Stephen Wolfram - Is Mathematics Invented or Discovered
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- čas přidán 21. 02. 2016
- Mathematics describes the real world of atoms and acorns, stars and stairs, with remarkable precision. So is mathematics invented by humans just like chisels and hammers and pieces of music?
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This is literally one of the most interesting questions in the world for me. It's a huge factor in how I got into physics.
+Cyrusislikeawsome, I'd like to hear more. How do you relate this to Phyics?
+TesserId So many ways :) The way my passion for physics really kicked off was almost 2 years ago now when I was learning about Paul Dirac for a piece of summer homework for my transition from secondary school to sixth form. I was amazed and dazzled by 2 things in particular: 1) what ARE imaginary numbers and matrices and how the hell can them come in the play when we're talking about things in the real world (bare in mind I'd so far only come into contact with GCSE levels maths and physics) and 2) this idea that you could, as Dirac had, find out new things about the world simply by rifling through and analysing and taking apart the equations in such a way that deep down in those tiny equations is incredible information about the universe that was COMPLETELY unknown to the people who first discovered the equation. The case of Paul Dirac was so amazing to me because (especially) by observing that his equation had solutions for plus OR minus energies (just as I'd just learned at school in GCSE that quadratic equations could also have a plus solution and a minus solution) then it must be possible that there could be a particle EXACTLY the same as the electron except it had PLUS charge instead of MINUS charge.
Now that brings in to question so many other possibilities about the relationship between maths and physics and, of course, go relate it to the original question, if we INVENTED maths how on EARTH are we getting these amazing results and if it is DISCOVERED then that's EVEN WILDER and EVEN MORE AWESOME because where does the maths come from? Could a theory of the universe one day include a theory of the origin of mathematics? It's so amazing I can hardly put it in to words.
+TesserId I mean I'll happily tell you I'm basically a philosopher at heart even though I'm going to study physics and want to be a physicist, my interest is really intellectual rather than worldly. I love learning for the sake of knowing and discovering and I want to know the fundamental nature of reality, that's my maths physics and philosophy are my bread and butter and why to me they are so wonderfully interlinked. As Einstein said, "I'm not interested in details. I want to know the mind of god, nothing more, nothing less"
+Cyrusislikeawsome I understand you very well. In fact, for me it is amazing not only the fact that so much of math has applications in nature, but also the fact that a big chunk of math doesn't have any connection with reality! The question always remains: what if in some distant future, those domains will also have some applications, like in the case of Gabriele Veneziano discovering the string theory based on some obscure, almost forgotten equations? To put it in a different way, the question is: is the connection between the nature and math totally biunivocal, or just...incidental?
+48acar19 It's nuts man XD I love it
The paradox is that we cannot invent any other new distinct mathematics. Any conceptually new mathematics will only be accepted as legitimate if it doesn't contradict the cardinal tenets of our old existing mathematics, and at that point it becomes immediately (logically) equivalent to our old mathematics. Kind of like an ideal in ring theory - anything from the ring upon coming in contact with any element of the ideal, becomes part of that ideal. Therefore a completely different distinct mathematics can only come from a different species. The frightening possibility is that if our mathematics is not the "right" one, however logically flawless and vast it will become, then our failure as a species in the universe is already predetermined. Our mathematics might never unlock a (the) key in correctly understanding (an aspect of) the universe necessary to move on to the next level.
Is not that ours might be wrong... But most likely that there are just better or more complete ones...
4:09 "My mathematicses ..."
- Gollum
Deserved top comment.
