Max Tegmark - Is Mathematics Invented or Discovered?
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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? Or is mathematics discovered-always out there, somewhere, like mysterious islands waiting to be found? Whatever mathematics is will help define reality itself.
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I love that at the very end the interviewer has the giant cheesy smile saying “we are now far more confused than when we started” as I now have a lot more questions and wonder if the original question even makes total sense
The richest answers to a question ARE those which leave us with more questions than when we started.
An "open and shut" slam-dunk of an answer, leaving no ambiguity, might feel more satisfying at first, but ultimately the most dramatic effect it has had is to take the wonder you felt peering through an open door -- and slammed it shut.
How much more wondrous it would be to lead you to that door, and in the space beyond, find five MORE doors beckoning with mystery!
Indeed. A very Socratic statement, the more I know, the less I know. :)
Read the wikipedia page on 'Philosophy of mathematics'
@@ASLUHLUHCE Thank you for that. I got mind fucked when the article kept using the phrase "mathematical entity".
I very much enjoy reading books by Tegmark as a physicist. This question, however, in its deepest sense, is best answered by not talking so much about physics. As someone else here commented, Tegmark answers the question in the best way in his first sentences where he distinguishes between our mathematical language (which we invent) and mathematics as such (which we discover). After that, I think the interview loses track in the hunt for the sought answer and deeper understanding of it.
I concur
I 100% agree with this. The realization that physics was not "equations that nature gives us" but rather mathematical models that model what we discover in nature really shook me and led me to this video. It's exactly what science is mostly: just humans trying to model our information about the world in an organized way that works for us.
"For every single physical entity, that we can think of something we can touch or measure with a detector, there is corresponding mathematical entity there in the mathematical structure."
Tegmark essentially answered the entire question in the first sentence. Mathematical language is an invention, a way of describing mathematical principles, confined to our known universe, which are "self-consistent" as Tegmark notes.
There are no mathematical principles confined to our universe. The physical world is not inherently mathematical even though we use mathematics to describe that world.
Jamaal Richardson Michael Beer Interesting and thx for the reply.. I find it rather amazing that our self proclaimed “intelligence” has literally been halted at a fork in the road.. A point wherein we are left with one unanswerable question... ..was mathematics discovered or invented? Surely in the quantum realm mathematics, physics, standard model and all theories dissolve.. Equations that are flawless in predicting the macro are rendered completely useless in the micro! Ironically, within such micro it seems scientific explanation is just as ridiculous as all religious explanations!
What do you mean "confined to our known universe"? Tegmark believes in a sort of multiverse, whereby all structures that exist mathematically also exist physically.
Yes, but later they both seemed to say that math was platonic "Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. Just as electrons and planets exist independently of us, so do numbers and sets" To me this implies that math is discovered and not invented since it was there all the time waiting to be discovered. I don't know though, this question seems like a very complicated one and I'm not making a strong stand either way.
@@BladeRunner-td8be You are right that's his pedantic point. He's the new Plato, the enlightened mathematician that will unveil the truth for us.
Numbers have emerged in our history as a simple tool for counting units around us. We could never imagine how many things these numbers would still show us.
The problem with insanely intelligent people is their ability to put their thoughts into a clear statement. The easiest answer to this question isn't explaining how math is real because of some multiverse theory but rather by giving an easily understood analogy. Say you have 1 marble in your left hand and 1 marble in your right hand. Now when you put both of those marbles in your right hand, you now have 2 marbles in your right hand. This proves math to be real. The word "two" is used to describe how many marbles (or whatever you're counting) you have. One + One = Two. So let's just say instead of the word "two" mathematicians decided the word should be "owt." So in this case One + One = Owt. So you can see that no matter how you describe how many marbles you have. You will always have a set answer that is manufactured by a predetermined set of laws in our universe. The language of math is invented. The laws of math are universal and discovered.
Arithmetic and mathematics are not the same thing!
@@usejasiri , mathematics is the cognition, arithmetic is the calculation.
@@thomashooper9148dude... that is literally what we are saying... this is what I'm trying to explain. People are too focused on the vocabulary of a subject when in reality it can be explained much simpler without all the added nonsense. Arithmetic is like getting from point A to point B and math is a like car. Getting from point A to point B is the same distance no matter how you decide to get there but we invented a system to get us there faster and easier even though the distance stays the same. So we can keep inventing new ways to get to point B faster but the distance is always the same. THE DISTANCE IS UNIVERSAL, THE CAR IS HUMAN MADE.
GreenEarthers Business , then we do disagree. Take infinity as an example, this is clearly a man made concept. One that has no real bearing on reality. Yet a mathematical reality!
