Is Math Discovered or Invented?
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Is math invented or discovered? We'll try to answer this age old question in today's video.
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Ok discovered vs invented is probably too simplistic, so share your full ideas down below.
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Both.
You invent the axioms and discover the consequences of your actions
The question is really, is truth invented or discovered
@@ryanthescion so what you’re saying is that mathematics is the ultimate version of _fuck around and find out_ ???
@@doctorscoot always has been
This is exactly what I was going to say and I don't understand how anyone who's thought about it could possibly think differently.
Yeah, this is just the only consistent way to look at it, imo.
To equate this to the species example, we invented the definition of a species, and then discovered things that fit that definition.
I vote for imagined.
As in "people imagine they understand it"?
We can invent new chemicals by combining different molecules, but we do not create the elementary particles out of which they are composed and we cannot just will them to have certain properties. Whether a chemical cures or causes cancer depends on the laws of nature. If we want to invent a cure for cancer, we must discover what combination of molecules will do so. We cannot just choose for a mix of salt and sugar to cure cancer.
Likewise, we can invent new mathematical objects and operations, such as matrices and their multiplication, by combining different definitions and axioms. However, this does not mean that we create the fundamental logical concepts out of which they are composed or can just will them to have certain properties. Once we define matrix multiplication as we do, the laws of logic determine that their multiplication is not commutative.
The "laws of logic" are completely made up too. We assert them because they are useful but they are certainly not discovered
Your personality and communication style is incredible, I wish you and your channel lots of success and thank you for the content :-)
Haven't watched the video yet, but here's my thoughts: An axiomatic system is invented in the sense that we choose it as a place to study. But then the results we find are logical implications, and classical logic itself matches the real world. So I feel that while our choice of study is a choice (so "invented" for the question at hand), what we find is discovered. As another related note, the math itself transcends all notations, while notations are an invention.
I like that explanation. The axioms are invented, and their logical consequences are discovered.
So, it is the logic that exists in and of itself.
I suppose math describes the real world only if the axioms describe the real world. If the axioms do not, then the math will not describe the real world.
Does that sound right?
Has anyone created axioms that don't describe the real world?
@@divermike8943 Well part of my explanation is I don't think the axioms are invented either. Sure, our exact wording may be original, but there's certain sets of logical axioms that are equivalent.
But it's our choice to study that set that is special. We could have chosen to study something else. But we chose to study this set. That choice is our invention. Not the logical consequences, and potentially not even the axioms either.
Enjoyed this a lot; particularly like the point you made about abstracting vs. inventing specific cases. It definitely feels like ideas like continuity or differentiability, while they seem very natural, have an element of invention to them. But it's hard for me to imagine a different version!
I had a discussion with my friend about this, recently, and as an (amateur) linguist - he had a very particular answer as to why it's inherently of our own creation, and humanity has always known this:
The very basic, fundamental concept of plurality itself (that mathematics is inherently based upon) is inherently of human creation, and humanity knows and understands this in a fundamental way. Humanity has always seen to recognise that the world/universe exists as itself solely based on all individual 'things' (dependant on scale) that exist as such, independent in time/space. Treating any such individual things as being the 'same/identical' is therefore a HUMAN created concept. Treating them individually, then, is seen to be more consistent with how the world/universe functions, which is how and why humanity has seen it's own equivalent as also being important and powerful - the use of creating individual names.
Mathematics inherently exists as human created 'content' (of/as thoughts) we apply to describe either the universe around us or other thoughts we've created, as a (collection of) system/s of measurement. At no time should it ever be confused for a) what is measured, or b) the specifics of its application/representation, (since its content inherently exists as part of existing langauges, anyway.) That it's so suitable to describe many things within the universe itself, is the entire point.
In the the first chapter of the book 'Is God a Mathematician' by Mario Livio, he paraphrases remarks concerning Roger Penrose's three worlds hypothesis. They are: 1) the world of our conscious perceptions, 2) the physical world and 3) the Platonic world. The first is how we view everything with all of our biases, the second is 'physical reality' and the third is the world of mathematical forms (prime numbers, laws of physics etc). 'Physical reality' seems to follow the norms of the Platonic world. Second our conscious perception emerged from the physical world. The circled is closed by the fact that our conscious perception allowed us to discover the Platonic world. All arguments become an infinite loop, as does the invented versus discovered debate.
To me, the most interesting part of mathematics is that pure mathematics so often becomes applied mathematics.
Human languages are tricky sometimes. In my native language, "to invent" and "to discover" have the same word.
Oh interesting, what language is that if you don't mind me asking?
@@DrTrefor Indonesian, both are "menemukan". So "Graham Bell menemukan telepon", but also "Watson & Crick menemukan struktur DNA"
Something unknown becomes known.
I was wondering exactly that. if "to invent" and "to discover" can be expressed ambiguously
Would love to see a video on simplec geometry! It sounds really interesting to apply geometric solutions to classical mechanics problems
It gets even stranger: if you take math as invented, you adopt that axioms are constructed, this itself is the "axiom of constructibility" (axiom V), leading to a universe where the continuum hypothesis holds true. On the other hand, if you take the perspective that "mathematics is discovered," the combinatorial, analytic, and topological consequences guide you toward Martin's Maximum or the forcing axiom-an axiom that is "true for as long as possible." In this universe, the continuum hypothesis is false.
So, the continuum hypothesis is false, great! 😅
I’m taking the perspective of V. Arnold “mathematics is the part of physics where experiments are cheap”.
Even the most basic mathematical axiom, like _1,_ has no meaning outside of a conscious creature's ability to conceptualize it. Whether or not the universe is out there spinning, expanding, vibrating, pulsating, and hinging back and forth endlessly on our mathematical ideals, nevertheless, mathematics as a descriptive and predictive tool still exists wholly and only within the mind. If the question is whether or not mathematics _means_ anything independent of the conscious ability to discern its patterns, then the answer is a stark NO.
I have several remarks about your line of reasoning:
1: Regardless of whether someone thinks about it, the universe seems to follow mahematical principles. So much so that some scientists say we live in mathematical universe.
2: Two independant thinkers can come up with / discovinvent the same math. But they can not come up with two maths, that contradict each other.
3: Meaning is THE fundamental social construct. Nothing in existance has inherit meaning by itself.
4: We are part of the universe, as are our minds, as are our thoughts. Thus: Math is part of the universe as well.