This comment is "precious"
-Gollum, Ph.D
Thanks so much for the excellence of yo programs.
absolutely correct and brilliant analysis
An example of other “mathematics” would have been lovely...
not sure I quite get this. Stephen has a picture behind him of Pascal's triangle mod 2 and we can see the way the nested sierpinski triangles form in this picture, jumping straight out of the natural structure of numbers. It doesn't matter what your axioms are, no matter who calculates that, they will always see those triangles. We didn't *invent* the fact those triangles appear, they just appear. Sierpinski triangle type forms appear on the shells of Conus and Cymbiola, are gastropods inventing their own axioms now?
this is so factual
I think Wolfram is saying that in effect we identified the discretely-specified set of conditions under which the fractal image appeared. The image itself was thus created by subjective human activity. It doesn't randomly appear in the clouds and thus force us to decree that human math is natural or emerging from nature. Math is a construct, a human-derived map of a universe that may reward even more richly a multi-map mindset. I get his point in that sense but I think it's easily debatable depending on how you define the foundations of the argument. In any case, to the degree that he is trying to urge us to open our minds and think more broadly about systems of rules, the point could be really effective.
Plato's Meno suggests that Learning is remembering. Perhaps recognizing is a better word.
do you have anything more recent on Stephen Wolfram ... also ... have you interviewed the mathematician Jonathan Gorard .. thanks in advance .. also .. it will be interesting to see if the 'discovery' of computation will allow us to go down to the level of machine code of our universe ...
every time he says "Mathematics", take a drink
darnit, now he doesnt seem as credible haha
If math is an artifact, then the idea of a possibility space is also an artifact, and one that grew out of the artifacts of math and logic. You can’t then use that possibility space to posit the existence of other, alternative maths. You either stay within your system, which keeps you blind to what’s outside, or you jump outside and lose all rigour.
Why can't that possibility space be used to infer the existence of alternative maths?
I think it can't be ruled out that we can find/choose other axioms, but then it will only expand our math, to go even further to the root so to speak. However, I think it's more likely that this is Wolfram's ego talking - he so desperately want to be one of the "Idea Makers" (www.amazon.com/Idea-Makers-Personal-Perspectives-Notable/dp/1579550037 ). Read the book and you'll see why this is not idle talk.
Now that is a high IQ comment.
possibility spaces aren't inferred from "mathematical thinking", possibility spaces can be denoted by mathematical symbols, but their existence itself wasn't intuited because we did "math".
He's saying that mathematics is not something we uncovered, it's something we invented to explain the world to ourselves. It is an historical artefact and could have evolved differently.
Saying it's an artifact does not definitively answer the question of whether or not that artifact has an existence independent of our discovery of it. Sure...we created it...but we don't have a clue HOW we created it (or how we create anything for that matter).
this blew my mind. wooooow
Math is a kind of shorthand, what is important is relationships. These relationships have qualities that vary with scale and resolution.
+Rick D. Interesting, please expand.
TesserId
The Fractal Nature of Living Systems
www.linkedin.com/pulse/life-fractal-rick-delmonico-1?trk=mp-reader-card
Math shorthand
pages.uoregon.edu/brundan/math261fall08/symbols.pdf
+Rick D., Thank you.
I enjoyed the piece very much. Some of the points fit nicely with some points I made in a long discussion under the recent Closer To Truth video Marvin Minsky - What are Possible Worlds? I've also tinkered with fractals a bit and found a cube-based branching fractal that projects into Sierpinski triangle (posted at a poor resolution on my channel).
Thanks again.
ok thanks
i am from bangladesh sir
It appears that mathematics is existing at the fundamental level of the workings of the universe but what mathematics we have developed is a crude approximation of that. So yes it seems that what we have is something we invented to approximate what is really going on. And that what is real mathematics of the workings of the universe might be very different to our invented one.
And the, is there an aspect of physical reality that is essentially mathematical, or is that just an illusion (e.g Plato's dualism, etc.)? And, I think this hints at it. Still, a very interesting discussion; very enjoyable.
Late reply, but i would say to expand the common notion of mathematics and find your answer. By mathematics you perhaps mean a realm akin to meta-theoretical, which I would also posit is essentially valid, but there is also the sense that we live in a reflexive universe. As in, if you lower the thermal energy of water to 0 degrees celsius, it will freeze. That is consistent everywhere in the universe, and at any time within the life of the universe. That consistency is a form of logic. Logic is mathematical. 1 plus 1 equals 2, so to speak.
