18 ms·

Roger Penrose – Is Mathematics Invented or Discovered? [video]

286 points by mmmBacon 7y ago

novalis78 7y ago
The platonic universe of the entirety of math sounds (this week at least) like Wolfram’s “Universe of Computational reality” of which the physical reality is just a subset.
DeathArrow 7y ago
Intuitionist mathematics claims that mathematics is purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles existing in an objective reality. [0] In intuitionist mathematics there is only potential infinity, no actual infinity. Constructive set theory differs from Zermelo set theory. That has many consequences in practice. Applying intuitionist mathematics to physics we can come to the conclusion that time flows and it helps reconcile quantum mechanics with general relativity.[1] [0] https://en.wikipedia.org/wiki/Intuitionism https://en.wikipedia.org/wiki/Intuitionism [1] https://www.quantamagazine.org/does-time-really-flow-new-clues-come-from-a-century-old-approach-to-math-20200407/ https://www.quantamagazine.org/does-time-really-flow-new-clu...
- hyw 7y ago
  > Applying intuitionist mathematics to physics we can come to the conclusion that time flows Can you expound on this a little more? I'm completely new to the idea of intuitionistic mathematics and much more so to its applications in physics; the constructive approach to thinking about objects and properties is very refreshing and I'd like to hear how you've related those principles with the paradox of time in the context of classical physics.
  - DeathArrow 7y ago
    >Can you expound on this a little more? Unfortunately no. I got the info from Quanta Magazine article. I am not knowledgeable enough to respond on this. Even though I did some high level math courses in University, I wasn't interested enough. I am sure other HN users understand this stuff better.
- tartoran 7y ago
  From the wikipedia source: In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality. And Im wodering why Im not looking more into the philosophy of math, i find this interesting because I am mainly an intuition driven person. Any ideas for courses online?
  - dnautics 7y ago
    The mathematical definition of intuition is not the colloquial definition. For example, in intuitionistic mathematics, you can't proof by contradiction, which means that some things that are "intuitively true" in the colloquial sense, are inaccessible.
- mikorym 7y ago
  The reason why intuitionist mathematics is useful in QM is because it can be viewed as resource based logic. But remember that (as far as I know) you can do intuitionist mathematics in classical mathematics, but not the other way around. So you can think of intuitionist mathematics as being embedded in classical mathematics.
  - h-cobordism 7y ago
    I think in your first paragraph you're confusing intuitionistic logic and linear logic --- intuitionistic logic still lets you "clone" and "delete" propositions freely. The second paragraph is somewhat true --- at its base, the modern use of the term "intuitionistic logic" refers merely to not accepting the law of the excluded middle. However, there are varieties of intuitionism that are very, very inconsistent with classical logic, like those adopting the idea that "every function between real numbers is continuous".
    - mikorym 7y ago
      When you separate internal and external logic, for example in topos theory, the external logic is still classical. For example, any topos has internal logic on subobjects of at least a Heyting algebra. In some cases you specialise to a Boolean algebra. However, when you write things down such as your arguments around functors and constructions, you argue according to classical mathematics externally. When you enter into the category (which is chosen not to be Boolean) and argue on the subobject structure, then you are entering intuitionistic logic. This is what I mean when I say that classical logic can be argued to have intuitionistic logic embedded in it. The cardinality of the reals being equal to the cardinality of the naturals is then something you have to construct inside the topos. So you need to construct along the spirit of a natural numbers object (perhaps a real numbers object) and then use the internal logic to argue about continuity.
    - mikorym 7y ago
      I think you are right about the comment about linear logic. I must be confusing A U A with A U ^A. I checked the paper that I was thinking about and the condition that I saw there is distributivity being dropped rather than double negation being dropped.
- hliyan 7y ago
  I've always thought math as we know it is a result of both deep underlying relationships between natural constructs and how we perceive them. For example, a species who has no sense of vision will not develop geometry the same way we would. A species that has a sense of vision that also includes a direct distance perception (as opposed to our stereoscopic vision) will probably come up with a very different form of geometry. Though I'd like to think that most species would come up with some version of calculus, even though the notation will be obviously different. Afterall, two of our own species did so independently.
  - Kaiyou 7y ago
    Combining the words "most" and "species" feels a little bit silly. Among all the millions of species on this planet only humans have displayed the intelligence necessary to contemplate any of this.
    - ableal 7y ago
      > on this planet Location was not specified ...
    - foldr 7y ago
      That's only true if you limit this to conscious thought. Lots of organisms perform impressively complex calculations. The desert ant odometer is one classic example: https://jeb.biologists.org/content/210/2/198 https://jeb.biologists.org/content/210/2/198
  - Phenomenit 7y ago
    I've always figured it's our visions ability to perceive wholes, one of something. A clear conceptual border between two entities.
    - hliyan 7y ago
      Yes, but it's possible to perceive distict objects without vision, except the separation is in time, rather than space. Or, depends on the number of sense organs available (e.g. a blind person can sense two different objects simultaneously by holding one in each hand).
- peterburkimsher 7y ago
  Tell me more about the difference between "potential infinity" and "actual infinity". Recursive functions in computer programming, or fractals in maths, are infinite but can be described. Each small part looks like the whole, but the whole can never be fully known within a finite universe. Is that "potential" or "actual" infinity? I've been thinking a lot about infinity this past week, mostly because of Conway's lectures. I'm trying to figure out whether the following logic makes sense. - Assuming that the universe is finite -> Infinity cannot be approximated with finite numbers (MIP* = RE) [1] -> Numbers cannot have infinite strings of digits -> The future can never be perfectly preordained (Gisin) [2] -> Free will exists (Conway) [3] This is not a proof that free will exists! But I think this is a consistency proof that free will is finite. Now you're telling me that there's another kind of infinity, and I want to know what that means for this thought. [1] https://www.quantamagazine.org/landmark-computer-science-proof-cascades-through-physics-and-math-20200304/ https://www.quantamagazine.org/landmark-computer-science-pro... [2] https://www.quantamagazine.org/does-time-really-flow-new-clues-come-from-a-century-old-approach-to-math-20200407/ https://www.quantamagazine.org/does-time-really-flow-new-clu... [3] https://www.youtube.com/watch?v=tmx2tpcdKZY&feature=youtu.be&t=712 https://www.youtube.com/watch?v=tmx2tpcdKZY&feature=youtu.be...
  - gowld 7y ago
    The video timestamp you linked does not say "free will exists". Is says "humans has free will implies element particles have free will".
    - peterburkimsher 7y ago
      His definition of free will is quite limited. He only suggests that free will is limited to a "free" choice between options within one's power. Conway said that there either is determinism or free will. If the future can be perfectly preordained (Gisin) due to infinite possibilities, then there is no free will. But if there are finite options, I think that there is "free will" for particles and people to choose between those options. https://youtu.be/tmx2tpcdKZY?t=1074 https://youtu.be/tmx2tpcdKZY?t=1074 https://en.wikipedia.org/wiki/Free_will_theorem https://en.wikipedia.org/wiki/Free_will_theorem
- naasking 7y ago
  > Intuitionist mathematics claims that mathematics is purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles existing in an objective reality. [0] The "mental activity" part isn't strictly necessary. A constructive mathematics will produce results that are largely the same. What matters most is that mathematical objects are proven by construction, and so proofs have a direct translation to computation.
  - maxiepoo 7y ago
    I think you and OP are talking about different things because the term "intuitionist" has been used for both the philosophy that EJ Brouwer espoused (which matches what OP said) and the large field of constructive mathematics that has been greatly influenced by Brouwer's philosophy but has embraced approaches like formal logic which Brouwer himself disliked.
    - naasking 7y ago
      I was merely pointing out that the "mental" dependency of intuitionism isn't strictly necessary. Take it out, and you're left with an equally valid constructive logic.
keiferski 7y ago
The SEP article is an excellent introduction to the philosophy of mathematics: https://plato.stanford.edu/entries/philosophy-mathematics/ https://plato.stanford.edu/entries/philosophy-mathematics/
mrtksn 7y ago
Recently I watched Sixty Symbols video on why light is slower in glass and there are multiple explanations on it. What struck me was what prof. Michael Merrifield said at the end of his explanation: https://www.youtube.com/watch?v=CiHN0ZWE5bk https://www.youtube.com/watch?v=CiHN0ZWE5bk On the question about what is the reality, what explanation is the true one, Merrifield said that the Math works on all of the explanations and what those explanations do is simply to model the behaviour of nature and not necessarily reflect the reality. That's how Newtonian physics and relativistic physics are both correct models, tools to model nature and simply can be used to whenever suitable. Wouldn't that mean that mathematics is just an invented tool to reason about physical models?
- iNic 7y ago
  Well yes and no. A lot of mathematics found its roots in trying to model the real world, however I would then argue that the truths we prove about these models are truly discoveries. And with these discoveries we can often generalize the setting to more abstract formulations, independent of the physical reality. I think that this can be seen in the fact that sometimes mathematicians and especially physicists can reason about objects that they are not sure about what the right definition should be. Many mathematicians reasoned about continuous functions long before we had a concrete definition of them. But when Cauchy introduced the definition and Weierstrass proved it is equivalent to preserving limits (which was the intuition at the time), we had not truly discovered something new and mathematical. This whole idea was then generalized to topology when it was shown that pre images of continuous functions preserve the "openess" of sets, i.e. we realized that no concept of distance was not needed to describe continuity, which is very surprising.
