Are Prime Numbers Made Up? | Infinite Series | PBS Digital Studios

[@clssgn]

I miss Kelsey and this series. This is so fun and open my mind widely about things I thought to be hard and boring. So sad that it is cancelled now.

[@alihesham8167]

yeah my mind: crys me: oh my god that annoying brain that keeps crying

[@travelservices1200]

I agree.

[@ScienceAsylum]

I'm definitely in the "math is something we made up" camp. Math is a language and language is something humans made up. It's relatively ordered and so is the universe, so they tend to correlate. But, as a physicist, I can tell you that correlation is /not/ perfect. It just sits firmly in the "close enough" category. It is not the language of the universe. If you want to say the universe even speaks a language, then math is just our best translation of it into "human"... but, like with any other language, some things get lost in translation.

[@levinunemaker9406]

That "close enough" sounds an awful lot like a limit. If you consider the philosophical implications, it seems an awful lot, if the universe isn't explicitly doing so, like the universe just wants us to believe it.

[@bobgreen9897]

Is Pi infinite because a circle has an infinite number of small straight sides? Infinities exist in math but I'm not too sure about in our observable reality because of quantum limits like the Planck length etc...? So if Pi really is infinite and real that means our universe if infinite in size!? 😲 Or maybe we look at circles wrong and true curvature is possible but can only be described with the correct description, which we don't have yet?

[@archangecamilien1879]

There might be an Einstein quote, forgot it, along those lines...that math doesn't perfectly predict the universe, it's only precise when abstract, etc, or something, forgot the quote...at any rate, I wonder what a universe that didn't somehow follow math rules would even look like, lol...if it's going to follow any rules at all, wouldn't we end up calling those rules "math" anyway?...Well, I suppose one could say that there are plenty of things that follow rules but which aren't as directly "mathematical", like biological beings, etc, one would have to zoom in to find the mathematical rules, etc. But still, lol...how would a universe that doesn't follow mathematical rules even be like?...Is it conceivable, would it be a universe that doesn't follow any rules at all?...

[@jdrmanmusiqking]

@The Science Asylum I know this is 5 years old but this is more wrong than anything can possibly be wrong. Math isnt *just* the language of the universe but it IS the universe Like... Its baffling how any thinker can even believe that humans calling a thing 'something' actually affects that thing in real life. Thats absurdity and reeks of ego Humans arent an integral part of the universe... Mathematical symmetries exist in nature regardless of what humans call it. Math is simply cause + effect, logic, shapes, and what the gradients of the universe are doing in the present time. Logic on paper man logic on paper Like... Did triangles not exist before Pythagoras "invented" his theorem? Euler "Invented" countess mathematical equations that are fundament to reality itself. So.... Did reality not exist before Euler was born? Were the rules of reality different before Newton "discovered" gravity? Scientist are the most egotistical people on the planet jeeeeeez. These people are simple OBSERVING whats already there not CREATING.

[@archangecamilien1879]

Lol...I don't know...there are infinitely many things we could define, our there, lol...there is a sense in which they only start existing when we define them...a sense...but anyway, lol...

[@kevinmerrick]

I'm a strong formalist. At its base, Mathematics is simply defining some rules (or axioms) and carrying them out to their conclusion. The reason it describes reality so well is that we tend to pick axioms that we feel define our reality. We could pick completely different axioms and get completely different mathematics, but it just wouldn't describe reality. I believe this is most noticeable when we discover independent axioms, like the continuum hypothesis. Math works no matter what version we choose, but maybe we will see that one version lets us describe our reality better. Okay, long speech over. Just want to say I love the series, and I'm really excited to see more videos! You all are amazing, keep it up!

[@CheCheDaWaff]

Would you not agree that the set of conclusions from given axioms forms some kind of reality? If there is only one outcome of the rules and every observer would measure the same outcome, how is the outcome any less real than say, a photon?

[@pbsinfiniteseries]

Nice! You might be interested in a branch of the philosophy of mathematics called "logicism". I also think the Axiom of Choice would be an even better example for your argument, since it's now (essentially) accepted because it proved so many statements that mathematicians felt were true.

[@lalonmallosson208]

Modus Ponens Now we get into the question of what is really "real". Through the philosophy of skepticism, you could make the argument that our objective reality is created through our minds, since the only way we can truly "prove" something exists is through our senses, and our senses have been wrong before. Therefore, through the eyes of skepticism, math could be made up in or minds, since our reality itself could have been created in our minds. Of course, there are fakes with this theory, but I feel like it's worth noting

[@Sarastro404]

+Kevin Merrick But you don't explain the problem posed by Wigner about the effectiveness of math. The thing is that simply claiming that we chose axioms to mimic reality doesn't brush off the idea that these mathematical axioms/objects exist independantly of human minds. If anything it just reinforces the idea because if our axiomatic system is simply "copying something" then by definition it means we didn't really invent them. Also If that "something" we're copying is just an other mathecatical axiomatic system then it would mean that math exist!

[@benjaminprzybocki7391]

Modus Ponens I am a formalist and I have thought before about the question you raise. I think you're absolutely right; the logical deductions do form a objective reality. So, how does this not make me a mathematical platonist? Well, a platonist would accept the axioms as having some objective ontological significance. I still think the axioms are just the "rules of the game" that we defined. I also believe that when we say we've proven something to be true, we're really just saying that if we accept ZFC (or whatever we're using as our foundation), then such and such theorem is necessarily true. Since I don't accept the axioms as ontologically objective entities, but I do accept the logical deductions that follow from a set of axioms as ontologically objective entities, I like to say I'm a "logical platonist" but not a mathematical platonist. Logic is definitely real and objective in my opinion.

[@alexseioo610]

Mathematics is based on logic. The vocabulary is freely invented, but the rules of grammar for this formal language are the logic. The rules for logical thinking are constructed in a way, that is consistent with the *physical properties of the universe*. It means, the rules of logic to apply mathematics, to proof conjectures, are specifically designed to set the *physical system "human brain"* (or computer or alien brain) into a *restricted state, which always produces the same consistent results independent* from the specific system used or any superficial conditions (like mood). That's why different mathematicians using the same assumptions and axioms obtain the same results and why students can relatively easily follow the way of thinking by reading a proof. If, on the other hand, they would read some complete nonsense, a flawed proof or fail to apply proper rules, they discover an inconsistency between the way their brains produce results and the material at hand. I.e. the proof is logically inconsistent or the applied logical rules were not stringent enough to force consistent results.

[@DustinRodriguez1_0]

I take it you do not subscribe either to the Copenhagen interpretation of quantum mechanics, or to an idea of discrete spacetime units? Also, what happens when large swathes of mathematics are based on things like hyperbolic or higher-dimensional geometries? Sure the rules of logic used are the same, but the axioms are switched up.

[@justsayin9180]

Dustin Rodriguez The Copenhagen interpretation doesn't interpret anything. It just states the observed effect in a more formal manner. Still want to know why.

[@badseed86]

Is English real ? It seems to have an unreasonable effectiveness describing my thoughts.

[@heisenmountainb6854]

so its real

[@josephang9927]

except english changes. Its words dont just accumulate, but evolve. There are also other languages that describe other things better. Math is special. Plato was right. Math is more real than us.

[@mmandrewa2397]

English and every known natural language is spectacularly poor at describing reality. We just aren't aware of this because part of what we call our thoughts is expressed internally as words. Since part of what we are describing as 'thinking' is built out of words, well then this is like saying something is itself. But this is an incomplete statement because we have other modes of thinking, that we experience, that are not built out of words. For instance what are dreams built out of? Or what does your brain do when it 'sees' something? For all these other kinds of thought, words are really bad at conveying, identifying, or describing what is going on.

[@joshuaosei5628]

Maths itself is abstract, however we can use it to define and manipulate real things in terms of physics. Physics is applied maths. Our buildings don't fall down, and we used physics to make them.

[@nonyadamnbusiness9887]

Circular argument. English is defining your thoughts, therefore it should be capable of describing your thoughts.

[@valshaped]

I'm more of a procrastinist when it comes to math

[@calvinyoung480]

This has already proven itself to be an incredible series. The metaphor of writing the first page of a book was very nice. Thank you for producing it so wonderfully!

