What is a Number? - Numberphile
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- čas přidán 13. 01. 2022
- Featuring Asaf Karagila.
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8:52 If numbers exist in a context. The question of were numbers" invented?" becomes a matter of "can a context exist human without our human contextualization?"
Well, the notation for numbers is definitely invented.
Like, we could totally come up with a completely different way of notating numbers, using different mathematical operations... and still have it all make complete sense.
But that's kind of the point. What's the definition of "makes complete sense"? Well, that our notation system conforms to the real, objective functioning of the universe itself... and humans didn't invent that.
I guess it's like asking "is a map invented or discovered?". Well, the map itself - the piece of paper with the symbols and lines on it - is 100% invented... but it conforms to real, actual physical places out there in the world.
The "abstract representation of quantities" is totally invented - it's abstract, after all - but quantities themselves are a real objective thing that exists, and mathematics is our invention for mapping that objective space.
It's drawing this distinction between the notation and the thing that's being notated.
Like, if I write sheet music, then that's not the music. It's just a "map" of the musical terrain that a musician can follow to recreate the actual music.
But a map of a place is not the place itself. The map is definitely invented. The place, though, is a real physical location that's discovered.
And, indeed, to draw that distinction, you can consider that you can make maps of non-existent places. Like Tolkien's map of Middle Earth in his books. You can use the invented notation system to notate something that doesn't actually conform to any reality. As the map is not the place, and the place is not the map.
But you can use the map notation to create a map that does actually conform to a real physical place.
So, it kind of depends how you ask the question. Mathematics is a human invention, but we invented it - like making a map of a place - to correspond with a real objective phenomenon that exists out there in the universe. And the mad thing is that we can extrapolate with our notation into some undiscovered realm, get an answer then check it against objective reality... and it actually matches up. Therefore, there really is an objective thing out there - a place - that we are mapping with our notation that's discovered, and not invented.
The problem that I see is it appears to me that infinity can be defined to a child as being a number higher than any other number, after it is established that there is no highest number. This implies to me that infinity is a concept relative to numbers... and I thought it was a different concept in regard to sets. If that is the case, it seems to me numbers must be fundamentally different than sets.
I don't buy the standard answer presented here.
@@essiw "However if you define the number 2 as 2 objects of a certain thing" However, this is certainly an invention and not a discovery that 2 objects are a certain thing. It's a convention, a thought pattern, to say, this thing is like that thing. Like the quantity of apples, to say what an apple is, is a convention (does it count less with the stem off or on, etc.) and especially like the quantity of fruit (what is a fruit). And going very physically basic, say quantity of electrons, it certainly makes practical sense, but on the other hand, no electron is actually like another (pauli principle)
PhD tryi ng to work out what number means? Did he complete his PhD in numbers to the satisfaction of kindy returdia?
This guy is a joy to listen to. Clear, precise, refreshingly willing to say "I don't know," and on top of that, cool accent!
He's Israeli :-) I love his positivity and enthusiasm!
@@Roarshark12 Palestinian, you mean...?
@@danyalajmal2715 No he mean Israel because that is where he was born
@@danyalajmal2715 🤔
“*_WE_* don't know”, comrade.
Where was this guy when I was doing set theory? Honestly, the idea of EVERYTHING being written as sets is something I just never grasped. He says think of it as coding and it just suddenly makes more sense in my head.
I like to tell my students that they can think of it like a file format. The string of zeros and ones that encodes the r g b values of the pixels in a bitmap in binary isn't an image but a representation of an image. The same image might be represented by a completely different string of zeros and ones when using another file format like jpg or png. In the same way sets are used to represent numbers (or any other mathematical object) in a format we previously agree on, but within the framework of set theory we can only use sets (like you can only use strings of zeros and ones on a computer). There are other frameworks like lambda calculus or category theory but the same idea applies: mathematical objects are represented in a specific "file format" within the given framework. The reason you use such a framework in the first place is to keep the number of axioms (basic assumptions) as low as possible to avoid intrinsic contradictions (or to be more specific: to make it easier to convice others that your framework is free from contradictions).
I wish people explained class theory instead :/
all theorem proofers switched away from set theory, but noone talks about the thing they switched to
@@NoNameAtAll2 Theorem proofers (proof assistants) mostly swichted to type theory, which is typed lambda calculus (because there "proofs" are represented within the framework and don't need to be encoded in an additional meta framework). Classes are nothing complicated, they are just like sets but with fewer axioms. So you can collect sets which have a common property (like simply being a set or being a set with 3 elements) to make a class but that's about it. For example you cannot put a class into another class. They are used to have a specific object (which is "to big" to be a set) which contains, say, all groups or all principal ideal domains.