@@xario2007 1: Regardless of whether the universe seems to follow mathematical principles, unless someone thinks about it, mathematics has no inherent meaning.
2: Mathematics' standardization has been hard-won. It does not have a perfectly harmonious history. Such a belief would be romanticizing. If I remember correctly, they wanted to stone Euler for trying to use the witchcraft of imaginary numbers (i=√-1) in a math competition. Our present mathematics, no matter how internally consistent it seems, has not been a gift from the cosmos; it has been the fruit of our intellectual labor. Mathematics is a refined description.
3: You're making my point for me: there is no meaning in the universe beyond that which has been ascribed to be meaningful by (we) conscious creatures. That goes for math, too; it means nothing without us to consider it meaningful.
4: I have no disagreement with that statement.
The stars had spun before mathematics. They just didn't stop to think whether their spinning was mathematical. Humanity shows up some several billion years late to the party and then designates-with our typical inflated self-regard-that we have now "discovered or invented" the stars' spin. We've done neither; we are merely observers. And we're unfair ones, at that.
@xario2007 The Universe follows it's own principles, math is defined to describe and model it.
I have the exact same opinion, and you explained it really well. Very good video!
this reflects the general propensity of realist positions in the general culture with regards to meta ethics and meta physics...
really in the deep end with this topic 😅 my thoughts are below:
i dont think maths is the squiggles on the paper or the sounds we make when we speak, it is really what happens in our minds. we can guess about the world and have models and whatever, but really thats just our ideas. "maths" ideas, specifically, are more abstract and general when compared to what youd call physics ideas, physics is describing things we experience very directly, maths is much much dreamier.
really what i mean to say is: i dont believe there is any "correct" way to reason. most mathematicians appreciate there is no ultimately "correct" set of axioms, but also i think there is no "correct" way to even imply anything from anything else.
ultimately this stuff leads me to believe maths is just our ideas, and they can be whatever you want. all of that said, maths is very useful, and attempting to reason is the best we have and is going okay so far i guess :D to answer discovered or invented, i say both are bad descriptions and miss a whole lot.
anyone who read this ty, if u think differently maybe tell me why :3
Everything in math is on a spectrum between discovery and invention. The question is, how much of this is the human responsible for, and how much of it fell out of the universe and onto paper on its own?
The logical structure of math exists outside of all of our constructs such as the language we express its ideas in and the paths we take to traverse it. And outside of the universe, even.
So I say that all math is a discovery to some extent. And over time, all math “inventions” will shift towards being considered discoveries, as we uncover uses for these idea and consider them to be more obvious and fundamental to reality than we did before we knew about them.
"Discovered OR Invented" I'd say TRUE.
graph --> hyper-graph
polyhedra --> abstract polytope
integer --> complex numbers --> inf dimensional complex-like number
integer --> surreal number
"inf dimensional complex-like number"
What's that?
@@bjornfeuerbacher5514 Complex numbers (2D) can be extended to quaternions (4D), octonions (8D) etc… and at the end of this series is an infinite dimensional type.
@@alan2here As far as I know, this sequence can be extended at most to 16D, afterwards, it breaks down?
@@alan2here what do you think about bjorns reply, I’m curious
@@bjornfeuerbacher5514There isn't anything stopping you from going further, there's just not really any point in doing so as it is uninteresting
As a PhD student who’s also doing algebraic topology coincidentally, I refer to the techniques that I use in proofs as things I invented, while the fact that I’m proving as discovered.
I think math is discovered but the applications of that math are invented.
Mathematical relationships exist- the patterns are simply uncovered by inquiry- we invent the signifiers and descriptions that describe the relationships just as in chemistry there are certain elements that exist due to physics and then particular chemicals that are created by describing their patterns of useful combinations and exploiting these relationships
The way I look at it is that these mathematical relationships, facts, theorems, axioms and whatnot are fundamental to our universe and that we discover them, but we invent the language and the way we communicate said relationships. An example I would use is a red apple, or that's what we call it in english but in spanish the way they describe that same object that has the same fundamental properties that don't change no matter how we choose to describe its existence. Just as languages come up with ways to describe objects and whatnot, we come up or "invent" the language we use to describe the relationships we "discover". And just as languages change and evolve over time, the rigor and way we describe these fundamental properties we call "math" using the language of mathematics will change and improve over time to best describe these properties, just as languages come up with new words when needed and how definitions can change over time.
First: great video and thanks for this topic! The more I think about it, the more I struggle with and it seems more diffuse. In the beginning I also was tempted to say it's clearly discovered. But like you showed with terms of "definition" or lets say "constructions" it's definitely more harder to decide that really clearly. For example any definition of an Integral like the Lebesgue ones. Does it really exist? Especially if nobody is there to define it? And what means Definition really? To me it's like filtering out some skills of patterns in nature. So that we are be able to work with. It means to shrink something (or more precise to idealise) but nevertheless it seems that it works, because we can apply those definitions to make forecasts and so on, only based on some small rules ( but those rules or behavior must be recognized or "discovered")And here's really the magic: despite of our restrictions, means: make definitions and so on there are consistent logical outputs that you can verify or prove. So that means those rules seems to be transcendent. Thats maybe the real core of mathematics. At the end maybe we can answer this question in that way: Is Mathematics invented or discovered? It's both.
Calculus was invented by Newton and discovered by Leibniz.
I’m stealing this
He's casually showing off his golden pi.
FINALLY someone notices:D
Does the universe perform any actual calculations somewhere in physical reality? If so, where? If not, what actually governs the behavior of the universe because math is just an observation of the effects and not the cause of behavior.
"observation of the effects"? I think "description of the effects" is more appropriate.
And math isn't only about calculating stuff.
I was about to write a piece on this, glad I chose a different topic!
Math is discovered. Math notation is invented.
Ya we can def agree on math notation I hope!
Math notation often looks just haphazardly and inconsistently made up.
It's not even a matter of opinion it's just a fact there's nothing to disagree about here.@@DrTrefor
A notation is our method of communicating the ideas and concepts of math concisely, and so long as there are enough people who agree to that notation then the notation is valid.