[@thomasr.jackson2940]

Formalist here. The reason that math works so well in science, engineering and technology is because those subjects have long been strong motivators for mathematics. And because they work so well, those are the math systems we teach and study. There is nothing unreasonable about the fit, any more than it is unreasonable that my door fits into my door frame. (Though I still puzzle why my refrigerator doesn't fit into the space that is purported to have been built for it in my kitchen). Our invented math only appears to describe nature. Oh, it comes close, but when we look closer, it fails. Newtonian physics and Euclid is all neat and tidy and was assumed to explain nature, but it doesn't. We go fast, and we have to find a new geometry to make things work. Or closer to home, the Euclidean stuff doesn't really work measuring things on the surface of the earth because the earth is not a plane, it is a sphere. So all those lovely triangle rules don't really work. So we looked around at some of the new ideas and picked one that seems to fit nicely... so far. We go small and find that there are no circles because matter is quantized and thus inherently fuzzy. The concept of a circle is still useful in approximating reality., but it isn't reality. And that is the crux of the matter. It isn't that there is some "ideal circle" that the real world approximates. It's the other way around. There is a real world with which we interact, and our math approximates it. Well, some of it does. Chess is another form of math, just like the Game of Life. Neither describes reality, but the latter especially gives us a lot of insights into how replicating patterns behave over time. New math. And more complex mathematical models have been built on this insight and do an even better job of describing the real world. But I understand why we are so tempted to think of our math as "real". That is our great human talent, to imagine things and make them real, even when they remain imaginary. Money, countries, laws, corporations,... none of these are real, they are all inventions of our imagination. But because we believe them to be real, because we see these figments of our imagination as real and tangible and are so utterly convinced that we act accordingly, these abstractions act as though they were real. In fact, in a certain sense, we make them real. Just like math. They are abstractions that sprung forth from our imaginations and we endowed with the gift of existence and reality.

[@ObjectsInMotion]

Thomas R. Jackson The problem with formalism is that it implies that their are other systems that we could have just as easily have invented that would also be able to describe the universe, but there aren't. Only one mathematics, our mathematics, exists and can described the natural world. You can explain particle physics with matrices or differential equations but it's all the same math.

[@MrDajdawg]

I'm sorry but 2+2=4 is a constant. Quoting someone really smart doesn't mean that their quote holds any water. No matter what, math is going to be a constant in our reality.

[@TheIzrafel]

That was a great post by youtube comment section standards. Thanks for writing this.

[@donach9]

MrDajdawg I think you're missing the point. No one is claiming that at some point 2+2 will equal anything but 4; it follows from the basic axioms. We use the maths we do because it works so well at describing the natural world, but certainly over the past few centuries, and probably millennia, mathematicians have explored maths that doesn't appear to describe the natural world. Of course, in some cases, some of that maths has turned out to be useful. Similarly, I can imagine an alien species on a planet with radically different material circumstances, different chemistry, biology, and society etc developing ways of understanding their world that use different sorts of concepts and maybe miss out lots of what seems basic to us. Part of the problem is that maths is so ingrained into our thought and perception that it's hard to think about understanding the world without it.

[@Euquila]

You say laws and corporations are not real, but how are these things different from an EM wave or a cell nucleus? Aren't all things just trying to adhere to rules? I think a relatively stable environment respects more rules and we can talk about "time" and "energy" or the realness that is intertwined with causality. But just because there are perceivable layers to the rules of a causal world, it doesn't mean that some layers are less real than others. If anything, I would say the only thing that is not real is an environment that has no "time" or way of measuring "energy", no "causality", no patterns or logical progression. No rules!

[@popcorn485]

2 + 3 sometimes equals 1 or 11. Depends what you “base” the rules on, or how you “mod”ify them 😁

[@davidwilkie9551]

Let a few students lose in the jungle without a guide of any kind, and they would soon develop a language to learn to live there. The real information comes from the environment and the language is used to sort it out, and adjusts continuously to fit change. Maths is language, and part of us like all tools. Plus, everyone is is allowed to have a hypothesis about what is possible and test number relationships until the rules emerge - and then find the real world correlation and or corrections. It's endlessly interesting.

[@kolacats]

at 1:32 why is 9 there, 9 is NOT a prime (it's 3*3 btw)

[@pietervannes4476]

and why is 2 not there (assuming they wanted to show the first primes)

[@lEGOBOT2565]

@@pietervannes4476 2 is generally regarded as "not prime". Don't know why

[@pietervannes4476]

@@lEGOBOT2565 no its not

[@lEGOBOT2565]

@@pietervannes4476 not what I heard from doctorate level mathematicians that I have met at a nerd convention

[@pietervannes4476]

@@lEGOBOT2565 that is very strange, as 2 is definitely a number that meets every requirement to be a prime and functions like primes too. Also at another point in the video it was shown with other primes

[@julianobitsgen]

I still haven't found a better alternative than the one which was proposed by Gottlog Frege in his Foundations of Mathematics: mathematical objects are existent, but abstract: they are 'found' when we are using our language on a higher level. While in natural language there are concepts in which the value of reference is empirically determined, mathematical objects are a second order concepts, they are concepts of concepts. Nevertheless, they are true and objective: their logic, relations and operations are valid for more than one person, and work independently of our opinions or taste about math. In a certain way, Frege was called a platonist, but if we consider that mathematical objects, although true and objective are only found when one is thinking on a higher level of language, his approach was much more kantian and formalist. Math seems to work as a tool and a language to describe nature only because we already perceive nature as mathematically ordered. Math seems to fit so perfectly not because of math itself nor nature itself, but because of how our own mind works.

[@ohmandamp]

You have ten fingers, I have ten fingers, bob has ten fingers. Abstract away who has ten fingers and you get the abstraction "ten fingers" Like wise, I have ten apples, you have ten apples, bob has ten apples. Abstract away who has ten apples and you get the abstraction "ten apples" So we have the abstractions "ten apples", "ten fingers", "ten toes" etc. What if we abstract away what we have ten of? We get an abstraction of an abstraction. The number "ten" Every culture developed its mathematics through this abstracting from simple counting of physical things like fingers or apples to these abstractions. Numbers. And then considering that "fingers" and "apples" are already abstractions themselves, numbers are 4th order abstractions. They are so abstract as to be universal. And all of mathematics is just abstraction upon the abstraction of numbers, so no wonder they work so well, because they are so bloody abstract!

[@jdrmanmusiqking]

@Juliano Bittencourt I know this is 5 years old but you just used a bunch of words to say a whole lot of nothing smh Its not that hard people overcomplicate things because of their massive god complexes. Humans arent gods in hat regard humans didnt create the universe which is founded on mathematical principles. The symmetries of the universe exist regardless of what we call them. Subatomic particles are the physical manifestation of 1+ (-1) = 0 as charges and spin cancel out. Plants grow geometrically regardless if we have a word for geometry The circumference of a circle will ALWAYS be 2πr and always have been even before humans existed.

[@sophieward7225]

I would argue that math is like a game, where we're following rules to their necessary conclusions, but the rules aren't completely arbitrary: they're rooted in nature. We didn't just decide 1 + 1 = 2, it's a natural fact. We didn't decide a circle was the shape where all points are equidistant from the center, the universe did. Mathematicians took the rules nature gave them and ran with them, which is why math, while game-like, describes the real world so well.

[@danialhaseeb1909]

Did the universe define the circle though? There exists no circle in the universe with those properties. Or a sphere for that matter. It was probably a mathematician who sat down a while ago and defined what an ideal circle is and then ran with it. :p

[@sophieward7225]

+Danial Haseeb Maybe the universe didn't define the perfect circle, but the way space is shaped suggested that a circle would be the right shape :p

[@danialhaseeb1909]

M.W. Vaughn But that is what our human brains think is the right shape. Maybe some other sentient being would differ with us on that. :p This can go on and on.

[@sophieward7225]

Man, I love Devil's Advocate :p You're absolutely right, it can. Maybe there's no good answer

[@joshstew85]

M.W. Vaughn humans tend to think of circles as perfect, but imagine trying convince a sentient bee that circles are better than hexagons.

[@naedolor]

With all these interesting discussions going around, I thought I'd share my own view. I'm a struggling musician with a passion for math, physics and philosophy. I've read the essentials of Platon, Hilbert, Russell etc. I especially enjoyed reading some of Russell's ideas, but in the end I believe that math is like the product of a compiler outside this universe. It's the underline structure, or the predetermined framework on witch our universe exists. It's created by something we simply cannot fathom. I like to think about it like a computer game. All games have rules and more or less the world created inside that game follows basic logic and it's self consistent in that framework. It can also have physics and gravity programmed in it. The player, or the AI will always obey those rules. Now, if the AI was sentient, he would be aware of all these rules and would analyze its world from his perspective, but he will never have a clue that he's the product of some functions and scripts written in C#, outside his world. The very notion of C# is utterly meaningless in the product of C#. Anyway, fruitcake or not, that's just my take. It's not like I'm gonna get better answers this lifetime.

[@jarahal-ajeeli474]

Hi, I'm a human from a simpler time. I fish for a living. I keep finding lesser fish in my basket than I caught, I can tell someone is stealing them but I can't keep track of them. one day I decided to give a different name to each amount of fish starting from a fish to many fish and increasing by a single fish each time. And that is how I invented numbers who do not exist on their own whatsoever and the only reason they are so good at describing how many fish I have is that that is exactly what I modeled them after. TLDR: numbers are only so good at describing reality because, in the beginning, they were concepts that were INVENTED to Describe reality, like how many fish someone had caught.

[@esra_erimez]

Brain, body and beauty all in one.