I think by "coding" he means it in the representational sense, not programming. It "encodes" the meaning.
There’s another, even wackier formalism, called λ-calculus, where everything is a function. It’s often quite useful in computer science.
Asaf is great, I hope we'll see more of him in the future! 😊
Dang! This was the 1st comment I read and I tried to figure out what the abbreviation "ASAF" stands for. 🙈 Really looked like one of those common on the Internet and in chats... 😂
Liked him as well! Smiles a lot as well
@@hamc9477 yes, me too! I had a separate comment on that in fact but it got deleted for some reason that I really can't even fathom. It was a totally benign comment. No links, no strong language, no controversial points of view :/
@@unvergebeneid aw it could've just been a glitch or something. Happens to me sometimes I think. Remember that after all all these things are mostly run by machines in the end.
This felt like the beginning of a 5-part series.
And i want all those parts. I never had a class on set theory, i feel like I'm learning something totally new here!
@@Phriedah I want to know how he gets to rational numbers from integers.
@@paulthompson9668 That shouldn't be complicated as rational numbers are just pairs of numbers or fractions. N, Z, and Q are all sets of the same size. What I'm much more interested in is how you construct real numbers, as R is a bigger set than N, Z, and Q.
Similarly getting to C from R is also easy, as it's just a pair of numbers again
@@PattyManatty What you said sounds right, but I think I'd need at least a 5-minute video for going from N to Z. Plus, I'd expect going from Z to Q to have its own challenges because the first is countably infinite and the second is uncountably infinite.
@@paulthompson9668 Rational numbers are actually countably infinite! You just have to get tricky with it. Still doesn't change the problem when you wanna go to R of course
I kinda chuckled how he gave a subtle distinct difference from a human and an Electrical Engineer. As an aspiring EE, I have to agree 😅
Sad sad he said "i" though. I was taught to use "j". 😉
@@Elesario bro, same 😆
I have degrees in math AND in EE. I would wager that most mathematicians and most EEs don't really understand what j actually means in an electrical context, they just get used to using it because it works.
Humans use i to represent the imaginary unit of the complex numbers, while Electrical Engineers use the letter j. This is the key difference that prevents interbreeding between the two species.
Electrical/electronics engineering uses “j” to avoid confusion with “I” for current, I believe.
By the way, Python also uses “j”, but you have to use it as a suffix on a numeric literal:
So essentially, a 'number' is any of a given well-defined category of objects that follow a given list well-defined logically-consistent rules, which are generally used to model and solve problems. A 'number' in that sense is just a basic building block of a method of problem-solving.
When the problem is "Can the hunters fight the mammoths", then one way to model that involves having some way of counting, of expressing the size of the groups involved - a 'simple' model, certainly, but still a model which can then be used to solve the problem: describe what 'counting' means as part of your model, count the mammoths, count the hunters, use the model to determine which count is larger. We don't think of it that way explicitly, because 'how to count' is so ingrained as if fundamental... but there is no real guarantee that you *can* count, unless you specifically are building a model which enables counting - that makes the concept of 'the next number' meaningful.
And there's no guarantee you can count, because at a certain point, you can't meaningfully say what 'the next number' means. If you're working with the Rationals, despite them being 'countably infinite', you'd be hard-pressed to get a useful answer to "What rational number comes next after 3/4?" - but at least there *is* a way to define the rationals that permits that question to make sense. When you start looking at the Reals, the idea of 'nextness' loses all meaning entirely. "What real number comes next after the square root of two?" feels like a nonsense question, because 'counting' has lost all meaning, though there is still some sense of 'order' (arranging them in some order from smallest to largest in a consistent way) among the Real numbers. By the time you get to Complex numbers, you no longer even have that sense of ordering any more, let alone 'nextness'; is 1+2i larger or smaller than 2+i in any way that has meaning, even though they clearly aren't equal?
Ultimately the things we casually call numbers are unreasonably effective when used to model and solve problems, to the point that we enshrine them in some special place of importance; in practice, any system of consistent logically manipulatable objects that can be used to model and solve problems are just as 'number-like' as what we all think of as numbers. With that understanding, it seems trivial that whether numbers 'exist' is no more a meaningful question as to whether 'wind' exists - some underlying phenomena or collection of object exists, and we are using our ability to describe and understand those things to talk about them, the patterns they form, and the interactions they have.
I think you haven't gotten a reply since you explained it so well. That was stellar man, clearly well thought-out
I like this, got me thinking about how symbols and modeling relate to words and communication.