jelliclesfarm 7y ago
Of course..we invented the ‘language of mathematics’. I am on a Penrose kick right now. Just downloaded The Emperor’s New Mind Audiobook.
sunstone 7y ago
Put me into the 'discovered' camp.
jeffdavis 7y ago
The expression is invented but the underlying relationships are discovered. It's silly to think that the sqaure root of negative one is something real to be discovered. It's just a stand-in to express complex relationships more succinctly. The same for negative numbers, for that matter. It would be equally silly to think that the underlying relationships are invented. Nobody invented prime numbers, they were discovered.
- Y_Y 7y ago
  It's just as silly to think of the "real numbers" as being discovered. They don't exist in the real world, after all.
  - mikorym 7y ago
    They are necessary for the continuum. If you throw away real numbers, then you lose major things in physics. I think for example you lose the wave function in QM. It is not a question of whether they exist in nature, but rather whether they are the more superior technique, or not, to explain nature.
    - Y_Y 7y ago
      Wavefunctions are in fact generally complex-valued. I don't know if that supports your argument or not.
      - mikorym 7y ago
        Complex numbers have the same cardinality as the reals.
        Y_Y 7y ago
        That's true, but irrelevant. Complex numbers aren't "necessary for the continuum" as you put it, and some realists might argue that they don't hold the same "discoverability" as the reals. I wouldn't. I think the reals and the complex numbers have the same "realness" and that neither represents any innate property of the physical world, despite how obviously useful they are in physical models.
BtM909 7y ago
I read this book about it: https://www.amazon.com/God-Mathematician-Mario-Livio/dp/074329405X https://www.amazon.com/God-Mathematician-Mario-Livio/dp/0743.... Won't spoil the answer, but was nice to go through the evolution of Mathematics.
deleted 7y ago
[deleted]
mykhamill 7y ago
What can be imagined is what can be discovered. The etymology of invent has the terms 'contrived' and 'discover' baked in. if we take the contrived root, rather than just dead-ending to say that invent is synonymous with discover, we find that it is rooted in the ability to 'compare' and 'imagine'. From this we can then formulate the opening statement.
DeathArrow 7y ago
I am not sure of the implications. Does it even matter if Math is invented or discovered? Maybe it's both, and it isn't a contradiction between invention and discovering? We describe things having a certain radiation wavelength as having the color yellow. In that sense, yellow it's an invention. That doesn't mean the radiation doesn't exist. But to complicate things a bit, some things don't exist unless we observe them. This is the case with states of particles described by quantum mechanics. Math is more than a science, is the sciences upon which most other sciences and tech are founded. You can model anything in a computer and running on a computer using math. You can describe logic, natural language, technology, biology using math. In that sense, being the building block of other sciences, math is more akin to a language. Two physicist use math in almost the same way two people use English to describe things and communicate ideas. But math has building blocks, too. Set of axioms upon which any mathematical object and theory can be constructed. The most popular as of now is Zermelo set theory. There are more such fundamental theories, sometimes very different between them. So, to see if Math is discovered or invented, the easy thing to do is to see if a set of axioms can be discovered or is invented.
- goatlover 7y ago
  > Does it even matter if Math is invented or discovered? It matters for philosophical inquiry. Does philosophy matter? It matters for people who find it valuable, the same way art, music or literature matters.
  - guerrilla 7y ago
    You think logic, the art of making distinctions, the study of critical thinking and the foundation of mathematics only matter in the same way art does?
    - inopinatus 7y ago
      I’d say they don’t matter nearly so much as art. The most cosmically significant export of humanity to date is Voyager’s golden record.
NautilusWave 7y ago
I initially read the title as "Is Mathematics haunted or discovered".
xtf 7y ago
Invented to describe what's being discovered.
- Koshkin 7y ago
  You have to add: discovered inside itself.
ukj 7y ago
The answer is: "both". Where do Mathematicians look to discover Mathematics? In the depths of their own minds. The part where you "look and think deeply" is discovery. The part where you "express your discovery in a coherent language" is invention.
justforfunhere 7y ago
Could the same be said about just about everything? Like Music. Is a Song just a combination of notes, beats, intervals, voice etc. waiting to be discovered? Or is it something that a musician invents in her brain through talent, experience, practice and trail and error. Or for that matter, a startup idea? A product/service that would bring immense value to its consumers, but it is not there yet, waiting to be discovered. I guess philosophers must have dwelled on such questions before.
- cpsempek 7y ago
  I think I get your point and I‘m open to ideas which drive home your point in a more precise way. However, the main difference between concerning ourselves with whether or not mathematical objects exists vs music or start-up ideas is their place in building a foundation for understanding reality. Science has been a productive program for understanding reality. It happens to be underpinned by math. And as a result, we have encountered some pretty interesting relationships between what the math tells us and what we observe. For instance, neutrinos and black holes were known to exist mathematically before they were ever observed and measured “in reality”. It’s not clear to me that music and start-up ideas hold up to math when it comes to being a tool for understanding or describing our reality, or perception the rig. Math is somehow fundamental to many successful enterprises which seek to describe and explain reality. Therefore, it becomes the point of focus of much philosophical inquiry when seeking to understand reality. Hence the foundational question, do mathematical objects exist? And not, does music or blockchain kittens exist?
  - kruasan 7y ago
    >do mathematical objects exist? Yes, they do, in an abstract sense. Existence in math means a very different thing from existence in physics. There are mathematical objects that exist, and those that cannot and do not exist. Someone posted a SEP entry earlier, which is a good start on this topic. >the main difference between concerning ourselves with whether or not mathematical objects exists vs music There is no 'vs', because there is no difference. Music is math, any song or sound is a mathematical object. A physical waveform that you hear can be encoded digitally in numbers: ones and zeros. So any given wav/flac file is just a bunch of numbers that give rise to the qualitative experience of sound, when interpreted in a certain way. For example, a digital waveform consists of samples, each sample takes 16 bits to encode. Sampling rate of 44.1kHz is 44,100 samples per second. So you have 16 bits per sample x 44100 samples per second per channel x 2 channels x 300 seconds = 2^423,360,000 possible permutations of a 5-minute audio file without compression, which is a number with over 127 million digits. A little percentage of these permutations would count as music (even if your tastes are really diversified), most of it would just be noise. But all these possible 5-minute audio files include not only every song and every performance that existed or will exist. They also include every possible sound recording: songs that will not be written, Paul Graham saying that he hates HN, Paul Graham saying that he loves me and the rest of the file is silence, you and me discussing this topic with Plato for 5 minutes, etc, etc. The data exists and can be discovered and listened to, even though some of these examples are obviously not physically possible (i.e. Plato is dead). So all music already exists mathematically, and it can be a useful mindset that your job is to discover it. A lot of musicians see it that way, Tessa Violet, for example: https://youtu.be/QzBoGVToWEo?t=342 https://youtu.be/QzBoGVToWEo?t=342
guerrilla 7y ago
I'll never understand how someone otherwise so apparently intelligent can be so religious. Weird, the systems that we spent massive amounts of energy designing to precisely describe reality do that better than all the ones that we threw out along the way!
- noodles_ftw 7y ago
  > I'll never understand how someone otherwise so apparently intelligent can be so religious. Why not? Religious people believe that God is the Most Wise. Mathematics in nature attest to that attribute of God (as well as to other attributes of Him). An intelligent and religious person would recognize that, knowing it is God who came up with all the rules that keep the universe in balance. I'll never understand why some people believe science and creationism can't go hand in hand.
  - guerrilla 7y ago
    If you want to understand why some people believe that science and creationism don't go hand in hand, then studying epistemology would help you do that. Most introductory texts will cover what you need.
    - noodles_ftw 7y ago
      Sounds like I got something new to learn during this lockdown, thank you. After a quick look at the wikipedia page on epistemology (very interesting), I'm not sure how this explains why some people believe that science and creationism don't go hand in hand.
      - guerrilla 7y ago
        A super super short version is about the question "how do you know X?" Its generally (although not universally) agreed that there are two types of knowledge, a priori and a posteriori. Roughly, in the first case, all you can deduce is implications from assumptions. For example, "if X and Y are true then Z is true." Such things don't require experience. The second case requires evidence gained from experience. Mathematical knowledge is of the first type, scientific knowledge is of the second type. The school of rationalism prioritizes the first type and the school of empericism emphasizes the second. That the universe was created by an omnipotent omniscient and benevolent god is not an emperical fact but requires some kind of a priori supports. Thats why you see so called proofs of god rather than scientific theories of god. There are other options of couree like mysticism, the claim that knowledge can be gained outside of these methods, or an emphasis on faith, that we don't take these things to be justifiable. I probably butchered this but it gives you some starting points to research and hopefully shows a model of the debate. Huge topic. Check out https://plato.stanford.edu/entries/epistemology/ https://plato.stanford.edu/entries/epistemology/ as a hardcore crashcourse or check "crash course philosophy" on youtube for the relavent sections at a more HS level. SEP has entries on god, faith and mysticism as well as rationalism and empericism too.
