1:53 He's saying the 5-dimensionality of a manifold does not stop it from being applicable to a world that is "3 dimensional" in any way, and that the 3-dimensionality of space may not be remotely related to the dimensionality of this applicable 5-dimensional manifold. A simple example, if I am manufacturing a product and there are 5 parameters that determine my sales, profit, etc. then the combinations of those parameters form a 5-dimensional vector space, and that 5d vector space is directly applicable to my manufacturing process, and this has _nothing to do_ with the three dimensions we move around in. Nothing. The mistake that people make is considering n-dimensional abstract objects without abstracting the notion of dimension itself. By talking about "visualization" of a 5d mathematical object, you're insisting on these being _spatial_ dimensions, IE, you are failing to abstract the notion of dimension.
As the great Richard Feynman said:"Mathematicians do not care what they study or if they even understand what they are studying, they only care about the structure of the reasoning" I cant say that many mathematicians will disagree with that sick burn.
@@mikhailmikhailov8781 It's not even a burn. It's exactly true. It's the logical structure, the abstraction, that mathematicians are studying. He's not intending it to be a burn, either.
Engineers use applied math and physics to build things that work. Pure math and physics are pure science and therefore much more difficult to understand to many of us more mortals.😁😁
Math is a way of defining relationships. When you discover relationships in nature you have to describe it using the representations/symbols of math. You can think of math as a very rigorously detailed language that he to be dense, concise, and accurate but its still a language that people invented to describe relationships we discovered.
One perspective is that all inventions and artistic works are discoveries. When we think we're creating something we're actually just uncovering links between concepts that already exist. For example, there is a link between force, mass and acceleration, and we have symbolically represented our understanding of it, but nowhere did a human "creation" occur, only discoveries. Similarly, when a composer writes music, he is discovering which combinations of frequencies evoke certain emotions or experiences. But a particular combination of frequencies is not created, it is found. The composer did not "create" the sound a triad chord makes or the emotions he feels when he listens to it. Same thing with inventions - they are applications of scientific discoveries that have been found to have a practical use.
I agree with this. It kind of reminds me of Karl Jung saying "People don't have ideas. Ideas have people!" If all concepts already exist and are just merely discovered or experienced, then you technically dont have created the concept yourself, you just discovered the concept yourself. Such thinking is actually quite nice because it encourages to look further and beyond, to be curious and to be willing to be proven wrong to gain more knowledge in the longterm. I think at the core of the Scienticifc Objectivity, such thinking is key (or at least very helpful) in order to remove as much personal, subjective bias as possible.
While I understand how on a deeper level you could argue that every invention is a discovery, I think you can define the two terms in such a way that they refer to distinct concepts with practical applications. In fact, I think this interview itself exemplified a way in which the distinction is quite practical. If it is true that math is mostly about discovery, then we can reasonably expect alien civilizations we will find to (partly) be using similar math to us. If it is true that math is mostly about invention, on the other hand, then that expectation becomes inherently less reasonable.
@@tudornaconecinii3609 The symbols used in math could be considered "inventions", so they most likely wont hold up to alien symbols, but the logic of math would be the same. Once we figure out the differences in each syntax, we will quickly see the same logic at work. Of course, that only works if the universe has the same fundamental rules across space-time. If the rules fluctuate or underlying constants are not actually constant, then stuff might be more weird to us than we can imagine.
I'm a Computer Science student with a background of philosophy and have listened to many talks on this very topic. Grant is honestly the most humble mathematician I've ever heard talk about this. By both Tegmark and Penrose especially I was taken back by their ardent neo-Platonism, and while passion is necessary, it's also refreshing to see that Grant is just as passionate without having that passion rooted in the presumption of ones own metaphysical system.
Axioms and postulates are invented and the resulting mathematics is discovered. The Pythagorean theorem was discovered, but it was discovered after the required axioms and postulates necessary to deduce that were invented. Mathematics is merely inventing rules and then "discovering" the logical results of those invented rules. It's just that some "rules" are better at producing logical results that we "discover" that reflect the physical universe, but there's no logical reason why Mathematics must "prefer" some axioms over others merely because we arrive at results that make physical predictions. It's simply because we like to be able to use Mathematics to describe the real world that there is any bias in favor of such axioms and postulates that would give us such results to "discover."
That's a very interesting way of thinking. It's easy to wrap my head around this because it feel almost obvious after you pointed it out. I'm gonna stick to that explanation.
Grant is one of the most thoughtful speakers I’ve ever heard. He might be better at language than math. Would love him and say Harris to have a non-political chat.
An approach I’ve been exploring for the compressibility problem, is that our capacity to think and to measure physical phenomena is constrained by insufficient computational depth to recognize deeper aspects of variation. Complexity hides behind the relatively low computational capacity we have for conceiving the true depth of variation. It’s a sort of signal-to-noise ratio problem: deep details hide in the noise and we reduce complexity to make it manageable. What if it turns out that deeper methods of computing and measuring details of quantum jitters, for example, opens up the details of the blur and there are profound intricacies we can’t consider.... We just need our minds to fuse to AI in order to extract the extra depth of order from the data.
If you think fusing human minds with computational intelligence will take is in a direction to discover deep details of the universe, I think you haven't been paying attention to the world
actually not as Grant said, you can understand the math of quantum mechanics and GTOR, etc its just people don't have intuition for that but it can be developed. After working for some time with these topics they will look like common sense.
I like it. I disagree with it but I still like it. Sounds cool. I could just meme the second part: Drops Acid......takes a brisk walk outside of what our mind can even comprehend...
An interesting question, but you didn't establish a definition for "math." As soon as you choose either the notation we use, or what the notation represents, then the answer starts to feel trivial imo.
