Do numbers EXIST? - Numberphile 

My congratulations to numberphile for making a philosophy of mathematics video. In my opinion, this is a very interesting side of math that gets almost completely overlooked and it should get more attention

I've told my bank I'm a mathematical fictionalist... they looked at my account and agreed, but still expect my bills to be paid on time.

Easter Egg: In the thumbnail: RIIDIICIIIPO R2-D2 and C-3P0

He starts with "we are gonna be thinking about three different schools of thoughts on this"! So he believes that the number 3 exists!

I ended up from A4 paper sizes to the square root of 2 to whether numbers even exist..i love youtube

This guy (who I'm assuming is a professor) is really knowledgeable about the subject. This video is the only place on the entire internet that explains the Platonist/nominalist/fictionalist battle concisely and well. Thanks, Numberphile!

This was one of the most fascinating videos I've watched in a long time. Thank you so much @Numberphile.

I used to be a nominalist before I entered university. Then linear algebra started, and this is when nominalism totally collapsed for me. It was a momentary aha moment. All it took was to reflex on the axiomatic definition of what a vector is. Vectors aren't unique concepts in this manner, but it was the thing that striked me. Vectors and vector space create a world completely detached from reality. You don't need anything real to define vector space, and you actually can't use anything real to define a vector. Some vectors can be discribed using numbers and some basis, but vectors aren't those sequences of numbers. And there are vectors that can't be fully described using numbers, for example vectors in spaces with infinite dimensionality. And everything can be a vector if operations are defined. Vectors are literally "anything, for which operations of addition and scaling is defined and well behaved". Look, I say, that I define operation multiply(my bed, any number) = my bed; add(my bed, my bed) = my bed, vector space = set, containing only my bed. Now my bed (a literal physical object that I am sitting on right now) is a vector, because it satisfies all the axioms of vector space! Not a description of the bed, not a representation of the bed, but the bed itself is a vector. And no, it's not like some vector is used to represent my bed, but my bed IS a vector in this context.

Do ideas exist? Is existence dependent on materiality? What defines existence?

“All models are wrong, but some are useful” - George E. P.

I really like the talks about the philosophy of numbers. I feel they could go in depth about each one and more about where it's troubles are, and maybe even how they've gone alone historically (with more resolution). It's an interesting topic.

Mathematics is just the study of abstract ideas, the properties and facts of abstract objects that obey axioms. The subject emerged out of studying physical objects, and in modern times, has now been generalised to study abstract problems. Mathematics exists in the realm of thoughts.

Numbers exist, the very act of knowing about it maintains it's existence so it really doesn't matter if they're a metaphysical object or just an abstract fiction it doesn't change the fact that the idea of them exists. So I guess I'm a Platonist?

I have a pencil. I have another pencil. Oh, two pencils!

Brilliant. Thank you Dr.Tallant. This is a fascinating lecture that clarified some questions I had about the subject. Although there are concerns and issues with each of the three philosophies, I am leaning towards nominalsim at this point.

I think the (truncated) platonist idea is that numbers have properties-like primeness-independently of how you conceive of them-hence you can misconceive of a number-and therefore they have mind independent properties. As such, they exist.

That portrayal of mathematical fictionalism was very far from neutral.

Thanks for this channel. I'm very interested in Maths, so this channel is the best opportunity. But my english isn't so good at all so I'm training with your videos my english to. The prounciation of all Numberphile members is very good/perfect. So I have no problems with searching for a translation if there is a vocab which I don't know. Thanks for all :*

This video made me think about this more than I thought I would.

The problem with nominalsim is, even for simple concepts like the number 3, if you have 3 apples on one hand and 3 oranges on another, what do these two sets of fruits have in common? Would we say they have the same quantity, but that there's no such thing as quantity?

This is extremely interesting from a self examination standpoint. With my very limited mathematical background (I watch these RU-vids out of interests, not application), I strongly relate to Nominalism (how I learned math) yet Platonism seems impossible to grasp. This helps explain why I can't warp my mind around there being 'imaginary' numbers or even something life infinity. For me, a nominalist, I need a real number to exist in my mind; for a Platonist, that number does exist regardless. VERY interesting indeed. PS I'm a soccer coach... math plays very little in my profession. I just enjoy Numberfile and Sixty Symbol videos a lot! thank you.