danck 7y ago
If you only look at mathematics I think it's simply: - Axioms are invented - Conclusions are discovered The magic part for me is that some axioms have been chosen so well that their conclusions are confirmed in the real world.
- vanderZwan 7y ago
  Isn't that simply because axioms that don't lead to consistent conclusions are rejected?
  - danck 7y ago
    You can invent and pick axioms in many ways that (probably) won't lead to inconsistencies. But they won't all be powerful enough or relevant in the real world.
    - vanderZwan 7y ago
      Well, then it sounds like your reasoning gets it backwards: the axioms that produce systems without significant consequences or connections outside of their own abstract realm end up being ignored. Or in other words: the constraints on maths are imposed from outside of maths.
    - gibspaulding 7y ago
      Doesn't this imply that, while you can invent all the axioms you like, you must discover which ones are consistent with each other and with experimental results.
- red75prime 7y ago
  Arithmetic existed long before its axiomatization. Arithmetic was useful and no one stumbled upon contradictions in it. So it was natural to suppose that it can be described by some axiomatic system. Peano found it.
  - kevin_thibedeau 7y ago
raghavtoshniwal 7y ago
Last week I came across another fascinating podcast by Penrose that you folks might also like: https://www.youtube.com/watch?v=orMtwOz6Db0 https://www.youtube.com/watch?v=orMtwOz6Db0
- peterburkimsher 7y ago
  Thanks for the link. At 1:02:57, Penrose proposes "infinite cycles of the big bang" https://youtu.be/orMtwOz6Db0?t=3777 https://youtu.be/orMtwOz6Db0?t=3777 Is that multiverse theory disproven by Wolfram's latest blog post? "could there be other universes? The answer in our setup is basically no." https://writings.stephenwolfram.com/2020/04/finally-we-may-have-a-path-to-the-fundamental-theory-of-physics-and-its-beautiful/ https://writings.stephenwolfram.com/2020/04/finally-we-may-h...
magicalhippo 7y ago
If B follows from A via logic, but A is invented, isn't B invented as well then? Just to take a recent example which was mentioned here, Geometric Algebra[1]. There you assume you have some objects which aren't numbers but which when squared equals a given number. By doing that a bunch of nice results have been discovered. However to me the basic premise, take some objects which aren't numbers but which square to a number, feels very much like an invention. So as such wouldn't the nice results be inventions as well? [1]: https://bivector.net/doc.html https://bivector.net/doc.html
- lmm 7y ago
  > If B follows from A via logic, but A is invented, isn't B invented as well then? Not really, because you have no choice about B - it's true (or rather it follows from A) whether you want it to or not. You could have picked a different A, but once you picked A then B was fixed.
  - magicalhippo 7y ago
    Yet if I pull the trigger we say that I killed someone, not that the bullet did.
    - lmm 7y ago
      People invented the electric chair because people discovered that high-voltage electricity can kill people. You might invent a particular axiom because it has an interesting mathematical consequence (e.g. the axiom of choice was invented to allow Hilbert's basis theorem to be proven), but what you invented was the axiom, not the consequence.
- analbumcover 7y ago
  It sounds to me like you are giving special precedence to "numbers" in your considerations, as well as drawing your intuition for "square" from working with the integers or the reals. A square is just the result of a binary (product) operation where both of the operands are equal. The operation can be defined on an algebraic object with much richer structure than the integers or reals, and the operation itself can be much more complex than, say, integer multiplication. So the geometric product of geometric algebra is just one animal in the zoo of examples. You might call the specifics of the geometric product operation an "invention," but not because the square of a non-number can be a number.
simplesleeper 7y ago
This is just the age old question of nominalism vs realism
rezeroed 7y ago
Fallacy of equivocation.
deltron3030 7y ago
Mathematics is just a language and likely a human discovery method for rules that encapsulate the universe and cascade down. A method invented to discover discoverable rules.
- roywiggins 7y ago
  You can build all sorts of bizarre structures in ZFC that seem to have no relationship with reality at all. You can prove that almost all real numbers can't be computed or even described. There is an entire tower of infinities bigger than the natural numbers that telescope up and up. The axiom of choice gives you all sorts of nonphysical results. The world of ZFC doesn't feel very constrained by the actual universe, is all. And if you don't like ZFC, you can pick another set of axioms entirely- there are several alternatives.
  - deltron3030 7y ago
    Sure, math/ZFC can describe potential and impossible universes including ours. But the motiviation for its invention as a language was very likely to understand relationships in our reality, physics, trade etc.
itvision 7y ago
There's one thing I cannot wrap my head around. Math is purely abstract science yet it describes the world around us to the utmost precision. Does that mean that this world is simply ... a math model which means everything around us is ... not real? I've long stopped believing in free will because everything points at it being an illusion of our brain because we're a product of this world and we had no chance of influencing the conditions which brought us to life, and even after our minds and consciousnesses form it's hard to believe they are fully autonomous and not simply a function of the processess in our brains we're simply not aware of. If you think about all of it, it becomes utterly depressing as you begin to realize you're a biological robot, a byproduct of the universe evolution which couldn't care less about our species and this little tiny blue planet.
- tirant 7y ago
  Mathematics has not been static throughout history. It has evolved at the same pace our science and discoveries have evolved, and as such it has incorporated all necessary elements to help describe new phenomena. To the point that now is able to describe never observed phenomena. So no, our world is not unreal because it can be described with maths. Maths have evolved to describe it. Same as human language has evolved to support our close environment and our human interactions.
- kd5bjo 7y ago
  > Math is purely abstract science yet it describes the world around us to the utmost precision. Does that mean that this world is simply ... a math model which means everything around us is ... not real? You can only reach this conclusion if you categorize math as "not real". If everything we interact with is a perfect mathematical object, I prefer to take that as evidence that all mathematical objects may be real, however abstract they seem.
- novalis78 7y ago
  What’s depressing about being a nanotchmological robot attempting to conquer the solar system and spread his technology and nanotechnology? It sounds rather exciting and fascinating. Perhaps if the depression is too severe you need to check out your programming ;-)
gingahbread31 7y ago
Thank you for sharing this, this an amazing interview series !
zeepzeep 7y ago
Java was invented, Agda was discovered. (Obviously I'm in the religion where math does exist.)
ebj73 7y ago
A vaguely, but tangentially related fact, is that Stephen King actually considers the stories in his books as discovered, rather than invented. He talks about it in his book 'On Writing'. He considers the stories to be sort of preexisting things, and his job as a writer to unearth and discover them, rather than invent them. It's about his frame of mind while writing, I guess, but still.
- ineedasername 7y ago
  It's a common theme among sculptors too: the shape is already in the stone, they just have to uncover it.
  - DagAgren 7y ago
    Is it? The only time I have ever heard this expressed is as a bad joke.
    - ineedasername 7y ago
      I know from a few general humanities classes I took that it was a common way of thinking about sculpting during the renaissance. And modern learning material on the topic presents it as a visualization aid. Let me be clear: It wasn't some mystical way of thinking there was, literally something that existed in the material. Again, it was more of an aid to thinking about the endeavor.
- RashadSaleh 7y ago
  I doubt that he actually said that but I could be wrong.
  - Trasmatta 7y ago
    This is actually a big plot point in The Dark Tower books.
  - ebj73 7y ago
    He describes stories as 'relics'. Here's a direct quote from the book: “Stories are relics, part of an undiscovered pre-existing world. The writer’s job is to use the tools in his or her toolbox to get as much of each one out of the ground intact as possible.” The book is pretty good, well worth the read if you're only slightly interested in the topic!
scrozart 7y ago
Phenomena are discovered. Mathematics are invented to describe them.
- Noughmad 7y ago
  Not all mathematics though. Math works by starting with a set of axioms. These are chosen, usually but not necessarily such that they will describe some kind of physical reality. That is the part which you invent to describe phenomena, like the Peano axioms which describe quantities of countable things. But then, you discover (and of course also prove) things that follow from these axioms. So a mathematician would find that if you take this set of axioms, then this statement is true. Is that an invention or discovery? I would call it a discovery, although different from a discovery of a natural phenomenon.
  - roywiggins 7y ago
    Geometry and arithmetic and calculus all existed before axioms and worked okay. You can do quite a lot of mathematics "naively". It's only in the 20th century that axiomatization became really important, because naive set theory turned out not to work very well.
- naasking 7y ago
  And what if all phenomena are intrinsically mathematical? Then mathematics is discovered as well.
naringas 7y ago
A bit both: e.g. The circle is an invention. But the number pi (it's value) is discovered
- nybble41 7y ago
  What do you mean when you say that the circle is an invention rather than a discovery? Are you drawing a distinction between the mathematical ideal and the arbitrarily close approximations which can be found in nature? Though by that criteria that case pi must also be an invention…
Reedx 7y ago
Same question also answered by Max Tegmark: https://www.youtube.com/watch?v=ybIxWQKZss8 https://www.youtube.com/watch?v=ybIxWQKZss8 Stephen Wolfram: https://www.youtube.com/watch?v=nUCwtLTUPQ4 https://www.youtube.com/watch?v=nUCwtLTUPQ4 Steven Weinberg: https://www.youtube.com/watch?v=NpMk9G-ddiM https://www.youtube.com/watch?v=NpMk9G-ddiM I particularly liked Max's answer who neatly makes the distinction between the structure (we discover) and the language (we invent). We're free to invent the names, but not the structures.
- 7thaccount 7y ago