Grant gave three approaches to math, which are close to definitions. I mean, you can start with the vague "Math is what mathematicians do" and go from there
in the higher dimentions or realms where we don't have physical objects to relate to ( like in pythagorean theorem) . we realize mathematics from the perspective of symmetry.
I would say its invented to make discoveries. You create a shovel to get to something underneath the ground to discover but you could also use something else other than a shovel to get there. But some equations make me think more about this question than others.
So glad I found others mulling over this fundamental question. Math can be used to explain the world as it is, or can be used to give "proof" of theories and ideas that can take humanity in the wrong direction. How do we determine true from false if the math is created to prove itself? If it is invented, how can the argument, "...But the math works." be allowed as an argument?
i think it's a really interesting question - how do we reason without logic? or are there forms of logic, independent of math, that allow us to bootstrap enough context to talk about math without using it directly
Hey Lex. I'd love to hear Joscha Bach's contribution to this conversation, in response to you and Grant. Ever thought about holding a round table? You could compile some big questions like this and have the world's greatest minds debate over them.
"Cupcell" a word I just created to define a cup next to a cellphone. Anything known now about this new definition is new knowledge therefore a discovery. "Cupcells" look nice, discovery. Like this?
Calculate the volume of a red ball Mathematician: Triple Integral Physicist: Displaces water Engineer: looks up serial number of red ball Math is neither discovered or invented, it emerges with imagination.
"Math emerges with imagination"? Wtf does it supposed to means? Seems like these beauty quotation that people keep repeating but in the end says nothing at all.
It's a good idea to keep Grant's very grounded take on math and physics when listening to Witten and others closer to the fields that pull physics into the direction of math. Grant's point at the end, about using math developed by our (possibly) simple biological brain gets something about the real world given successes of engineering, etc. - strikes me as profound.
I would say that there are lots of cycles at play here: - Discovering things in the real world and inventing math to model them - Inventing abstract concepts and discovering applications to the real world - Discovering abstract patterns and inventing abstract concepts to describe them - Inventing abstract definitions and discovering abstract properties of them
On the last topic of it being surprising that the world seems to be describable through relatively simple equations, two thoughts: 1) It could a matter of us inventing compact notation and sophisticated concepts so that whatever is complex about the universe ends up appearing in a simple form on paper. The example is how the standard model equation seems to be surprisingly simple to describe everything in the universe, but actually behind each symbol is some insane amount of concepts buried within it. 2) It could be that the world actually isn't describable through simple equations - only some parts can. E.g. we think F=ma, which is simple, is the law of motion, but it is only valid to good approximation. The special relativity formulation captures more scenarios, but it still isn't everything, and we go to more and more complicated frameworks. In fact, many physicists have been looking for a "theory of everything" that can merge theory of gravity and the 3 other fundamental forces, on the basis that there must be a simple model that underlies everything. The fact that we haven't been successful could either be that we haven't tried hard enough, or that the world simply cannot be described simply!
"Why do you think the fundamentals of quarks and the nature of reality is so compressible into clean, beautiful equations?" It's about 5% the axioms we postulate, 5% having a bias in favor of elegantly simple resulting laws, and 90% *notation.*
Spaces are mathematical abstractions that represent information about objects in terms of distance functions measured along dimensions. Abstract spaces of higher dimension can represent many things that are applicable, they simply are not needed for representing the position of a single physical particule in relation to an observer, and that makes them hard to see as our visual cortex is optmized for that specific application.
Chaos theory feels sooo fulfilling to me. All information matters. We have the illusion of free choice because we can not process all the factors leading up to it. I think that we are trying to make sense of the world retrospectively through our senses, but in fact we're always a fraction behind of the present. And it _still_ feels like I live my own life, that's amazing.
The idea of Mathematics and how we conceptualise it as individuals is discovered but the way we notate it as a tangible representation of the abstract discovery is an invention.
I tend to think that nature must have at its core something that's 1:1 congruent with a subset of mathematics, but then we extend that core, creating objects and abstractions way outside of it. Even though I can point to mathematical objects that I believe are not 'real' [like infinitely divisible _real_ numbers], I don't know where the line between the two is exactly. But assuming that extension is generally consistent in some ways with the core, no wonder a lot of it is useful even when it has no physical representation. Metaphorically speaking, it would be like inventing a map that has higher resolution, more dimensions, and more features in general, than the thing that's being mapped.
What is the difference between a discovery and an invention? What you discover already existed, you just had to find, in contrast to an invention. Nothing precedes the axioms of mathematics, they must have been invented. But depending on the set of axioms we choose, every result that follows from it is already determined. So I would consider everything except the axioms as discovered.
I know very little about mathematics. But Grant just explained how it can be both very well. Somehow I followed along and I understand his point of view. Nicely stated sir.
Interesting to compare to Andrew Wiles answer to this question. Wiles said in an interview that he didn´t know a singel mathematician who didn´t think it was discovered.
Math stems from the observation of duality, or seperation. If you can perceive two things being two separate things, you now have the first understanding of quantity.
Mathematics is the 7'th layer of our neocortext, because the 7'th layer is the highest possible abstraction layer of known creatures. Dolphins for example have only 3 layers. DL experts may approve this idea. "As the number of layers increase the abstraction ability increases". Thus, mathematics is invented by our brains, as our brains discover the world and ask questions about its mechanisms.