[@ΖήνωνΕλεάτης-δ7κ]

Sex appeal, as well!

[@constancepassas5386]

I'm already hooked on this series! Totally looking forward to more episodes :D

[@pbsinfiniteseries]

Glad to hear it! We're looking forward to delivering more episodes. :)

[@bestnocture]

PBS Infinite Series please do one on Complex Functions and their Graphs!

[@johembrey3616]

I just realized I wasn't even subscribed... I've watched so many of the vids cause they appear in suggested and home pages... time to change it I think!

[@B3Band]

*If A Perfect Circle exists, how do you interact with it?* _With a Tool_

[@oliverhoare6779]

Found the engineer.

[@lineikatabs]

Maybe the key to the puzzle is the origin of mathematics. Just like you said math is a book that write itself but we have to write the first page. And it's not difficult to see why 1 + 1 = 2, just count your fingers. We agree on the basic rules of adding, subtracting, dividing etc. because they make sense in the real world - I still remember learning about division with X apples and Y plates to put them in... So, math isn't real but it's not unreal either. And it's effective because the real world is in itself consistent, just like math is. It's fascinating that our logical brains can extrapolate unknown real world properties by just looking at the math - I'm talking black holes, dark matter, time dilation and so on - all real world objects which were predicted by the math.

[@lineikatabs]

***** yeah, I guess I should have mentioned I don't know shit about maths and I'm talking out of my ass... well, mentioning it now.

[@lineikatabs]

well this has been the nicest youtube argument I've ever had :D

[@d0themath284]

In science, the simplest solution that explains the most amount of things always prevails. Certain types of math are purely platonic, but other types explain reality so simply and generally it would be crazy to say that that math only existed in our heads.

[@jdrmanmusiqking]

For some reason youre in the vast minority. When i look at the ratios, most idiots think humans made up math. Even so called intellectuals believe such an easily disproveable thing Like BRUH THE UNIVERSE EXISTS! Humans didnt create the universe and math is fundamental to the forming of the fundamental particles that make up reality and then some

[@norielsylvire4097]

A phisycist and an engineer were talking. The engineer told the phisycist: You phisycists think everything is easy and quick, you just sit down with your equations and computers and wait for instruments to pop up. And the phisycist replied: We phisycist think of engineers as a magic lamp; you rub it's surface, tell them what you want and telescopes and instruments pop out of it.

[@MrDajdawg]

Math isn't like science, where you gain enough evidence and declare something true, you have to have a precise logical proof. Greatest sentence ever uttered by a human being ever.

[@Redflowers9]

All languages have effectiveness in everything, like the one I'm using right now to tell you this, which I couldn't do nearly as effectively with mathematics.

[@jmw1500]

I'm a formalist. The effectiveness of math is not unreasonable as long as perfect knowledge can exist.

[@jarlsparkley]

I never understood this debate. To me the answer seems obvious: just like you explained, we invent the axioms and discover the emergent properties of the system defined by our axioms. Maybe the real meat of the issue is what is considered math: the axioms, the theorems, or both? Well, in that case are you arguing semantics or philosophy? As for the "unreasonable effectiveness" of mathematics in describing the natural world, neither does this seem to be a good counterargument for formalism, nor does it seem at all mysterious. A mathematician typically works with axioms that were intentionally selected to resemble how we understand our world to work, so of course the emergent properties would likely also resemble our world. Yes, chess does not describe our world but that's because the rules of chess were not devised to resemble our world... Smells like an inverse error to me. Just because a game does not describe reality does this mean that no game can? I simply fail to see any real distinction between the rules of a game and mathematical axioms besides the level of rigor you might typically expect. For instance, I can describe axiomatic systems that do not model our world any better than the chess game you describe that nobody would hesitate to call "mathematics", and suddenly the "unreasonable effectiveness" argument falls flat on its face. Anyway, I enjoyed this video and like things that make me think. I am looking forward to more.

[@EntropicNightmare]

Okay, but that doesn't really solve the problem. Sure, there exist axiomatic systems which do a poor job of describing the universe. We could pick one of those to explore, but we choose rather to pick the one that describes reality. That doesn't explain why there exists an axiomatic system which does describe reality. Why should the universe follow coherent rules at all? And if the universe does follow coherent rules, doesn't that mean that those rules exist independently of a system that we invented?

[@larryp5359]

As for why the universe should follow coherent rules, I'd like to apply the anthropic principle to state that any universe that didn't follow coherent rules almost certainly wouldn't support life intelligent enough to ask the questions we're discussing. Of course, this just changes the discussion to why our universe has all the nice properties that it does so that we are able to be here to observe it.

[@Silverizael]

If multiverse theory applies, wouldn't it be the fact that there has to exist universes in the infinity that do meet those properties and intelligent life could only ever exist in those universes that do have those properties? Thus, our ability to have intelligence by necessity dictates we are in one of said universes with said properties?

[@jmitterii2]

I don't think it's even necessary to invoke anthropic principle... we can see our axioms on mathematics change to suit the needs of reality. Newtonian mathematics breaks down on very close orbits like Mercury. But Relativity using new tensor geometry works well for all orbits including Mercury. Axioms of mathematics are infinite, and only those that are useful get any attention. The rigidity of a mathematical systems is what gives the sense of independence. But the fact we have to invent new forms of geometry and probability of which axioms come from the outside world and old mathematics that fail due to their own rigidity shows even the rigidity is an illusion.

[@ohmandamp]

jmitterii2 You have ten fingers, I have ten fingers, bob has ten fingers. Abstract away who has ten fingers and you get the abstraction "ten fingers" Like wise, I have ten apples, you have ten apples, bob has ten apples. Abstract away who has ten apples and you get the abstraction "ten apples" So we have the abstractions "ten apples", "ten fingers", "ten toes" etc. What if we abstract away what we have ten of? We get an abstraction of an abstraction. The abstraction ten. If we have ten, 1, 5, 9, etc. and we abstract away the magnitude from each of these things, we get the abstraction numbers. Every culture developed its mathematics through this abstracting from simple counting of physical things like fingers or apples to these abstractions of numbers. And then considering that "fingers" and "apples" are already abstractions themselves, numbers are 4th order abstractions. They are so abstract as to be universal. And all of mathematics is just combunations upon the abstraction of numbers, so no wonder they work so well, because they are so bloody abstract! So much for the unreasonable effectiveness argument. One might as well ask about the unreasonable effectiveness of the laws of logic, or of abstraction itself, to pick the few things that are even more abstract than math. But isn't the rules of chess also an abstract system? So why does it not work to describe reality? The reason is because it is not abstract enough. Chess is a lower order abstraction than mathematics, because mathematics is needed to describe chess. You need the abstraction of numbers to describe how many pieces there are on the board, but you don't need the abstraction of a stalemate to describe any concept of mathematics. This is also why many mathematical fields have no bearing, no connection to anything in the real world, because they are lower order abstractions than those of numbers.

[@DavidDagninoV]

the proof that math is real will be when we manage to contact aliens and see if they math the way we do.

[@davidwuhrer6704]

Do plants count?

[@eduardolopes243]

That should be astonishing! It would prove that Math is actually the language of the universe, and that instead of us having invented it, it would tell us we understood and written it. Altough I believe it's not the case here, it would be nice...

[@Macisordi]

It could prove that we have discovered only a part of it or that we both have discovered the same part, if, as me, you think math exists besides humans.

[@GLPentAxel]

Not prove, but strongly indicate! What if there is some alien race that doesn't math? What if 50% "maths", while the rest doesn't?

[@Macisordi]

It will be thrilling to know how they think. You use math even if you don't know it!

[@Math_oma]

Since there's a lack of platonists on this comment section, let me come out of the closet. One argument for platonism that should be taken seriously is the _indispensibility argument_ which says, as a principle, we are ontologically committed to the objects about which our best scientific theories (i.e. physics) make statements about and since our best theories make statements about mathematical objects, we're committed to abstract mathematical objects, as they are _indispensible_. Perhaps if someone someday comes along with physical theories that work just as well and don't make statements about mathematical objects, we can ditch this argument.

[@Math_oma]

+Keating Allen I don't understand how this _gridline objection_ touches any of the premises of the argument.

[@goldjoinery]

Penelope Maddy addresses P1 - we ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories - of Quine's indispensability argument in Indispensability and Practice (1992), Naturalism and Ontology (1995), and Naturalism in Mathematics (1997). Her strategy is to undermine Quine's naturalist and holist justification for P1 of the argument, by showing that the former is incompatible with the latter. Maddy forwards three objections: 1. Naturalism says we ought to respect the attitudes of working scientists. Holism says there shouldn't be significant variations in those attitudes. Yet, working scientists often disagree heavily on existing theories. 2. Maddy suggests mathematics exists not within the true elements of scientific theories, but rather those idealised elements used to model reality. Water is assumed to be infinitely deep in analysing water waves; matter is continuous in fluid dynamics. 3. According to holism, logicians mathematicians should be assessing and revising the foundations of mathematics (the ZFC axioms) in accordance with the needs of physicists and other scientists. Given that such arguments for/against ZFC are almost always intra-mathematical, and we should respect that, this demonstrates that naturalism is favourable compared to holism.