  Max Tegmark has some fun books. Apparently he got to be at the forefront of some discoveries as he and his wife knew C++ and would do all the after the fact data analysis. If you've ever been around a campfire with a good storyteller that has their audience at the edge of their seat, I felt like this in some spots.
- throw0101a 7y ago
  Entire playlist: * https://www.closertotruth.com/series/mathematics-invented-or-discovered https://www.closertotruth.com/series/mathematics-invented-or... Gregory Chaitin was interesting as well: * https://www.youtube.com/watch?v=1RLdSvQ-OF0 https://www.youtube.com/watch?v=1RLdSvQ-OF0 The (apocryphal?) story about Godel not believing in empirical science and only in a priori truths was interesting.
- nikofeyn 7y ago
  > I particularly liked Max's answer who neatly makes the distinction between the structure (we discover) and the language (we invent). We're free to invent the names, but not the structures. i came here to give my opinion, but this is pretty much it. i need to read max tegmark's books. the idea that how we got to the structures and concepts we find is undoubtedly invented, because we've already seen that we've re-invented our methods multiple times. but the structures and concepts don't really change. we extend them, but there seems to be some set of structures and concepts that get settled to. if we found some alien species that has mathematics, we'd probably find that there's some mapping between our structures modulo small choices that were made.
theaeolist 7y ago
Like in any game, you invent the rules (axioms and logical systems) then you discover the consequences (theorems), just like you invent a maze then you discover the way out.
lisper 7y ago
The question assumes that there is a difference between invention and discovery. There isn't. Invention is a kind of discovery. The apparent distinction arises from our separation of physical and mental processes. But this is a purely artificial separation, a human conceit, because (unless you're a dualist) mental processes are physical processes. Specifically, mental processes are computational processes, which are physical. So it's not surprising that the structure of the physical world should be reflected back on itself in the structure of mental processes. It's universal Turing machines all the way down.
- koheripbal 7y ago
  This all seems like semantic masturbation.
- yters 7y ago
  So, if there is a difference between invention and discovery, then dualism is true?
  - lisper 7y ago
    Yes, though that's a bit of an odd way of putting it. It's kind of like saying that if matter can't be broken down into smaller constituents beyond a certain point then atomism must be true. It's more of a definition of dualism/atomism than a logical entailment.
    - yters 7y ago
      One man's modus ponens is another man's modus tollens.
- pdonis 7y ago
  > Invention is a kind of discovery. Some inventions can be thought of this way, but I don't think all inventions can. For a relevant example, human mathematical notation is a human invention that I don't think can be usefully thought of as a kind of discovery. (Another poster upthread mentioned Tegmark's response drawing a distinction between the structure of mathematics, which we discover, and the language we use to describe it, which we invent.)
steve76 7y ago
I would go with invented. I think this ends with humans changing the universe to fit their needs. The math is right because humans made it. I think of satellites and solar orbiters made from matter waves that encompass everything. Medical applications built around them. Humans manipulating their photoelectric properties. They still see a sun. Everything else sees the sun as a black hole because of what we did to ourselves. Distant stars focus into existence because of telescopes on Earth. Everything that existed already leaves us alone, so to a degree they must see it too. There's some contest to see how insignificant we can make ourselves just for petty license. Convince no one. Start building. Odds are, other more powerful people thought of it and did it before. That telescope in geneva to me is not just some far off irrelevant land to look at. It's in the material, a grain structure of our Sun. Serves as great motivation.
Wistar 7y ago
Dr. Hannah Fry's 3-part BBC documentary series, "Magic Numbers: Hannah Fry's Mysterious World of Maths," explores this very question in the first episode, "Numbers As God." The episode description reads: "Documentary series in which Dr Hannah Fry explores the mystery of maths. Is it invented like a language, or is it discovered and part of the fabric of the universe?" https://www.bbc.co.uk/programmes/b0bn6wtp https://www.bbc.co.uk/programmes/b0bn6wtp She interviews a number of prominent mathematicians and scientists, such as Brian Greene, and they certainly don't agree one way or the other in the invented/discovered question. (Alas, I now see that the series is listed as unavailable on the BBC site but I watched it, I think on Amazon Prime or maybe youtube.)
lavp 7y ago
I find mathematics to be the most beautiful thing in the universe, because it is the only thing I can truly think of as being perfect in every way. Existing only as an abstract concept, it still manages to find its way into every single aspect of our lives. Two completely remote civilizations will still have the same mathematics. Sure one may have a more developed understanding, but if both civs wondered about how to get the hypotenuse of a right-angled triangle, they would both end up with Pythagoras' theorem. The only thing invented in math is our language and representation of such concepts. Many people believe that mathematics becomes invented the further up you go like pure mathematics and I can understand why this perspective would come through, but take the example of G.H. Hardy who famously said “The Theory of Numbers has always been regarded as one of the most obviously useless branches of Pure Mathematics”. It could be argued at the time that these branches were simply invented mathematics because there was no practical application of it, but 30 years following his death came breakthroughs in cryptography. All of a sudden, this was no longer an invention. Calculus is not seen as an invention but a discovery because of all of its insane number of applications in physics and other areas. Many areas of math that have yet to find practical use will always come under the scrutiny of being 'invented' but that simply isn't the case. I firmly believe that all areas of mathematics are practical to the universe and its wonders (and therefore discovered), whether or not we can achieve a level to use such mathematics (or experience it) is a different matter. I believe that at its core, mathematics is the universe. To say that we have invented it is arrogant and completely strives it of its beauty.
bane 7y ago
In most fields, you have to invent tools that enable you to discover things. My guess is that this is also true in mathematics. We get confused sometimes because the tools often happen to look like the discoveries because of all the fancy greek symbols.
raincom 7y ago
It is a dispute in ontology. Do mathematical objects (say, numbers, sets) exist in the world? Some say, yes; others say, no; some others say, they exist in another world--called Platonic world. We see similar disputes in philosophy of (natural) sciences: for instance, instrumentalism doesn't subscribe to the ontology of realism.
amarte 7y ago
Imagine an interview with a great painter. The painter is asked about the nature of painting and he responds that when he sets out to work in his studio and his brush strokes a canvass, he is discovering the fundamental nature of reality. He doubles down and exclaims that, in fact, reality is actually JUST lines and curves and shades and colors, and his proof is, well, look at how accurate his paintings are! Let's pretend that he actually is a very skilled painter, and many critics have marveled at the extent to which his paintings are indistinguishable from his subject matter. Still though, the interviewer clears his throat after an awkward pause and continues on to the next question. To me, math is a medium of description much like painting or writing. It's units are not colors or words but points. A point, or "that which has no part", is much finer and carries with it far less baggage than something like a word. Points can be assigned numerical values and played with in clever ways. You can even sprinkle a fine dust of them over anything observable and create a copy of it to arbitrary degrees of precision. I know math is far more than just the study of points, but I'm not convinced that math is anything more than our capacity to distinguish and describe extended to its limit. I also don't mean to belittle the accomplishments of mathematicians and theoreticians, I just think it's more reasonable to say that math is JUST the limit of description than it is to say that reality is JUST math. After reading through some of the comments, I think many of us are on the same page.
- bloomberg2020 7y ago
  Agreed The crux of it is obvious: math is true from the perspective of a human consciousness It’s defined after cognitive measure. I won’t say it’s whittled down. I would say ... it’s sight, sound, touch... attempt at quantifying our qualified experience All the best math nerds were often deeply gifted and practiced forms of human art: poetry, music, sculpture... and many artists good at math, Brian May, for one example IMO to get math one needs to build all those abilities It’s really the people I know that don’t do that see math as a boring toolbox of rules and points Not saying you’re not that Math is not a thing like a rock because our ability to behave like more than a rock gave us the ability to define math. It has to be more than a simple this or that
hyperpallium 7y ago
Operational theory of mathematics: you define (invent) some rules, of starting points and ways to change them. Like the rules of a game. In a sense, all possible outcomes already exist (only to be discovered).
Koshkin 7y ago
The linguistic conundrum here arises due to the recently adopted mindless use of the word 'invented' instead of 'created' or 'designed.' (Indeed, Linus created Linux rather than 'invented' it as many would say today.) Invention is a form of discovery. So, the question should be, "is mathematics created or discovered?"
Koshkin 7y ago
My take on this is that Mathematics is such a vast area of investigation - much larger than Nature itself - that there is in fact a mix of both. In that regard it is close to Engineering, where in an attempt to design something on might stumble upon things or perform some types of research and discovery, and vice versa.
monadic2 7y ago
This is a silly semantics game and a boring one at that. the presumption that the words are contradictory, or that mathematics is a monolithic entity that plays a single role of invention or discovery, is a poor way of exploring the social and cultural (co)evolution of mathematics and methodical experimentation and observation.
- Koshkin 7y ago
  You can't just say this to people who keep asking this question, and have been for ages.
  - monadic2 7y ago
    Why not? It’s not like they have any obligation to respond, and someone needs to point out the bullshit smell wafting through the room.