It's not that black and white. Math means a lot of things, like theorems, proofs, constructions, definitions, etc. Math is composed by both invented and discovered things
@Lolwutdesu9000 We can definitely invent math. For instance: let there be an algebra constructed by two operations, one that takes the maximum of two numbers, the other that multiplies these two numbers. By these definitions, I just invented a new algebra, or a new "math".
i personally sit in the boat of "some math is invented, most math is discovered." like. complex numbers, irrationals, negatives and the general types of numbers were discovered.
but the collatz conjecture and whatever it affects, is invented.
Invented~Discovered is a dual concept, like Rights~Responsibilities or Wave~Particle, it's both
If you don't understand how it can be both, think about this: I am lying
Hahaha! You are a funny cat!!!
We are yet to discover/invent a word that means "to discover/invent".
Incover? Disvent?
@@gcewing I think discovent rolls easier off the tongue 😃
As hoc I would claim all the following (though I can't proove all of that):
We try to discover all (or art least as many as possible) real existing structures (structures that are instanciated in reality) by guessing some basic truth in the form of true propositions (which we call axioms) and try to discover as much derived truth (also in form of true propositions, which we call hypothesis, theorem, ...) as possible. We use definitions to describe structures that we want to study and we use an invented notation to reference and/or represent objects.
We call a specific system of basic, derived truthes, definitions and notation an '(sub)area of Mathematics' and name it, to be able to differ between multiple ones. We call all areas of Mathematics combined Mathematics.
If all axioms are real (in the sense that they are instantiated in reality), then we discovered all axioms, and discover all derived truthes.
If at least one axiom in a set of axioms is not real (in the above sense), then we might derive truthes that are not instanciated in reality; I would call those truthes (axioms and derived) invented, though we might also discover real truthes (and we may not be able to differ between invented and discovered ones). However, I believe that invented axioms always lead to contradictions; that also seems to be an historical assumption, because as far as I know, whenever axioms lead to contradictions, mathematicians seemed to have either completely dropped such bad axioms (at least those they could identify) or they have replaced it with a more restricted axiom (which often came as a 'you can't do that specific thing').
I would say methodologies are invented, while associations are discovered.
Inventions are solutions to problems (sometimes problems that were imagined out of nowhere, but problems nevertheless). When you discover a connection, inventing the method to express that association is the task you have before you, and vice versa; create a methodology and discover the way that generates associated data.
Ultimately, I'd say it's 50/50 on the nose.
Hey thats me!
math is discovered by invention and invented by discovery
I wonder about origins and how things came to be. I've been thinking of Euclid's axioms and why he would think of such a thing, the Axiomatic Method. He was solving a philosophical problem. For centuries, the Greeks have proved interesting and important theorems. A simple example would be:
To prove A, you ultimately have to assume B. When you prove B, you assume C. It turns out that C relies on A. You have circular reasoning. Euclid's genius was the decision to choose a set of "truths" to break the loop. So, he invented the Axiomatic Method but discovered appropriate axioms.
Maths to me describes the relationships between things. As long as we can agree that there is an object to talk about, then there is a language or framework that the object exists in. The axioms we choose may be our choice but choosing different axioms doesn't make the relationship between the things that make up the object not exist.
The mathematical abstraction are invented, and their properties, discovered.
There’s an interesting connection to Turing-Completeness here. When a computer is Turing-Complete, it can simulate any other Turing-Complete system. So I think that while we may start in a particular axiom system, many other axiom systems will emerge from it or be dual to it. If our universe is itself a Turing-Complete system, then we are involved in this interlinked network. That’s why math is so effective in physics.
Very interesting idea. Cannot attest to the veracity of it but definitely interesting
The question really comes down to a philosophical question (a very real one). Are _Plato's Forms_ real objects? One issue that empirically minded people might have with the idea of the _Forms_ being real, is that you pretty much run straight into the problem of God, or the divine at least -- the _Forms_ themselves are perfect abstractions that don't really exist in the Universe. Take almost everyone's favourite number: π (pi). It's an irrational number, never-ending. Yet, with only a _finitely many number of digits_ (I believe it's less than a hundred), you can calculate the circumference of the observable Universe to the accuracy of less than a proton's diameter. That, to me, implies that there's a limit, a finite bound, beyond which the mathematics of π are effectively an _abstraction_ -- mathematics is invented. It's a game, we invent the rules and we play out the game.
The only question left for me, is then, why is this game so damn _useful_ ?
Just looked it up; you only need 63 digits of pi to calculate the circumference of the universe to the precision of a Planck length (the smallest length that can physically exist)
I'm reminded of Turing's Halting Problem. Build a program which when given the text of another program, return 1 if it terminates and 0 if not. He gave a wonderful proof that there is no such program. He was not worried by non-terminating programs. His machine was invented to model what a mathematician does on a piece of paper or on the blackboard. The mathematician can always stop when tired of calculating. Newton considered the calculation of π to sixteen digits was a sufficient waste of time.
It's useful because, for thousands of years, less useful formats get replaced.
Agree 💯 that it can be likened to inventing rules for a game, then discovering strategies to win.
Discovery feels linked to 'searching', and mathematicians are often times searching for a system/prime or something like that, which makes it feel like discovery.
Contrast this to invention were only parts of the invention are usually discovered, but the invention as a whole isn't.
There is a book about this topic, it's called "Pi in the sky", it ends with the words: "We are the children as well as the mothers of invention." To fully get this quote I recommend to read the book.
To take your example of the iPhone. It certainly wasn't discovered (unless there is something Apple haven't been telling us) but invented doesn't seem like quite the right term either. An iPhone is based on many separate technologies that were each invented, but overall I would say that the iPhone itself was designed. That isn't quite the same as invented - there can be many different designs that do more or less the same thing using different sets of technological inventions.
Maybe the same is true of some mathematical systems? Eg Zermelo-Fraenkel with or without the Axion of Choice are not really separate discoveries or inventions, they might better be described as different design variants.
3:16
2×3×5×7×11×13+1 = 30031 = 59×509
Yes. Euclid's proof says the product of all primes + 1 is either a new prime or a compositive number with a prime factor which is not in the list. I don't know why Dr. Trefor said the theorem in that (wrong) way...
Oh yes I misstated that - oops, I'll add a pinned comment
And in coontrast, 2x3x4x5x6x7x8x9x10x11x12x13 + 1 is a large number which is prime, since it always has reminder 1 when you divide by a whole number up to 13.
I see them both correct, it depends on the perspective :
Invented : we invented a way to manipulate the abstraction of nature rules, logic,laws, patterns...