[@benjaminprzybocki7391]

Mathoma Isn't it the other way around? Don't our best scientific theories make statements about the universe oftentimes *using* mathematical objects? If science was making statements about mathematical objects, it would be math.

[@Math_oma]

+Benjamin Przybocki I think it's pretty obvious the theories are making statements most directly about mathematical objects. We then say, secondarily, that the statements have something to do with physical objects. Whether they are directly or indirectly about mathematical objects, we both recognize that mathematical objects are needed in this picture _somewhere_.

[@Math_oma]

+Communisation This is a complex comment so I'm not sure I can address it all at once. Let me just give my reactions to Maddy's objections to P1: the ontological commitment premise. Actually, a note on P1: why do we have to say _all and only_? I was initially under the impression that we should, at a minimum, be committed to such objects, not exclusively the objects needed for the best scientific theories. Perhaps this goes against Quine's aesthetic need to have a sparse ontology, but I don't really care about being profligate for the time being. 1. It seems obviously correct that scientists have major disagreements on existing theories but one thing I've never heard a disagreement on is whether the theories should be mathematically phrased. 2. I don't see what's left of a scientific theory once you've removed the mathematics, so how could it not be a true element of a scientific theory? Furthermore, the scientific theories often use mathematical terms right in their statements. For example, Faraday's law makes a statement about a _derivative_ of a magnetic _flux_, both terms already being mathematical concepts. 3. I don't understand this objection well enough to see how it's actually an objection to P1. Nonetheless, I'll have to check out some of Maddy's work.

[@ranevc]

It is as simple as this: Math is ideas extracted from the real world. What is so difficult to comprehend?

[@ArseneGray]

YAAY coem from the PBS Space Time, now another PBS show to subscribe. very happy

[@oarevalo21]

Formalist. A platonic interpretation cultivated my love of mathematics, but was also a HUGE OBSTACLE to clearly thinking about the deep problems in mathematics, like foundations.

[@darksid007]

infinite series, what is your limit?

[@alminhelex]

darksid007 I tend to converge to pi but occasionally break loose and diverge

[@fossilfighters101]

+

[@98bransonfun]

It converges to infinity

[@abstractapproach634]

@@98bransonfun it does indeed, but if you watch an episode of Khan academy at the same time you can use la'hopitals rule to show it converges to 4

[@typo691]

"la'hopitals"

[@Bakasan16]

Formalist here. To address your point, the reason math tends to describe things in reality is because we tailor the rules to model objects we see in reality. Math's usefulness in physics relies heavily on the accuracy of this model. Also, there is plenty of math that doesn't describe reality; it just doesn't get as much press.

[@Cubinator73]

Formalism FTW :) I never understood (and probably never will understand), why some mathematicians think that every concept/idea has some physical object attached to it. At some point in time humans started counting sheep and made up _numbers_. When we started thinking about shapes, humans made up concepts like _distance_ and _angle_ by defining them. There might be big circles floating around in the universe, but the concept of a perfect _circle_ is just a definition a human made up (possibly based on real-world-objects). Integrals were made to calculate the area under a curve, then a definition came by and suddenly we can do much more with integrals. Great video :) PS: A prime number is not defined to be only divisible by itself and one, because this would include one as a prime number (because one is divisible by one and also divisible by one), but one isn't a prime number. There is a similar definition though: Every number greater than one is called _prime_, if and only if it is divisible by itself and one. Another definition (I personally like the most) is this: A positive integer is called _prime_, iff it is divisible by two different positive integers exactly.

[@Math_oma]

+Cubinator73 But was anyone claiming that concepts and ideas have physical objects attached to it? People who think mathematical objects really exist have to accept that they are non-physical abstract objects. Ironically, it's the naive empiricists who want to say all mathematical statements are tied up with physical objects that would want to draw this link between mathematical objects and physical objects (even though that was a failed philosophical project).

[@ΖήνωνΕλεάτης-δ7κ]

Ideal forms of humanly perceived, elementary objects! Simplified by stripping them down to their bare minimum, but still keeping them recognizable (e.g. a circle)! Numbers are tools for counting objects (1,2,3,4,.. etc), AND also NAMES, assignable to objects, even non-sequentially!

[@donach9]

"non-physical abstract objects" don't exist in the way I use the word 'exist'. I don't accept that because we think about something that means it has an objective existence. If what you are saying is that things exist because people think about them, then that implies God exists simply because people think about the concept of God, which is an even weaker argument than St Anselm's. Is that your position? While there has been some sloppy thinking on this thread, I think the reason antirealists (certainly in my case) have drawn links between physical & mathematical objects is to show that Wigner's effectiveness is not unreasonable at all. Rather, we've invented the abstract objects in order to explain the phenomena we see in the natural world, and we've got better and better at inventing and refining the abstract tools we use for this over the millenia.

[@Cubinator73]

+Donach mc kenna Nope. I don't say things exist, because people think about them. I can think of a perfect circle and yet there is no perfect circle anywhere. But the idea of a perfect circle exists (not as physical object) and is denoted / described with language / remembered with electrons buzzing around in my brain. I can think and talk about God, but that doesn't make him suddenly exist as physical entity. I can even think about perpetuum mobiles, but that doesn't make them possible in reality. Even though, there is no real Harry Potter, he still exists in books, in movies, ... The (probably idealized) idea of some thing is not the thing. The thing *might* exist as physical object. The idea cannot exist as physical object, but it exists as description that might be physical letters in a book, electrons in our head / in a computer...

[@donach9]

Cubinator73 My comment was directed to Mathoma. I'm not sure why the +Mathoma disappeared from it. I largely agree with you on this.

[@gertwillems4456]

As a physicist I'll let you guys in on a secret: we have set up all our best theories as a game too... Let's take general relativity as an example. At the very heart we find the following basic assumption: empty space can be described as R_ij = 0 with R being the Riemann matrix. This is a very abstract concept since there is not such thing in real life as empty space so we have no way of confirming this so Einstein simply posited it as being true. Now if that isn't a game rule, I don't know what is. The fun part is that once you start playing go get to make these predictions like, add some masses and start calculating how space warps around them. Then you simply claim that what you calculate is also what you will see if you do a measurement of say a star nearby (actually behind) our eclipsing sun and you challenge some people who like to travel to go and measure this and lo and behold their measurements and your findings are within reasonable error flags. Now has humans we like to jump to conclusions that we found the underlying principle of what governs reality. But in fact we are kidding ourselves, we simply found a game that closely matches what reality does. We should never forget that part. All we are doing is trying to model what we see in some game rules but we can never get this complete because of what Gödel proved in 1931, there will always be true things out there that won't be connected with our game rules that we invent.

[@AxiomTutor]

What a wonderful video, I'm so glad to see a mathematician taking the Philosophy of Mathematics so thoughtfully. I tend to think of mathematics as "schematically true". That is, it is true for anything adequately described by the premises of the structure. If an object's motion is close enough to f(t) = 2t+1 then you can predict closely enough that after the stop-watch strikes 10 seconds the object will be located at 21 meters away. 2+3 works well for describing trucks and apples up to a point, unless you dump 2 and then 3 apples into a blender. Does 2 exist? Only as a schematic place holder for two objects that will persist over time long enough for the mathematical structure to tell us something useful.

[@Ussurin]

I study IT and what "we" think math is is basically "the rules describing relations between measurements". Numbers, etc. are real, but they aren't objects, only describtions of some facts about object or theoretical objects. It's mostly realist view of maths, but without requiring "interaction" with numebrs, cause you only need to interact with objects on which this "rules" are described upon and after getting this rules (for example: 2 apples + 2 apples = 4 apples), you no longer need these objects to extrapolate logically these rules to different objects (2+2=4, no need for apples).

[@omargoodman2999]

A lot of people conflate "concrete" (as in concrete vs abstract) with "real". They view concrete things like "matter" and "energy" as "real" while abstract things like "love" and (potentially) "math" as "unreal" or "made up". But there's no sufficient evidence that indicates that being "real" is contingent on being concrete. There's also no sufficient evidence that what we understand to be concrete is necessarily "real". Descartes's famous statement, "je pense, donc je suis" is abstract in nature but it's also tautological; it cannot be untrue. Or, in other words, it absolutely *must* be "real". But this concept is outside the scope of science because it does not yield to empirical analysis. In other words, there are lots of things that Science is equipped to handle, but just because science can't handle it doesn't automatically make it unreal. Hence, even if there is no empirical evidence for the existence of math, no concrete manifestation thereof, it can still be considered "real".