In my opinion, math indeed was discovered. I think of the math that we use as a hammer, the hammer was invented but the metal was discovered. The metal is the math, itself. The handle of the hammer is the medium through which we perceive math i.e equations, notations, diagrams and whatnot. If we have the length of the shadow of a tower, the length of our shadow and our height, we can easily find the height of said tower (basic trigonometry). All the relations were already in existence, like the metal, but they weren't really useful until we equipped them with a proper handle. Math in our world is limited to real world applications (for the most part, I believe) but there are so many kinds of 'metals' which are yet to be discovered, and even if we do find them, there'd be those metals which the average human could never comprehend, we could never equip them with a proper 'handle', you can say. There's so much of math that already exists that we're sleeping on. TL;DR : Math was discovered
Why doesn't anybody make the argument that math can definitely be invented as for example game developers can make up their own weird mathematical rules that will lay the foundations of the physical world inside the game itself. In that world you can have cases where 1+1 doesn't not equal 2 and the Pythagorean theorem is not valid. In short one can make up their own mathematical rules and apply them inside a world of a video game.
A comparison of Mayan Maths to Western Maths shows that underlying ideas can be represented in mutually unrecognisable ways. The principles are discovered and the representations are invented.
This is probably wrong, but how we ask ourselves how the universe is works is probably why math works so well. We are essentially able to conceptualise the ideas about the universe in such a way that we are comfortable with. These puzzles aren’t inherently “natural” in the physical world, we just have a way of overlapping them on what we perceive. It’s essentially like making analogies which closely resemble an idea and then applying formulations with what we have already figured out, probably wrong and nobody asked but 🤷🏾♂️
this guy just casually dropped 4 or 5 such entireley new concepts that i would’ve bought a book to get to know each of them. i haven’t heard a person like him this before.
Math is like logic. The principles exist abstractly, waiting to be discovered, but the EXPRESSION on paper and in writing is invented by the culture using them.
There are 4 types of answers (or mix of these four)for the question according to 4 types of mathematic doctrines: a) Realists or Platonists: math is discovered, "mathematical concepts and properties exist in some objective sense and [...] can be apprehended (detected) by human mind" (Kline); b) logicists: math is discovered, Russel and Whitehead believed in the physical truth of mathematics and they tried to use logic to obtain analytically that knowledge, avoiding paradoxes (or at least trying); b) Intuitionists: math is an invention, the concepts created are not from experience (don't exist objectively) but are synthetic contributions of the mind; c) Formalists: math is invented, Hilbert said that maths should not be treated as factual knowledge but as a formal discipline, that is, abstract, symbolic, and without reference to meaning (though informally meaning do enter).
Nowadays we recognize ''a2 + b2 = c2'' as Pythagorean. As an older person I still remember the old way I was taught Pythagoras, ''The square on the hypotenuse of a right angled triangle is equal to the sum of the squares on the other two sides''.
Just maybe, so many interactions can be labled simply, because in their basic form they literally are basic interactions. Like your example planets. It's 2 large bodies moving around each other, and they have their inertia and the gravity pulling on them. But once you get to 3, it gets immensely complex and complicated.
Once I have done with all my exams(medicine) I got bored- then I found Mathematics( or it found me)- after 1 year I realized what I have got into! something I can learn as long as I live! That is the beauty of it.
Kinda, it uses artificial intelligence that learns from videos you watch and duration for which you watch them etc... to predict more or less what you want so in a sense it read your mind.
Hard to say. Math explains digitally a phenomenon which is discovered. The ability to describe this must be created with the terms available at the time. This is what's invented, the language and means of explanation But what math is describing is what is truly discovered
The issue is that the equations we have that govern our universe do break down at some point. Physicists have traced back the timeline of our universe to fractions of a second after the Big Bang, which is the point at which we no longer understand the physics of what was happening. In the world of quantum chemistry, there are a lot of things that we don't have accurate physical/mathematical laws to represent. We still cannot analytically solve for the wavefunctions of systems with more than one electron. We only have numerical solutions for such things. So I would say that we only have simple, elegant laws for topics which are "easily" understood, and as we branch out into more complicated topics the mathematics grows in complexity to reflect that.
he would be genuinely delighted at that question and tell you why he thinks she might have thought of it that way, which is exactly what our attitude towards the 'dumb' questions should be
Nature too has defined distance. That's why nature has patterns and ratios. That's why constants occur in nature. That's why synchronisation is found in nature. Nature does it with the help of forces. Interestingly, humans too need forces to define length, distance etc.
What about Axioms? Were Axioms of (say Logic, a Subset of Axioms of math) Deduced? Or they were invented Purely as abstract form? Or maybe even Axioms were based on observation from the external world?
@@UnworthyUnbeliever try to invent an axiom that contradicts what we already know to be true. For ex, invent some axiom where you can deduce a theorem that proves that 1 + 1 = 2365.
@@fexcasanova 1. Definition of inductive reasoning in philosophy =/= definition of proof by induction in mathematics 2. Mathematical system(s) that humans currently use is established, it doesn't makes much sense to add an Axiom to a already established system with thousands of deduced theorems. 3. Informally speaking, complex statements shouldn't be Axioms. 4. Product statements (things you normally deduced from Axioms) shouldn't be forced to become Axiom. 5. I cannot make a 1 + 1 = 2365 an Axiom because it contradicts existing Axioms (plus mathematics assume law of non contradiction, therefore, it cannot be done) 6. I Assume a mathematical system in which 1 + 1 = 2365. Since i dont care, i leave all of used symbols undefined.
@@UnworthyUnbeliever 1 - can you show the difference plz?u can copy and paste. 2 - im not saying for you to add an axiom to already established system. U did ot get my point. If an axiom is invented, you would be able to invent a new one where you can derive what i was asking. 3 - "informally speaking, complexo statments shoudn't be axioma" Invent a formal one then. 4 - read what i wrote again. I did not say for you to force 1 + 1 = 2365 as an axiom 5 -" i cannot make a 1 + 1 = 2365 an axiom" I DID NOT ASK U TO DO THAT. I Asked you to invent an axiom where you can derive a theorem to prove that 1 + 1 = 2365(1 + 1 = 2365) 6 - what about the theorem that proves your mathematical system?? I ask you to try to invent this axiom, based on your own question
Cristian Proust I think you are right here, I believe math excited without us inventing it but it’s a pure form, it is not numbers or symbols, it’s just a thing that we explain with math, the concept exists without human but we use symbols to understand the concept
@@maniebrahimi7952 Not just understand the concept but also help communicate it between subjects. We create standards and definitions to help us build on top without having to recreate everything at each step over and over again.