If maths is invented and not discovered, then it *seems* like all mathematical things can be described. But some mathematical things cannot be described, since the set of all numbers is uncountable but the set of all describable numbers is countable (since the set of all English sentences is countable). Contradiction?
Why do you assume that if math is invented then it seems like all mathematical things can be described?
"Mathematicses" You gotta love that word.
So it's an invented artifact, but he concludes by talking about "exploring the space of possible mathematics-s"..... which means it's discovered. OK then.
How come nobody notices he just contradicted himself?
Architecture structures are artifacts, but we can still say that there is more kind of structure to discover yet. I'm not sure that "exploring the space of possible mathematics-s" contradicts the fact that they are man-made. Art is man-made, however, there is still more art form to "discover" (which means "invent" in that case, and not "find"). Maybe I'm wrong and didn't get your point, but I tried to make sense of Wolfram' saying.
Imagine all the cake recipes in the universe. There are probably lots of recipes we haven't thought of right? But if someone creates a new cake recipe, you would probably say he invented the recipe, not that he discovered it. It's the same here. So when he talks about the space of possible mathematicses, he talks about the space of potential mathematicses.
Las matemáticas son un artefacto histórico.
Progresiva generalización de la aritmética y la geometría, más una idea metodológica: uno puede hacer teoremas y pruebas abstractas de esos teoremas.
Si uno mira a los sistemas formales, ¿tendrán el desarrollo de la matemática? No: diferentes tipos de matemáticas (diferentes axiomas(?)).
En el futuro: explorar esos otros tipos de matemática.
Razonamiento circular: nuestras matemáticas nos han servido para entender el mundo. Sí, pero esas cosas que hemos entendido son precisamente las nuestras matemáticas nos han permitido entender.
Does this discussion address the question in the title?
Nope.
100%
To learn more Wolfram thoughts on about college, AI, and the Computational Universe. Watch our interview with him.
But if mathematics is consistent, then a sufficiently intelligent and devoted observer can develop it once instructed in some of its basic premises, notation, language, logic, and so on. For example, the integers are probably known to all highly intelligent beings in this universe, and other universes. Although, different models of the integers may be in used by different beings. If integers are known, then so are rational numbers, real numbers, euclidean geometry, and so on. So mathematics may be something we discover, rather than invent. By the way, bees are known to use a version of polar coordinates; of course not as sophisticated as we human use, but certainly develop-able to an equally sophisticated apparatus given sufficient intelligence.... Can we believe that binary arithmetic is only a human development, and not something of an universal nature probably known to all sufficiently sophisticated civilizations who use machine assisted computational technology?
Here is the problem, one that Wolfram deeply understands. Is that Integers are based on our perception of what "things" are. You look at a rock right...well in your perspective that is one rock. But now look at something to you that is less identifiable, like a gust of wind. How exactly does one count a "gust of wind."
So imagine you are an alien, and you go to our planet for the first time, you look at things, and you count them in a particular way...whereas the alien might not count that thing the same as you do. While you are saying "This is one dog" The alien might be saying this is "30 trillion cells"
If you extend this argument out, the notion that there is any concrete sense of "one, two...three..." is arbitrary based on our relative point of view as to what we distinguish as well...distinguishable things. The argument gets alot deeper when you consider JUST how similar we are as humans, to the ancestry of our own planet. We all grew up here on planet earth, so we all have the same perception of time...the same perception of weight and gravity...so it makes sense that other creatures will develop similar axiom systems based on those commonalities. Aliens however might have a completely different sense of gravity...time, scale... Wolfram goes so far as to say that Aliens might be so alien that they could be based on atomic axiom systems that aren't organic (non-carbon) and function in a way we simply can not possibly understand.
