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Is Mathematics Invented or Discovered? (2020) [video]
- Mimmy 3y agoI 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.
- pa7x1 3y agoIt'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.
- random3 3y agoAre the theories beyond axiom fundamentally different if axioms are changed, though? And if not, aren't then axioms merely props or placeholders for invariants?
- sweezyjeezy 3y agoYes they are different - example: https://en.m.wikipedia.org/wiki/Parallel_postulate https://en.m.wikipedia.org/wiki/Parallel_postulate
- sweezyjeezy 3y agoI'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'.
- pa7x1 3y agoWithout entering into an endless debate about semantics and metaphysics I would simply say that if you want to use the word discovered for the axioms then you must acknowledge that the theorems are not the same kind of discovered.
- Mimmy 3y agoI would agree the axioms are chosen, but what’s the connection between choosing something and inventing it? Choosing to study molecular biology doesn’t mean cells are a human invention.
- mensetmanusman 3y agoThat the universe chose axioms is indeed the mystery.
- hurryer 3y agoWhat about natural numbers? I've read that Godel's result and diagonalization procedure shows that they exist (not invented).
- openquery 3y agoProving 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.
- ezequiel-garzon 3y agoWho was the Fields medalist?
- threatofrain 3y agoI 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.
- photochemsyn 3y agoDifferent 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.
- datameta 3y agoCan 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).
- mtreis86 3y agoKeep 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.
- nyssos 3y agoIf you want to understand the incompleteness theorems you need an introductory model theory textbook. GEB doesn't really have anything to do with it.
- markisus 3y agoI recall that GEB gives an informal proof that systems that can formalize arithmetic are incomplete. But a standard textbook would certainly give a terser and more rigorous proof.
- a_wild_dandan 3y agoGödel 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.
- lupire 3y agoIs a new pharmaceutical invented or discovered? How about a new iPhone? This is linguistics, not science.
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- mensetmanusman 3y agoit is for imprecise ideas... math is too precise.
- svara 3y agoAre 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.
- mensetmanusman 3y agoNature being consistent to 10 significant digits in an abstraction means that math is more than a language.
- turtledragonfly 3y agoWhy is that? (genuine question) Just because an approximation is really good in some cases doesn't mean it's more than an approximation, does it?
- mensetmanusman 3y agoI believe because all other domains of low entropy information exchange (language) are incredibly imprecise, contradictory, and incomplete. 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.
- cscheid 3y agoI 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)".
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- dang 3y agoUrl changed from https://abakcus.com/video/roger-penrose-is-mathematics-invented-or-discovered/ https://abakcus.com/video/roger-penrose-is-mathematics-inven..., which points to this.
- dang 3y agoDiscussed at the time: Roger Penrose – Is Mathematics Invented or Discovered? [video] - https://news.ycombinator.com/item?id=22896671 https://news.ycombinator.com/item?id=22896671 - April 2020 (311 comments)
- abnry 3y agoWhen was this filmed? Penrose looks considerably younger.
- sgdpk 3y agoMy view is that there is a landscape of mathematical truths, but we can only explore/discover precisely those things that our imagination allows. In other words, we discover what we can invent.
- yarg 3y agoIt's emergent.
- loughnane 3y agoDon’t have time to check it out but this smacks of Plato’s theory of forms. 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.
- esahione 3y agoThat's pretty different though: the structure of mathematics seems to be more related to truth than a chair. Notice that even the universe follows that structure.
- loughnane 3y ago> a chair I didn't mean "a chair" i meant the [universal idea](https://en.wikipedia.org/wiki/Universal_(metaphysics) 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.
- hurryer 3y agoWhat's the consensus on natural numbers? I've read that they exist in a non-tautological way, which also matches common sense (obviously not enough in math).
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- Almondsetat 3y agoSuppose God descended upon us and either told us maths is discovered or invented (whatever that means). What would change? Nothing, because it’s fundamentally a useless question
- fuoqi 3y agoWhen 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".
- narinxas 3y agoNumber "one" is invented. Zero is discovered the circle is invented, the value of PI is discovered
- WheelsAtLarge 3y agoHere's a version that NOVA did on the invention or discovery of math: https://www.youtube.com/watch?v=WWb8gfoDDhs https://www.youtube.com/watch?v=WWb8gfoDDhs After watching it, I fall on the side of invention.
- slowmovintarget 3y agoMathematics is language powerful enough to help us discover properties of the universe. The language is invented, the properties are discovered.
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