(x,y,z,phi,theta,r)= a 6d entity that defines gradient for all of space. AI: An oil lamp except the clear liquid is a gradient and the bubbles are parameters that perform an action. Then its p>p' goes to q, then flipflops p q>q'. if the AI improves at a particular task, p'=p or q'=q . two randoms cancel each other out on an error function and acts like an implicit rolles theorem without explicitly stating d/dx=0.
Maths is similar to musical notes. musical notes exist in frequencies found in nature and humans have invented the way to describe that note on a scale i.e. ABCDEFG#maj etc. Therefore maths is just the human explanation for what is obvious to nature.
If physical laws were different at every point in space, physicists would write that as a differential equation and abstract it away so that the theories are still about as simple
I'd say it's a translator from universe to what we understand and use. You discover something in the universe and you express it so it's some what easily read by others. In a world where there are like 6,500 languages (according to quick google search lol) each with different names referring to the same thing, math is just a common ground for everyone, regardless of what language you speak, think, whatever. Heck, i'd assume that early humans observed other humans from different part of the earth, and by observation and interaction, slowly learned what each other calls each object. And to think that humans from different places of the earth speak different languages, indeed makes you think how different would an alien species be...
2:16 "For us humans, it boils down to 3- dimensional world". Brilliant statement, and the foundation for and other number of dimensions. For instance E8, It's still based on 3-D.
Lex is like Joe Rogen, but chill and smart. They are both super inquisitive, and that makes for a good interviewer. But Lex actually understands the responses lol. And he is humble. And likeable. And not Joe Rogen.
Math is a distributed, evolving, not strictly synchronized object. The host of which is the social network of the brains of mathematicians. Which is part of the social network of human brains. (In other words, mathematics is a subculture of the culture of mankind.) More precisely, this object is a model of mathematics. Which is part of the model of the world.
This dude helped me through linear algebra.
I will forever be grateful.
Same for me.
CannibalWarthog *grantful
I am currently watching the series.
Same. I would fail my exam without him but I wrote a 10/10 thanks to him
1:53
He's saying the 5-dimensionality of a manifold does not stop it from being applicable to a world that is "3 dimensional" in any way, and that the 3-dimensionality of space may not be remotely related to the dimensionality of this applicable 5-dimensional manifold.
A simple example, if I am manufacturing a product and there are 5 parameters that determine my sales, profit, etc. then the combinations of those parameters form a 5-dimensional vector space, and that 5d vector space is directly applicable to my manufacturing process, and this has _nothing to do_ with the three dimensions we move around in. Nothing.
The mistake that people make is considering n-dimensional abstract objects without abstracting the notion of dimension itself. By talking about "visualization" of a 5d mathematical object, you're insisting on these being _spatial_ dimensions, IE, you are failing to abstract the notion of dimension.
Agree
Sure....
As the great Richard Feynman said:"Mathematicians do not care what they study or if they even understand what they are studying, they only care about the structure of the reasoning" I cant say that many mathematicians will disagree with that sick burn.
@@mikhailmikhailov8781 It's not even a burn. It's exactly true. It's the logical structure, the abstraction, that mathematicians are studying. He's not intending it to be a burn, either.
Wow never thought like that. Thanks now it makes sense
Mathematician: "Math is the language of the universe."
Physicist: "Math is the language of physics."
Engineer: "sin(x) = x."
Toss a coin to your engineer.
Immanuel Kant: Math is the language of the mind
lol, there's a weird assumption that engineers don't deal with large angles
g = 10.
Engineers use applied math and physics to build things that work. Pure math and physics are pure science and therefore much more difficult to understand to many of us more mortals.😁😁
This guy looks exactly as I thought he looked
.
Bold of you to think he wasn't a transcendental mathematical being with no human form
every educational narrator using video with a black background is going to look like Salman Khan in my head forever
Damn. Thought he was black this whole time...
This isn't what I thought at all
I love Grant. Imagine if everyone was only a bit like him. The world would be such a better place.
And a more handsome place lol
P vs. NP just gave you the finger.
id feel very stupid in a very short time
Grant is the best math teacher I’ve ever had
Imagine 1 percent of the world was like him that would be enough tbh
When you become depressed that the internet is full of poison, you find things like this to relax your troubled mind. Great podcast. Great guest.
sheesh man, sending you some hug vibes, spread the love.
@@dmitryduryagin6980 thanks buddy but I'm good! I'm just saying, the internet can be toxic and easy to forget that it is also a place of wonder.
The second Grant opens his mouth you can tell he knows exactly what he’s talking about.
10:12
@@peezieforestem5078 lol
And he says it with smile on his face
Math is a way of defining relationships. When you discover relationships in nature you have to describe it using the representations/symbols of math. You can think of math as a very rigorously detailed language that he to be dense, concise, and accurate but its still a language that people invented to describe relationships we discovered.
One perspective is that all inventions and artistic works are discoveries. When we think we're creating something we're actually just uncovering links between concepts that already exist. For example, there is a link between force, mass and acceleration, and we have symbolically represented our understanding of it, but nowhere did a human "creation" occur, only discoveries. Similarly, when a composer writes music, he is discovering which combinations of frequencies evoke certain emotions or experiences. But a particular combination of frequencies is not created, it is found. The composer did not "create" the sound a triad chord makes or the emotions he feels when he listens to it. Same thing with inventions - they are applications of scientific discoveries that have been found to have a practical use.