286 points by mmmBacon 7y ago

The platonic universe of the entirety of math sounds (this week at least) like Wolfram’s “Universe of Computational reality” of which the physical reality is just a subset.

DeathArrow 7y ago

Intuitionist mathematics claims that mathematics is purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles existing in an objective reality. [0] In intuitionist mathematics there is only potential infinity, no actual infinity. Constructive set theory differs from Zermelo set theory. That has many consequences in practice. Applying intuitionist mathematics to physics we can come to the conclusion that time flows and it helps reconcile quantum mechanics with general relativity.[1] [0] https://en.wikipedia.org/wiki/Intuitionism https://en.wikipedia.org/wiki/Intuitionism [1] https://www.quantamagazine.org/does-time-really-flow-new-clues-come-from-a-century-old-approach-to-math-20200407/ https://www.quantamagazine.org/does-time-really-flow-new-clu...

hyw 7y ago
> Applying intuitionist mathematics to physics we can come to the conclusion that time flows Can you expound on this a little more? I'm completely new to the idea of intuitionistic mathematics and much more so to its applications in physics; the constructive approach to thinking about objects and properties is very refreshing and I'd like to hear how you've related those principles with the paradox of time in the context of classical physics.
- DeathArrow 7y ago
  >Can you expound on this a little more? Unfortunately no. I got the info from Quanta Magazine article. I am not knowledgeable enough to respond on this. Even though I did some high level math courses in University, I wasn't interested enough. I am sure other HN users understand this stuff better.
tartoran 7y ago
From the wikipedia source: In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality. And Im wodering why Im not looking more into the philosophy of math, i find this interesting because I am mainly an intuition driven person. Any ideas for courses online?
- dnautics 7y ago
  The mathematical definition of intuition is not the colloquial definition. For example, in intuitionistic mathematics, you can't proof by contradiction, which means that some things that are "intuitively true" in the colloquial sense, are inaccessible.
mikorym 7y ago
The reason why intuitionist mathematics is useful in QM is because it can be viewed as resource based logic. But remember that (as far as I know) you can do intuitionist mathematics in classical mathematics, but not the other way around. So you can think of intuitionist mathematics as being embedded in classical mathematics.
- h-cobordism 7y ago
  I think in your first paragraph you're confusing intuitionistic logic and linear logic --- intuitionistic logic still lets you "clone" and "delete" propositions freely. The second paragraph is somewhat true --- at its base, the modern use of the term "intuitionistic logic" refers merely to not accepting the law of the excluded middle. However, there are varieties of intuitionism that are very, very inconsistent with classical logic, like those adopting the idea that "every function between real numbers is continuous".
  - mikorym 7y ago
    When you separate internal and external logic, for example in topos theory, the external logic is still classical. For example, any topos has internal logic on subobjects of at least a Heyting algebra. In some cases you specialise to a Boolean algebra. However, when you write things down such as your arguments around functors and constructions, you argue according to classical mathematics externally. When you enter into the category (which is chosen not to be Boolean) and argue on the subobject structure, then you are entering intuitionistic logic. This is what I mean when I say that classical logic can be argued to have intuitionistic logic embedded in it. The cardinality of the reals being equal to the cardinality of the naturals is then something you have to construct inside the topos. So you need to construct along the spirit of a natural numbers object (perhaps a real numbers object) and then use the internal logic to argue about continuity.
  - mikorym 7y ago
    I think you are right about the comment about linear logic. I must be confusing A U A with A U ^A. I checked the paper that I was thinking about and the condition that I saw there is distributivity being dropped rather than double negation being dropped.
hliyan 7y ago
I've always thought math as we know it is a result of both deep underlying relationships between natural constructs and how we perceive them. For example, a species who has no sense of vision will not develop geometry the same way we would. A species that has a sense of vision that also includes a direct distance perception (as opposed to our stereoscopic vision) will probably come up with a very different form of geometry. Though I'd like to think that most species would come up with some version of calculus, even though the notation will be obviously different. Afterall, two of our own species did so independently.
- Kaiyou 7y ago
  Combining the words "most" and "species" feels a little bit silly. Among all the millions of species on this planet only humans have displayed the intelligence necessary to contemplate any of this.
  - ableal 7y ago
    > on this planet Location was not specified ...
  - foldr 7y ago
    That's only true if you limit this to conscious thought. Lots of organisms perform impressively complex calculations. The desert ant odometer is one classic example: https://jeb.biologists.org/content/210/2/198 https://jeb.biologists.org/content/210/2/198
- Phenomenit 7y ago
  I've always figured it's our visions ability to perceive wholes, one of something. A clear conceptual border between two entities.
  - hliyan 7y ago
    Yes, but it's possible to perceive distict objects without vision, except the separation is in time, rather than space. Or, depends on the number of sense organs available (e.g. a blind person can sense two different objects simultaneously by holding one in each hand).
peterburkimsher 7y ago
Tell me more about the difference between "potential infinity" and "actual infinity". Recursive functions in computer programming, or fractals in maths, are infinite but can be described. Each small part looks like the whole, but the whole can never be fully known within a finite universe. Is that "potential" or "actual" infinity? I've been thinking a lot about infinity this past week, mostly because of Conway's lectures. I'm trying to figure out whether the following logic makes sense. - Assuming that the universe is finite -> Infinity cannot be approximated with finite numbers (MIP* = RE) [1] -> Numbers cannot have infinite strings of digits -> The future can never be perfectly preordained (Gisin) [2] -> Free will exists (Conway) [3] This is not a proof that free will exists! But I think this is a consistency proof that free will is finite. Now you're telling me that there's another kind of infinity, and I want to know what that means for this thought. [1] https://www.quantamagazine.org/landmark-computer-science-proof-cascades-through-physics-and-math-20200304/ https://www.quantamagazine.org/landmark-computer-science-pro... [2] https://www.quantamagazine.org/does-time-really-flow-new-clues-come-from-a-century-old-approach-to-math-20200407/ https://www.quantamagazine.org/does-time-really-flow-new-clu... [3] https://www.youtube.com/watch?v=tmx2tpcdKZY&feature=youtu.be&t=712 https://www.youtube.com/watch?v=tmx2tpcdKZY&feature=youtu.be...
- gowld 7y ago
  The video timestamp you linked does not say "free will exists". Is says "humans has free will implies element particles have free will".
  - peterburkimsher 7y ago
    His definition of free will is quite limited. He only suggests that free will is limited to a "free" choice between options within one's power. Conway said that there either is determinism or free will. If the future can be perfectly preordained (Gisin) due to infinite possibilities, then there is no free will. But if there are finite options, I think that there is "free will" for particles and people to choose between those options. https://youtu.be/tmx2tpcdKZY?t=1074 https://youtu.be/tmx2tpcdKZY?t=1074 https://en.wikipedia.org/wiki/Free_will_theorem https://en.wikipedia.org/wiki/Free_will_theorem
naasking 7y ago
> Intuitionist mathematics claims that mathematics is purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles existing in an objective reality. [0] The "mental activity" part isn't strictly necessary. A constructive mathematics will produce results that are largely the same. What matters most is that mathematical objects are proven by construction, and so proofs have a direct translation to computation.
- maxiepoo 7y ago
  I think you and OP are talking about different things because the term "intuitionist" has been used for both the philosophy that EJ Brouwer espoused (which matches what OP said) and the large field of constructive mathematics that has been greatly influenced by Brouwer's philosophy but has embraced approaches like formal logic which Brouwer himself disliked.
  - naasking 7y ago
    I was merely pointing out that the "mental" dependency of intuitionism isn't strictly necessary. Take it out, and you're left with an equally valid constructive logic.

keiferski 7y ago

The SEP article is an excellent introduction to the philosophy of mathematics: https://plato.stanford.edu/entries/philosophy-mathematics/ https://plato.stanford.edu/entries/philosophy-mathematics/

mrtksn 7y ago

Recently I watched Sixty Symbols video on why light is slower in glass and there are multiple explanations on it. What struck me was what prof. Michael Merrifield said at the end of his explanation: https://www.youtube.com/watch?v=CiHN0ZWE5bk https://www.youtube.com/watch?v=CiHN0ZWE5bk On the question about what is the reality, what explanation is the true one, Merrifield said that the Math works on all of the explanations and what those explanations do is simply to model the behaviour of nature and not necessarily reflect the reality. That's how Newtonian physics and relativistic physics are both correct models, tools to model nature and simply can be used to whenever suitable. Wouldn't that mean that mathematics is just an invented tool to reason about physical models?

iNic 7y ago
Well yes and no. A lot of mathematics found its roots in trying to model the real world, however I would then argue that the truths we prove about these models are truly discoveries. And with these discoveries we can often generalize the setting to more abstract formulations, independent of the physical reality. I think that this can be seen in the fact that sometimes mathematicians and especially physicists can reason about objects that they are not sure about what the right definition should be. Many mathematicians reasoned about continuous functions long before we had a concrete definition of them. But when Cauchy introduced the definition and Weierstrass proved it is equivalent to preserving limits (which was the intuition at the time), we had not truly discovered something new and mathematical. This whole idea was then generalized to topology when it was shown that pre images of continuous functions preserve the "openess" of sets, i.e. we realized that no concept of distance was not needed to describe continuity, which is very surprising.

jelliclesfarm 7y ago

Of course..we invented the ‘language of mathematics’. I am on a Penrose kick right now. Just downloaded The Emperor’s New Mind Audiobook.

sunstone 7y ago

Put me into the 'discovered' camp.

jeffdavis 7y ago

The expression is invented but the underlying relationships are discovered. It's silly to think that the sqaure root of negative one is something real to be discovered. It's just a stand-in to express complex relationships more succinctly. The same for negative numbers, for that matter. It would be equally silly to think that the underlying relationships are invented. Nobody invented prime numbers, they were discovered.