"is 1+2i larger or smaller than 2+i in any way that has meaning"
IIRC, for complex numbers you first compare the real part, and then the complex part. So 1+2i is smaller than 2+i (because 1i).
You're onto something here man.
Next to 3/4 there is 3/4 plus an infinitesimally small quantity, etc.. as numbers are uncountable... theories from Cauchy, Dedekind...
2+i and 2i+1 are IMHO the same size provided we look at the vector length that is the same
I feel like this should have been your first video
Love the video, but the expanded representations of numbers at 3:35 are incorrect for 3 and 4. For example 3 = {0, 1, 2} = {Ø, {Ø}, {Ø, {Ø}}} if I'm not mistaken.
You're not mistaken, they *are* incorrect.
I saw that too.
@@sanderspoelstra8072 Is that a compliment?
With how the naturals are defined at 3:15, you are absolutely correct. However, as stated at 9:41, the coding doesn‘t really matter, and (if I’m not mistaken) that representation for the naturals is also sometimes used. I think there is a list on wikipedia of some common ways of defining the naturals in ZFC?
Thank you, I thought I had missed something and that's why it didn't work.
Finally asking the real questions.
And the answers are not that complex 😸
And getting complex answers
It‘s natural to ask philosophical questions if you want to see the whole picture
@@Alexander_rekaX Может ты и прав.
It's only natural of us humans.
"Beliefs leads you to being sure you are right, and you can't really know" is now written on my whiteboard in my home office. Brilliant quote! Thanks Asaf! Also thanks for a very interesting video!
But note that bit about "you can't really know". "Knowledge" is typically defined in epistemology as "justified true belief" (after Quine, with a lot of tricky stuff around the edges of that definition), and "belief" as "holding something to be true". So if you've written the quote on your whiteboard because you hold it to be true, you have a belief and you've fallen into its trap. It's actually pretty much a statement of epistemological solipsism, which is a tenable position but might not be what you wanted.
@@digitig Excellent response.
I disagree with him but only slightly. I have some beliefs but I am not in love with them, I recognize that they are only axioms. If I were to choose different axioms I would be led to different results, and thus I have somewhat equal faith in both results since I have no actual proof of my axioms. When he said he was agnostic on it, that struck a chord for me.
I would like to hear more from this guy, his example was on point but it was very brief.
@@digitig Quine himself is disputed on that point sooo...
@@emmanueloluga9770 That's what I meant by "some tricky stuff around the edges" - Gettier counterexamples. But they don't change my point - I'm not aware of anyone who argues one can know something without holding it to be true.
I still remember my college roommate explaining this to me while I was high.
I've read so many of Asaf's enlightening answers and thoughtful questions on MSE and MO for years. What a treat to see him on numberphile!
At 3:40, there's a mistake. It seems that we are using the von Neumann ordinal encoding here to represent the numbers as sets, but in that case, it should be 3 = {0, 1, 2} = {∅, {∅}, {∅, {∅}}}, instead of {∅, {∅, {∅}}}. And similarly, 4 = {0, 1, 2, 3} should be {∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}}}.
To be fair, it also is a construction that satisfies the Peano Axioms as far as I can tell, with S(n) = {∅, n}
@@timseguine2 Yeah, correct! I think the beauty in von Neumann encoding is that you can implement some common operations on numbers directly with common set-theoretic operations, such as "greater-than" as "subset-of" etc. From efficiency standpoint, both the Von Neumann encoding and a "direct" nested-sets encoding you present are quite bad; a binary (or any base n where n is greater than 1) encoding would help.
Here's an attempt to do that: let's define a list. An empty list is an empty set: [] = {}. A list of one element a is a set that contains the element and the empty list: [a] = {a, {}}, a list of two: [a, b] = {a, {b, {}}}, and so on. Then we can define two digits, 0 and 1: {{}} and {{{}}} (defined so to avoid clashing the definition of zero with the definition of empty list). Now we can have numbers as list of binary digits, a lot more efficient encoding: [0, 1, 1, 0, 1]!
I find it interesting that we're driven to extend the number systems by insisting on closure under inverse functions. If you just want closure under addition, multiplication, and raising to natural number powers (except 0^0), lots of number systems will do. But if you want subtraction, the inverse of addition, to have closure, you end up with negative numbers. Wanting multiplication to have an inverse pushes you into rationals, and wanting integer powers to have inverses pushes you into algebraic and imaginary numbers.