All of our meaning is derived from our surroundings so the sort of alchemy of that is really interesting
I agree with this.
It kind of reminds me of Karl Jung saying "People don't have ideas. Ideas have people!"
If all concepts already exist and are just merely discovered or experienced, then you technically dont have created the concept yourself, you just discovered the concept yourself. Such thinking is actually quite nice because it encourages to look further and beyond, to be curious and to be willing to be proven wrong to gain more knowledge in the longterm. I think at the core of the Scienticifc Objectivity, such thinking is key (or at least very helpful) in order to remove as much personal, subjective bias as possible.
While I understand how on a deeper level you could argue that every invention is a discovery, I think you can define the two terms in such a way that they refer to distinct concepts with practical applications.
In fact, I think this interview itself exemplified a way in which the distinction is quite practical. If it is true that math is mostly about discovery, then we can reasonably expect alien civilizations we will find to (partly) be using similar math to us. If it is true that math is mostly about invention, on the other hand, then that expectation becomes inherently less reasonable.
@@tudornaconecinii3609 The symbols used in math could be considered "inventions", so they most likely wont hold up to alien symbols, but the logic of math would be the same. Once we figure out the differences in each syntax, we will quickly see the same logic at work.
Of course, that only works if the universe has the same fundamental rules across space-time. If the rules fluctuate or underlying constants are not actually constant, then stuff might be more weird to us than we can imagine.
*"concepts that already exist"*
concepts are a product of the mind.
KEvron
I'm a Computer Science student with a background of philosophy and have listened to many talks on this very topic. Grant is honestly the most humble mathematician I've ever heard talk about this. By both Tegmark and Penrose especially I was taken back by their ardent neo-Platonism, and while passion is necessary, it's also refreshing to see that Grant is just as passionate without having that passion rooted in the presumption of ones own metaphysical system.
That's not to discourage metaphysics. God, if such a being exists, only knows the degree to which modern society already has for the past century.
@@Michael-Hammerschmidt Would love to hear your take on Edward Frenkel's view about this topic. Thank Q
It’s fun to watch the interviewer slowly fall asleep as he asks each question
This podcast is quite a gift.
Perfect comment
Axioms and postulates are invented and the resulting mathematics is discovered.
The Pythagorean theorem was discovered, but it was discovered after the required axioms and postulates necessary to deduce that were invented.
Mathematics is merely inventing rules and then "discovering" the logical results of those invented rules. It's just that some "rules" are better at producing logical results that we "discover" that reflect the physical universe, but there's no logical reason why Mathematics must "prefer" some axioms over others merely because we arrive at results that make physical predictions. It's simply because we like to be able to use Mathematics to describe the real world that there is any bias in favor of such axioms and postulates that would give us such results to "discover."
I believe exactly this
Math is invented, but an alien will probably "discover similar math"
That's a very interesting way of thinking. It's easy to wrap my head around this because it feel almost obvious after you pointed it out. I'm gonna stick to that explanation.
@Sailor , is this irony?
Philosophy first then mathematics.
The way he smiles makes him seem like he knows some hilarious secret. Crushing
Grant is one of the most thoughtful speakers I’ve ever heard. He might be better at language than math. Would love him and say Harris to have a non-political chat.
Is it just me or does Grant kinda sound like Mordecai from Regular Show?
Artkotix Art oh no... I can’t unhear
To See The Face Of This Guy(Grant) Is Like Discover The Real Face Of A Super Hero!! #Respect
This is literally *big brain time*
An approach I’ve been exploring for the compressibility problem, is that our capacity to think and to measure physical phenomena is constrained by insufficient computational depth to recognize deeper aspects of variation. Complexity hides behind the relatively low computational capacity we have for conceiving the true depth of variation. It’s a sort of signal-to-noise ratio problem: deep details hide in the noise and we reduce complexity to make it manageable. What if it turns out that deeper methods of computing and measuring details of quantum jitters, for example, opens up the details of the blur and there are profound intricacies we can’t consider.... We just need our minds to fuse to AI in order to extract the extra depth of order from the data.
Nice comment
If you think fusing human minds with computational intelligence will take is in a direction to discover deep details of the universe, I think you haven't been paying attention to the world
"math in the 20th & 21st century...takes a brisk walk outside of what our mind can even comprehend... LOVE THAT!!
actually not as Grant said, you can understand the math of quantum mechanics and GTOR, etc its just people don't have intuition for that but it can be developed. After working for some time with these topics they will look like common sense.
I like it. I disagree with it but I still like it. Sounds cool.
I could just meme the second part:
Drops Acid......takes a brisk walk outside of what our mind can even comprehend...
Your mind probably can comprehend it if you put in the time and effort.
An interesting question, but you didn't establish a definition for "math." As soon as you choose either the notation we use, or what the notation represents, then the answer starts to feel trivial imo.
Lex is better then most in terms of trivializing topics. That being said, you are spot on.
Counting things is independent of a notation.
Grant gave three approaches to math, which are close to definitions.
I mean, you can start with the vague "Math is what mathematicians do" and go from there
in the higher dimentions or realms where we don't have physical objects to relate to ( like in pythagorean theorem) . we realize mathematics from the perspective of symmetry.
Finally a face to the voice that made me pass all my introductory math courses
I would say its invented to make discoveries. You create a shovel to get to something underneath the ground to discover but you could also use something else other than a shovel to get there.
But some equations make me think more about this question than others.
So glad I found others mulling over this fundamental question. Math can be used to explain the world as it is, or can be used to give "proof" of theories and ideas that can take humanity in the wrong direction. How do we determine true from false if the math is created to prove itself?