[@kephalopod3054]

Goldback conjecture for 2n: for any positive integer n, there is at least on k, k >= 1, k

[@sophiathekitty]

i always thought of math as a logical language for describing and considering stuff in the world around us. it works so well because we've been building up the rules that math uses based on useful observations. numbers started as counting things that exist. then measuring things like distance and size.

[@DustinRodriguez1_0]

Except... we explicitly and definitely do NOT do what you said. What you described is science. Mathematics is different. Mathematics absolutely is not built on rules based on useful observations. In fact, no observation of reality could ever disturb a single mathematical fact. Mathematics is a product of pure reason, totally divorced from reality except in its respect of basic logic. (And reality seems to perhaps not respect that logic so much, such as in the Copenhagen interpretation of quantum mechanics.) For instance, there is substantial physical evidence to suggest that spacetime is not continuous in the way the real number line is continuous, there being an infinite number of distinct positions between any two selected points regardless of their distance from one another. This means that the circle mathematics deals with can not exist in our universe. Every circle must have an exact answer for the ratio of its circumference to its diameter expressed in that 'smallest unit'. But, this does not cause any problems whatever for the definition of Pi, the real number line, or the geometry of circles. Those are intact because they were never built upon physical observation or definition to begin with. They were derived purely from a small set of basic axioms, and logical extensions thereof. Nothing more.

[@thomasr.jackson2940]

Dustin Rodriguez I am not convinced that math is a product of pure reason. Much of our math, including numbers, is rooted in practical problems, such as counting stock, making trades, measuring fields. Sure, abstract logic and constructions were used, but they were inspired by the real world. Sure, an abstract world of reason developed from this, and often took a life of its own. But whenever it bumped into a problem with the real world, new math was developed, or picked up off the shelf. That is the root of the "unreasonable" correspondence of math with science. Science, business, engineering, all inspires a great deal of math, whether mathematicians like it of not. Archimedes, Newton, Noether, all mathematicians that developed their math based on scientific problems.

[@DustinRodriguez1_0]

Thomas R. Jackson Things like the natural numbers (not including 0) historically originated from counting stock and such, but that has nothing to do with mathematics. Mathematics is, by definition, exclusively the abstract arguments built upon reasoning from axioms through use of proofs. And while, yes, mathematics was expanded in ways to address physical problems, they never broke from the pure basis of it. The fact they never needed to is why the correspondance is 'unreasonable'. And in many cases, the math came first. Complex numbers and complex geometry are the usual example used. They were developed with no practical application, simply an outcome of pure reason. Then Einstein came along, and they made explaining relativity simple. Why? There's no good answer for that.

[@thomasr.jackson2940]

Dustin Rodriguez Yours is an interesting perspective. Tradesmen learn to count and add, and group, but that is not math because... I am at a loss. They may not have published a paper, but they did reason it out. It seems to me that the "pure" basis of math developed over time. But its very "purity", the idea that you can reason these abstract concepts, even invent new ones, perhaps inspired by the physical world, but maybe not, makes my point. These are abstractions, dependent upon our own thoughts to make them out. If they are "real" , then they are real in the same way that we make so many of our abstract thoughts real. Money, nations, laws,... these are all "real" in a sense too. They have measurable effects on our behaviour. That is empiric reality, something the proper subject of science. But if humans disapeared, so would these things because they have no independent reality. So to math, in a sense. The sense is that so many of our concepts in math were founded on our observation of the natural world, as seen through our perceptions. So that will live on beyond us. Will mathematical "truths" remain true when we are gone? I think the question is nonsensical because we invented the whole concept of truth as well. If we destroy a computer and its software, it does not exist. We might rebuild it, and perhaps the new code we construct will develop the same concepts and behaviours, and exist again, perhaps not. But the bottom line is that information has no existence apart from its physical matrix. Remove that, and information is gone. I think the idea of our abstract ideas having an existence apart from us or our creations, or records, is mysticism, pure and simple. It is a story our brains give us, a perceptual illusion if you will, to help us respond to our world. It is like the idea of truth, or self, or soul, or or any of the other things we conceive of as distinct from actual matter. Philosophically you may disagree, and I can not "prove" you wrong. But neither can you prove your spiritual conception right.

[@donach9]

It's a matter of historical fact that the first mathematics came from trying to understand and manipulate the natural world. Then we humans, being as intelligent as we are, were able to abstract, refine and expand the knowledge we had so gained. But although parts of maths had followed axiomatic systems as far back as Euclid, it was only in the late 19th century that there was a systematic attempt to derive the whole of maths "from a small set of basic axioms, and logical extensions thereof" So, in seeking to understand Wigner's "unreasonable effectiveness" (the objection to antirealism) the historical roots must be kept in mind. If someone had come up with ZFC set theory purely recreationally,, then its effectiveness would be astounding, but instead ZFC was an attempt to systemise and axiomatise abstract and logical procedures we (humans) had developed primarily (tho not exclusively) for understanding the natural world. Don't get me wrong, I understand we have abstracted that knowledge and then run with it; and the 'running with it' doesn't necessarily have to have anything to do with the natural world any more; and abstractions, pretty much by definition, do not exist in the real cconcrete world.

[@jcortese3300]

Minor quibble about science -- we don't declare things "true." We declare things as "they seem to explain all the available facts, and the machines we build using them appear to work, but it'll all probably get overturned in 150 years or so." That's quite a different concept from "true." Physics doesn't have much to say about "true." That's a math concept. In physics, we always just assume that the applecart's going to get upset at some point. "Planets move in epicycles with the Earth at the center of the universe!" "Nope, planets move in perfect circles with the Sun at the center of the universe!" "Nope, planets move in ellipses, with the Sun at the center of the universe!" "Nope, planets obey a simple law of gravity that results in orbits that look like conic sections!" "Nope, planets and everything else move under the influence of distorted spacetime, and by the way, there is no center to the universe!" And on every step of that path, ironclad math was unreasonably effective in describing it. Much as I love these videos, I have to say that in every single one, I'll run into something that reminds me of why my MS is in Physics and not Math. :-) Turns out watertight math can be used to describe a lot of wrong sh*t just as well as sh*t that seems to be right, but like Newtonian gravity, only until it gets toppled a few centuries later only because our idea of what constitute available facts changes. We get a glimmer of something, build a theory that allows us to make a machine that lets us see slightly further, until we catch a glimmer of something previously inaccessible, and then we build a new machine that gives us some more insight ... it's like a constant game of leapfrog. So anyhow, "true" has little to do with physics.

[@Darkenedbyshadows]

So many Formalists here! I would say I'm a Platonist, but I mean it kinda goes with the territory when commits to studying Number Theory. :D

[@Math_oma]

+Darkenedbyshadows Yeah, I definitely have more sympathy with platonism. It's very odd because pure math students that I talk to in real life are always platonists, never formalists, but nearly everyone online seems to be a formalist.

[@SmallSpoonBrigade]

I think that speaks about the environment in which we're using the mathematics more than anything else. I'm a math and sciences tutor and on any given workday I can be using math for physics, chemistry, statistics and levels of math from arithmetic up to the calculus series and sometimes beyond. And if things are exceptionally slow, sometimes computer science. It's more or less impossible to work that range of material and not be forced into formalism. And that's not even considering the instructors at the college I work at who insist upon bringing their pure mathematics into the classroom and the myriad approaches I see to the same basic problems. Formalism has the rather significant advantage that you can focus more of your attention on the mathematical validity of what you're doing while you do the problem and then after you have a result, you can consider whether or not the result is one that applies to the application. You can also consider at that time whether or not the mathematical model even applies as a lot of that information comes from the form of the problem.

[@stevepittman3770]

It seems to me that formalism is the best explanation for math, because math is just a logically self-consistent system of precisely describing relationships. Any such system would be equally useful, whether you used numbers or colors or sounds, as long as the definitions are precise.

[@xhelloselm]

> Any such system would be equally useful, whether you used numbers or colors or sounds, as long as the definitions are precise That statement actually makes you a platonist, because you agree that the underlying mathematical concepts exist beyond our brains or our selected numbers, symbols and rules. They are "there", waiting to be discovered, no matter what names we give them.

[@blairnelson2149]

When 1+1=2 I'm a formalist. When 1+1=10 I'm just a realist(ic) old code slinger.

[@christopherender8164]

1+1=11

[@rursus8354]

When 1+1=3?

[@dhopkinsyt]

out of all of the videos on youtube, this one made me think about what math is, the most.

[@AndyWebster232]

I'd love to see a response video on this from Hank Green over at CrashCourse: Philosophy.

[@kephalopod3054]

Goldback conjecture for 2n: list [odds in] 1 to n in top row and [odds in] n - 1 down to n in bottom row, at least one prime is above another. For 32: 01, [03], 05, 07, 09, 11, [13], 15 31, [29], 27, 25, 23, 21, [19], 17

[@meeraa1941]

i am already in love with this series.