Y_Y 7y ago
It's just as silly to think of the "real numbers" as being discovered. They don't exist in the real world, after all.
- mikorym 7y ago
  They are necessary for the continuum. If you throw away real numbers, then you lose major things in physics. I think for example you lose the wave function in QM. It is not a question of whether they exist in nature, but rather whether they are the more superior technique, or not, to explain nature.
  - Y_Y 7y ago
    Wavefunctions are in fact generally complex-valued. I don't know if that supports your argument or not.
    - mikorym 7y ago
      Complex numbers have the same cardinality as the reals.
      - Y_Y 7y ago
        That's true, but irrelevant. Complex numbers aren't "necessary for the continuum" as you put it, and some realists might argue that they don't hold the same "discoverability" as the reals. I wouldn't. I think the reals and the complex numbers have the same "realness" and that neither represents any innate property of the physical world, despite how obviously useful they are in physical models.

BtM909 7y ago

I read this book about it: https://www.amazon.com/God-Mathematician-Mario-Livio/dp/074329405X https://www.amazon.com/God-Mathematician-Mario-Livio/dp/0743.... Won't spoil the answer, but was nice to go through the evolution of Mathematics.

deleted 7y ago

[deleted]

mykhamill 7y ago

What can be imagined is what can be discovered. The etymology of invent has the terms 'contrived' and 'discover' baked in. if we take the contrived root, rather than just dead-ending to say that invent is synonymous with discover, we find that it is rooted in the ability to 'compare' and 'imagine'. From this we can then formulate the opening statement.

DeathArrow 7y ago

I am not sure of the implications. Does it even matter if Math is invented or discovered? Maybe it's both, and it isn't a contradiction between invention and discovering? We describe things having a certain radiation wavelength as having the color yellow. In that sense, yellow it's an invention. That doesn't mean the radiation doesn't exist. But to complicate things a bit, some things don't exist unless we observe them. This is the case with states of particles described by quantum mechanics. Math is more than a science, is the sciences upon which most other sciences and tech are founded. You can model anything in a computer and running on a computer using math. You can describe logic, natural language, technology, biology using math. In that sense, being the building block of other sciences, math is more akin to a language. Two physicist use math in almost the same way two people use English to describe things and communicate ideas. But math has building blocks, too. Set of axioms upon which any mathematical object and theory can be constructed. The most popular as of now is Zermelo set theory. There are more such fundamental theories, sometimes very different between them. So, to see if Math is discovered or invented, the easy thing to do is to see if a set of axioms can be discovered or is invented.

goatlover 7y ago
> Does it even matter if Math is invented or discovered? It matters for philosophical inquiry. Does philosophy matter? It matters for people who find it valuable, the same way art, music or literature matters.
- guerrilla 7y ago
  You think logic, the art of making distinctions, the study of critical thinking and the foundation of mathematics only matter in the same way art does?
  - inopinatus 7y ago
    I’d say they don’t matter nearly so much as art. The most cosmically significant export of humanity to date is Voyager’s golden record.

NautilusWave 7y ago

I initially read the title as "Is Mathematics haunted or discovered".

xtf 7y ago

Invented to describe what's being discovered.

Koshkin 7y ago
You have to add: discovered inside itself.

ukj 7y ago

The answer is: "both". Where do Mathematicians look to discover Mathematics? In the depths of their own minds. The part where you "look and think deeply" is discovery. The part where you "express your discovery in a coherent language" is invention.

justforfunhere 7y ago

Could the same be said about just about everything? Like Music. Is a Song just a combination of notes, beats, intervals, voice etc. waiting to be discovered? Or is it something that a musician invents in her brain through talent, experience, practice and trail and error. Or for that matter, a startup idea? A product/service that would bring immense value to its consumers, but it is not there yet, waiting to be discovered. I guess philosophers must have dwelled on such questions before.

cpsempek 7y ago
I think I get your point and I‘m open to ideas which drive home your point in a more precise way. However, the main difference between concerning ourselves with whether or not mathematical objects exists vs music or start-up ideas is their place in building a foundation for understanding reality. Science has been a productive program for understanding reality. It happens to be underpinned by math. And as a result, we have encountered some pretty interesting relationships between what the math tells us and what we observe. For instance, neutrinos and black holes were known to exist mathematically before they were ever observed and measured “in reality”. It’s not clear to me that music and start-up ideas hold up to math when it comes to being a tool for understanding or describing our reality, or perception the rig. Math is somehow fundamental to many successful enterprises which seek to describe and explain reality. Therefore, it becomes the point of focus of much philosophical inquiry when seeking to understand reality. Hence the foundational question, do mathematical objects exist? And not, does music or blockchain kittens exist?
- kruasan 7y ago
  >do mathematical objects exist? Yes, they do, in an abstract sense. Existence in math means a very different thing from existence in physics. There are mathematical objects that exist, and those that cannot and do not exist. Someone posted a SEP entry earlier, which is a good start on this topic. >the main difference between concerning ourselves with whether or not mathematical objects exists vs music There is no 'vs', because there is no difference. Music is math, any song or sound is a mathematical object. A physical waveform that you hear can be encoded digitally in numbers: ones and zeros. So any given wav/flac file is just a bunch of numbers that give rise to the qualitative experience of sound, when interpreted in a certain way. For example, a digital waveform consists of samples, each sample takes 16 bits to encode. Sampling rate of 44.1kHz is 44,100 samples per second. So you have 16 bits per sample x 44100 samples per second per channel x 2 channels x 300 seconds = 2^423,360,000 possible permutations of a 5-minute audio file without compression, which is a number with over 127 million digits. A little percentage of these permutations would count as music (even if your tastes are really diversified), most of it would just be noise. But all these possible 5-minute audio files include not only every song and every performance that existed or will exist. They also include every possible sound recording: songs that will not be written, Paul Graham saying that he hates HN, Paul Graham saying that he loves me and the rest of the file is silence, you and me discussing this topic with Plato for 5 minutes, etc, etc. The data exists and can be discovered and listened to, even though some of these examples are obviously not physically possible (i.e. Plato is dead). So all music already exists mathematically, and it can be a useful mindset that your job is to discover it. A lot of musicians see it that way, Tessa Violet, for example: https://youtu.be/QzBoGVToWEo?t=342 https://youtu.be/QzBoGVToWEo?t=342

guerrilla 7y ago

I'll never understand how someone otherwise so apparently intelligent can be so religious. Weird, the systems that we spent massive amounts of energy designing to precisely describe reality do that better than all the ones that we threw out along the way!

noodles_ftw 7y ago
> I'll never understand how someone otherwise so apparently intelligent can be so religious. Why not? Religious people believe that God is the Most Wise. Mathematics in nature attest to that attribute of God (as well as to other attributes of Him). An intelligent and religious person would recognize that, knowing it is God who came up with all the rules that keep the universe in balance. I'll never understand why some people believe science and creationism can't go hand in hand.
- guerrilla 7y ago
  If you want to understand why some people believe that science and creationism don't go hand in hand, then studying epistemology would help you do that. Most introductory texts will cover what you need.
  - noodles_ftw 7y ago
    Sounds like I got something new to learn during this lockdown, thank you. After a quick look at the wikipedia page on epistemology (very interesting), I'm not sure how this explains why some people believe that science and creationism don't go hand in hand.
    - guerrilla 7y ago
      A super super short version is about the question "how do you know X?" Its generally (although not universally) agreed that there are two types of knowledge, a priori and a posteriori. Roughly, in the first case, all you can deduce is implications from assumptions. For example, "if X and Y are true then Z is true." Such things don't require experience. The second case requires evidence gained from experience. Mathematical knowledge is of the first type, scientific knowledge is of the second type. The school of rationalism prioritizes the first type and the school of empericism emphasizes the second. That the universe was created by an omnipotent omniscient and benevolent god is not an emperical fact but requires some kind of a priori supports. Thats why you see so called proofs of god rather than scientific theories of god. There are other options of couree like mysticism, the claim that knowledge can be gained outside of these methods, or an emphasis on faith, that we don't take these things to be justifiable. I probably butchered this but it gives you some starting points to research and hopefully shows a model of the debate. Huge topic. Check out https://plato.stanford.edu/entries/epistemology/ https://plato.stanford.edu/entries/epistemology/ as a hardcore crashcourse or check "crash course philosophy" on youtube for the relavent sections at a more HS level. SEP has entries on god, faith and mysticism as well as rationalism and empericism too.