If it is invented, how can the argument, "...But the math works." be allowed as an argument?
What do you mean by "take humanity in the wrong direction"? Which proofs have done that in the past?
i think it's a really interesting question - how do we reason without logic? or are there forms of logic, independent of math, that allow us to bootstrap enough context to talk about math without using it directly
Me and my friend going to the bar: “let’s not nerd out at the bar”
Me and my friend 3 beers later:
Hey Lex. I'd love to hear Joscha Bach's contribution to this conversation, in response to you and Grant. Ever thought about holding a round table? You could compile some big questions like this and have the world's greatest minds debate over them.
my brain be like : "Aight, imma head out."
I love your podcasts man! They're always so insightful and interesting!
It is shocking that Lex is the only one doing deep interviews with intelligent approaches
Definition=Invention
Discovery=Previously Unknown consequence of the definition
"Cupcell" a word I just created to define a cup next to a cellphone.
Anything known now about this new definition is new knowledge therefore a discovery.
"Cupcells" look nice, discovery.
Like this?
I meant the definition of a concept/idea and not just naming objects but your example works too, I think.
Lex seems like a smart man and does a great job on his interviews.
The appeal to the weak anthropic principle always shuts down the most interesting lines of inquiry.
yet self bias is often the most overlooked fallacy
Im jealous of Lex, he gets to have conversations with Kasparov and Grant
Calculate the volume of a red ball
Mathematician: Triple Integral
Physicist: Displaces water
Engineer: looks up serial number of red ball
Math is neither discovered or invented, it emerges with imagination.
If it emerges with imagination. then it is invented.
Only one calculated. Discovery is empiricism, invention is rationality.
"Math emerges with imagination"? Wtf does it supposed to means? Seems like these beauty quotation that people keep repeating but in the end says nothing at all.
i came here to check how dumb i am.
It's a good idea to keep Grant's very grounded take on math and physics when listening to Witten and others closer to the fields that pull physics into the direction of math. Grant's point at the end, about using math developed by our (possibly) simple biological brain gets something about the real world given successes of engineering, etc. - strikes me as profound.
Outstanding video, thanks for posting, much appreciated. Love your channel and work Lex : )
I would say that there are lots of cycles at play here:
- Discovering things in the real world and inventing math to model them
- Inventing abstract concepts and discovering applications to the real world
- Discovering abstract patterns and inventing abstract concepts to describe them
- Inventing abstract definitions and discovering abstract properties of them
And stumbling across magnetism despite not having any way to sense it.
On the last topic of it being surprising that the world seems to be describable through relatively simple equations, two thoughts:
1) It could a matter of us inventing compact notation and sophisticated concepts so that whatever is complex about the universe ends up appearing in a simple form on paper. The example is how the standard model equation seems to be surprisingly simple to describe everything in the universe, but actually behind each symbol is some insane amount of concepts buried within it.
2) It could be that the world actually isn't describable through simple equations - only some parts can. E.g. we think F=ma, which is simple, is the law of motion, but it is only valid to good approximation. The special relativity formulation captures more scenarios, but it still isn't everything, and we go to more and more complicated frameworks.
In fact, many physicists have been looking for a "theory of everything" that can merge theory of gravity and the 3 other fundamental forces, on the basis that there must be a simple model that underlies everything. The fact that we haven't been successful could either be that we haven't tried hard enough, or that the world simply cannot be described simply!
"Why do you think the fundamentals of quarks and the nature of reality is so compressible into clean, beautiful equations?"
It's about 5% the axioms we postulate, 5% having a bias in favor of elegantly simple resulting laws, and 90% *notation.*
Spaces are mathematical abstractions that represent information about objects in terms of distance functions measured along dimensions. Abstract spaces of higher dimension can represent many things that are applicable, they simply are not needed for representing the position of a single physical particule in relation to an observer, and that makes them hard to see as our visual cortex is optmized for that specific application.
Chaos theory feels sooo fulfilling to me. All information matters. We have the illusion of free choice because we can not process all the factors leading up to it. I think that we are trying to make sense of the world retrospectively through our senses, but in fact we're always a fraction behind of the present. And it _still_ feels like I live my own life, that's amazing.
dude's channel is bonkers, love this guy
"Pure puzzles and abstraction" are the loves of my life.
OMG Two of my favorite people discussing my favorite question!
The idea of Mathematics and how we conceptualise it as individuals is discovered but the way we notate it as a tangible representation of the abstract discovery is an invention.
I tend to think that nature must have at its core something that's 1:1 congruent with a subset of mathematics, but then we extend that core, creating objects and abstractions way outside of it. Even though I can point to mathematical objects that I believe are not 'real' [like infinitely divisible _real_ numbers], I don't know where the line between the two is exactly. But assuming that extension is generally consistent in some ways with the core, no wonder a lot of it is useful even when it has no physical representation. Metaphorically speaking, it would be like inventing a map that has higher resolution, more dimensions, and more features in general, than the thing that's being mapped.
What is the difference between a discovery and an invention? What you discover already existed, you just had to find, in contrast to an invention.
Nothing precedes the axioms of mathematics, they must have been invented. But depending on the set of axioms we choose, every result that follows from it is already determined. So I would consider everything except the axioms as discovered.
We may have invented partial interpretations of the existing unspoken mathematical axioms of the universe.
Dunno much about axioms but the classic "theres nothing new under the sun" phrase says it all.
I thought I had a dirty screen, but it's dark line on your wall.
I know very little about mathematics. But Grant just explained how it can be both very well. Somehow I followed along and I understand his point of view. Nicely stated sir.
That 'and land' at the end is underappreciated
I feel like I’m listening to Joe Rogan on a DMT trip.
I cheer for the mention of Vladimir Arnol’d!