@@thomashooper9148 space is infinite. words invented to describe things that people see or measure doesn't discredit the concept? If I see a blue flower it doesn't matter if I call it indigo, sky blue, baby blue. The flower is blue. We invented the word blue to describe the color we see, just as we invented addition to describe how many things there are when you put things together. This + that = those. It's been this way since the dawn of time and it will continue to be this way until the end. Math is all around us but how we choose to perceive it and measure it is up to you
I've always enjoyed when Max says: "We physicists..." he loves saying that in all his interviews haha
More than Michio Kaku?
@@RunnerBrain lol
Incredibly Interesting. Thank you.
'Nature prefers simplicity....this is a deep mystery' My favourites are E=mC^2, Euler's Identity ad the beauty and complexity of a simple fractal.
Thank you prof Max. Mathematics us not a single thing to decide whether it is invented or discovered. Prof's 8 mt lecture clarifies how to go about finding out what maths is already in nature and what we have invented.
Max is a hero. Love his attitude and intelligence.
My fav archdolt😊
@@ligayabarlow5077 🤓
the hosts confused smile and nod at the end says it all
Are you back to making new interview videos or is this a repeat of an older post?
To me, mathematics has always been a language used to describe the physical world and the intrinsic laws of nature. But all I have is first year college math to back that assertion.
If a mathematician says: "The symmetry of mathematics is beautiful", I immediately have to think of observers bias - Still, I agree :-)
When he said they discovered the five shapes but cannot discover NOR INVENT the sixth... that was compelling. Sounds so simple, even basic, in retrospect - but Ive listened to a lot of these without hearing such clarity. From there his identification of what exists vs doesnt exist platonically (spaces but not each number etc) needs a lot of working out.
Thing about math being discovered is: we would first have to agree on an ontology of math to meaningfully discuss this question. Otherwise we can’t tell if math pre-exists or is created by us, because we didn’t even specify what existence means for math. The ontology of math has been highly controversial for centuries though.
Fascinating
Thank you Spot on, Doug:).
I really like the attitude and passion Max has in his interviews as much as I enjoyed his first book. Nonetheless I have to say that in this interview he really dodged the question several times, whereby the interviewer was clearly concerned about the cardinality of the platonic realm (or level 4 multiverse in Max's terminology).
Why do you say he dodged the question?
The mathematical multiverse contains only Gödel-complete structures, in fact all computable functions, and we know that there is a relation between complexity and decidability, all mathematical formulas more complex than d(E)=K(E)-lenght(E) are indecidable, where K stands for Kolmogorov complexity and E is a mathematical statement.
There is a deep relation between formal systems and computability.
His hypothesis makes sense and is probably true, reality has to be necessary in order to exist, math is the only answer I can imagine.
@@omega82718 It could be the world is indeed a multiverse where everything than can exist, exists. But that doesn't mean that the world is necessarily such a multiverse.
@@cube2fox Unless mathematics is a metaphysically necessary being. I don't see how a tautology could fails to exist, and math is just a bunch of tautologies.
@@omega82718 It's quite a jump from "all mathematical statements are necessarily true" to "all non-contradictory statements are necessarily true (in some sub-universe)".
We discover mathematical truths and invent our ways of understanding them.
I prefer the question: are the truth-conditions for mathematical statements dependant on our naming them as such?
That's a pretty interesting question
In some cases yes, for example there are certain mathematical statements that are only true for a base 10 number system. Or structures that depend on a particular orthonormal base we choose in a vector space. Those statements that are true regardless of these arbitrary choices are truly fundamental.
The way I explained it to the pupils I tutored was that relationships between parts of material reality pre exist mathematical expression . There is an internal logic to material reality and maths also has internal logic which is potentially capable of expressing possibly all material reality in mathematical form. Platonic Idealists would say the opposite ! As a keen follower of science and philosophy I know that at the extreme opposite ends of reality the subatomic and the infinity of the whole universe uncertainty prevails ! That is challenging for those that expect universal certainty and consistency ! But not for a dialectical materialist for whom all reality is in dynamic and contradictory movement ! But only very curious and in-dependant minded pupils think and ask questions that cross discipline boundaries ! Which I encourage them in, mostly by defying their expectation that I give them solutions rather then posing puzzles ! In school they get trained to mechanically perform to produce ‘correct’ answers ! That kills curiosity and limits meaningful thought. Computers calculate, minds should do far more !
Mathematics at it's core has nothing to do with our universe.
To quote Roger Penrose loosely, the mathematical fact that there are infinitely many prime numbers holds forever and everywhere, even if there was only nowhere.
It is entirely inadequate to refer to physics to give an answer to the question if mathematics is discovered or invented.
The fact thatour universe is that seamlessly described by Mathematics - from the mathematical perspective - is a mere coincidence, and honestly has no meaning for Mathematics itself.
Thus the fact that our uinverse contains uncertainties and flaws does not allow for any deductions about mathematics.
