." There's just a lot of interesting stuff to be discovered in the (failed) attempt at answering it, in the meantime.
The scientific method is a method for testing of hypotheses, yes, but that is simply one way of discovering what is true. Logic, testimony, written accounts and records, mathematical proofs, and so on are all other ways of discovering truth.
For instance, you cannot prove who was the Emperor of Rome at a particular date in the past with experiments. You must use historical record to do so. Unless you call your experiment opening a book - but that's not a controlled experiment. And even if all books said that Marcus Aurelius was the Emperor of Rome on March 16, 180 you still wouldn't be able to prove it mathematically or using logic. You're using inference to the best explanation in a form of Bayesian probabilities.
Even though it is a matter of fact that either he was or wasn't, there's no experiment you can run on that hypothesis that could tell you the truth.
On your point 2, "X=>Y" indeed means X implies Y, so given X is true, Y is true as well. It could mean causality, but it could also mean necessity. So saying "X is true means Y is true" can be applied to numerous different contexts, including one of causality. I let go of the apple, and the apple falls. This last statement being equivalent to every body that is not subject to forces follows the geodesic created by the spacetime manifold. Which can be put in a logical form X=>Y.
And on your first point: the natural world is clearly following mathematical truths, and that is the entire point of the conversation. The ask of "Why is nature consistent" implies something much much deeper about the nature of reality than what scientific experiments can show. It is a metaphysical question, not a scientific question.
Well here I soundly agree with you. I believe that's what I was trying to communicate at the very start, when I said "those 'why' questions are exactly what math and science do not answer."
I suppose our difference relates to how we conceive of other kinds of "why" questions.
> Mathematics is constantly proving things true.
Also agreed with you there. That's one key difference between math and science, in my mind. For that reason, I suppose I'd put theoretical physics more in the "math" realm than the "science" realm, at least w/respect to the theorem you mention, and similar.
Small aside: thank you for the tenacious-but-respectful discussion — I always worry when a thread goes 3+ replies deep that it'll just become an angsty flamewar, so it's nice to have a counterexample in my training set (:
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.
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.
I've read that Godel's result and diagonalization procedure shows that they exist (not invented).
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.
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.
In turn, introduction of a new (useful) axioms set can be called an "invention".
After watching it, I fall on the side of invention.
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.
https://en.wikipedia.org/wiki/Foundations_of_mathematics#Phi...
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.
Roger Penrose – Is Mathematics Invented or Discovered? [video] - https://news.ycombinator.com/item?id=22896671 - April 2020 (311 comments)
In other words, we discover what we can invent.
I've read that they exist in a non-tautological way, which also matches common sense (obviously not enough in math).
What would change?
Nothing, because it’s fundamentally a useless question
the circle is invented, the value of PI is discovered
This is linguistics, not science.