So that's the idea...and i would recommend checking out Wolfram's Lecture "How Universal is the Concept of Numbers" which digs deep into this topic and really changed my perception of how we treat numbers and mathematics.
Note to self, wolfram is saying that our mathematics is a single mathematics out of a larger possibility space of mathematics.
That integers, imaginary numbers, whatever, are a construct ('artifact') of the human mind is an inane observation. However he didn't mention the necessary incompleteness of any his 'quintillion' mathematics he could have said more there on the sigificance of Godel's work.
I wonder why he never mentioned Godel's work. It seems to me it is pertinent to the discussion. This appears to be an extract from an interview that is longer, so perhaps he mentions it elsewhere.
all life is territorial ,and given sufficient time and luck will evolve into an organism that will require mathematical tools to map and record the space it controls. it simply must progress through the simple math of defining right angles as the most efficient way of dividing volume to the more esoteric knowledge of ratio . life only requires cognition to seek define and react to sensorial patterns, and if capable describe laws with which it can both predict and reorder that reality. these base level equations will all be equal.
as to the hyper complex multidimensional stuff mr wolfram is probably discussing that will be superceeded by a.i. mathematicians and i'm sure any other alien computational system would translate its work and comprehend its beauty with ease, so i doubt there are other advanced math frameworks that different from our own as biology will always have similar limitations and desires no matter where and how its brain is first illuminated so as to finally reflect upon the fractals ;)
Interesting theory. My measly human brain till this day believed that the mathematics today was the universal language of the cosmos.
Really brilliant contribution. How does the human subconscious make sense of the world and solve problems, or a dolphin with its echolocation, or a human being when it learns by intuition alone to seek out mushrooms when foraging? None of these use mathematics or logic in the sense we understand, but they do solve problems critical to the organism. Now we have AI which may organically invent its own ways to solve problems, totally alien to mathematics, and beyond the scope of our reasoning.
Intelligence overloaded
Level Up ^.^
Is the "space" of all possible mathematic(es) infinite?
Dr. Wolfram is a real life genius.
Might want to read up more on him before coming to such a strong conclusion, just saying. His book has been ridiculed for taking others ideas and being mostly hot air.
It's an exploration of how the human mind functions. It's inward. That it happens to rationalize what's out there, well now we're cooking with gas.
It would be kind of absurd for the thing which raises us from the dust to imbue us with a language that holds no correlation to the larger structure. Actually, thats logically impossible. It would imply a fundamental disconnect.
And of course, im synthesizing language and perception in a basic sense
Math is invented to describe discoveries. Math describes more then we would ever be abe to discover, i call that its fantasy part making it ,to me, clear that it is a human thing.
Universally 1 plus equals 1... we discovered the math not created imo
Totally what he just said 😬
The history of mathematics is full of people questioning axioms and adding new axioms to see how it changes the system. That's one of the main ways math progresses other than finding new proofs. So I disagree that current math is a single historical artifact. Plenty of people have thrown out the standard axioms before, and inconsistent or trivial sets have been discarded, while interesting sets are investigated. Really useful ones become taught as standard math, to then be questioned and expanded...are they then to be disregarded as mere "historical artifact?"
Besides, Cantor and Gödel showed that no mathematical system (of sufficient power to be useful) can be both consistent and complete...regardless of axioms chosen. So the fact that many things are unsolvable in the standard math is no knock against it, the same would be true of any alien math they care to invent. Mathematicians are well aware some theorems will be unprovable in certain systems, but that's hardly a reason to throw out useful axioms entirely. It's also impossible to know which theorems are unreachable by which axioms, so might as well try.
Wolfram here gives very little credit to current and past mathematicians. His argument here would have been relevant before Gödel perhaps. Before the invention of imaginary numbers, topology, and hundreds of other innovations in maths that blew away previously hard held dogma.