danck 7y ago

If you only look at mathematics I think it's simply: - Axioms are invented - Conclusions are discovered The magic part for me is that some axioms have been chosen so well that their conclusions are confirmed in the real world.

vanderZwan 7y ago
Isn't that simply because axioms that don't lead to consistent conclusions are rejected?
- danck 7y ago
  You can invent and pick axioms in many ways that (probably) won't lead to inconsistencies. But they won't all be powerful enough or relevant in the real world.
  - vanderZwan 7y ago
    Well, then it sounds like your reasoning gets it backwards: the axioms that produce systems without significant consequences or connections outside of their own abstract realm end up being ignored. Or in other words: the constraints on maths are imposed from outside of maths.
  - gibspaulding 7y ago
    Doesn't this imply that, while you can invent all the axioms you like, you must discover which ones are consistent with each other and with experimental results.
red75prime 7y ago
Arithmetic existed long before its axiomatization. Arithmetic was useful and no one stumbled upon contradictions in it. So it was natural to suppose that it can be described by some axiomatic system. Peano found it.
- kevin_thibedeau 7y ago

raghavtoshniwal 7y ago

Last week I came across another fascinating podcast by Penrose that you folks might also like: https://www.youtube.com/watch?v=orMtwOz6Db0 https://www.youtube.com/watch?v=orMtwOz6Db0

peterburkimsher 7y ago
Thanks for the link. At 1:02:57, Penrose proposes "infinite cycles of the big bang" https://youtu.be/orMtwOz6Db0?t=3777 https://youtu.be/orMtwOz6Db0?t=3777 Is that multiverse theory disproven by Wolfram's latest blog post? "could there be other universes? The answer in our setup is basically no." https://writings.stephenwolfram.com/2020/04/finally-we-may-have-a-path-to-the-fundamental-theory-of-physics-and-its-beautiful/ https://writings.stephenwolfram.com/2020/04/finally-we-may-h...

magicalhippo 7y ago

If B follows from A via logic, but A is invented, isn't B invented as well then? Just to take a recent example which was mentioned here, Geometric Algebra[1]. There you assume you have some objects which aren't numbers but which when squared equals a given number. By doing that a bunch of nice results have been discovered. However to me the basic premise, take some objects which aren't numbers but which square to a number, feels very much like an invention. So as such wouldn't the nice results be inventions as well? [1]: https://bivector.net/doc.html https://bivector.net/doc.html

lmm 7y ago
> If B follows from A via logic, but A is invented, isn't B invented as well then? Not really, because you have no choice about B - it's true (or rather it follows from A) whether you want it to or not. You could have picked a different A, but once you picked A then B was fixed.
- magicalhippo 7y ago
  Yet if I pull the trigger we say that I killed someone, not that the bullet did.
  - lmm 7y ago
    People invented the electric chair because people discovered that high-voltage electricity can kill people. You might invent a particular axiom because it has an interesting mathematical consequence (e.g. the axiom of choice was invented to allow Hilbert's basis theorem to be proven), but what you invented was the axiom, not the consequence.
analbumcover 7y ago
It sounds to me like you are giving special precedence to "numbers" in your considerations, as well as drawing your intuition for "square" from working with the integers or the reals. A square is just the result of a binary (product) operation where both of the operands are equal. The operation can be defined on an algebraic object with much richer structure than the integers or reals, and the operation itself can be much more complex than, say, integer multiplication. So the geometric product of geometric algebra is just one animal in the zoo of examples. You might call the specifics of the geometric product operation an "invention," but not because the square of a non-number can be a number.

simplesleeper 7y ago

This is just the age old question of nominalism vs realism

rezeroed 7y ago

Fallacy of equivocation.

deltron3030 7y ago

Mathematics is just a language and likely a human discovery method for rules that encapsulate the universe and cascade down. A method invented to discover discoverable rules.

roywiggins 7y ago
You can build all sorts of bizarre structures in ZFC that seem to have no relationship with reality at all. You can prove that almost all real numbers can't be computed or even described. There is an entire tower of infinities bigger than the natural numbers that telescope up and up. The axiom of choice gives you all sorts of nonphysical results. The world of ZFC doesn't feel very constrained by the actual universe, is all. And if you don't like ZFC, you can pick another set of axioms entirely- there are several alternatives.
- deltron3030 7y ago
  Sure, math/ZFC can describe potential and impossible universes including ours. But the motiviation for its invention as a language was very likely to understand relationships in our reality, physics, trade etc.

itvision 7y ago

There's one thing I cannot wrap my head around. Math is purely abstract science yet it describes the world around us to the utmost precision. Does that mean that this world is simply ... a math model which means everything around us is ... not real? I've long stopped believing in free will because everything points at it being an illusion of our brain because we're a product of this world and we had no chance of influencing the conditions which brought us to life, and even after our minds and consciousnesses form it's hard to believe they are fully autonomous and not simply a function of the processess in our brains we're simply not aware of. If you think about all of it, it becomes utterly depressing as you begin to realize you're a biological robot, a byproduct of the universe evolution which couldn't care less about our species and this little tiny blue planet.

tirant 7y ago
Mathematics has not been static throughout history. It has evolved at the same pace our science and discoveries have evolved, and as such it has incorporated all necessary elements to help describe new phenomena. To the point that now is able to describe never observed phenomena. So no, our world is not unreal because it can be described with maths. Maths have evolved to describe it. Same as human language has evolved to support our close environment and our human interactions.
kd5bjo 7y ago
> Math is purely abstract science yet it describes the world around us to the utmost precision. Does that mean that this world is simply ... a math model which means everything around us is ... not real? You can only reach this conclusion if you categorize math as "not real". If everything we interact with is a perfect mathematical object, I prefer to take that as evidence that all mathematical objects may be real, however abstract they seem.
novalis78 7y ago
What’s depressing about being a nanotchmological robot attempting to conquer the solar system and spread his technology and nanotechnology? It sounds rather exciting and fascinating. Perhaps if the depression is too severe you need to check out your programming ;-)

gingahbread31 7y ago

Thank you for sharing this, this an amazing interview series !

zeepzeep 7y ago

Java was invented, Agda was discovered. (Obviously I'm in the religion where math does exist.)

ebj73 7y ago

A vaguely, but tangentially related fact, is that Stephen King actually considers the stories in his books as discovered, rather than invented. He talks about it in his book 'On Writing'. He considers the stories to be sort of preexisting things, and his job as a writer to unearth and discover them, rather than invent them. It's about his frame of mind while writing, I guess, but still.

ineedasername 7y ago
It's a common theme among sculptors too: the shape is already in the stone, they just have to uncover it.
- DagAgren 7y ago
  Is it? The only time I have ever heard this expressed is as a bad joke.
  - ineedasername 7y ago
    I know from a few general humanities classes I took that it was a common way of thinking about sculpting during the renaissance. And modern learning material on the topic presents it as a visualization aid. Let me be clear: It wasn't some mystical way of thinking there was, literally something that existed in the material. Again, it was more of an aid to thinking about the endeavor.
RashadSaleh 7y ago
I doubt that he actually said that but I could be wrong.
- Trasmatta 7y ago
  This is actually a big plot point in The Dark Tower books.
- ebj73 7y ago
  He describes stories as 'relics'. Here's a direct quote from the book: “Stories are relics, part of an undiscovered pre-existing world. The writer’s job is to use the tools in his or her toolbox to get as much of each one out of the ground intact as possible.” The book is pretty good, well worth the read if you're only slightly interested in the topic!

scrozart 7y ago

Phenomena are discovered. Mathematics are invented to describe them.

Noughmad 7y ago
Not all mathematics though. Math works by starting with a set of axioms. These are chosen, usually but not necessarily such that they will describe some kind of physical reality. That is the part which you invent to describe phenomena, like the Peano axioms which describe quantities of countable things. But then, you discover (and of course also prove) things that follow from these axioms. So a mathematician would find that if you take this set of axioms, then this statement is true. Is that an invention or discovery? I would call it a discovery, although different from a discovery of a natural phenomenon.
- roywiggins 7y ago
  Geometry and arithmetic and calculus all existed before axioms and worked okay. You can do quite a lot of mathematics "naively". It's only in the 20th century that axiomatization became really important, because naive set theory turned out not to work very well.
naasking 7y ago
And what if all phenomena are intrinsically mathematical? Then mathematics is discovered as well.

naringas 7y ago

A bit both: e.g. The circle is an invention. But the number pi (it's value) is discovered

nybble41 7y ago
What do you mean when you say that the circle is an invention rather than a discovery? Are you drawing a distinction between the mathematical ideal and the arbitrarily close approximations which can be found in nature? Though by that criteria that case pi must also be an invention…

Reedx 7y ago

Same question also answered by Max Tegmark: https://www.youtube.com/watch?v=ybIxWQKZss8 https://www.youtube.com/watch?v=ybIxWQKZss8 Stephen Wolfram: https://www.youtube.com/watch?v=nUCwtLTUPQ4 https://www.youtube.com/watch?v=nUCwtLTUPQ4 Steven Weinberg: https://www.youtube.com/watch?v=NpMk9G-ddiM https://www.youtube.com/watch?v=NpMk9G-ddiM I particularly liked Max's answer who neatly makes the distinction between the structure (we discover) and the language (we invent). We're free to invent the names, but not the structures.