The existence of the mathematical entities is absolute and not dependent of the existence of a universe or even a mind to think about.
As a platonic idealist I would think that mathematics help us to uncover a reality which is ultimately identical to the physical reality. Mathematics are a window into the fundamental structures of the universe. But we, as limited and sentient beings we will never be able to cross the divide between intellectual forms and matter.
So this goes further than Platonism in asserting that not only do all mathematical objects exist, but nothing else does. All structures that exist mathematically also exist physically.
The physical is identical to the mathematical structure.
@@BugRib Personally, I wouldn't say that the universe is literally maths, but just that mathematics describes aspects of the universe
Anonymous - Lately, I’m kind of leaning towards mathematical structures being literally identical to physical structures-but maybe these structures are only “actualized” when they produce conscious observers (whatever “conscious observers” even are).
Sounds kind of “woo”, but it feels like a reasonable possibility to me.
The thing is, mathematical notation may be a human invention, but what is it actually describing? There really seems to be a deep significance to math. I think mathematical truth _is_ base reality.
No. I don’t have a shred of empirical evidence to back this conjecture up... 🤷🏻♂️
@@BugRib Hi again. So after more thought, I think that perhaps the "unreasonable effectiveness" of mathematics is explained by a relation between the nature of consciousness (and thus its mathematical intuition) and the nature of physical reality (from which the conscious mind comes from).
@@BugRib Hi again. So now a year later, I completely disagree with what I wrote above
What's the most important thing to understand is that there are 2 things we call mathematics. There is the mathematics with small m, our human invention which some of us hate for understandable reasons. The other kind of math, is Mathematics with capital M. Our human made mathematics was invented to help us understand the order and properties of Mathematics. Mathematics is just pure logical relationships which have to exist between certain matehamtical structures. They just exist. They're timeless. They have to, otherwise reality would contradict itself. Once you really understand "Mathematics", everything else will start making sense.
Great analogies.
To help the interviewer see why "5" is not a mathematical structure but "the integers" is, the words words "set", "operations" and "axioms" would have been far more useful than the words "Pseudo-Riemannian manifold"...
I was expecting some further explanation when it abruptly ended. I wasn’t watching it so I was surprised. I wasn’t satisfied at that point.
Mel Gross maybe set your expectations lower next time. Problem solved.
Mel Gross he could just say what he said from 1:50 to 2:05 in so many words, the things we discovery about the physical world are same as we discover in mathematics. Which came first chicken or egg. Obviously the egg.
There seemingly are strictures put on reality, which enable stuff to exist and we discover their limits.
Amazing..new sub👌
A=A is an axiom, and mathematics is a system of restating that in more complex ways in order to describe and leverage what we observe with our senses.
Action = Equal and opposite reaction.
That's precisely the problem. All axioms are subject to choice. A different axiom could've been chosen. For example A != A.
Formally and computationally speaking, such a Mathematical universe can exist. Like so: repl.it/repls/EnchantingMindlessSource
It is but a definition, not even an axiom.
Lol no it fucking isn’t. Please tell me where you’ve observed 21 dimensional shapes? What senses did you use when you ‘observed’ the power set of the set of real numbers? I’m sticking to very basic mathematical objects here.
@@massecl A definition can also be axiomatic. Can you attempt to disprove A=A without relying on that in your argument? That's what makes it axiomatic.
Didn't understand much of it. Just restarting myself in this field after shitty school education. Hope to change some perspectives and try and learn something
I love Max...but everytime I see him, I hear in my head, "Say hey-low tu mah lit-tel freh!"
Amazing
When this aired?
Max always looks intoxicated to me.
🤣🤣
I find Max Tegmark the most fascinating mathematician physicist in cosmology. He really is one of a kind. I admire him but also regard him as 'too much'. Not even I, a Mathematical Realist (commonly known as a Platonist) can call myself a super-Platonist like Max Tegmark. For him, every modal potentiality is actual, e.g. every possible (i.e. self-consistent) universe exists physically. He literally fuses mathematical reality with physical reality. All possibilities, in this precise sense of the word, are real, and actual. Real means that it’s possible to find a physical structure that matches it, and actual means that it’s not only possible but already existing. Thus, his multiverse is the set of all possible mathematical and physical realities. Not surprisingly, this is a tautological statement in itself, since, for Tegmark, possible = actual = real. Or, in his own words: “all structures that exist mathematically exist also physically". In summary, Max Tegmark is a genius, perhaps too intelligent to be restrained by mere common sense.
I think he's saying the opposite. That for every physical phenomenon there is a corresponding mathematical object. Which is a pretty much just that the world is intelligible, not super platonism.