It's fascinating as well as depressing to see something so much powerful and magical can't even define it's existence.

Right. Got it. Abstract, outside of space and time. Yup, I got a term for that. Imagination.

I like Seven Up, more than I like sprite

I'd rather see numbers as the linguistic filter through which humans interpret their physical reality. If that makes sense to anyone else? Doesn't make mathematics "absolutely true" or "absolutely false". Since they occur to us in our thoughts, numbers are conceptually true, but since they are physically intangible, they can just as easily be called "false", if you wish to see it that way.

This was a terrific video with a terrific interviewee. Thank you! I want to point out that (being something closer to a mathematical fictionalist myself) that part of the problem in explaining these things has to do with confusion between the words being used, namely "true," "exist," and (barely used here at all) "real." These all stand to be delineated from each other to prevent probably equivocation (a logical fallacy) due to their colloquial synonymy. true ≠ exist ≠ real It's worth taking the time to parse these out. A fictionalist indeed won't agree that numbers offer "true" descriptions of things, but that's in part because of one of two things (depending on the sort of fictionalism you belong to): (1) nothing is real anyway (a.k.a. "anti-realism," which I do not endorse), or (2) what is real is impossible to access directly by any means available to humans and non-humans alike (a.k.a. speculative realism). As a speculative realist, the problem around numbers is that they exist (notice by saying they exist, I'm not necessarily saying they're true nor that they're real) as a type of intentional objects (which has a specific meaning derived from medieval philosophy and, more recently now, phenomenology) which give literal descriptions of real *qualities* of objects (real ones), without actually being real objects themselves nor able to "tap into the truth" of real objects. It would take a whole other video or essay to really get into this, but I think it is inaccurate to say that fictionalism is necessarily extreme. What's extreme to me would be denying that numbers exist at all (which as I've indicated, isn't what all fictionalists would argue) or that they are "merely useful" as opposed to being both useful and aesthetically interesting in their own right.

loved, loved, loved! just thanks!!!

excellent intro to philosophy of mathematics, or rather to the different views on it!

I don't agree with any of those completely, but I see pieces of each. Math doesn't really exactly describe objects. If you have 2 apples, those 2 apples are different, therefore you don't have an exact two apples, you have two sets of mass that are separate from each other that are relatively the same. And therefore an approximation of what you actually have. There is no physical depiction of a circle that has a surface, volume, or area that can be accurately calculated by pi. In fact, a circle itself is just another abstract object. I don't have a circle for a mirror, I have a mirror shaped approximately like a circle. Mathematical objects are just tools, like a hammer. The hammer strikes a nail, but the full surface of the hammer doesn't meet the full surface of the nail, there are grooves that hide surface away from each other, but at the end of the day, the job is done just as well.