[@royhedine5153]

Here’s my guess: Our mind is a pattern recognition machine, it’s how neurons work. We sometimes make simplified analogies of commonly observed patterns and articulate them. Culture accumulates the most useful of these observations and weeds out inconsistency. Five rocks are a common pattern, but coarse sand and very small rock fragments confuse things in nature on the beach, so we grab five roughly the same pebbles to demonstrate “five” to others. We filter, or simplify ideas, out the patterns that interest us. The idea of five can be used to build other simplified analogies of commonly observed patterns, such as five groups of five, so on. The ideas and symbols of commonly observed patterns represent something both in and outside of our minds. We can say very precise things about “five”, that we could never say about a small pile of rocks, sand and dirt.

[@morgengabe1]

This is an important video.

[@kuonirat]

I think the mention of the "unreasonable effectiveness of mathematics in the natural sciences" is the most important point of this video. Not the "real vs unreal" debate, which I really don't consider "serious" in any sense - rather just a play with words. On the other hand, the effectiveness of mathematics is something that is stumping me for a long time and I think getting into bottom of this might be one of the most important tasks human kind can (if) ever deal with. Is the physical world a natural consequence of math? If so, why? And can we exploit this "causality" to create another physical realm similar to ours? This is the big question.

[@ΖήνωνΕλεάτης-δ7κ]

Do not forget that our models are seldom, if ever, entirely complete or correct! We typically start with a crude approximation, enough to give us some rudimentary insight and ability to predict physical outcomes! And then we start a tedious model refining process, in order to make the answer fit closer and closer to the outcome of the physical process! Sometimes the necessary math, happily turns out to be wonderfully simple and astonishingly elegant, but for the most part, it is a female dog! And many times, we are forced to adopt entirely new mathematical structures, in order to achieve some success in our efforts! You don't just throw some numbers together, and out comes the perfect model, miraculously, as it is presented here!

[@photosinensis]

Team descriptivism, standing by.

[@Roxor128]

Ready to describe reality and take names.

[@jcfreak73]

I am an antirealist, but not a formalist. I'm a figuralist where the way we TALK about mathematics include figures of speech. Numbers are not really nouns in the sense that they are objects. Rather they are adjectives we use as nouns. 2+2=4 is not a description of 2s and 4s, but is a simplified way of saying "if I have two things and then have two more things, I then have for things". Math is fundamentally the study of the relationships of ideas, whether it be spatial ideas (geometry), measured ideas (arithmatic), or logic.

[@adizmal]

Near the end... "knowing the rules of chess doesn't help scientists launch rockets into space" .......K, but a game about launching rockets into space would. Easy semantics, gg close.

[@OrchidAlloy]

Agreed

[@Vincent-sk1tb]

Kerbal

[@maxinator2002]

I’m a realist. One of my favorite explanations why has to do with stars. When asked how much longer is the distance around a star than the distance through its center, I can guarantee that it’s just over three times as long. I can also guarantee that if you were asked this same question for every star in the universe, the average answer would converge to about 3.14159265... Yes, the notation used to represent this answer is made up, but the number is very real. This is because the answer represents a scientific fact, and yet, it is a number: a quantity, just like having three apples and a tiny sliver of another apple. Therefore, math (and numbers) are just as real as science is, and science certainly seems to be very real. Furthermore, we (the human race) would never be where we are today without mathematics, so apparently we’re doing something right by investigating it.

[@AbeDillon]

Mathematics is the study of patterns. The symbols and notation of mathematics is simply our language for describing different patterns. Pondering whether numbers are real is like pondering whether words are real. The word "banana" is used to communicate about something real. The word "unicorn" is used to communicate about something not real. Mathematics can be used to describe patterns we observe in the universe and patterns we don't observe in the universe. When we study mathematics we are essentially exploring our ability to describe all different kinds of patterns. Prime numbers are interesting because they are constructs of our language of patterns, yet their distribution is so difficult to characterize with that language.

[@aashraynarang7220]

we are not talking about the representation of numbers we are talking about their meaning. you may represent 1,2,3 by a,s,d or any other symbol but their relationship won't change.number are highly consistent with natural phenomena even if we don't intend them to be which is weird and suggests the independent existence of numbers

[@AbeDillon]

Aashray Narang​ I'm not talking about the representation of numbers either. I'm talking about their meaning. We intend math to describe patterns. The reason it's so effective at describing the universe is that there are patterns in the universe.

[@aashraynarang7220]

Abe Dillon there are patterns in universe only when you see them through maths. there are no patterns if there is no maths. we didn't create maths to see patterns because we notice patterns when we have maths (don't consider abstract patterns like constellations) and this application of maths in physical world is weird.

[@AbeDillon]

"there are only patterns in the universe only when you see them through maths" That's absurd. Plants adapted to seasonal patterns long before humans invented math. Pulsars still pulse periodically even without math. Atoms would still bind to form molecules in specific geometric structures even if pattern language didn't exist. "We didn't create maths to see patterns because we dnotice patterns when we have math." My claim is that we created math to *study* and *describe* patterns. It's ridiculous to say that we need math to "notice" patterns. The brain is an exceptionally good at pattern recognition. It has been long before we came up with math. Simply knowing the complex behavioral pattern required to reproduce is evidence enough for patterns existing before we developed a system for studying and describing them. The universe is clearly not 100% chaotic. You don't see TV static every time you open your eyes. You see a universe with lots of predictable, repeating phenomenon.

[@aashraynarang7220]

Abe Dillon i want to clarify that i am not talking about patterns like plant seasonal pattern or abstract patterns and i am also not saying that objects in universe know maths and then they show patterns , i am saying that we can predict the behavior of objects in universe by using maths take for example hydrogen, energy of electron in hydrogen atom is given as E= (2π^2×m×e^4)/(n^2×h^2) put n =1 and you get the ionisation energy of electron that is 13.6eV per atom provide electron this much energy and ya it leaves hydrogen atom, provide energy less than this the electron do not leaves hydrogen atom this predictability in the behaviour of electron is weird (don't say its chemistry and not maths) and there are thousands of other examples which predict the behaviour of physical objects using maths and i am talking about such patterns

[@nwh57]

at 6:52 , did you guys just find that graphic somewhere or did one of you animate it?

[@themonkeyonyourback]

2+3 is always gonna be 5? Tell that to George Orwell

[@stardude692001]

2+3 could be 10 or 11, and that's without even changing the definitions of what the numbers are.

[@ΖήνωνΕλεάτης-δ7κ]

The base, is part of the definition!

[@stardude692001]

2 represents the same thing in base 4 as base 10. I would say only 2 digit numbers have their base as part of their definition.

[@ΖήνωνΕλεάτης-δ7κ]

stardude692001 And that is precisely why, the sum can be 10 (base 5), 11 (base 4), or 12 (base 3). Of course, in the case of base 3, one would have to have the 3, of the addends, expressed differently, to avoid confusion, since the numeral 3 could not be written as is, in that base, or it would be meaningless!

[@stardude692001]

Ζήνων Ελεάτης opps, you got me there. I'm a bit rusty on the old maths.

[@closair]

Math is like writing a paragraph and finding a book that starts with it.

[@Scy]

I'm so distracted by whatever is on her right shoulder.

[@Jimpozcan]

I'm distracted by her atrocious vocal fry.

[@bestnocture]

Scy I'm distracted by whatever is her

[@pacinpm2]

Nah, it's just a part of her bra or skirt.

[@NoActuallyGo-KCUF-Yourself]

It is not behaving like a piece of clothing. It might be a tattoo or a wire to a worn microphone.

[@vaishaliasthana5905]

It's a microphone wire I believe

[@LordZeebee]

I'm in the camp that maths is a language we made up to describe reality. A perfect circle doesn't exist somewhere but the idea of one does. We noticed that there are circle-like shapes and asked ourselves "how would i best describe all of these as a whole? What is the average circle?" Like how every cat is an individual and inherently different from every other cat but in order to talk about them we use the word cat. We're likening it to the idea of "a cat" even tho that average cat doesn't actually exist.

[@fosforos7]

The answer is that we didn't simply made up rules that mathematics it built upon. We discovered them while observing natural world. Yes, math is like rules of a game, but those are rules embedded in the universe we live in. Mathematical rules govern all natural phenomena, including very first moments of our life. We see perfect mathematical division in the development of human zygote, which after four divisions consists of exactly 16 blastomeres. Golden ratio determines proportions between individual bones in our hands. We see logarithmic spirals in shells of snails and arms of the galaxies. There's really nothing surprising in the fact that mathematical rules we uncover from the world around us help us accomplish scientific miracles.

[@shellypooper]

Great content. Who produced the music that is heard in the background during the explanation ?

[@wlan246]

3:03 I have it on good authority that 2 + 2 = 5, for extremely large values of 2. ;-)

[@briankelly5828]

Yes, if your value for 2 is 2.499, then 2+2 = 4.998 or 5.

[@glyphix42]

what is the intro music? is that custom or part of a song?