That is so cool
Then try to describe geometry and you get the constructables, until someone makes a curve and you get the transcendentals
Woah
I’m still trying to find a similarly concise way to motivate the p-adics. Haven’t had much luck so far!
In short: dividing things up has the potential to create new things.
"Philosoplically speaking i'm very agnostic, i don't want to have any concrete set of beliefs because beliefs lead you to being sure that you're right and and you can't really know". Asaf Karagila
After having just staunchly claimed 0 is a natural number.
@@JM-us3fr I find it very natural
Really wise words!
@@juanausensi499 So do I. I just find it funny that he’s weary to have beliefs, but is adamant about that particular belief
@@JM-us3fr That's not a belief, it's an opinion or preference. Zero being a natural number is not some truth waiting to be discovered, it is only a matter of definition and convenience.
So what is a phile?
"beliefs lead you to being sure that you're right"
What a wonderful quote. I love it
We discussed set theory in my computer science classes. My professor explained that there are infinite infinities thanks to set theory. And with how you are describing how to build all of the subsequent sets of 0, it’s so clear.
As someone who really enjoys the philosophy and foundations of mathematics, this kind of video is pretty refreshing, because it cuts rather deep in a very intuitive concept to mathematics (as least, as people usually understand it).
Acho q te vi no canal do Porto
deep in what sense ?
@@Shaolin-Jesus well, one thing is making an "analytical definition" of numbers, which just amounts to agreeing by convention to replace number words with some appropriate symbolic expressions which supposedly don't invoke numbers to be understood (such as expressions standing for certain sets in ZFC). another matter entirely is "logically analysing" numbers, as in, breaking down the concept itself (to be strictly differentiated from the expressions that denote it) into simpler concepts, or displaying in some way that this can't be done, which entails a lot more knowledge about how numbers are to be understood than mere know-how of symbolically manipulation of numerical expressions would (though some philosophers have said this know-how is all there is to know in the first place, but I digress). though there are definitions of numbers in set theory, the analysis itself of numbers as sets can hardly be said to be intuitive or conclusive, as questions about "which of these sets really *are* numbers?", put forth by anyone who knows more than one such definition, display
@@Shaolin-Jesus the insight at the end that sets are just as natural to thought as numbers, is itself very deep and one that is very often neglected even by modern mathematicians and philosophers, because famously they have supposedly shown the concept of set to not be well-defined (see Russell's paradox, though preferably elsewhere other than just Numberphile), which led to attempts to make it so that ultimately rob it of its naturality (just contrast naive set theory with any axiomatic theory of choice, in complexity of content and presentation). this is an insight shared by the forefathers of modern philosophy of mathematics, however: features such as Gottlob Frege and Richard Dedekind, and of course the authors of the famed Principia Mathematica, namely Bertrand Russell and Alfred N. Whitehead
@@jan_kulawa I agree with your outlook very much. You also make a fine conjecture upon the nature of mathematics. it leaves me only with wanting to speak with your more as I see you have been kind to your mind to grant it thought upon the deep nature of reality.
regarding what you said, Modern mathematicians seem to have very little regard for the philosophy of mathematics and I feel this only becomes a disservice to mathematics itself.
I feel once you acknowledge its philosophical relevance you understand its numerical relevance much better than one (most people) who have been conditioned to perform a mathematical task without regard for the beauty of mathematics itself.
most people can multiply, but if you were to ask them why 2+2 and 2x2
equal the same number they wouldn't be able to tell you
This is evidence of the fact that mathematics is often performed without being understood
I am actually currently writing a dissertation on the 'origin of mathematics' and have arrived at some theories i would love to discuss with you
ironically my research touches on expressing number without number like you mentioned
leading to the postulation that one is the only natural number since every other number is essentially just a 'set of multiple ones'
are you on any other social media platform such as instagram or facebook
feel free to follow my Instagram @weildingfire
looking forward to hearing from you
This has some interesting similarities to the set-construction of the surreal numbers. I love these topics a lot.
Asaf was my instructor when I studied set theory as a freshman, so cool seeing him pop up on Numberphile! Hope you're doing well in the UK!
I love how CZcams subtitles transform the Zermelo-Fraenkel Axioms into Tomato-Franco Axioms. ^^'
all my homies hate ZFC
TFC 4 life
@@gasun1274 TFC gang reporting in!
It's just what you get when you abstract ZFC through the vegetable set
As a layman and Numberphile fan, normally I have at least a small understanding of what's going on. But here I haven't a clue haha! That said an engaging guest and really interesting philosophical discussion near the end.