@@blakemcalevey-scurr1454 I read the Mathematical Universe several times, and I think @jit1881 understands it correctly, that is why this is such a rare idea. The level 4 multiverse is actually the proposal that the existence of a mathematical structure that is complex enough to describe a dynamical system (not sure, which universe corresponds to just the integers with addition) is equivalent with the existence of that universe. For our universe this of course needs to be a very complex structure, for example a pseudo-Riemannian manifold Tegmark talks about in the video describes a curved spacetime, but with nothing in it, so definitely not our universe.
To me the answer to the question is tritely obvious: mathematics is a reality awaiting discovery. What interests me more, though, is how it comes to pass that there is often more than one way to describe what is essentially the same mathematical reality. The most obvious example that springs to my mind is Newton's fluxions. Those fluxions, which provide a way of expressing the same notion as calculus, were cumbersome to work with. Coming from a different angle, though, Leibniz conceived of a language of mathematics that looks much more like modern calculus. Both geniuses saw the same essential problem (how to address change over time), but from two distinctly separate viewpoints. Reality, then, can be conceived in different, though yet compatible ways. Is this distinction merely semantic - a matter of nomenclature - or does it of itself reveal some further mystery about the nature of mathematics? That is to say, do mathematic truths cast out from them the shadow of perspectives of meaning, expressed in different, though arguably identical, ways? And are those shadows of meaning mathematical in nature? If not, what are they?
I think the way we try to understand the things we use to make sense of things is created. But, we don’t necessarily create how we think. We just do. And math is the embodiment of thinking. It’s like looking in the mirror asking why your reflection shows every time. They do because you know it makes sense that they do. So I guess it’s more of the language of abstraction that you discover as universal since all people think.
How about you wait to hear the full answer to your last question before you ask the next?
The fact that the 20 or so so called universal“constants” are not in fact constant, the best our mathematics is is an approximation.
i think mathematics is the language in which the physical approximations are described in, but mathematics itself is not an approximation as it can exists independently. Even if that means that it would be pointless without a physical world. But i guess everything would be pointless then.
That was not the question.
The question was: Is Mathematics invented or discovered?
How does your statement relate to this question in *any* way?
Mathematics does not "depend" on physics.Mathematics is a world perfect and eternal.
If you want to express a relationship between Mathematics and Physics, then it is our physical world that is a crude approximation to the perfection of the mathematical one.
Did he manage to answer the question on whether every individual integer exists in the mathematical world or the idea of integers exists?
As I understand him: In the mathemathical world the system of integers exists as system without further details, and when we understand that system we can, if we want or need for some reason, calculate/infer individual integers that are like signs/representations FOR US of a virtually/potentially implied relation within that system. In the physical world (or in any specific one of the multiple physical worlds) not all but some of these relations are/have become "real" as a kind of physical double of the counterrelation in the mathematical world. So in the physical world these specific realized integers have come to exist. - My question here is: What then is the difference between the mathematical and the physical world, what is the specific physical or actual about it? - In one interview Texmark seems to say, it is the being perceived or at least has some connection with its being perceived by a perceiver (that, as he sais there, could be an animal as well). I guess we won't get around reading his book.
No, I don't think he did. He sort of referred to the concept that the integers exist. I would describe it by saying "Oh, look at that!" That eureka moment means you've found a _thing_ . Then, five minutes later, you surprise everyone by saying, "Look! There's another one!" which means you've invented the concept of "two". That kind of implies that the concept of "the same as" is required to go from "one" to "two", and subsequently from "two" to "three". This is verging dangerously close to the idea of "the set of things which look like that." You can't count things unless you've chosen _which_ things you want to count.
He evaded the question
Could have just admitted that he hadn't thought about it
@@rjd53
This explanation is not self consistent
How much detail already exists
and how much is "created" when for eg humans want to calculate
Who decides and what is the level of detail ?
Do real numbers exist already ?
Rationals ? Irrationals ? Complex numbers ? Does zero exist?
@@abhir7823 In the meantime I have read his book "The Mathematical Universe". Now I know that what I've written in the comment above is wrong. What he means is: reality is not described by, reality IS the various mathematical structures. Different kinds of such structures exist, that is one of the reasons why different kinds of universes exist. Math'l structures do not consist of numbers at all, they consist of pure relations. Numbers do not exist. Numbers are just the way WE represent these relations for us. So, the relation pi exists, but not the number we come up with, when we calculate the relation as division. That the number cannot be calculated, even by best computers, is OUR problem, but not a real aspect of the universe itself. - By the way Stephen Wolfram would not regard this as just our problem, but he also thinks, that numbers are not necessary to do math.
What a perfect idea, Jeff Goldblumb cosplaying Steve Jobs just WORKS FOR ME
We invent our myths and maths, and sometimes some of them, but by any means never all, find partitions of correspondence in reality. It's not a one-way road, it's a reciprocal circuit of creation/discovery moved by what is. Often, we do take our mental creations for the reality, as in those many forms of idealisms the history of philosophy is chock-full.