[@shadfurman]

That's kinda a false dichotomy isn't it? We defined math, for certain. But we defined aspects of math based on natural observations. We did just THINK up numbers, we thought up numbers to label counts of actual objects. Newton didn't invent calculus and then realize it magically mimicked the motion of the planets, he specifically defined calculus to represent the motion of planets. The fact that calculus can be used to describe other natural phenomenon is more unsurprising than it is surprising, many things in nature have a relationship between states similar to the relationships that cause planetary motion. Math exists in natural form only in the sense that things can interact in physics and have causal relationships that can be described with math, that doesn't mean there is some universal calculator that exists independent of the human (or alien) mind. (unless we're in a simulation). It's quite the opposite, the particle interaction of nature that allows for causal relationships is what allows computation of the human brain and computers. It does kinda turn into semantics, semantics of definitions. Is a tree a house? Depends on how we define house, does it keep off the rain? Does it keep out the wind? A house without a windows or doors (but holes for them) doesn't keep out the wind, is it still a house? Is a cave a house? If we define the relationship of natural phenomenon to be mathematical, it is, because we defined it. But the similarities between abstract numbers and their relationships of mathematical functions are distinct from the natural relationships. The similarities are not surprising, because we've defined the rules of math to mimic natural phenomenon and as such, the fact that it mimics natural phenomenon in unexpected ways, isn't surprising at all. It's like if you make a life simulator and it creates lifelike behaviors that seem to defy the original definition, it might be surprising emotionally, but it's not unexpected, because it's what you defined it to do.

[@drjeopyl]

Not sure which camp I fall into, I am 37 and have just begun to find math interesting for the first time in my life. Great series!

[@adityakossambe5265]

love your hair as much as the concepts you explain 😍

[@RonJones37568]

Some philosophers might argue that this question is "just" another version of "The Tree Falling in the Woods" conjecture that the English philosopher George Berkeley raised in his "A Treatise Concerning the Principles of Human Knowledge".

[@Math_oma]

Although there are many non-platonists in this comment section, can anyone offer an actual philosophical argument for why they hold their position? It seems everyone is giving their opinion, e.g. _math is invented_, _math is just a tool_, ... without giving an argument for their philosophical position.

[@wsmith49]

Lovely! Is this your personal discovery, or is this one of those arcane facts known only to a small circle of intimates, or is this a commonly known fact that I am naively ignorant of? Thanks for pointing it out, you have made my day.

[@abd.137]

@@wsmith49 It is just the solution of the equation x^2 = x + 13

[@abd.137]

@@wsmith49 which is, by definition, defining a number 'x' with the said property.

[@theedgeofyourfeather]

To me, one of the most uncanny aspects of the frontiers of mathematics is how often the most brilliant minds develop "conjectures" like the Goldbach conjecture mentioned here, and how often these are proven to be true decades or centuries later. That means that the mathematician had a hunch or feeling that some statement must be true, rigidly and absolutely true, even though he/she was unable to prove it. There is an area a given mathematician works in that consists of what has been proved, what is true, and beyond that is a void. The mathematician somehow (unconsciously?) feels something out there in the void, but just cannot built the path to reach it.

[@priscillaborges]

I'm very sorry to say but there is a mistake at 1:32. 2 is a prime number and 9 is NOT! I'm disappointed... The first few prime numbers are 2, 3, 5, 7, 11, 13.

[@Mohanchous]

I'm a formalist as well. To me, the subject of mathematics is theories which are collections of statements (strings of atomic symbols) generated by production rules (axioms, schema, and logic). These statements have no intrinsic meaning, the goal of applied mathematics is to assign meaning to and extract meaning from mathematical statements by interpreting symbols as ideas (e.g. numbers, shapes) or physical quantities. The fact that physics, properly interpreted, seems to obey mathematical laws is less a reflection on the unusual effectiveness of mathematics and more a reflection on the unusual order of physics. In a less ordered discipline, say psychology or politics, mathematics is not so effective.

[@umbraemilitos]

Saying that mathematics exists independent of human minds is like saying the rules of chess exist independent of human minds.

[@Macisordi]

Math is the language in which it's written reality.

[@umbraemilitos]

Maci Sordi Math is a language, sure, but this language is a tool humans invented to solve problems in reality, which is why it has the ability to model reality.

[@Macisordi]

I don't think so, we are part of this reality therefore ( a fortiori) we think in a mathematical way. But this are just conjectures...

[@umbraemilitos]

Maci Sordi I don't think I am making conjectures. For example, there are tribes whose adults cannot understand numbers, because their language never invented them. We can try all sorts of visual comparison methods, but they cannot tell the difference in quantity between a group of 5 sticks, or a group of 6 sticks. They nevwr needed to invent a number system, brcause numbers didn't serve any purpose for them, and didn't solve any problems for them. The way we think is based on our evolutionary psychology, and societal development. We know humans create languages, and we know humans create tools to solve problems. Our species didn't have language or civilization until primitive agriculture began to make civilization possible. Mathematics is a creative part of modern languages, and was invented to solve specific problems by simplifying reality into a useful fiction, allowing for trade, taxation, and engineering to be done more consistently. There's no magical universal form that humans are psychicly reading from.

[@Macisordi]

This doesn't mean that their ears cannot do Fourier Transform to listen... Our brain is made by things that follow the rules of our universe, mathematical rules. We discover(ed) our "special mathematic", but it is only a limited part of the total mathematic that exists apart from us. There is no magic in what i say.

[@jonathanjones770]

"You can make up new rules for checkers, but 2+3 is always gonna be 5" That's because the rules of checkers aren't representations of the logical absolutes, while "2+3=5" is essentially a very basic, valid syllogism in numerical form- it's a tautology. Our definition of "2+3" is identical to our definition of "5". Plus we can also verify "2+3=5" in the real world(becsuse that where the concepts come from). I don't see why numbers should be considered objectively existent objects- as far as i can tell, they're symbolic representations of the logical absolutes and their implications on relationships

[@omersapir6781]

That makes you a realist. :)

[@while_coyote]

Why not a perfect superposition of both camps?

[@maxweber06]

An interesting thing to consider is how some mathematical constants have so much abstraction that the base or root definition for them doesn't even have a platonic form or it does but is hopelessly abstracted away into meaningless. Take for an example e, it's not based on other constants or even idealized geometric shapes. It's based on the rate of growth approaching infinitesimal iterations. It describes much more than interest from banks, like radioactive decay, which exists outside of human interaction. That's why I feel like math is simply a tool that describes how the laws of the universe behave and interact. Those laws and interactions existed long before us, we just made a language that eases how we understand them.

[@juandominguezmurray7327]

Well... 2+3 is not always 5.. you are a mathematician so you know that.

[@rikkathemejo]

she quickly made the point... maybe too quickly for some of us

[@bloodyadaku]

Please show me an instance where 2+3 is not equal to 5.

[@juandominguezmurray7327]

bloodyadaku in Z/4 for example 2+3=1. In Z/5, 2+3=0. In fact, in every Z/n the sum 2+3 where "n" is not equal to 1 and is less or equal than 5, the result would be a different value than 5. Magic right? haha

[@bloodyadaku]

what is Z/4? What are you defining as Z? ℤ In mathematics represents all integers. Not sure if you're referring to that.

[@juandominguezmurray7327]

+bloodyadaku Yes, you are right, Z is the integers. Z/n would be the algebraic space where the number is the remainder of dividing the result, calculated like we normally do, by n. This are other spaces or classes within algebra. If you ask me, it changes the standard definition of what the operators "+", "-", etc, do. And this is a far as I understand it. I am sure that a mathematician could give a much better and clear definition than what I did. I am an engineer, but my wife is a mathematician. Z/4 I think is the notation that is used in english, in Argentina (spanish language) she always used the notation Z_4, or more generic Z_n. But that is just a notation, although I would have to say that is much more clear the english than the spanish one. When many afternoons in the bar are filled with researchers in math this sort of things come up lol. In the end it is just another definition that is useful for them in some way that goes beyond my ability to understand what they say. It is like when they ask "what is the sum of all possitive integers", I answer "infinity" and they say "that is one possible answer, but it depends of what you are talking about". Standard maths will yield the result infinity, but if you change only one definition (how you calculate the limit of an infinte series), you get the increidible answer "-1/12". This last result is greatly explained by mathologer (just type "mathologer ramanujan" in youtube and you will find it). To be honest, I was really unconfortable with all of this at first (and was really grumpy about it). For my engineer head it was extremelly annoying to say the least hahaha. But I guess math is all about logical consistency. It is not that they wanted to make "2+3=0" or "2+3=1" or "1+2+3+... = -1/12", it is just a result that is the logical conclussion when you just change a definition. I guess math is much more amazing and even much more complicated (and maybe much more annoying) than we all non mathematicians realize at first glance.

