Is Math Invented or Discovered?

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Rigid Designator

When Pythagoras first reached, grasped and brought to earth what we call the Pythagorean theorem, he also, like Jack up a bean-stock, seized upon a bit more and was so excited, so happy that when he got back down to earth he threw a huge party. A hectatomb of animals were sacrificed and roasted for his guests! * Who does that now a days? Such a feast and celebration would not be raised for a mere invention and vainglory. The Pythagorean theorem was an acquisition to be shared with the world. It lives on in every superposition and linear operator H | (Kinetic Energy + Potential Energy) > = H | KE > + H | PE > . * William Whiston tells this story in his introduction Euclid's Elements (1747 sixth edition). Whiston introduces the reader to a pantheon of mathematicians from Greece to Alexandria. In Euclid's time, the pyramids still had smooth alabaster like surfaces that gleamed, even in starlight, like the faces of a Platonic solid. The Sphinx brooded at their feet and then got almost got covered in a drift of sand. And then Napoleon arrived . . . . .

Scott Ready

I really like the work you're doing. The topic of the video is great. Unfortunately, I still haven't found time to watch it, because my video-watching habits involve adding new videos to my "watch later" list on youtube, and then listening to videos on that list when it's convenient (like when I'm washing dishes). Not having this video on youtube means I almost never think to watch it when I would have time. One thought about the topic of your video: I read somewhere that discovery versus invention is actually on a spectrum, and the distinction between the two is related to David Deutsch's idea that a good explanation is 'hard to vary'. The more a particular invention/discovery is 'hard to vary', the more it is like a discovery. I wish I could remember where I read that, but it sounds right to me. Here's Deutsch's talk about good explanations being hard to vary: https://youtu.be/folTvNDL08A

Michael McGuffin

I knew the video was going to be good but I’m honestly blown away! Can’t wait for more

Great topic! Great presentation! In my opinion, discoveries and inventions form an interdependent cycle that details a problem solution more and more. Example: Problem: We want to fly. Solution: 1.) Discovery of the function of bird wings, 2.) Invention of aircraft, 3.) Discovery of turbulence at the wing tip, 4.) Invention of winglets, 5.) ... smaller refinement of winglets. the same with mathematics: problem: representation of cumulative sums: solution: 1.) discovery of positive countable numbers, 2.) invention of negative numbers, 3.) discovery of problem root of -1, 4.) invention of i, .. ..

Woah. Super great video! At first I thought I was firmly in the "invented" camp but mostly due to the way I like to define what math actually is. All the things about math that we say "have always been there" are not really math itself, but the subjects that math describes. (It's sometimes hard to distinguish between the equation and the physical phenomena it describes. Or a special arrangement of matter in the real world vs. the concept of a "shape" we ascribe to it.) Math is a way to describe things about the universe, a way that we invented. But several times during the video I had to reconsider and go "hmm!". It was really eye-opening in a way. Added some new ways of thinking about stuff I hadn't considered before, and that's great! "If one thing holds all the information of another, can we really say that the two are different?" (!!!) And then we reach into the subject of the aspects of reality (physical, mental, math, ... etc??), and it starts to get really interesting. I've always disliked when people say a thing is only "real" if you can touch it. How are concepts and ideas not "real" in their own way? (More than just signals in our brain or ink on a paper, because things can be more than the sum of their parts.) *** Minor point, but I'm a bit unsure about how non-euclidean geometry or negative numbers don't exist in the real world while euclidean geometry and positive numbers do, I think that's mostly down to interpretation... I mean do positive integers really "exist in the real world"? I don't think so, they're just a tool we can use to describe something like a collection of things. Similarly, negative numbers can be a tool to describe taking away or lacking a subset of things. And non-euclidean geometry can describe the angles between the bodies of several worms climbing on the surface of an orange (maybe? sounds ridiculous but I think it works, and it's the "real world"). Anyway, this isn't really criticism, just more food for thought! *** Sorry for the rambling comment, all in all I just want to thank you for making this video, it's really good! *** [By the way, us Hungarians can be awfully proud of our large host of world-renowned scientists and inventors, so it's nice to see two of them (Bolyai and Erdős) mentioned here :p (as the Finnish would say, torilla tavataan)]

Parachuting Turtle

Indeed it does

The Raven

Great video! Well worth waiting for -- thank you! I love these kinds of questions. The hypothetical idea of communicating with aliens using mathematics is quite a revealing thought. Whether or not aliens exist, if you *believe* that aliens would come to similar mathematical conclusions as us (which I do), then that reveals an implicit belief that the information about maths can be drawn from the universe. But having said that, I don't feel that every conceivable piece of mathematics would be recognizable by aliens, especially for the maths that turns out not to correspond to any patterns in reality. The space of patterns that mathematics could describe is larger than the space of patterns we find in our universe. So perhaps math is like a garden of branching paths that you can walk along ("discover"), making a choice at each junction about which fork (which "axioms") to follow, and then uncovering where the path leads from those choices. While you could walk along any path in the garden for the sake of intellectual curiosity, not every path corresponds to something in reality, so it's more scientifically useful to use scientific discoveries as footprints to see which way reality has walked through the garden before us, giving us clues about which further paths reality must have taken (did it take the string theory fork, or some other?). Does my metaphor make sense?

Michael Hunter

I fall in the discovered group. The relationships, even the imaginary ones, exist regardless of humans. They are fundamental to the universe as it is. Math is the language we "speak" to communicate. It's just in this unique language we all agree on the words, tenses and syntax on this planet. That is unless you don't agree that two plus two is four because sometimes it's deux plus deux font quatre. :-D How's that for wishy washy?

The Raven

Thank you for this exploration. What struck me is this quote from you in the last minute of the video. “It is ultimately a deeper question of the nature of information.” – Jade, Up and Atom As any good question should this lead me to many others. This question is one of how is math categorized. Perhaps this is the wrong question. Is categorization a subset of math, like set theory, for example? Do categories exist independently? What is a category? How could it be recognized as existing independent of human categorization? What tests can be performed to determine answers to these and related questions? - “Is the universe mathematical or are we?” > If the universe is mathematical so are we. Conversely if we are not mathematical then this tells us nothing about the nature of the universe. > If we are mathematical, but not the universe, then can we observe / explain / describe / predict the universe in a non-mathematical, but provable fashion? This may be testable. > If math is dependent upon the nature of the universe, could a tweak in the nature of the universe change math? If so, then is such a math discoverable or verifiable? Does such a math already exist in multi-verse theory? - Can there exist a mathematics that is independent of existing mathematics that is provably true, but incompatible? > If yes, then what does this mean? It suggest that math can be invented, but does this imply that math is not discovered?

Moose Thompson

the link works fine Jade, great job on the video!