Don't over complicate mathematics> It is a system for human beings that helps them under their existence. It is nothing more.
2:24 what’s with the Beavis and Butthead impersonation?
Dark Matter i knew i heard that somewhere before lol
This is what your brain comes up with?
@@chrisw7347 Haha why not? Mike Judge is an engineer.
@@ungoyone Your answer being complete nonsense tells me that you're an AI
@@chrisw7347 Ha! Mike Judge is the creator and does voices for both B&BH. So AI that!
I get it. The author of physics and math is the same dude and the author of the bangs! Take out the author and it becomes a really interesting conversation, for me at least. Forgive me. I'm a literature major. But I'm often compelled to investigate all things related to the origin of mathematics. I simply don't have a vocation for it, so these videos help immensely. Y'all's comments are helpful as well.
10:20 My thoughts exactly! 🤣
Our "scale" of reality is dependent on certain mathematical Newtonian rules, therefore it was here before we discovered it. I think a good example is the Monty Hall problem. Intuitively makes no sense yet it works 100% of times.
The mystery is whenever we look for the truth, we end up in mathematics. Reminds me of "word problems" in grade school -- the goal is always to reduce it to a mathematical statement, remove the obfuscatioins and get to the meat of the issue, and then solve. Now, that makes me think of the opposite -- suppose you took a word problem, and then added to the story, turning a two sentence word problem into a three paragraph story, adding all kinds of new, possibly irrelevant information -- wait -- that's policitics.
Kind of fun when he talks on a metalevel that the other guy cant grasp.
Because the argument aginst it would be like:
- we set the rules and without them there would be nothing to discover.
Tegmark is moore like:
- there are infinit rule sets that could give you some kind of math, and sometimes we discover one simple form that we like.
So what if it is the mind which is Mathematical (or logical) in nature, and it is these available concepts and limitations which leads us to these mathematical conclusions?
For neoplatonists, there is our own mind and a more "universal" mind, the "intellect". The intellect is related to our mind and when we explore reality we discover both physics and the structures of the intellect as two ways reality appears to us but are ultimately the expression of a unified reality.
Amounts & shapes are discovered. Math is a language used to approximate perceived patterns.
amounts are values. values are a product of valuation, which is a reasoning process. patterns describe sets, and sets and their descriptions are products of the mind.
KEvron
Δωδεκα (dodeca) means twelve in Greek and Εικοσι (icosi) means twenty. Εδρα (hedra) means side (sort off). So the dodecahedron and icosaedron literally means the shape with 12 and 20 sides respectively. The name given by the Greeks was anything but random!
It’s not about the name being random, the point is that the name is arbitrary. The form exists irrespective of the name.
@@JD-cf4or yeah, you are right. But i wanted to clarify that and share the info...
why does dodeca mean 12 in greek?
you dont have to answer (because you can't) In fact the word is arbitrary
I like the dodecahedron. It’s fun to say too. I ordered a Rubik’s cube dodecahedron but it hasn’t arrived yet. I better check on it. Thanks 🙏🏻
Language but not the structure? I don't know I think the jury still out but I like this guy
I think the is finite and isnt infinity...i thin all the math structure is what is called the string landscape ..its 1 and 500 zers next to it...its huge but finite....
the math discovered by other aliens could start with postulates different from ours and so the definition of what is considered a simple concept such as integers maybe something that is only apparent after complex derivations in their system. but the overall idea that they would be just exploring different parts of the same larger mathematical universe is completely valid and I agree with him
How do we really know that that overall idea is 'valid'? Isn't it more plausible that it has to do with explanatory power (cf. S. Weinberg)? So we do discover things, but we can so far only say that we are discovering things about our own way of describing things. That is to say, we are discovering "ourselves". I am not sure, but it seems to me to go a step too far to say that mathematics is the only 'valid' way of describing the world. Seems that we simply don't really know.
First The symbol is discovered, converted into mathematics and explained by the words chosen by the interpreter ?
When discussing mathematics, and this specific question, I think what troubles people is the idea that anything could be eternal. First, it's a weird concept for finite organisms to get comfortable with - there's something that is always there but it's not us. (Although whether "life" is eternal is a question to consider.) Second, I think people fear that a discussion about any eternal aspects of the universe ultimately will point to some concept of god. And it's likely not "god" that freaks people out, it's the interpretations of god that come from the monotheistic religions - those are not especially inviting descriptions of god. Once you understand that there are really solid, rational and non-moralistic ways to discuss god, like found in Plato, Spinoza and Advaita Vedanta, the eternality of math becomes a lot less frightening.
Well the problem is once you settle the reality of God, alot of people will realize how much could be on the line in terms of morality. And people are selfish, so they don't want that sort of threat.