Numbers, letters, and symbols are all abstract units we have devised over the ages to better communicate and rationalize about the world as we experience it. Language is one of the earliest attributes we developed, but it took a while before we started writing down what we thought our language sounded like. How we decided to represent the sounds of our language in symbols draws from our creative faculty, which is linked to our senses and perception, which differs from person to person. Words are special concepts we recognize and associate with objects, ideas, emotions, etc. Numbers are a lot like words in that they allow us great freedom in describing the world; on the other hand, whereas numbers are infinite, our ability to communicate ideas, and especially to convey an experience, is limited. Our use of language as a primary means of communication is still far from perfect. Language is only our best attempt at conveying our thoughts and emotions. For example, you cannot absolutely convey your current state of being using language. Likewise, you cannot concretely define 'happy'. Granted, there are some experiences that a majority of us can agree we would say tend to make us 'happy', but 'happy' is just a string of letters. That's it. It's an identifier, and just like any other word we readily recognize, it points us to number of other things stored in our memory banks that we associate with the term. But what about when we see a word or symbol we don't recognize? It has little or no meaning at all to us, and no switches in our mind are tripped because we don't have any images or experiences to attach to the word. On the other hand, we aren't restricted by the inherent obstacles of language when engaging in mathematics. Numbers can completely and absolutely describe any physical quantity we would possibly want to record to any degree of precision we want, and then some. Numbers are only as ugly as we need them to be, but the more beautiful side of mathematics lies in an ideal world where we can say that we have a circle of radius 4 or an infinite sum of integers or even such abstract concepts as dividing zero objects among zero recipients. However, in the real world, it means very little to me if someone says they measured a sandbox to be 6 feet by 8 feet with a depth of 6 inches, so they know they can fit exactly 24 cubic feet of sand in it. The molecular make-up of the sandbox tells me that the width varies microscopically due to the uneven surfaces, and no matter how precise a measurement you give me, I can always go one decimal place further, and we can play that game on and on. Though really we're limited by the tools we use to measure said quantities, but I think we can all agree that there comes a point where a measurement is "close enough" for our purposes. We could instead say that there's a certain maximum amount of sand particles we could fit in the sandbox. It turns out that units of measure are ideal, fixed quantities we can envision, and we apply them to the best of our ability, but they really don't "fit" in the real world though they've worked wonders for us throughout many fields of study. Likewise, numbers exist in theory and are at the core of science, but the way I see it is that number is a property that describes either the ordinal ranking of something, or how many of something there are. That is, . Anything beyond that and you're treading in the hypothetical land of mathematics, where solids have faces that are perfectly flat, and we can say we know a quantity exactly. There's only so far you can go applying mathematical principles to the world we live in. As far as numbers existing, they all exist in theory: the imaginary unit, i, Euler's number, e, pi, the golden ratio, even the Fibonacci sequence, prime numbers, perfect numbers, real numbers, infinity, one, zero, rational numbers, irrational numbers, etc. They are all at the root of many natural phenomena and play major roles in many applications, but I truly believe that only natural numbers can be seen as tangible, existing concepts in the real world. I can hold three bricks, sixty thumbtacks, one shovel, and I can be number 1 in a list, number 5, number 1000000, etc. I cannot hold zero staplers or be number zero on a list. Neither of those statements makes sense. Using a number outside the natural number set, or using natural numbers for anything other than ordinal ranking or counting is to apply abstract concepts from an ideal world to a world that's not ideal, and there are plenty implications to consider when making such a move.

Very interesting. It also helps explain, why students have great difficulty calculating with negative numbers, with fractions, etc. They are very abstract indeed [unless you mind-numbingly just manipulate the symbols]

Excelent presentation!

Well isn't mathematics sort of a language? A language can't be true of false, its type isn't a boolean far as I know. It can be used in any environment to "describe" a problem and build up frameworks to generalize and simplify the system we interact with (which could be real life, a computer program, observations basically anything we can study and break down), this is how I interpret mathematics, many toolsets within mathematics lets us express ourselves better than we could do orally. Neither of these philosophical statements proposed on the video make any sense.

0:13 plato's theory of forms

this is such a mind blowing concept

This channel is awesome.

Is 'blue' real? One can define a range of frequencies as being blue - but that definition would be arbitrary; that is to say that if some children were raised without being told that frequency range, and then asked to define blue, they likely would not come up with the same range. Blue is just a concept we use to encapsulate a commonly perceived phenomenon. Are our perceptions 'real'? Let the ontological debate begin!

Platonism: mathmaticians are so reliable because unlike all of us in different realms, they have thoughts that are pretty clear in each step, there is no use of the subconscious mind, you might intuitively get a correct math answer but not always and the same is for all kinds of thinking (math or others) but we still use intuition in others.

Through my "career" as a elementary school kid, to A-levels and now university I've went through all three stages of maths and now at the end of my bachelors degree/soon starting masters degree, I'm switching through all three every few weeks xD

I’ve gone down the rabbit hole and it’s now after 7am. Night spent well.

I seem to be somewhere between Platonism and nominalism. I do like the notion that numbers are mathematical objects, which did start off as interpretations of real phenomena but aren't restricted to it. The fact that numbers can only interact with each other in a finite number of ways, to me, seems eerily similar to how physical objects too can only interact with each other in finite ways. That doesn't mean that numbers exist like physical objects. But they're different from simple abstractions because they're not subjective. Does that make sense?