[@Orgyilkos]

As a physicist I think mathematics is "just" a language to express the properties of the universe. Maths was never about the objects it uses. It is only the collection of consequences of the validity of a given ruleset. Maths is "pure" in the sense of it never describes reality, it just describes "if this than that" relationships. An example with the most trivial concepts: Mathematician (the geek type, who just likes to solve puzzles): If something is perfectly round (I'm not saying this object should be imagined to exist, just that this condition can be applied), then that and that is true. Let me investigate the properties of this CONDITION even more!

[@Orgyilkos]

Here is the rest of the story: Physicist (the theoretical one, who just wants to be able to describe things): I've got this thing called static electricity, the experimentalists told me how it looks like. Now I wonder how can I describe it (and usually does the actually difficult part of math)... Oh, now I see, everything works out if I choose spheres (these spheres obviously don't exist either) for equipotential areas for each point, and if these potentials simply just add up, so I can integrate over any kind of solid (this one can both be a concept and the real deal, it should not matter, so he does not care about this either). [By the way, would it have not worked out, he would just used a different hypothesis as a CONDITION to test.] Experimental physicist (who does most of the painful labour to collect data): Okay, now we got this theory, we should test it and verify or deny it. I've got this sphere (finally a real one), on which I can place a point-like charge anywhere. Let me see how the theory fares... It seems to hold up reasonably well... Now I have to do the most dificult calculations, the theoreticians have forgotten about. I have to provide an error. (Now, providing an error is always the hardest calculation of an experiment). I wish those mathematicians had thought about this too... But they are usually only interested on giving if then than that rules, not "how much of that, when this is this much true". No wonder, it would be too difficult to calculate for all the existing maths, so we only do it to the parts we use. (months pass...) Okay, I got it drilled, I just have to numerically solve these integrals. I'm lazy to do it, and anyway, it should be the mathematicians job to figure out solutions to this... Diff. equation specialist (who is in it for... I genuinely never knew :D): Ah, here are some new challenges for me. I don't even care what it is about, but I am the only one who is able to solve it in less time than a century (spheres never even come into his/her mind).

[@frenchmarty7446]

There is a total false dichotomy here. Math is either a made up game or about imaginary objects? Those are the only two possibilities worth mentioning? Why did you not consider the idea that math is about _real_ objects _viewed from a certian perspective?_ Take the example of a "perfect" circle. If you look around you, you can find countless examples of circularity in many forms. If you take those examples and _abstract away_ their differences, you are left with the concept of a circle (with each example being a variant on the same theme). There clearly isn't a "perfect" circle somewhere out there waiting for us to discover it. It's also clear that the concept of a circle wasn't just made up out of nowhere (how would you even come up with the idea without first looking at reality?).

[@rursus8354]

Very well stated!

[@rursus8354]

Very well stated!

[@nonameforyouokpeterrodney9051]

The solution to factoring any composite integer is equivalent to determining the non-trivial, positive real-value of a variable (k) I call the 'Coefficient of factorization' used in a unique, integer-factorization expression of mine which I've termed the 'Transformula'.

[@TheGokki]

I love her hair!

[@ImAllInNow]

This is how I felt when I studied Grafting Numbers. That I was discovering something that was already there and the connections that I found to Catalan Numbers were completely unexpected!

[@DheerajBhaskar]

The intro sound track and the host herself are so classy 😀 At 3:30 saw her shoes which don't at all go with her dress. You broke my opinion 🙈

[@kindlin]

Check that dress strap. It bugged me all episode.

[@steve-ks9df]

there is a real sense in which something like a number is "made up", because if humans did not exist, the universe would be indifferent to its inner workings, the ratio of the circumference of a circle to its diameter, etc. But the fact that we can communicate at all using abstract concepts like circle means that there is something that we can call objective reality, but not a Platonic reality in which perfect circles exist

[@travisjohnson6676]

one of the hottest mathematicians I've ever seen. smart and sexy

"If a tree falls in a forest and no one is around to hear it, does it make a sound?"

[@JamesSpeiser]

Talk about fruitless over analysis.

[@ozicell]

Not a mathematician here as such but love the sciences without any formal education of the subjects! Eg. Frequency - sound and light, heat and cold are all related and our understanding of them relies on our abilities to calculate. Heat and cold, sound and light are all observable. Understanding what they are relies on our use of maths! One of my favorite math concepts is Infinity - a line an inch long technically only has 1 dimension - length and that length of 1 inch is infinitely divisible. A line a million miles long also has only 1 dimension - length, and that length of million miles is also infinitely divisible, so is it just a perception that makes the difference between the two or is our perception limiting us to consider distance as a factor which may turn out to be an illusion? Then you have the axis of a rotating body - this axis also only has one dimension - length. The further out from the center of the axis, the more observable the rotation is but as you get closer to the center of the axis - is there a point where the rotation ceases to exist or maybe becomes or exists purely as a void like a nano-sized black hole!? Whichever is the case, the axis may be infinite or infinitesimal in length and have only that one dimension yet hold almost unimaginable power as it seems that is lays at the heart of almost everything we can observe! As I said, I am not a mathematician or a scientist, I am more of a philosopher and so I love these questions which often keep me awake at night. Thanks for your channel, I have subscribed and look forward to future episodes. Cheers Jeff

[@whatthefunction9140]

Do I see a hint of a hot tattoo under that dress?

[@whatthefunction9140]

Cool poem, I'm sure Kelsey's looks as well as her intellect are what landed her this hosting gig. Both are on display and I'll comment on both. If she doesn't like it she should do voice overs.

[@DeePal072]

Dylan T​ Watch her left forearm, Sherlock! 🔍🤔😱😳😅

[@Diggzist]

"If she doesn't like it she should do voice overs." "My name's Dylan and instead of respecting women I think they should just not show their face to prevent me from disrespecting them."

[@whatthefunction9140]

We dont know what might offend someone. We can't live in a world where no one comments on others for fear they might possibly be offended.

[@Diggzist]

Yea you're totally right but your commenting on her body is completely irrelevant and off topic and women have been undermined in the STEM fields since they originated and you making it about her body rather than the subject she's talking about is pretty disrespectful and gross. Not to mention the fact that what you just did happens to women of high esteem constantly. Female actors, singers, business owners, and just in general women are asked about their physical appearance rather than their accomplishments and your comment is the exact same. Plus your idea about how we should be able to comment on each other is more for friends telling each other if they're being dicks or if they should stop doing something. Not for a stranger online to discuss whether or not an intelligent woman has a tattoo under her dress or not.

[@adamkey1934]

Why have I only just found this channel? Subscribed

[@Psd863]

If any mathematician could interact with the platonic form of numbers, it would have been Ramanujan

[@only1gameguru]

numbers are properties of matter and maths are the interactions. numbers are like happy, you can't touch happy

[@deldarel]

I'm a realist. I think of it as 'we observe two things, look what they have in common, and then compare the objects to find out how much this property aplies' and we do this everywhere. 'this guy looks way more american than this other guy' or 'this ball is way rounder than an american handegg'. We now pick then two arbitrary points and add a number line to it. Since this line doesn't have established absolutes, they're degrees. This is pretty much how temperature also got measured. However, much like how the absolute of temperature was eventually found (0 K. Also, since it's an absolute, you don't use 'degree' with kelvin), the absolute of a sphere has been found. Neither 0K nor a perfect sphere physically exist in the universe, but that doesn't mean they don't exist at all. You start formally describing the world around you, and come to very real results that don't exist physically, but still exist because anyone can get to this very result, preferably in multiple different ways. We're walking through a room blindly, bump into stuff, and we examine these things with our hands trying to figure out what it is, and the more we try, the better we can distinguish what we have. Sometimes we feel stuff we think is real, but upon closer examination, it's actually part of something we roughly touched upon earlier. If someone else were to describe that same object by touching from a different side, he would eventually come to the same conclusion if both are right. I think that, if every piece of evidence points to the same result after a certain amount of retries, then it's safe to assume it's real, even though we can never know for sure.

[@z-e-r-o-]

Yesterday I read your article about this issue on Scientific American Japanese Edition. It was a very interesting and clear explanation. Thanks!

[@91722854]

merry christmaths, everyone!

[@accidentalengineering]

The definition of a prime number is NOT "divisible only by 1 and itself". That's just a property.

[@-dennis3755]

Realist here.. well sorta. Essentially mathematics IMO is the way you program the universe and going farther in it's how you also program what's in the universe, and even farther it's how you program the things in there to know how to program in math. For instance, If you were to build a self aware AI it'd code itself with the same code you used. Much like how we make up AND discover math to create things that bridge our minds with our universe. But really I'm just rambling

[@DrB1900]

Mathematicians need to stop hanging around with philosophers. I would like to see a proof that this kind of discussion isn't a colossal waste of time.

[@KarnKaul]

This channel is insane! Subscribed!

[@cookieDaXapper]

Goldbach is a trick; math rules: adding two even numbers makes an even, two odds, an even,but an odd and an even an odd. So using primes there is only one even prime, 2, and adding it to any prime makes an odd number.