Discovered : we dicovered how quantities interact with eachothers ( numbers system is invente)
Sorry my english is bad.
I think the universe *is* mathematical in nature, and what we can currently describe are just emergent systems that arise from that model. This is consistent with what we observe and follows Occam’s Razor neatly. Whether we can adequately or even perfectly describe that underlying model-practically or in principle-is difficult to say.
Pre-watch statement of position: "Discovery" and "invention" are both concepts that refer to human activity in the physical world, and applying them to mathematics necessarily requires an element of metaphor. Once the question is framed as "which metaphor is better", the expectation that any objective answer exists to be settled can be put to bed. (Which, coincidently, is where I'm going now, otherwise I'd watch the video right away.)
Had this discussion with a friend a while back, wonder what your perspective is on the topic
As a language, used to describe the many multiple functioning variables within the cosmos.
It is just in physics on gravitation. You remove humanity/sentient species on earth but earth will still follow the law of gravitation. Meteors colliding with planets, planets rotate around the sun etc...
I think you argue very strongly for compatibilism. Aka. while all things are discovered (given the library of Babel) none the less upon using our focus we decide that which is meaningful
and therefore invented
I come from a physics/engineering background and I've always looked at this question in the light of Wigner's comment on the "Unreasonable Effectiveness of Mathematics in the Natural Sciences." We discovered that if we take an object and put it with another object then we can exactly hold an object in each hand (assuming we have the normal number of hands for a human). Then we invented a notation for this, the numbers 1 and 2 and the fact that 1+1=2. We then discovered that this generalizes to more than just 1 and 2 in the physical world so we invented more notation. And so on. Eventually, the discoveries depend upon notation that was invented earlier.
So my answer is both. We discovered that the universe seems to run according to some rules and then invented the terminology to be able to reason and discuss those rules of the universe. And this loop continues.
A lot of mathematics are fundamental realities and are discovered. For example all numbers being products of prime numbers , the number e, and pi were discovered.
Other maths like those dealing with approximations, precision ,and error rates are invented since we are not capable of infinite precision.
kinda crazy this video was recommended to me when my chair broke...
The algorithm knows all!
It's framed. Neither invented or discovered. Basically, it just gets more and more clear the more we learn about it. Every equation gets more focused the more advanced the mathematics become, or the more we need to do with it, so it's like framing a picture, and cropping parts of it. Like, all principles in math are just built from other smaller principles, starting with the basic operations, all the way to a shape that could describe Reiman's Hypothesis. You don't discover the shape, or invent it, but describe it.
Did the questionnaire determine the background of the filler inners?
I am a retired UK academic, having had a career in theoretical physics/computational chemistry.
In my experience Mathematicians like to imagine that they discover Maths - so consider themselves scientists, whereas Physicists consider Maths to be invented, so consider Mathematicians to be more artists/engineers than scientists, so can look down on them.
2:22 I reject that there being infinitely prime numbers is a _fact_ : it is only so after the _concept_ of prime numbers has been invented, and the _concept_ of infinity has been invented, that we can deduce the _concept_ of infinitely many prime numbers. Concepts cannot be discovered, but their consequential deductions can, but that is not the same kind of discovery when we discover a fact that comports with reality. The consequential conceptions are functions of the axioms, not of reality - that which can be measured.
(My study seat is from a Jaguar XJ-8, complete with heating and electronic controls, so no FlexiSpot for me!)
I agree with the comment you showed, that we invent axioms and then discover what they lead to. But if someone asks me that question, I answer "invented" to counter the common idea that math is meant to describe the real world. It was in the past, but not anymore; math is its own imaginary world that just sometimes has some parallels with the real world.
I think the words invention or discovery themselves are too vague to describe or relate to something like maths.
However, I contend maths has originally evolved alongside natural languages.
There is no difference among English words table, man, five, addition, for example; they are all just normal English words; but five, addition also belong to what we later used to call as Maths.
Then we naturally and gradually developed mathematical concepts - I could narrate like "first came counting numbers (up to some upper limit, like 4999 as in ancient Roman number system), then addition, then subtraction, then negative numbers (hence positive negative distinction), ..."
This is mainly arithmetic. Mathematics is CONCRETE in its origin by necessity. Later it was developed and more and more ABSTRACT concepts were INVENTED to "handle" (interpret) the norms or physical world.
I don't think "discovery" has a formal place in the story of history of maths, because it has been going through a kind of evolutionary process.
Ya it is a bit awkward trying to take math - where we are always super precise - and apply vague english language connotations.
Down with platonism!
lol, we need some signs!
I am to some kinda platonic
Down with nominalism!
Just because you got friendzoned by your crush doesn't mean you have to take it out on the rest of us
I would say math describes relationships, the relationships we find is what we discover. But the symbols and the way we approach addition, subtraction, multiplication, division, etc. I would say are invented, since you can use the Roman numerals for example, or a different method to represent addition.
I like to think of it like Celsius or m/s. They are not the only way to describe reality, so they must be invented, we have for example Kelvin instead of Celsius. But their relationship to reality is real, and one could say is discovered.
Another way to think about it is like natural language. The word Rock describes something real, something discovered. But the word itself is invented and a multitude of other words can be used from different languages to describe the same thing that the word Rock describes.
And that word "rock" has multiple meanings e.g. it can be a music style just as well as type and/or size of stone.
And many other words also have multiple meanings.
While the aliens argument seem’s to work well with simple concepts, in my opinion it can’t be extended to complex/specifics things.
Imagine there are countless Aliens cultutre and, by chance, one of them invented the chess game before us.
Well, it’s hard to say we haven’t invented chess as well, exactly as both Newton and Leibnitz invented calculus at the same tine, in different places, and their independence is shown by the greatly different formalism and notation they used to develop it.
Besides that, even without take aliens in the game, we see the result of numbers and a lot of complicate calculations in nature (while it’s unlikely the the nature actually counts), but we don’t see in nature things like chess or the result of the Zermelo’s axiom of infinity, merely because we don’t know if the universe has infinite parts and in any case we can’t check infinite things.
Math is the result of defining a set of objects, operations, and rules, and the result is anything you can create with those 3 things.