What strikes me, having just watched this again, is that all these theories are based on a correspondence theory of truth. They are true, and they are criticised, based on how well they refer to the world. Even the fictionalist's argument is based on a concept of true= referring to 'the world'. I've started to consider maths as being an example of a coherence theory, where something is true because it is consistent with the other statements in the set. Of course this comes to problem of 'if there is an infinite number of numbers, can we ever check that every number is coherent with any other'. The answer, of course, is no. But in broad terms, maths seems to work because statements are consistent with each other. Disproving a mathematical theory involves finding it to be inconsistent with other, accepted, theories. It's is probably true that nominalisation brought maths about, but for these abstract terms it seems more prudent to focus on it's consistency.

Very well explained and entertaining

This is super interesting! as some one qualified in nothing my thoughts tend towards maths been a language.

I think the fictionalism is the most accurate, since Mathematics are simply models of the things we consider true… Mathematics uses true logics, but isn't always using accurate representation of the true world… Mathematic is a platonic/nominalist science. Real sciences are connected to the real world, not to fictional models.

Plato to the core!

I personally adopt a stance of mathematical nominalism and only dabble in mathematical Platonism when it aids me in learning new concepts/ideas. It is much easier to treat a new thing with old rules than learn new rules and all that entails.

Nominalism is the one I agree the most. We start mathematics describing simple things that exist in the concrete world and the more complex numbers, like the irrational or the complex themselves are just descriptions of things we cant measure (like Pi), things in more than 2 or 3 dimensions, or just descriptions about abstract concepts that are not concrete, but depend their existence from beings that think or believe is such concepts.

For something to exist outside space and time seems to be like saying something exists no where and never.

I think the 'discovery' of mathematics is like the creation of a language, albeit one that's a lot more rigorous (i.e. logically deductive) in its creation than that of a spoken language. And it's a language for describing the patterns/ structures we can see or imagine. I don't think we know enough about the conscious mind to understand what it means for a structure/pattern to be mathematically describable by it (and not just by our mind, but one that's maximally 'intelligent').

Complex numbers are very useful describing and handling the properties of dynamic systems in which there is an internal delay in response to stimuli, so they describe real characteristics of real objects. It just so happens that at times the mathematics is found before meeting the objects it describes... which would be troublesome, if we did not live in an universe where this kind of things tends to happen on a constant basis (see the single photon diffraction experiment, with the later choice addendum).

I think the nominal way to define pi is to call it the ratio of the circumference of a circle to its diameter. In exactly the same manner we define “two” to be “a singularity plus a singularity”. We use an approximation for pi, sure. But the concept of its existence/definition as a ratio is as real as any other “number” which can be called “true” by definition under the rules of mathematics.

How about nothing but numbers exist? Basically, every particle has its values, its location, its properties etc. All the numbers we make up are ways to connect those values. Numbers don't exist in the sense of 1, 2, 3 but as relations. So this is twice that, and this is 3 times that and so on. The complex numbers like phi are simply infinitely long connections. In the end, if we'd take away the human interpretation of the universe, it's all just trillions times trillions times trillions times trillions of data points with interaction with one another. Just my view on the topic, be welcome to share your thought.

Numbers is just a language used to define logic. Is language non-existent too?

well done on breaking down these viewpoints into a digestable form for ppl like me.

MORE OF THIS!!

Numbers are similar to language, intellectual property, thoughts, ideas, memories, and other brain states. In this regard, they are as real as the concepts of a chair, a bed, or a house. These are different from the objects (chair, bed, and house), but the concepts (i.e. brain states) are needed for humans to construct chairs, beds, and houses from real resources (wood, plastic, cotton, etc.). This does not imply that people could not sit on rocks, sleep on the ground, or live in caves without these concepts, but we would not be as comfortable doing so. We can extend this logic to another level. Wood, for example, results from Nature's DNA memory of how to grow and to reproduce trees. Rocks result from Nature's memory that SiO2, Al2O3, etc. can form crystalline grains and that gravity and rain can pack these grains together into dense solids. There is a theory, in fact, that the fundamental physical particles are merely the knowledge of how to form these superstring states from space-time fabric. Given these facts (I know the theory is not a fact..) it is reasonable to ask which is more real? Is the chair more real or the knowledge of how to build a chair and the knowledge of how it will make us more comfortable?