I look at my computer screen and I see a rectangle. I look at the door to my office and I see a rectangle. I look at the rug on the floor and I see a rectangle. A lot of people will look at the same objects and see three different things. They will never see the abstraction that is the rectangle.
Now, I am not a religious person at all but if the rectangle could talk and claimed that "I am the computer screen, I am the door and I am the rug. All three, jointly and severally!" I will have no problem in believing it! Go figure. The church has been trying to explain the concept of Trinity for 2,000 years and have only managed to confuse the faithful. A mathematician could explain it in simplest terms such as the example of the rectangle above.
The church has of course dealt with such a simple trait-likeness as shape or anything remotely like that. Problem is more like the rug, the screen, and the door is essentially (whatever "essential" is) the same thing eternally and in perfect union, still apparently three different things. The church has pondered on this for hundreds of years with various proposals. A simple trait-likeness would never do as an explanation.
As Einstein did, I believe math, is empirical.
Very interesting that his answer to the question posed in the title is the opposite of the answer given by Roger Penrose to the same question.
"Is Mathematics Invented or Discovered?" I propose we're over thinking this. We simply need to recognize the bedrock of mathematics. There are distinctive objects in the world. There for one plus one equals two. Math is discovered, not invented.. We are confusing the methodological with the discovery.
+ThePetachu
Your problem begins with "one". What is "one"? You can't just say "I know it when I see one of something".
Math is invented. It's a big invention, so we're discovering parts of what we invented.
+Christopher Hall
So, how do I define the circle I'm drawing, without math?
You see your circular logic, I hope.
But you are correct that the speaker does need to demonstrate his claim. I suspect due to the brevity of the clip that he has done so elsewhere, so I'm off to find it.
Or…is it both simultaneously?
without examples of what he is talking about, I get convinced of the opposite
If Aliens come to earth (or communicate), that will be a world-shocking event. (Hopefully not dangerous.) So it will be interesting to see how their math differs from ours, if at all. And it may help prove "Ancient Astronauts" if we got math from those storied people.
Is the thoughts I am having now the only possible thoughts out there? Or are there other possible thoughts? Is the sandwich I ate today the only possible sandwich or is there a sandWitch out there?
This interviewer is like whatttt the fuck
Thank you sir, this neoplatonism in physics is turning naive approaches to math into a religion. Its really sad to see extremely qualified scientists fall into that circular logic which I cannot help to percieve as arrogant. Knowledge's metanarratives will always be fresh and practical. That things can be explained through concepts doesnt make our language inherent to the universe (in particular when out syntax is necessarily flawed).
So... invent a new mathematics.... or do we lack the intelligence and imagination to do that?
Probably we lack intelligence.
Haven't we done it many times already? Arithmetic, geometry, boolean logic, topology... any one of them could stand without others as mathematics in it's own rights.
Stephan Wolram is very right here.
We invent mathematical calculations and extend mathematics to include calculation and proof. We learn from evidence that a set of axioms can be deduced in a special way with the help of certain rules. However, this path of proof does not exist before our construction.
Ludwig Wittgenstein, first described in his Tractatus.
No, not at all. Hume wrote about the problem of induction in the 1700s. And it doesn't follow from this problem that there should (or would) be "other mathematicses", only that our methods of induction beg the question with regard to the justification of induction.
Plato vs Aristotle. Greek vs Babylonian. The world of math has been dominated by Plato/Greek style of thought, and math is worshipped as an ideal.
"Invented" is the wrong term! "Developed" step by step".
Both invention and discovery are incremental, so it's not a distinguishing factor.
It depends on how you define Mathematics. If we're talking about the language that describes laws, then it's invented. If we're talking about the laws themselves, then it's discovered.
I disagree. When you're talking about the "laws", you are most likely talking about the physical laws, not mathematics. In this case, you might have "discovered" the laws via observation, but the mathematics you apply to formally describe these laws is still an invention. If you are talking about the laws of mathematics, i.e. the axioms, then it is an invention, because there are many axiomatic systems that can be used to formalize mathematics. This is Wolfram's point.