I can see and sense Robert’s frustration with Max’s responses..Max explains what he’s been taught and has learned but has great difficulty explaining exactly why something is..Reminds me of the kid that knows math well and you ask him/her how they arrived to the right answer and they say “hell, I don’t know, I just know that when you apply such and such it just works out”...lol lol lol
Mathematical universe is such an underrated idea and hypothesis .so much umdeserved criticism.
No one can disprove it and no one can prove its existence either.
No one can prove the existence of your brain, either. Where do you get this shit?
It doesn't say anywhere close to everything about the world. We are discovering parts of our own mind. That's most definitely negligible in the grand scheme of things.
To answer this question, I think it helps to think of chess. Chess is a game that was invented by humans. Everything from there is discovered: all possible games, strategies, etc.
Well, like chess, the foundations of mathematics were invented. We set the foundations, but all the theorems, proofs, etc were discovered thereafter.
Then how do you go about explaining the fact that the entirety of the universe is bound to the mathematical concepts invented by the human mind? That's what made the observation of the higgs boson particle so spectacular; it was proven mathematically to be in existence long before it was actually discovered. You would not be able to use math to predict with incredible accuracy such complex things about the universe if it was merely invented by the human mind.
jdm11060 mathematic models could very well just be good approximations for the physical world as opposed to the definitions of it. In that case they are just tools we have developed, and as the op pointed out we often start with a framework and then discover different manipulations within it
02:24 to 02:30 : My reaction while listening to this with no clue what Tegmark is talking about
Go back to sleep
“As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” - Albert Einstein
I didn't know Bruce Dickenson knew that much about mathematics.
The word "discovered" implies "human" (the discoverer) The word "invented" implies "human" (the inventor). The human mind evolved within nature, and cannot be separated from it. If a system such as the brain evolved to accurately represent reality utilizing a specific type of language (math) then it is reasonable to state that to some degree mathematical language is natural.
When they say "Discovered" in this context, they mean (or should mean) that the mathematics already exists in the nature, we just identified it. So, yes, both verbs imply human activity, but that is not the issue. I would say that if nature behaves as if math is accurate, even when nobody is observing, then we "discovered" the math: we found something that existed. That seems to obviously be the case. The precise symbols, etc, of course are human inventions. But those symbols represent an underlying reality.
(Edit: I just re-read the original post by Michael Cameron, and I agree with it more than I thought the first time. I think I was reading something into it that wasn’t there.)
Math happens in nature regardless of humans.
because the physics were invented through evolution. so like finding a new planet it's a discovery
discovery produces new knowledge, thus it's inductive. invention draws on existing knowledge, thus it's deductive.
KEvron
Thank you for this. Maybe the best comment
Mathematics is the language by which we comprehend our reality. Or, by which the nature of our reality can be revealed. Sometimes we conceptualize the math first , and say it must be reality. sometimes we conceptualize the reality first, like Einstein’s thought experiments. Then That reality concept can be explored, maybe proven through mathematics. Math is the language of our universe. Good topic.
👌👌👌👌👌
Since we are living inside the mathematical structure, all we can do is describe what it is, to really know what it is, we would have to be the programmer.
You are...you just forgot 😉
Oh no. You've employed that moving cameraman again. I'm dizzy.
Pussy
It is a little distracting and unnecessary
Lol
numbers are a tool for mapping an underlying mathematical structure. the structure has some characteristic symmetries. the structures are discovered. the tools are a mental construct.
I think it’s a bit like Chess. The game itself is invented, but once you’re playing it every possible game that can be played is discovered.
Amazing!
Yes.
This is the classic error of redefining something and thinking you've discovered something new. In this case they have refined what it is to exist. This definition is different from the way the word is used in everyday speech. And it leads to a contradiction. We usually say that given a thing that exists, we can think of it not existing. For example, we can think of an apple existing on a table and we can think of an apple not existing on a table. But can we think of a "two" (the abstract object in the Platonic realm) NOT existing? If a "two that does not exist" is meaningless, then this "two" is a tautology. It is just a concept. You can, of course, expand the idea of "existence" to say that concepts exist, but by doing so you are changing the meaning of the word. This will only cause arguments where none need to be when you start talking to people who use the usual meaning of the word.
The question is what is the utility of this new definition? What does it help you explain that the usual definition does not. In this case, as in many cases, this only thing this redefinition does is blow your mind.
But it is meaningful to talk about mathematics being discovered...in the sense that new chess strategies can be discovered. Does that mean we need to consider that chess moves "exist" in a sort of game realm?
Whoa. I think I just blew my own mind.
Jimmye Winburn Yes ,they exist and wait to be discovered.