An integer is an object that supports multiplication, addition, and subtraction, etc. any operation which produces an integer from integer inputs basically. However, division is an operation that can only be defined in the rationals, and comes with the definition of a rule that you cannot divide by zero (at least, in our normal number system). Irrationals and transcendental comes with the introduction of further operations and expanding the definition of our “number” object.
If you ever don’t have a broad enough definition for your object, or want to add or remove rules, you can do so as you wish as long as they continue to be consistent with their definitions. Real numbers not enough for you? However about we add the square root of the negatives and make the complex numbers? Want zero divisors? Go ahead, that’s possible. Maybe you’d like to differentiate between two different infinities based on its rate of growth? Sure, rules exist for that. And nobody ever said numbers were the only types of objects, there’s vectors and matrices, or sets, or you could even define math objects as a physical object such as a cake that you wish to divide with a cut being the operation.
Simply put, math is both discovered and invented. Some objects are defined as real things that physically exist, others as various levels of abstract ideas that are invented to fill the gaps in other systems. Some operations are naturally obvious to the universe, like things that affect quantity in an obvious way, others are more abstract in real value, like the totient function. Some rules inherently make sense, some we see as soft limitations caused by badly defined objects and operations that can simply be resolved by expanding an existing definition. Math is just systems of objects, operations, and rules, and whether it’s discovered or invented purely depends on if was made BECAUSE of the real world, or created arbitrarily just to screw around.
Invent a proceedure, use it to Discover truths in mathematics.
The true math was Aways there, we discover it by Inventing ways to do so.
We invent mathematical tools so that we may discovery the beauty of maths ❤
I think the denial of the existence of discovery is a recent phenomenon that a good sleuth would be able to bloodhound down to an original midden at the heart of someone's political faith.
Because rigor in words would lead to rigidity, there's some plasticity in most of our words. They come fitted with a wide error bar to avoid narrowing the scope of their metaphoric connectivities - or the range of concepts they can be used to make "real" (as if tangible) by intuitive associations of the kind we naturally use, even if we're primitive people, barbarians, cave men, and other forbidden descriptors such as these. A good example of such metaphoric word use would be the currently fashionable use of the word "tool" to mean not something you can hit, twist, or machine something with, but as an intellectual device which can fruitfully be imagined as hitting, twisting, or precisely machining some conceptual framework (itself a narrative device to tell such stories, since we find stories intuitively instructive).
So the word "discover" is something you know from experiencing the various ways we use it these days, and even how it's been used in various ways for, say, 300 years (where the lack of rigidity of meaning allows it the meaning to move around a bit).
Of course to have a historical perspective of the meaning of the word, it's necessary not to have too strong a degree of contempt and ridicule for the absurd little childrens passtime called "history".
What does "discover" mean? To some extent that depends on what you know and what you are ignorant of. To some extent (it also turns out), it depends on what you try to force it to mean, and on how ignorant you either are or pretend to be (of things like its range of meanings.)
This ignorance is not stupidity, it's just a by product of the fact that nobody knows everything. ("Ignorance" is a word that can be used in a variety of ways, too. And some people say that's just because it's a word.)
Anyway, what does "discover" mean? Or since this is a quite concrete thing, let's say what does it mean to "discover" the Americas, for instance? Well for many centuries that meaning was pretty simple to anyone not having some need to feign ignorance of it (or perhaps just be a scholar whose analytic toolbox is a language, but who somehow failed to know how this little spare part in that toolbox is generally treated as functioning - a language peddlar whose language skills are deficient, like unto a calculus teacher who only knows how to add and subtract some of the integers - which would be a strange thing, no?).
Let's pretend to know nothing and explain it as if to a very stupid child (the way lots of highly educated people explain things to the stupid adult children they share their wisdom with). "Discover". To encounter something (generally by a search motivated by the surmise that it exists) that the "discoverer" didn't know existed before.
Yes, duh. And sorry for that, too. But how do you go along with the game of feigned ignorance without being reduced to some kind of spelling out of the obvious like this?
In a cognitively whole-minded world, I suspect that maybe we need to be able to move back and forth between loose and tight thinking? I mean how do you begin to formulate even a deep-mathematical suspicion you have, where maybe there isn't even an existing notation that's going to be useful to you? Maybe you reach toward the idea using words either a bit poetically (just in that direction - metaphor and crap like that), or the way you might use the drawing of some rough sketch curves you made (which is maybe more "linguistic" than mathematical - at least in this way).
And how do you explain things to even intelligent novices? Just pile in like an old fashioned textbook, speaking sentences in tight symbols that maybe once first had to be explained to you in a quite loose manner when you were young? Or do you draw pictures? Or draw more abstract "verbal pictures"?
But what do we mean by X discovered America?
What we obviously mean, using either the ordinary, natural language of thirty years ago, that we intuitively understand, and don't use formal definitions to narrow down to some accurate but narrow meaning, or by simply not pretending that English is a bunch of single words in a dictionary, all open to appropriation and redefinition in ways that constricts these words to meaning what we want to narrowly force them to mean.
By common sense dang it! By a bit of nous. Know by feel when to tighten up in areas where we've found restriction gives us enough sharpness to make up for the narrowness that comes with that, and when to loosen up when fixing on one meaning is just an attempt to make a point that isn't really a point. (It just feels like one. Like taking the word "discover" and making as if ordinary common sense is somehow inferior to arbitrarily imposing a narrow definition on it.)
We mean "we" didn't know about America, and then it was (don't play stupid, please) discovered. We found out about it. But we don't speak like little kids, so we can use words like "discover" to mean that. It's too simple a concept to require simplification.
OK that's a thousand words, so I've now answered the question enough. :D
I see it as invented. A language that describes things in a specific way. It was invented to figure out how many things we had to sell and how many we could buy, and how much each costs etc. One apple plus one apple is two apples. So it's a great tool. But that's it.
Could topos and higher topos theory replace FZ as a foundation for math ? Math invented to accommodate certain ways of conceptualising and generalisation? Asking. I Don’t know. I want one of those chairs !
Ya I definitely think there is significant choices in the way we create mathematical foundations
My personal take is nuanced but I think it covers all bases.
Mathematics is in a sense "applied logic" - logic tells you "if X then Y", but can never claim True/False. Mathematics is the act of picking some propositions to consider True by default (called axioms) and then finding out what web of truths unfolds given your choice. Therefore, mathematics is "invented" in the sense that _your choice of axioms is a choice, and not fundamental to reality._
Logic may or may not be invented as well, but I think it's safe to say that reality certainly appears to have consistent rules. Nothing in science has ever been inconsistent so far (note: probabilistic doesn't mean inconsistent!). We'll assume that consistent logical rules are not invented, but you are free to disagree of course.
