Grant Sanderson (3Blue1Brown): Is Math Discovered or Invented? | AI Podcast Clips
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- čas přidán 14. 05. 2024
- Full episode with Grant Sanderson (Jan 2020): • Grant Sanderson: 3Blue...
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Grant Sanderson is a math educator and creator of 3Blue1Brown, a popular CZcams channel that uses programmatically-animated visualizations to explain concepts in linear algebra, calculus, and other fields of mathematics.
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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
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.😁😁
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
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
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
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.
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
It’s fun to watch the interviewer slowly fall asleep as he asks each question
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
I love your podcasts man! They're always so insightful and interesting!
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.
Outstanding video, thanks for posting, much appreciated. Love your channel and work Lex : )
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.
"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.
This is literally *big brain time*
Is it just me or does Grant kinda sound like Mordecai from Regular Show?
Artkotix Art oh no... I can’t unhear
Finally a face to the voice that made me pass all my introductory math courses
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
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.
This podcast is quite a gift.
Perfect comment
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.
To See The Face Of This Guy(Grant) Is Like Discover The Real Face Of A Super Hero!! #Respect
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.
Me and my friend going to the bar: “let’s not nerd out at the bar”
Me and my friend 3 beers later:
OMG Two of my favorite people discussing my favorite question!
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.
That 'and land' at the end is underappreciated
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.
Lex seems like a smart man and does a great job on his interviews.
Two of my favorite voices.
7:27 that's why I love this podcast haha
dude's channel is bonkers, love this guy
my brain be like : "Aight, imma head out."
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
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.
Im jealous of Lex, he gets to have conversations with Kasparov and Grant
I cheer for the mention of Vladimir Arnol’d!
"Pure puzzles and abstraction" are the loves of my life.
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.
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.
It is shocking that Lex is the only one doing deep interviews with intelligent approaches
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.
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.
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
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 ❤
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!
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.
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.
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.
There needs to be a Part II for this interview---or maybe a full series. 3B1B and LX FM are earnest in their explorations.
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 appeal to the weak anthropic principle always shuts down the most interesting lines of inquiry.
yet self bias is often the most overlooked fallacy
"The fact that we can fly is pretty great... and land." -Lex Classic.
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 feel like I’m listening to Joe Rogan on a DMT trip.
Brilliant Guest…
Machine learning can be used to learn new physics or math. For instance. Can you learn the gravity value of each planet from the motion of all solar planets? Use a all planets as the training set and ?pluto as the test set. To see if the model predicts it right.
If by the "gravity value" you mean G then we already know it to be a fundamental constant of the Universe. If you mean "gravity value" as g then that can easily be calculated using kinematics.
Additionally if G was wrong then General Relativity would also be quite wrong and Einstein wouldn't be so famous.
Although if you mean "gravity value" as in the acceleration due to gravity on a planet due to all the other planets, then you're talking about finding solutions to the N-body problem. Perhaps the advent of Machine Learning could help understand these problems' solutions, but it'd be meaningless since ML might not give us a rigorous approach to solving these solutions, rather only values.
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
his voice is so soothing
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.
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.
Why is Grant's voice so damn pleasant??
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.
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...
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.
I must today evening in my reality with a distance of ~100 Meters on a trainstation heared a group of guys talking about gravetation.
I smiled about this fact.
I strive to articulate those thoughts. They speak really well.
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.
I have asked in some forums that given that general relativity models the universe as a 4-dimensional coordinate system, whether the universe actually _is_ 4-dimensional, or whether the 4-dimensional math is simply used as a tool to make calculations easier, and the universe isn't _actually_ 4-dimensional. I haven't got a clear definitive answer.
"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.*
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
This hole is very deep man!
How about perhaps a third alternative. Perhaps maths is understood. I have no real idea what I mean by that. It came to mind when thinking about the film The man who knew infinity and Srinivasa Ramanujan. From what I remember he had some or should I say many mathematical ideas but he could not communicate them to his teachers and peers. I'll have to watch it again to see what I may be thinking about.
Does CZcams 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.
Wait. About when you were talking about the Lp spaces, why did nature choose an L2 metric?
Could an unconscious observer, recorder or universe observe or record anything?
I assume so, unless the "observer" and "recorder" are really poorly named.
@@uncleswell It's the same oxymoron from the Bohr-Einstein measurement problem debate just repeated today, where Einstein apparently commented he thought the moon's existence did not depend on his having to observe it.
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.
I think we are attracted to compressing information into neat equations because our brains are biased towards that. We want to represent the world in the simplest ways where each symbol packs the most informational value because it made us better survivors as we evolved.
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.
Counterpoint: The usefulness for any given law of physics is directly proportional to the number of geometric instances that are germane to a given universe that operates with those premises. Relevant information compression is a utility goal for any person that learns about its environment.
- When a person computes their world via lossy compression, but with a high compression ratio it's called "intuition."
- When a person computes their world via lossless compression, but with a low compression ratio, it's called "experience."
- When a person computes their world via lossless compression, and with a high compression ratio, it's called "physics."
Can you expand on this?
What is the meaning of 3 blue 1 brown and what is r2?
IMO and based on my very limited exposure, mathematics may be simple and beautiful for the mathematician or physicist or philosopher, but expand the compressions that looks beautiful, and it can quickly become overwhelmingly complex (hence not beautiful) while still not being able to model/describe all reality or even all of the subfield a formula is applied to.
Would you consider the idea that uniformity in physics across the universe and simplicity in the smallest units of the universe would suggest a creator of the universe?
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).
My I.Q. increased a few points just by watching this video.
it's entirely possible there are aspects of reality that have no descriptive analog, but how would you know one if you ever came across it? is it godel incompleteness? is it quantum gravity? also, how would that information be useful to your every day epistemology? kind of just sounds like a non-starter question to ask.
This is mind blowin
Invented obviously. The people who say otherwise are usually invested in saying that math is more valuable that it really is because 99% of it has no applications. Even the more abstract math that has found applications in physics can be used per se. You can take those specific tools that improve predictions without the rest. This should seem obvious, but people act like that part being useful implies some abstract truth quality about the rest of math which is unfounded
I think that we encounter physical phenomena and our thinking forms abstract concepts which tries to grasp this discoveries but every person itself will come across other abstract concepts and the one who seem to describe nature the easiest way will push through but overall everything is only a categorization and relation between them build by human minds.
Patterns are discovered, the symbols used to describe them are invented. But since everything we perceive is something that already happened, an update in the system, however shortly (or long, as in the case of light from the stars), just before we became aware of inventing it, so in a way we're discovering everything, even thoughts we think we generated.
What if, claim of invention was a mere interpreted form of discovery?
I agree with Grant about math invention/discovery, under the assumption that math logic is right. I wonder what are Grant's thoughts about that.
If the logic behind math was discovered then all of our "inventions" in math are in fact bounded by the possibilities generated only through this logic.
In particular, if we refuse law of excluded middle we would not be able to proof a huge number of theorems, including those with the biggest impact in our lives.
Greetings from Poland! :)
Grant destroyed the false dichotomy with his answer. The man is a treasure
It’s not weird at all that “our” universe obeys relatively simple and elegant laws of physics. Chances are is that the vast majority of phenomena are extremely chaotic and so naturally, are ill-conducive to stable interactions that form observably emergent phenomena in the first place.
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.
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