I've discovered that I was fundamentally a mathematical nominalist, in the sense that I always thought numbers were descriptions of concrete objects. Explains why I've been so baffled by imaginary numbers and irrational numbers. Thanks, Numberphile!

Great work people. Could you people do a video about the Euler's formula and the transformation between imaginary and real numbers. Ontological mathematics.

In defence of mathematical Platonism: the reason that people themselves can interact with the abstractions in mathematics is simply because consciousness is "the same type of thing" as those abstractions, which I call _information_ (i.e. consciousness and abstractions are both forms of information). The mind and the body are a fantastic link between the material world (i.e. matter) and the immaterial world (i.e. abstractions and thought).

Can't we just say that "mathematical truth" is derived from the set of axioms what we've defined which are the rules of the game? It's really no different to chess it seems like. No one would seriously claim that chess moves exist outside of space and time, there are a merely a set of rules which govern the game. So the case of irrational or imaginary numbers... they just emerge from the rules of the game. The epistemology here is Mathematics and the truth or any mathematical claim is then determined by verifying it against the axioms of the epistemology... no?

Two plus two is four minus one that's three quick maths

As a nominalist, pi is easy: it's a relative value to circular calculations. Just like the square root of -1, yeah it's more complex, but ultimately it has relational value to other numbers.

Seeing a lot of passionately written walls of text, but not seeing anything that does not fall into one of the three philosophies described in the video.

I'm with plato on this one

Nominalism supplemented by substitution of the concept of truth by a state of being uncontradicted should bring it closer to everyday life practice. That can shed light on the interaction of the conciseness of models of thought and the fallibility of the mind.

You should make some calculus videos, or ones about vectors. Those are interesting.

If you asked me, "What is the most absurd combination of three words you can make in the English language?" I would have to say it'd be, "Math is false."

The numbers in our maths are logical concepts which at first best described the physical world around us and the abstract understanding is developed and refined over time. However, those numbers have original ties to the physical world and then became abstract describing reality that may or may not exist. At the end of day, they are just representations and not the actual thing being represented. Also, human concept of numbers might be totally alien to another creature and that creature might say, ""Wtf are you are guys talking about, there's a more logical and easier way to represent these things" Like numbers, truths are concepts -- something that we've agreed on over time.

for a nominalist, sqrt(-1) is the solution to the equation x^2+1=0, which is equivalent to the question: if you have x^2 objects arranged in a square with lengths x, and you add 1, what length is needed so that the resulting number of objects is equal to zero?

In eleven seconds into this video and I can already tell I'm nowhere near stoned enough for this topic.

DEEEEP. give this man a cookie :D

I'm pretty sure mathematics is a language- a system used to describe the relationship between different elements. The number one is no more or less real than the word "one", it's just more flexible as a tag. Irrational numbers are just hypothetical relationships that are not logical, and do not exist in our world.

I think an argument a nominalist can make with complicated numbers is that they describe a property of a real object, for example six lots of seven being the size of the set of objects. It seems a trivial distinction, but it opens the door to say the square root of negative 1 exists, because it is number which allows the schrödinger equation of particle properties to be represented in exponential form (for example, of which there are many) and pi, although we dont know its exact value, saying that it is the ratio of circumference to diameter for any circular object is sufficient to define its existence.

The Platonalists are the Algebra people; The Nominalists are the Numerics people; The Fictionalists are the ones, who ran away from maths in the school and ended up in philosophy thinking there were no maths.

Based on this video, I'm a nomalist.

Fantastic video! I think I’m on the end of Fictionalism in general but never considered it before now. Thanks for putting in the effort of making great videos 😁

this video made me sub to this channel a few years back... lol

I think this is similar to asking: Does beauty exist? Does justice exist?

