Roger Penrose â Is Mathematics Invented or Discovered? [video]
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
[1] https://www.quantamagazine.org/does-time-really-flow-new-clu...
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.
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.
The SEP article is an excellent introduction to the philosophy of mathematics:
Same question also answered by
Max Tegmark: https://www.youtube.com/watch?v=ybIxWQKZss8
Stephen Wolfram: https://www.youtube.com/watch?v=nUCwtLTUPQ4
Steven Weinberg: 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.
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.
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.
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 âThe Theory of Numbers has always been regarded as one of the most obviously useless branches of Pure Mathematicsâ. 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.
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
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.)
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.
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.
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
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?
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!
Last week I came across another fascinating podcast by Penrose that you folks might also like: https://www.youtube.com/watch?v=orMtwOz6Db0
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?
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.
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.
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.
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.
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.
I read this book about it: 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.
Put me into the 'discovered' camp.
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.
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.
I initially read the title as "Is Mathematics haunted or discovered".
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).
Phenomena are discovered. Mathematics are invented to describe them.
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.
The platonic universe of the entirety of math sounds (this week at least) like Wolframâs âUniverse of Computational realityâ of which the physical reality is just a subset.
Invented to describe what's being discovered.
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?"
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.
A bit both:
e.g. The circle is an invention. But the number pi (it's value) is discovered
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.
This is just the age old question of nominalism vs realism
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Of course..we invented the âlanguage of mathematicsâ.
I am on a Penrose kick right now. Just downloaded The Emperorâs New Mind Audiobook.
Java was invented, Agda was discovered.
(Obviously I'm in the religion where math does exist.)
Thank you for sharing this, this an amazing interview series !
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.
Fallacy of equivocation.