05092020 Review
So he's basically saying math is discovered because of the "artifacts" .
So invented or discovered? *Wolfram:* yes.
No, in the sense of the invented vs discovered question, he is definitely team invented. That's what he means by saying mathematics is an artifact of our world. Aliens could have come up with a completely different system. So it's not like we the discovered the one fundamental system in the universe. Instead we have invented a system that happens to work very well in describing certain areas of the world. Did you even listen?
It's like saying that He is making no sense but perhaps in another universe, he is in fact a genius. Yep makes a lot of sense.
Everything we know is discovered.
Everything we know is invented.
@@UnworthyUnbeliever
We invent with the knowledge we discover.
@@pauljasmine353
Knowledge is not discovered, percieved external phenomena with seemingly repetitive and connected properties is.
All percieved-material forms of knowledge, like science of physics, are human inventions to make abstract representation of the so-called external material world.
@@UnworthyUnbeliever
It seems you are a new ager. Welcome to Alice in Wonder Land.
All that there is to know in our universe, all knowledge, already exists. In order to reveal this knowledge, science must use incrementally aquired knowledge to invent detection instruments like
Spectrophotometry, Carbon dating, X-Ray dating, Mass Spectrometry, Light Scattering Detectors, advanced satellite telescopes, etc...
It's all about being able to observe.
@@pauljasmine353
way to project your baseless assumption unto others. if there was a factor for all the kabbalah-flavored new age beliefs out there, shared concept of "there is a G*d made of laws of mathematics and Logic, determining human endeavor to unearth the truth of it all and bring about a new age." would make a core definition.
belief that "mathematics is language of the universe" is the most new age postulation one can make, alongside determinism and 'unity of being' or, whatever.
and no. math is still a human construct. so is all so-called natural sciences. if in doubt, ask yourself: can you formally prove why we chose the axioms we did from the set of infinite possible axioms that we could have chosen? can you?
The host looks like a dear in the headlights
I want to be able to think like he can haha
Baloney. Math is discovered, not invented. The Platonists are right.
Anyone who does Euclidean geometry will discover that C = 2 x PI x R, whether it's the Greeks 2,000 years ago, us today, or Alpha Centaurians a billion years hence.
Mathematics is discovered and ALLAH PAAK is the Creator of all the universes
Bollocks.
This is refuted, by the simple observation that Mr Wolfram, when he encounters one other of his "mathematics'", will be forced to name it. And he will call it mathematics. He fails to understand the meaning of the word.
The meaning of "mathematics" that he's using here is simply "an axiomatic system of some kind". It's a dumbed down approach for the general public, but the unfortunate consequence is that commenters here think that Wolfram is a philosophical genius. :) And, of course, the title of the video has very little to do with the actual subject matter.
what nonsens
Stephen Wolfram has either not heard of, or doesn't actually understand, Kurt Gödel's incompleteness theorems. Because if he did, he simply wouldn't describe it in this way.
These guys don't know our Creator or how He used His program called the Beast to get the characters in His simulation to build things with their human hands. This means our Creator taught man everything including mathematics which was necessary to build objects with their hands. Anything He taught us to build has to have a plan with measurements.
The language of mathematics was also used to discover quantum mechanics which helped man to understand that visible objects are only illusions within the simulation program we're involved in.
Shut up you idiot.
carl green
You're the one who doesn't understand how we're created so it's very unwise to be calling me the idiot.
+Brad Holkesvig Brad. No offense but i think you need to leave this channel alone and let us normal people exchange reasonable ideas.
DP ie
I'm the only one on this planet who knows how our Creator created everything and how and why He taught His characters the language of mathematics.
This channel is open to those who understand all the questions scientists are confused with.
Brad. Can you please troll somewhere else. You are clearly delusional.
Intelligence overloaded
Intelligence overloaded