I should very much like to hear your evidence for the "existence" of math instead of a simple refutation of my argument. :)
People often discuss about existence of different things, but they rarely think about what existence is. If they look close enough, they will notice that existence is a construct of consciousness, which enabled grasping phenomena. Existence is not an inherent nature of things, it's a mode of perception.
@@NeoShaman But what is a perception. We perceive things in dreams and do not believe they exist - for logical reasons that determine what counts as existing. But does that logic "exist"?
Texmark sais that we discover the SYSTEMS. The details we infer from it might be just our conceptions, ways WE conceive of the systems. So, in analogy, what exists are the rules of the game. That makes sense, because they do not depend on you, you did not make them up, you have to understand and obey them, or you don't play the game. The concrete moves in a specific game exist as well, but in a different way, in another realm of existence. They do depend on you, your decicions, your action.
Not only he's smart
But he communicates his smartness thro a second language just like that
He didn't answer the question asked twicely: is every number in the platonic space or "just algortithm".
This guy is my idol
You must have a very good dad, Max.
Language vs structure. Answered the question 🙋
Math consists of Axiomatic Systems which have Undefined Terms and Axioms and meaningful statements some of which become Theorems when we can prove them. We invent the Systems and then discover some of the meaningful statements we can prove as Theorems. Then sometimes we can use an Axiomatic System to create a Math Model of some physical system. In modern times we have created Axiomatic Systems just for their beauty and intrinsic interest. Indeed, many mathematicians don't really care about any physical system. The E8 exceptional Lie Group he mentions is a good example. I'm pretty sure the inventor wasn't thinking of some physical system. E8 is very abstract and we still don't understand it very well. But, some physicists are using E8 to create a model that extends the current standard model. Lisi for example. My guess is that there are some Axiomatic Systems we haven't invented yet that will create much better models. My guess is that there are pretty simple processes that create very complex processes we have no good Math Models for. Indeed, if we do live in a discrete world where the Planck length is true then we may need a completely different number system we haven't invented yet to create a good Math Model for physics. Who knows? I'm just trying to teach some high school students some math that will help them and there are a few videos on these topics on my personal website: craighane.com
Its natural people to gets their thoughts fragmented while doing a philosophical discourse about already complex and abstract topic like how mathematics relates to reality.
Thats what it is, Max said that though mathematical structures are complex, and there are many of them, it goes to infinite, but our reality, he said the nature is very simple.
I think that it is because nature wants us to see a simple reality so that we can interact perfectly and pass our genes for evolution. Its natural selection. Nature has put some aspects in our sight and deleted the others not needed. And we only see and experience that much reality which is needed for our survival in turn hiding the complexity of reality. Now as intelligent species if we wanna know everything, we have to go beyond our perception. Now the only way is that we have to study what we experience, and the things we dont experience or cant imagine, we have to study those with the help of maths. Thats it. Maths is fundamental. But its sometimes illogical..........😩
Now THIS is the kinda shit I'm interested in
That which is, exists, whether we know of it and can describe it or not. Since Pythagoras, we have developing our knowledge of what is, and the language that we use to describe it is mathematics. Today we are a point whereby we discover things by observing the physical reality, but will not accept as reality unless we can describe it mathematically. It begs the question, which came first the chicken or the egg; the math or the universe?
Nature prefers simplicity like a builder prefers simplicity. These things build themselves first and in an infinite universe are going to be most common.
But why does the physical world behave in a way such that it remains in lock step with our creation of symbols and expressions of quantity? What dictates this obvious symmetry that allows us the use of mathematics?
I don't have an answer for why or how it is, but reality seems very fractal to me. And so a description of something on one scale tends to have an analogy on another.
Because our universe appears to obey laws (recurring behavior). It would be hard to imagine a physical reality which contains stuff but has no repeated behavior, and even more difficult to imagine intelligence developing there.
jakejakeboom yes and thus this “recurring behavior” can be thought of as a fundamental fingerprint of the universe which can be mathematically quantified. But the question remains if there exists a substrate for what we may regard as fundamental or could our consciousness perception of the symmetry of the universe be a mirage? Where we have placed order and meaning onto that which is inherently devoid of such a truth.
We don't know how the physical world behaves. We only extract information that is pertinent to us, by projecting mathematical structures upon it. We have no mathematical formalism that describe the world in full details and in a consistent way. We must use patches that work for particular domains, according to the type of information we are interested in. The set of all these patches is not the description of a behaviour.
Maths is form without content: all that matters is the relationship between its elements, not the actual elements themselves. Physics is form with content: not only do the elements of the model satisfy certain relations, but they also matter in their own right, and furthermore, they correspond by some consistent dictionary to elements of reality. The word that professor Tegmark is searching for is “model” : physicists use mathematics to model the physical world, and the success of the model depends on how closely its properties imitate the properties of the real world.