So the rules of logic are discovered, while mathematics - the application thereof - is invented.
If this is the case, how do we explain the "unreasonable effectiveness of mathematics in the natural sciences" as Wigner put it? The answer is simple: _we didn't pick arbitrary axioms!_ Most mathematics done today falls under ZF(C) set theory. Certainly other frameworks exist, but they are almost never needed for the sciences. So we'll look at ZF(C) for now.
How did ZF(C) come about? Even though today it forms the theoretical basis of mathematics, it wasn't the first mathematical notion humans came up with. We started reasoning mathematically by dealing with "what felt intuitive" and "what was useful for everyday tasks" - counting, geometry, algebra, etc. We didn't consider the axiomatic base of these fields at first, we just "did what felt right". Naturally, then, these fields slowly formed axioms that _resemble those followed by our universe_ - Euclid's axioms of geometry are all followed by our reality (though the 5th breaks at large scales, it is true on human scales).
Millenia later, mathematics had become much more advanced and developed, yet remained "intuitive" in this manner. People then decided "wouldn't it be nice if this was all unified under one rigorous framework?". This line of reasoning eventually led them to the realization that _most existing areas of mathematics could have their axioms founded on Set Theory,_ which itself could be made rigorous using a small collection of axioms (~8, depending on formalism). These are the axioms of ZF(C) we use today.
So we see that _contemporary mathematics was designed to fit our intuitive expectation of how the universe ought to work!_ It can be considered a "rigorized form of intuitive reasoning". This leads to another question: _why were our intuitions seemingly quite accurate?_ This comes down to evolution: beings that could predict the behaviour of the world around them were more easily able to spot threats, find food, build shelter, and in general _being good at science makes you better at survival!_ So, by natural selection, humans evolved a strong intuitive sense for causality and pattern-recognition.
In summary:
1. The universe appears to follow consistent logical rules (cause-and-effect). This means that creatures who can reason with logic are at an advantage, as they can plan ahead rather than purely react in the moment. For this reason, humans (with our large brains) evolved a strong and reasonably accurate intuition regarding logic and (human-scale) physics.
2. This intuition developed into mathematics, which is the application of logic to a given set of axioms. The axioms we chose are specifically designed to match our physical intuition when it comes to things such as counting, geometry, and algebra. Since these were evolved via natural selection, they obviously reflect reality quite well (as intuitions that didn't would cause creatures to make inaccurate predictions and not survive as much).
3. Therefore, our formalized mathematics reflects reality quite well!
I would thus argue that logic is "discovered", mathematics (in a broad sense) is "invented / chosen", while contemporary (ZFC) mathematics is "founded on discoveries" (in the form of human intuition / physics).
I've always hated this debate because it's not one or the other, it's both.
Math structures can be seen in real world but math itself is theoretical representations of them. A language, a tool to analyze and explain them. Guys you dont discover tools and languages, you invent them
But one can discover the consequences or properties of such inventions.
@landsgevaer literally opposed with yourself.Those two you discovered still arent results of or as inventions individually nor together
Forget all that, if true; thing you just said will still be the continuance of our language invention math
Math is a language (of sorts...) that can be used to describe certain objective things in the universe (and therefore has some aspects of "discovery") and can also be used to describe certain non-objective things that are the fancies of the human mind (and therefore has some aspects of invention). It really is not that complex not is it necessary to be in any way reductive about the question at all.
I would argue that invention and discovery are two sides of the same coin. Lets take the anaconda you mentioned for example, you have to invent a method to differentiate 2 snakes apart to even discover that they are different species and to create such method you have to discover that different species can look quite similar at first glance. From my perspective invention is focused on the act and method while discovery is focused on the end result. When you invent something, you discover many paths that don't work and some that do, when you have discovered paths that work and you put them together then you have invented something which in turn is just a discovery of an interaction between different discoveries.
We invented the game of chess, we didn't invent the universe. We set axioms to explore possibilities that fit reality
The moment one jumps into the exclusivity of the “or” is the moment one limits the probability of being correct. - Me.
lol I like quoting myself too
@@DrTrefor ... you get quoted all the time (deservedly so) ... I didn't think anyone other than me would quote me, so I decided to be quoted at least once. 😁
As long as it's "or", but not "xor", I wouldn't say that limits the probability of being correct. )
I agree when in programming (or even formal logic), but respectfully diverge if in conversational language inquiries.
@@xyz.ijk. So if someone says in conversational language "it will rain or snow today", and if it afterwards rains _and_ snows on that day, then you will say that he was wrong? Seriously?
Haven’t watched but… I believe math is a way of interpreting the world and investigating the relationships between different things. While the notation we use is invented, the underlying logic behind what we’re describing is discovered.
Axioms are invented. Anything that follows are discoveries.
Ya that’s for sure a common answer
The more you learn about math the more you understand how it is discovered, it may feel invented if you don’t know where something comes from e.g quadratic formula, but we can only invent the notation and our perception of the concepts that exist without our presence.
Wow "Algebraic Topology"!. I asked Microsoft Copilot and it states "Algebraic Topology" by Allen Hatcher is a standard textbook. I suppose also Greenberg and Harper "Intro to Algebraic Topology". But I have been neglecting to study or review or look at Algebraic Topology. I convinced myself (perhaps erroneously) that Algebraic Topology tries to tell whether two shapes are equivalent (isomorphic) using algebraic structures but given the subject of computers and something like an Alias Wavefront .obj file one wants to know whether the two shapes in an .obj file are different. I think that led to believe that computational algebraic topology was more important and then to the subject of creating a texture map for a 3D .obj file. I just left my curiosity there not seeing any way to explore the topic further on the computer. I wondered then if Algebraic Topology had a real use outside of mathematics or applied to the real world. I guess I am silly so I do not review my Algebraic Topology textbook much being not motivated to review it. I still like my textbook, but I read Neuroscience textbooks more than Mathematics because I use my brain all the time. I just figure thinking is more applicable to the world than Mathematics unless I combine both topics or each topic individually.
If math were invented, then Dirac's correction/extension of the Schrodinger equation would in part (1/2 way) seem to be no more 'real' than imaginary/complex numbers when you try to take a 'real' measurement with them.