7thaccount 7y ago
Max Tegmark has some fun books. Apparently he got to be at the forefront of some discoveries as he and his wife knew C++ and would do all the after the fact data analysis. If you've ever been around a campfire with a good storyteller that has their audience at the edge of their seat, I felt like this in some spots.
throw0101a 7y ago
Entire playlist: * https://www.closertotruth.com/series/mathematics-invented-or-discovered https://www.closertotruth.com/series/mathematics-invented-or... Gregory Chaitin was interesting as well: * https://www.youtube.com/watch?v=1RLdSvQ-OF0 https://www.youtube.com/watch?v=1RLdSvQ-OF0 The (apocryphal?) story about Godel not believing in empirical science and only in a priori truths was interesting.
nikofeyn 7y ago
> I particularly liked Max's answer who neatly makes the distinction between the structure (we discover) and the language (we invent). We're free to invent the names, but not the structures. i came here to give my opinion, but this is pretty much it. i need to read max tegmark's books. the idea that how we got to the structures and concepts we find is undoubtedly invented, because we've already seen that we've re-invented our methods multiple times. but the structures and concepts don't really change. we extend them, but there seems to be some set of structures and concepts that get settled to. if we found some alien species that has mathematics, we'd probably find that there's some mapping between our structures modulo small choices that were made.

theaeolist 7y ago

Like in any game, you invent the rules (axioms and logical systems) then you discover the consequences (theorems), just like you invent a maze then you discover the way out.

lisper 7y ago

The question assumes that there is a difference between invention and discovery. There isn't. Invention is a kind of discovery. The apparent distinction arises from our separation of physical and mental processes. But this is a purely artificial separation, a human conceit, because (unless you're a dualist) mental processes are physical processes. Specifically, mental processes are computational processes, which are physical. So it's not surprising that the structure of the physical world should be reflected back on itself in the structure of mental processes. It's universal Turing machines all the way down.

koheripbal 7y ago
This all seems like semantic masturbation.
yters 7y ago
So, if there is a difference between invention and discovery, then dualism is true?
- lisper 7y ago
  Yes, though that's a bit of an odd way of putting it. It's kind of like saying that if matter can't be broken down into smaller constituents beyond a certain point then atomism must be true. It's more of a definition of dualism/atomism than a logical entailment.
  - yters 7y ago
    One man's modus ponens is another man's modus tollens.
pdonis 7y ago
> Invention is a kind of discovery. Some inventions can be thought of this way, but I don't think all inventions can. For a relevant example, human mathematical notation is a human invention that I don't think can be usefully thought of as a kind of discovery. (Another poster upthread mentioned Tegmark's response drawing a distinction between the structure of mathematics, which we discover, and the language we use to describe it, which we invent.)

steve76 7y ago

I would go with invented. I think this ends with humans changing the universe to fit their needs. The math is right because humans made it. I think of satellites and solar orbiters made from matter waves that encompass everything. Medical applications built around them. Humans manipulating their photoelectric properties. They still see a sun. Everything else sees the sun as a black hole because of what we did to ourselves. Distant stars focus into existence because of telescopes on Earth. Everything that existed already leaves us alone, so to a degree they must see it too. There's some contest to see how insignificant we can make ourselves just for petty license. Convince no one. Start building. Odds are, other more powerful people thought of it and did it before. That telescope in geneva to me is not just some far off irrelevant land to look at. It's in the material, a grain structure of our Sun. Serves as great motivation.

Wistar 7y ago

Dr. Hannah Fry's 3-part BBC documentary series, "Magic Numbers: Hannah Fry's Mysterious World of Maths," explores this very question in the first episode, "Numbers As God." The episode description reads: "Documentary series in which Dr Hannah Fry explores the mystery of maths. Is it invented like a language, or is it discovered and part of the fabric of the universe?" https://www.bbc.co.uk/programmes/b0bn6wtp https://www.bbc.co.uk/programmes/b0bn6wtp She interviews a number of prominent mathematicians and scientists, such as Brian Greene, and they certainly don't agree one way or the other in the invented/discovered question. (Alas, I now see that the series is listed as unavailable on the BBC site but I watched it, I think on Amazon Prime or maybe youtube.)

lavp 7y ago

I find mathematics to be the most beautiful thing in the universe, because it is the only thing I can truly think of as being perfect in every way. Existing only as an abstract concept, it still manages to find its way into every single aspect of our lives. Two completely remote civilizations will still have the same mathematics. Sure one may have a more developed understanding, but if both civs wondered about how to get the hypotenuse of a right-angled triangle, they would both end up with Pythagoras' theorem. The only thing invented in math is our language and representation of such concepts. Many people believe that mathematics becomes invented the further up you go like pure mathematics and I can understand why this perspective would come through, but take the example of G.H. Hardy who famously said “The Theory of Numbers has always been regarded as one of the most obviously useless branches of Pure Mathematics”. It could be argued at the time that these branches were simply invented mathematics because there was no practical application of it, but 30 years following his death came breakthroughs in cryptography. All of a sudden, this was no longer an invention. Calculus is not seen as an invention but a discovery because of all of its insane number of applications in physics and other areas. Many areas of math that have yet to find practical use will always come under the scrutiny of being 'invented' but that simply isn't the case. I firmly believe that all areas of mathematics are practical to the universe and its wonders (and therefore discovered), whether or not we can achieve a level to use such mathematics (or experience it) is a different matter. I believe that at its core, mathematics is the universe. To say that we have invented it is arrogant and completely strives it of its beauty.

bane 7y ago

In most fields, you have to invent tools that enable you to discover things. My guess is that this is also true in mathematics. We get confused sometimes because the tools often happen to look like the discoveries because of all the fancy greek symbols.

raincom 7y ago

It is a dispute in ontology. Do mathematical objects (say, numbers, sets) exist in the world? Some say, yes; others say, no; some others say, they exist in another world--called Platonic world. We see similar disputes in philosophy of (natural) sciences: for instance, instrumentalism doesn't subscribe to the ontology of realism.

amarte 7y ago

Imagine an interview with a great painter. The painter is asked about the nature of painting and he responds that when he sets out to work in his studio and his brush strokes a canvass, he is discovering the fundamental nature of reality. He doubles down and exclaims that, in fact, reality is actually JUST lines and curves and shades and colors, and his proof is, well, look at how accurate his paintings are! Let's pretend that he actually is a very skilled painter, and many critics have marveled at the extent to which his paintings are indistinguishable from his subject matter. Still though, the interviewer clears his throat after an awkward pause and continues on to the next question. To me, math is a medium of description much like painting or writing. It's units are not colors or words but points. A point, or "that which has no part", is much finer and carries with it far less baggage than something like a word. Points can be assigned numerical values and played with in clever ways. You can even sprinkle a fine dust of them over anything observable and create a copy of it to arbitrary degrees of precision. I know math is far more than just the study of points, but I'm not convinced that math is anything more than our capacity to distinguish and describe extended to its limit. I also don't mean to belittle the accomplishments of mathematicians and theoreticians, I just think it's more reasonable to say that math is JUST the limit of description than it is to say that reality is JUST math. After reading through some of the comments, I think many of us are on the same page.

bloomberg2020 7y ago
Agreed The crux of it is obvious: math is true from the perspective of a human consciousness It’s defined after cognitive measure. I won’t say it’s whittled down. I would say ... it’s sight, sound, touch... attempt at quantifying our qualified experience All the best math nerds were often deeply gifted and practiced forms of human art: poetry, music, sculpture... and many artists good at math, Brian May, for one example IMO to get math one needs to build all those abilities It’s really the people I know that don’t do that see math as a boring toolbox of rules and points Not saying you’re not that Math is not a thing like a rock because our ability to behave like more than a rock gave us the ability to define math. It has to be more than a simple this or that

hyperpallium 7y ago

Operational theory of mathematics: you define (invent) some rules, of starting points and ways to change them. Like the rules of a game. In a sense, all possible outcomes already exist (only to be discovered).

Koshkin 7y ago

The linguistic conundrum here arises due to the recently adopted mindless use of the word 'invented' instead of 'created' or 'designed.' (Indeed, Linus created Linux rather than 'invented' it as many would say today.) Invention is a form of discovery. So, the question should be, "is mathematics created or discovered?"

Koshkin 7y ago

My take on this is that Mathematics is such a vast area of investigation - much larger than Nature itself - that there is in fact a mix of both. In that regard it is close to Engineering, where in an attempt to design something on might stumble upon things or perform some types of research and discovery, and vice versa.

monadic2 7y ago

This is a silly semantics game and a boring one at that. the presumption that the words are contradictory, or that mathematics is a monolithic entity that plays a single role of invention or discovery, is a poor way of exploring the social and cultural (co)evolution of mathematics and methodical experimentation and observation.

Koshkin 7y ago
You can't just say this to people who keep asking this question, and have been for ages.
- monadic2 7y ago
  Why not? It’s not like they have any obligation to respond, and someone needs to point out the bullshit smell wafting through the room.