Numbers exist. Every application they have, every property, has always existed. From the time the universe was created, the concept of a number, say, "7" was real. Base 10 also did. We gave 7 that shape and took Base 10 as our main guide. Even the newest theorem, has been true all along. Or was it something none existant at some time? You think we humans could've invented this vast, perplexing, ever expanding world of mathematics by ourselves? You see, we don't invent numbers and properties, we discover them .

i (as a student of math) agree most with Fictionalist but what annoys me is that existing and being true or false is getting mixed up here. for me they are totally independent...

The question about Platonism comes from someone who has not studied Plato, and is really quite easy to address. In a Platonist standpoint: Anybody can interact with the Forms, or "abstract objects", and in fact the world would not exist as it does without this interaction. The difference is that one interacts with a chair with the senses - you see, touch, smell, etc. a chair - while one interacts with the Form of "Chairhood" through the mind. Mathematicians can interact with numbers in just the same way a person interacts with Justice, or Freedom, or Goodness, or Chairhood; through the mind. Through Reason.

I always have this video playing while writing a paper late at night.

why didn't he mention Rorty and pragmatism when he was talking about fictionalism. which reminds me of bentham idea of truth as a fiction or as an agreement.

Numbers are Binary Code for our Universe if you want put it that way

It all depends on your definition of exist. To me exist means has observable, measurable physical properties. This definition excludes such things as love, hate, ideas, numbers etc.

Great video.

What do you mean by "fictionalists" think math is false? That seems to imply that they think it's incorrect in some way. By false do you mean not real? or do you mean incorrect? I'm leaning to think that numbers aren't real. They're just a concept we humans use to explain things. And with things like Pie and square of -1 we are beginning to see our concept, aligned with proper logic, has the power to show us things we don't understand and will never be able to. I think the very fact that imaginary numbers exist and we can prove it with math proves that there are certain un-explainable things that exist somewhere outside of our reality.

0:29 "so [numbers] don't exist in the same way that tables and chairs do" 1:37 "so in the same way a table exists, a chair exists, a number exists" Talking about Platonism both times -_-

I think it is more useful to think of wherefrom the truth of mathematical statements come. Immanuel Kant says a great deal about this but strict quantitative relations are analytic (counter to what Kant thought). However when we state things such as "the area of a circle is equal to the radius squared times pi" it becomes synthetical. However the quantitative relations are strictily define by us. It depends on our definition of quantities and such. And as such Mathematics is true but only as so far as the statement is synthetical or as so far as we define the outcome.

This has a tremendous potential to help us promote Intellectual Nutrition thanks!

"What use....."? Was my question. I know the answer but I get the feeling you are maybe missing the point buddy?

Numbers are a language for communicating quantities.

Please do a video on complex numbers!

im sort of a combination of a nominalist and a fictionalist. We started out with nominalism and then extrapolated and created things like imaginary numbers that were useful in describing physical patterns in the world and creating models.

If I draw the blueprints for a house that hasn't been built, are those blueprints "false"? Certainly the house doesn't exist, but nor is it an abatract name for houses. If I were to build a house, I could use those blueprints I carefully planned out and thought about, but the blueprints do not become "true" if I build a house closely resembling the abstraction I put on paper. So my position is that numbers are not real, they are a mental tool, but that doesn't make math somehow "false."

Isn't existing outside space and outside time a bit of a contradiction? As what you are really saying is that they exist at no place and at no time. Which is the same as they do not exist at any place and do not exist at any point in time. Which start to sound an awful lot like the definition of not existing.

What determines how one would respond to this question depends more on your definition of existence, rather than the description of the theories themselves. If we discuss this, we can see that every theory can be correct, depending on your definition of existence.

"Ontology is mathematics." - Alan Badiou "Dasein already exists." - Martin Heidegger "Our external physical reality is a mathematical structure." - Max Tegmark (Mathematical Monism) "The only kind of verification of which ontological claims are capable is phenomenological." - Paul Gorner "The Universe is the complete spatio-temporal logic of the self." - Robert Lanza (7th Principle of Biocentrism)

So many pretentious 13 year olds in the comments...