BUT.... there really are positrons!
The first person to discover that making something round reduces friction on a flat surface must also be the person who invented the wheel
I am not a mathematician. My only credentials are that I had a perfect score on my BC Calc Exam (our teacher made us write out our exam twice, so that he could personally grade them), and a 100% on my multi variable Calc exam in college. Then I quit math. This is not to brag. This is only to say that I at least think I have a leg to stand on.
At some point in high school, I was shown was root 2 is irrational and later why the log function is not assymptotic. I didn’t discover either of these things. Much like I never discovered oxygen.
But it SEEMS to me: ALL of math extends from the binary. There is existence, and we call that 1. There is non-existence and we call that 0. From there, all arithmetic follows. While we had to “build” geometry, it’s just arithmetic with direction. From there trig and Calc just happen. Sure, we needed to “invent” the concept of Cartesian and polar notation. But i always existed, and we discovered it based on that notation. But if we had a different notation, we would have either still discovered i, or we wouldn’t have, but i would nonetheless exist.
It is posited (in Planet of the Apes, I think) that we can demonstrate intelligence to alien races by drawing out the grid illustration proof that a 3-4-5 is a Phytahgorean triplet. Which seems fine enough for literature. But one of my dear HS math teachers countered, what is you encounter an aquatic society? It’s not that the concept of a right triangle doesn’t exist underwater, just that it is not possible to represent. This society in exchange would likely have a bunch of intuitive ways to understand fluid dynamics that we simply don’t. And yet fluid dynamics and right triangles are not locally constrained.
Yes, we build different edifices, but the total possibilities of edifices built already exists. We need to discover it.
This is UNLIKE an iPhone, in that, an iPhone does not follow from thing and nothing, 1 and 0. An iPhone NEEDS a creator. A sine wave does not.
1:27 this is exactly why math is discovered. in the case of chess or a painting you could argue that the inventor chose the game or the painting from the set of all possible game rules and brush strokes, but you can't argue the same way about math, pi is pi and the riemann is.
Nobody says we invented pi though. But we invented numbers, the euclidean metric, and such, and then discovered the special properties of pi.
I generally prescribe to the idea that the axioms are invented, but the theorems are discovered. Yes there is an arbitrarity in deciding what to focus on proving, but that’s the same with say geographic exploration. Just because you arbitrarily choose to move in a direction does not mean that you invented any of the land that your find as a result. Just like the land you discover, the logic built up around a set of axioms is already there once you create the axioms
(Commenting before watching) You know, if you had asked me this question in a subtle way that didn't make me think about it, I would have said discovered, obviously. But after seeing your video title, I thought about it, and I'm gonna change my vote to invented.
Why? Because if I say 'discovered', and then follow the implications of my reasoning, it quickly approaches what Scott Alexander once called "proving too much". The way I thought about math being discovered, I realized, would essentially imply that 'invention' isn't a real thing.
Why? Well, if you think about it, just about any invention out there, is just someone figuring out that you can combine elements of reality in a way to do something interesting or useful, and that's no different than mathematics. Take imaginary numbers, for an off the cuff example. Were they invented or discovered? Well, I think it's pretty clear that imaginary numbers were there all along. They fall out of the fundamental rules governing reality as well as the fundamental rules on which math is based. But I say that this is not discovery, but invention.
Because imaginary numbers aren't just a facet of math that we have discovered. They're a facet of math that we have operationalized to do mathematically useful things that we want to do. "Imaginary numbers" doesn't just literally mean i, the concept encompasses all of the various ways we can use them, as well as all of the various problems we can solve with them. Indeed, their original conception as I understand it, was motivated by the desire to solve a specific problem (finding roots of polynomials that do not have real roots), and they accomplished that goal handily.
Consider by way of analogy, computers. Were computers discovered or invented? Well, the behviour of transistors is a fundamental element of reality, albeit reality in a very specific arrangement. But, if humans didn't exist, and through the infinite possibilities of quantum randomness, matter spontaneously configured itself into an NPN junction, that would be a transistor no different than the ones inside my laptop right now. And then, computers themselves. Arrange a set of switches in a particular order, and you get logic gates. Again, fundamental element of reality. For a particularly funny example, cs.stackexchange.com/questions/64680/why-is-the-video-game-braid-turing-complete. The mechanics of Braid are Turing-complete. Computation is fundamental to it's in-game reality.
But again, while all of these elements are fundamental to reality, I don't think it's fair to say that someone simply discovered them. Someone set out with a specific goal in mind, a specific problem they wanted to solve. They attempted to use and manipulate the laws of reality in order to get reality to do a specific thing, for a specific purpose, that is to humanity's benefit. I think it is reasonable to call that invention.
For comparison, consider the explorers of the age of sail. They didn't invent North America, they discovered it. To stretch my example, they discovered it while trying to invent a trade route to India. They didn't set out with the specific goal of finding North America, which they intended to use to solve a specific problem. They just set out to see what they could find. And I think that's the fundamental distinction between discovery and invention. Discovery is the result of open ended exploration. Invention is the result of focused exploration dedicated to achieving a specific goal.
Maybe there is some mathematics that qualifies as discovery under my rubric, but as I watch all these math videos and learn more and more about the history of mathematical development, how all of the random facts I memorized in highschool are not arbitrary but have coherent stories behind their development, I see that most mathematics counts as invented under my definition.
EDIT: for another example that pops to mind, math channels love doing videos about the Lambert W function. I think that by any reasonable definition, the Lambert W function was invented. I mean, hell, we can't even write it as a combination of more elementary functions. We simply said, the inverse of we^w is this function. We then set about characterizing the traits such a function would have to have to be meaningful. And we developed it for the specific goal of solving equations of the form we^w. If we didn't care about solving those equations, or if we had some other way of doing it, would the Lambert W function even exist? Probably not. So well yes, in some sense, it was a fundamental mathematical reality that was sitting out there in conceptspace waiting to be discovered, I don't think it's useful to place it's 'discovery' in the same category as more conventional discoveries. Doing so implicitly elides all of the context around its 'discovery', as well as the hard work it took to 'discover' it
For more food for thought, I forget where I was reading this, but I was reading something that was absolutely fascinating to me. And that was that, all of the technological preconditions for the invention of modern mass-production textile tools and processes (eg cotton gin, automated thread spinners, that kind of stuff) already existed at the time of the Roman empire. And yet, they weren't actually invented until the 1700s. All the fundamental elements were already there, just waiting to be 'discovered', and it took over a _thousand years_ before someone discovered it.