Interesting to compare to Andrew Wiles answer to this question. Wiles said in an interview that he didn´t know a singel mathematician who didn´t think it was discovered.
I love the fact that science is mainstream now and its cool to be curious of science ❤
Math stems from the observation of duality, or seperation. If you can perceive two things being two separate things, you now have the first understanding of quantity.
Mathematics is the 7'th layer of our neocortext, because the 7'th layer is the highest possible abstraction layer of known creatures. Dolphins for example have only 3 layers. DL experts may approve this idea. "As the number of layers increase the abstraction ability increases". Thus, mathematics is invented by our brains, as our brains discover the world and ask questions about its mechanisms.
In my opinion, math indeed was discovered. I think of the math that we use as a hammer, the hammer was invented but the metal was discovered. The metal is the math, itself. The handle of the hammer is the medium through which we perceive math i.e equations, notations, diagrams and whatnot. If we have the length of the shadow of a tower, the length of our shadow and our height, we can easily find the height of said tower (basic trigonometry). All the relations were already in existence, like the metal, but they weren't really useful until we equipped them with a proper handle. Math in our world is limited to real world applications (for the most part, I believe) but there are so many kinds of 'metals' which are yet to be discovered, and even if we do find them, there'd be those metals which the average human could never comprehend, we could never equip them with a proper 'handle', you can say. There's so much of math that already exists that we're sleeping on.
TL;DR : Math was discovered
Math is a game in which the rules are discovered, but strategy is invented
Nice
Why doesn't anybody make the argument that math can definitely be invented as for example game developers can make up their own weird mathematical rules that will lay the foundations of the physical world inside the game itself. In that world you can have cases where 1+1 doesn't not equal 2 and the Pythagorean theorem is not valid. In short one can make up their own mathematical rules and apply them inside a world of a video game.
A comparison of Mayan Maths to Western Maths shows that underlying ideas can be represented in mutually unrecognisable ways. The principles are discovered and the representations are invented.
Perfectly put
Two of my favorite voices.
This is probably wrong, but how we ask ourselves how the universe is works is probably why math works so well. We are essentially able to conceptualise the ideas about the universe in such a way that we are comfortable with. These puzzles aren’t inherently “natural” in the physical world, we just have a way of overlapping them on what we perceive. It’s essentially like making analogies which closely resemble an idea and then applying formulations with what we have already figured out, probably wrong and nobody asked but 🤷🏾♂️
this guy just casually dropped 4 or 5 such entireley new concepts that i would’ve bought a book to get to know each of them. i haven’t heard a person like him this before.
Math is like logic. The principles exist abstractly, waiting to be discovered, but the EXPRESSION on paper and in writing is invented by the culture using them.
abstractions are a product of the mind.
KEvron
@@KEvronista think about this way: the principles is a tree, the EXPRESSION is the chair. The invention is the manipulation of the principles.
@@fexcasanova
think of it this way: principles are descriptive, thus are products of the mind.
KEvron
@@KEvronista so if there is no mind the moon is the moon at the same time the moon is not the moon??
@@fexcasanova
*"the moon is the moon"*
the actual is not mind-dependent in its existence, nor is the abstract actual.
KEvron
"The fact that we can fly is pretty great... and land." -Lex Classic.
There are 4 types of answers (or mix of these four)for the question according to 4 types of mathematic doctrines:
a) Realists or Platonists: math is discovered, "mathematical concepts and properties exist in some objective sense and [...] can be apprehended (detected) by human mind" (Kline);
b) logicists: math is discovered, Russel and Whitehead believed in the physical truth of mathematics and they tried to use logic to obtain analytically that knowledge, avoiding paradoxes (or at least trying);
b) Intuitionists: math is an invention, the concepts created are not from experience (don't exist objectively) but are synthetic contributions of the mind;
c) Formalists: math is invented, Hilbert said that maths should not be treated as factual knowledge but as a formal discipline, that is, abstract, symbolic, and without reference to meaning (though informally meaning do enter).
his voice is so soothing
Nowadays we recognize ''a2 + b2 = c2'' as Pythagorean. As an older person I still remember the old way I was taught Pythagoras, ''The square on the hypotenuse of a right angled triangle is equal to the sum of the squares on the other two sides''.
Congratulations, you just realised that algebra and geometry are fundamentally connected
Just maybe, so many interactions can be labled simply, because in their basic form they literally are basic interactions. Like your example planets. It's 2 large bodies moving around each other, and they have their inertia and the gravity pulling on them. But once you get to 3, it gets immensely complex and complicated.
Clearly math was discovered. I am interested to see what these two have to say about it. I am not sure there is much of a debate on this idea.
It's basically a philosophical debate about what constitutes discovery and invention, which in some sense is a more interesting topic.
7:27 that's why I love this podcast haha
My I.Q. increased a few points just by watching this video.
Once I have done with all my exams(medicine) I got bored- then I found Mathematics( or it found me)- after 1 year I realized what I have got into! something I can learn as long as I live! That is the beauty of it.
Does RUclips read my mind I was literally just thinking about this.
It doesn't read your mind but it does influence your thoughts.
Kinda, it uses artificial intelligence that learns from videos you watch and duration for which you watch them etc... to predict more or less what you want so in a sense it read your mind.
Hard to say. Math explains digitally a phenomenon which is discovered. The ability to describe this must be created with the terms available at the time. This is what's invented, the language and means of explanation
But what math is describing is what is truly discovered
The issue is that the equations we have that govern our universe do break down at some point. Physicists have traced back the timeline of our universe to fractions of a second after the Big Bang, which is the point at which we no longer understand the physics of what was happening. In the world of quantum chemistry, there are a lot of things that we don't have accurate physical/mathematical laws to represent. We still cannot analytically solve for the wavefunctions of systems with more than one electron. We only have numerical solutions for such things. So I would say that we only have simple, elegant laws for topics which are "easily" understood, and as we branch out into more complicated topics the mathematics grows in complexity to reflect that.