Mathematics is necessary to be able to do physics at all, how will you do physics if you don't first assume the existence of numbers (quantity and difference), mathematical truths such as 2+2=4 and the like, so it can't just be things that they do not exist; Tegmark is very smart and realizes how ridiculous the materialist paradigm is that mathematical entities are just a fiction even though I may not agree that all reality is just mathematical.
Many say mathematics is an infinity unto itself. And we are great for touching it. Many say it's a description of infinity. And we are not great. And the war goes on.
The structures are limited by the axioms. All statements logically consistent with the axioms is the totality of what he is describing. They may be infinite, but limited in the richness of their structure. Logical structures which are consistent within themselves but not with the axioms are not inside this set.
Read about structuralism in the analytic philosophy of mathematics if you wish to know more about Tegmark's position.
Is a song an artistic creation or a discovery. The possibility of the melody has always existed.
Tegmark seemed to evade the last question...
Whether each mathematical element such as each integer exists or is it just the general idea or algorithm of producing integers
He could have just said actually I haven't thought about it
Maybe in higher dimention we could invent it, but best answer to this question ,
The foundation of any branch of mathematics consists of abstract entities and relations between them, called “axioms” or “propositions”. That foundation is *invented.* So long as we have reason to believe that it has a _logically self-consistent_ structure we can then go on to deduce from it, by logical inference, certain statements called “theorems”. Theorems are not “invented”, they are *discovered.*
Though the foundations of a mathematical discipline are invented they are not arbitrary. They are *chosen* by mathematicians as worthy of study because the relations have some correspondence with experience of the "real" world. Or sometimes simply because they "seem interesting" to those who enjoy mathematical thinking!
The rules of math are invented. The ways the rules play out are discovered. It's the same with chess. The rules of chess were invented, but the strategies for winning were discovered.
@Adam Southworth I could say the rules of chess existed in some other realm before we invented them, but that doesn't change the fact that we invented them, unless these men are suggesting that we gained our rules of math from this other realm and didn't invent them?
@@momentary_ that's exactly the point. Is math a construct of humans or a part of nature itself?
@@everything777 We may have gotten the inspiration for mathematics from nature, but the idea of mathematics is a human invention. Counting, arithmetic, geometry and so forth may have started out as ways to represent nature, but much of the mathematics derived from those starting points have no representation in nature. They exist only hypothetically. It's safe to say that mathematics extends far past only nature and as far as we know, only a subset of nature follows mathematical law. There is no proof that all of nature follows mathematical law. We press on with math not because it will work, but because it has worked. It's the best we got right now.
@@momentary_ I don't think you've understood his answer very well or at least your answer differs from his. Also your original point is somewhat nonsensical. You say the rules are invented but how they play out is discovered, this seems incompatible. How the rules play out ARE the rules. What name you give them is irrelevant.
@@momentary_ That's a poor analogy that doesn't work well with mathematics. You've chosen an artificial human law which can be broken at any time as opposed to a law or rule in the mathematical sense which describes reality and cannot be broken.
What he is saying is that in a certain mathematical structure for example vectors, addition is commutative. The property of being commutative is the rule, we could have called it any name we wanted but that property exists for that structure. Physical phenomena that are described by vectors follow that rule whether we know about it or not or what name we have for it.
To go back to your analogy it would be like going to strange country where nobody drinks and discovering that prohibition was in effect there but people didn't even have the concept of breaking it.
SQUIRRELS discovered that they can jump from post to post using observed distance quantities using brain computation. That is how they use math structures and not human-like math language. The information about distance goes in and the forces, and directions to direct their muscle-skeleton system are computed. Many disagree for a limited number of reasons, e.g. there are no squirrel schools. An implication is that organisms discovered math. A second example of an organism using math is the hook-beak raptor diving for a moving field mouse, calculations involving speed, distance and geometry allow it to intercept the mouse with its talons.
I didn't know that Michael J. Fox was a physicist, he should've built the time machine himself.
I think discovery forces itself upon the observer to freeze into rules, what we confined our selves into it...to feel safe..
"...nature prefers simplicity." very important concept.
a dumb agreement of a dumb statement. literally fuck all about the observed universe is simple. the fact that after some 200+ years of life even the brightest minds still remain utterly ignorant of fundamental principles of the universe stands testament to that.
Mathematics are human invention, just like a PC or an abacus (which in itself uses properties that we didn't invent, just learned to control), it's a tool we use to simplify things and predict, etc. The properties that Math and Physicists try to predict/understand are what we didn't invent.
Great videos but guys.... Sort out your audio levels. One video is way up and the next is way down... You need to get some consistency here.
Math was created, built into the very creation. We are composed of it,part of it. When you discover this, you discover a part of yourself, who you are. So yes, in a sense math, which is part of YOU is a discovery.