Why? Well, the paper proposed that nobody 'discovered' it because nobody had any reason to look for it. Slave labour was plentiful, populations weren't large enough to require such scales of mass production, and markets didn't exist that required those kinds of machines to satisfy. Once conditions changed to motivate the 'discovery' of machines like this, inventors attempted to do so.
And so, the cotton gin was _invented_. It was a concept that was just lying there, in conceptspace, waiting to be discovered. But it's discovery was the result of a focused, motivated process, intentionally undertaken by people, to satisfy a specific goal. Invented
Man this is an interesting video
5:46 "Axioms are invented and everything else is discovered" the way I'm conceptualizing of it is almost exactly the opposite of that. 😅. Axioms are in a sense the base facts that existed before our system (at least, as a sort of philosophical theory of why they're justified), and in some sense count as discovered. Everything else is _constructed_ from those axioms, with some level of intentionality, and count as invented
Why the dichotomy? Why just "discovery" or "invention"? Why not something else? Why not both at the same time, or neither? Does this dichotomy apply to other subjects? Maybe discovery and invention aren't the right things to ask.
I think it is equivalent to asking whether mathematical concepts exist before anyone has ever known about them (if so, discovered; if not, invented). And the proposition "concept X already existed before any mathematician knew about it" is dichotomous afaik.
It feels like it is discovered. But that begs the question; Where does math come from? If discovered, then it doesn't come from our own minds. If invented, why does it describe the real world so well? Even pure mathematics that at first seems to have no practical application later is found to describe a real physical phenomenon.
I think is discovered , even ideas already there , is just about creating an instance of that idea that could be called invention
Many different human societies independently developed sea-faring vehicles with similar shapes. Did we discover kayaks?
As you defined a discovery, it's a matter of universe you consider the concept/object wasn't there before, or was there before but nobody noticed! If we restrict that universe to our real universe, math is all invented, but if we set it to the be the math world itself, whatever you find was there before, so all discovery!!
Math, as we know it, is a human construct. It does try to abstract ideas beyond the human experience and in the process we discover new results inside this framework, but it is human made, not merely human made, but it is a eurocentric construct (not saying that europeans have all the credit, on the contrary, many things in math comes from eastern civilizations), and saying that it is "the language of nature" and that we're just "discovering it" puts eurocentric civilizations on a high pedestral. Yes, it is extremely potent as a tool to describe reality, but is not perfect, and neither is universal, believing that aliens would arrive at the same mathemathical structure as we did is assuming eurocentric culture is somehow superior for "figuring out nature's language"
Edit: no, this is not the opinion of someone outside the highly mathemathical sciences, this is the opinion I've arrived as a theorethical physicist who works with quantum gravity, so I do study A LOT of high-level abstract math, and the more I study the more clear it is that this is a human-made language as any other
If you consider logic or geometry to be representative of math, then you can think about the situation in Plato's "Meno". There, Socrates shows that Meno's uneducated slave really already knew the Pythagorean theorem, if you just ask the slave the right questions. Math is part of who we are as humans. It was 'invented' (i.e., created) with the 1st person's creation, and then is 'discovered' by each of us humans within the normal operations of our minds.
This might be a trite answer, but I think you invent questions and discover answers. Maybe in relation to actual proofs it's more like you invent strategies to reason to a discovery, but there isn't necessarily a clear distinction there. In a sense, the question under this is really "what is reasoning?".
I feel like it's very unhelpful to lump all math into a single category to try and figure out if it's invented or discovered. Some math, for example arithmetic, seems to be quite rudimentary and in essence nothing more than a more rigorous way of representing reality, in the same way that a language would be. In that sense, asking if math is invented/discovered is like asking if language is invented/discovered. It's not clear if the question is referring to the words themselves or what the words represent. And even then, not all words are equal. The word "water" itself may be arbitrary, but the thing it's referring to is not. But we can also have words for things that don't exist. That's what I've settled on. Math itself is just a language. What you do with it is up to you. It could correspond to reality or it could not 🤷♀
Concepts are inventions, and theorems are discoveries
The mathematical realm exists as its own entity, even when humans aren't present, where we are able to access it through invention.
This discussion is essentially pedantic. You can argue all inventions are just discoveries of particular sequences of logic, as the construction of a physical machine, for example, can in itself be thought of in that way. A lot of times when you’re building something, it feels like you’re trying to “find” the proper way to meet your goal.
I think it’s more useful to think about things in terms of the rigidity of axioms; which ones we feel are truly fundamental to a working construction of math, and which ones come under debate. It’s also enlightening to consider how much of a creative process math and logic can be. Proofs may be inevitable “discoveries”, but in many cases, their steps are not at all obvious to a thinker based on the knowledge and techniques they’ve developed by that point. Sometimes you just go in a random direction, and suddenly you have yourself a Nobel prize
It seems to me that the root of the problem of discovery vs. invention is the ambiguity of language and nothing to do with Math per se. Take language for example. It apparently starts out with verbal grunts, screams, laughter, etc. and gestures and perhaps drawings in the dirt to communicate between humans. Later the grunts evolve into acoustic patterns that turn into words. Independent groups of humans evolve different acoustic patterns for the same concept or object (e.g. chair, chase, fire, ...). Some languages include tonality. In the end, for a specific group, a common vocabulary and grammar develops, all passed to future generations. Is this vocabulary invented or discovered? Or would you just say it evolved out of the necessity for better, more efficient, communication in the real world.
Eventually the idea of recording spoken language as writing developed so humans could leave written messages for each other and record information for later generations. Is developing an alphabet and then words or other symbols an invention or a discovery?
Later humans "invented" stories and wrote them down. Some were true experiences some were fiction. Or was this "discovered"? Or ...
It seems to me that encapsulating all this human activity by two fuzzy words from the English language is just inadequate. I see Math as just another language but with additional operational properties above and beyond "communication languages".
I'd say Invented. My reasoning being, Math is a language which we use to understand the universe. So just like an apple is discovered but the word "apple" Is invented, similarly all the laws of universe are discovered, but the numbers and formulae we use to represent then are invented.