Grant destroyed the false dichotomy with his answer. The man is a treasure
Imagine Katy Perry asking Grant Sanderson this question "Is math related to science?"
he would be genuinely delighted at that question and tell you why he thinks she might have thought of it that way, which is exactly what our attitude towards the 'dumb' questions should be
The reason is because the way we think mathematically and how math works are very different.
They went from Pythagoras theorem to 5d manifolds directly
Nature too has defined distance. That's why nature has patterns and ratios. That's why constants occur in nature. That's why synchronisation is found in nature. Nature does it with the help of forces. Interestingly, humans too need forces to define length, distance etc.
math is deductive. the pythagorean theorem was deduced. deduction does not create new knowledge, it only reorders existing knowledge.
KEvron
What about Axioms?
Were Axioms of (say Logic, a Subset of Axioms of math) Deduced? Or they were invented Purely as abstract form? Or maybe even Axioms were based on observation from the external world?
@@UnworthyUnbeliever try to invent an axiom that contradicts what we already know to be true.
For ex, invent some axiom where you can deduce a theorem that proves that 1 + 1 = 2365.
Math is also inductive. Proof by induction is a thing.
@@fexcasanova
1. Definition of inductive reasoning in philosophy =/= definition of proof by induction in mathematics
2. Mathematical system(s) that humans currently use is established, it doesn't makes much sense to add an Axiom to a already established system with thousands of deduced theorems.
3. Informally speaking, complex statements shouldn't be Axioms.
4. Product statements (things you normally deduced from Axioms) shouldn't be forced to become Axiom.
5. I cannot make a 1 + 1 = 2365 an Axiom because it contradicts existing Axioms (plus mathematics assume law of non contradiction, therefore, it cannot be done)
6. I Assume a mathematical system in which 1 + 1 = 2365. Since i dont care, i leave all of used symbols undefined.
@@UnworthyUnbeliever 1 - can you show the difference plz?u can copy and paste.
2 - im not saying for you to add an axiom to already established system. U did ot get my point.
If an axiom is invented, you would be able to invent a new one where you can derive what i was asking.
3 - "informally speaking, complexo statments shoudn't be axioma"
Invent a formal one then.
4 - read what i wrote again. I did not say for you to force 1 + 1 = 2365 as an axiom
5 -" i cannot make a 1 + 1 = 2365 an axiom" I DID NOT ASK U TO DO THAT. I Asked you to invent an axiom where you can derive a theorem to prove that 1 + 1 = 2365(1 + 1 = 2365)
6 - what about the theorem that proves your mathematical system??
I ask you to try to invent this axiom, based on your own question
We humanos have evolved so we can learn math to explain the world around us as much as we can learn a language so we can communicate
If grant Sanderson were my teacher for all 8 hours of school, I'd actually want to go instead of ditching
Why is it that when I say this, I’m a conspirator, hill-billy, but when he says it, it’s right? Lol
math is invented to effectively measure consistency....the universe just happens to be very consistent so they mesh well.
Cristian Proust I think you are right here, I believe math excited without us inventing it but it’s a pure form, it is not numbers or symbols, it’s just a thing that we explain with math, the concept exists without human but we use symbols to understand the concept
@@maniebrahimi7952 Not just understand the concept but also help communicate it between subjects. We create standards and definitions to help us build on top without having to recreate everything at each step over and over again.
What about singularities and black holes?
(x,y,z,phi,theta,r)= a 6d entity that defines gradient for all of space.
AI:
An oil lamp except the clear liquid is a gradient and the bubbles are parameters that perform an action. Then its p>p' goes to q, then flipflops p q>q'. if the AI improves at a particular task, p'=p or q'=q . two randoms cancel each other out on an error function and acts like an implicit rolles theorem without explicitly stating d/dx=0.
Maths is similar to musical notes. musical notes exist in frequencies found in nature and humans have invented the way to describe that note on a scale i.e. ABCDEFG#maj etc. Therefore maths is just the human explanation for what is obvious to nature.
For the people who think cows say "moo" and live happy lives on green pastures, maths is discovered.
If physical laws were different at every point in space, physicists would write that as a differential equation and abstract it away so that the theories are still about as simple
Am I the only one that felt the tension in the room?
I'd say it's a translator from universe to what we understand and use. You discover something in the universe and you express it so it's some what easily read by others. In a world where there are like 6,500 languages (according to quick google search lol) each with different names referring to the same thing, math is just a common ground for everyone, regardless of what language you speak, think, whatever.
Heck, i'd assume that early humans observed other humans from different part of the earth, and by observation and interaction, slowly learned what each other calls each object. And to think that humans from different places of the earth speak different languages, indeed makes you think how different would an alien species be...
Why is Grant's voice so damn pleasant??
I think you were citing Immanuel Kant and not Chomsky at the end of your video. Maybe both, I haven’t read up on Chomsky. Nice vid tho.
Holy shit, I think we found someone smarter than Lex.
2:16 "For us humans, it boils down to 3- dimensional world". Brilliant statement, and the foundation for and other number of dimensions. For instance E8, It's still based on 3-D.
Lex is like Joe Rogen, but chill and smart. They are both super inquisitive, and that makes for a good interviewer. But Lex actually understands the responses lol. And he is humble. And likeable. And not Joe Rogen.
Math is a distributed, evolving, not strictly synchronized object. The host of which is the social network of the brains of mathematicians. Which is part of the social network of human brains. (In other words, mathematics is a subculture of the culture of mankind.) More precisely, this object is a model of mathematics. Which is part of the model of the world.