To explain philosophically what’s going on, mathematicians construct numbers in such a way to require the fewest conceptual commitments. Sets are abstract objects which are very useful to higher mathematics, and so by constructing numbers utilizing tools of set theory, it means we only need to assume set theory rather than set theory *and* arithmetic.
If sets feel kind of abstract, think of them like Venn Diagrams. Each region of a Venn Diagram represents types of “things.” These “things” could be numbers, functions, sets, or any concept we care about. When we represent a set by a list (rather than a circle in a Venn Diagram) it looks something like {1,2,3}.
@@JM-us3fr To expand and clarify on this idea, sets can be viewed as venn diagrams, but I think nesting dolls are a better analogy. Each prior set is contained in its subsequent set.
This is by far my favourite Numberphile video. Been subscribed for many years and love many of the previous videos. But this one is the best :)
I’m more than down for more videos on topics with these incredibly abstract concepts, let’s get one on category theory
What I like thinking about is how the operations on numbers make the next kind of numbers be needed. substraction leads to negatives, division leads to fractions, and so on.
The reals are the tricky one. Exponentiation leads to (irrational) algebraic numbers and complex numbers, but the entire set of real numbers is something that isn't brought about by arithmetic operations, as those tricky transcendentals get in the way.
Here is how I have come to view this question over time: Numbers are operables, that is, anything that can be operated upon by an operator. So anything can be a number if it can be operated upon in some way: digits, letters, matrices, atoms, universes, etc. This shifts the question from asking about nouns (numbers) to verbs (operations). We can then ask, what are (mathematical) operations? This lets us progress to some more interesting questions.
Relational operators < and > are indeed called 'operators', and hence it's better to think of them as verbs, processes, than as nouns and objects. Increases-decreases, amplifies-attenuates etc.
Building a formal language starting with just relational operators has been an interesting hobby. In my investigation the formal touched first the numerical world in terms of (a variety!!!) of Stern-Brocot type structures, rationals in their basic coprime forms.
IMHO Wolfram's most interesting and important key finding is that computational reversibility comes with two basic relation types: additive and nesting. The numerical object-oriented math has been traditionally all about the additive relation, on which the notion of 'field' is founded. Nesting is mereological part-whole relation, and can't be simply reduced to any "basic unit".
PS: the so called "real numbers" are not operables, as they are non-demonstrable, and hence don't form a field.
@Santeri Satama, What definition of a field are you using?
The real numbers definitely form a field when using the normal addition and multiplication operations.
I would like to be more precise, sets, matrices, vectors, tensors and functions can be operated upon, so are also operables.
In what way can numbers be operated upon in a way that is unique?
I have tried to answer this, see my comment to the video for details, and my comment to Tbop3 (or Tbob3, I don't remember the name).
As for the question of whether humans have discovered or invented math, that question simply shows anthropocentric hubris. Many studies have demonstrated number sense, counting, and algorithmic ability exist in diverse species as apes and other mammals, crows and other birds, octopuses and other cephalopods, and even in plants, slime molds, and microbes. Humans certainly did not invent math. Did mammals invent it? Did birds? Did plants? Did microbes? Indeed, the limits of what organisms use math is set more by our inability to understand other organisms than by the innate abilities of those organisms. Some might argue that such organisms are only following natural rules and not really using math, but the same argument can be applied to humans as a part of nature. Humans could have neither discovered nor invented math when it has been around far longer than even the most distant ancestors of humanity. We can only say that we have perhaps developed mathematical (and non-mathematical) tools beyond our awareness of the capabilities of other known species.
@@edomeindertsma6669 The standard definition, in wich rational numbers form a field by mathematical definition.
As for "reals", practically all of them are *non-computable* whatever, without even any finite algorithm to write and demonstrate the claimed number.
The thing about non-computable "numbers" is that they don't compute. Basic arithmetics as the definition of a field is a form of computation.
We've been waiting for this video for ages
Great video would love to see more from this guy !
Loved this guy!! Hope we see more of him in this channel.
I love this. This shows how even something I thought was obvious and fundamental is still the result of unexamined (in my case) assumptions.
This is great. Especially the philosophy of maths at the end. Would love to see more philosophy of maths episodes with philosophers. There are a couple, but I feel like there is so much philosophy of maths to explore!
aljazeera did a nice philosophical documentary on mathematics if you would like to know....
why do you feel there is so much philosophy of maths to explore ?
I spent half of last semester constructing the reals in my analysis class. It’s still mind blowing that with nothing but the empty set and some set operations provided by ZFC, you can construct all of the real numbers
Except that you can't construct non-demonstrable, non-constructuble "numbers".
@@santerisatama5409 Not really sure what you mean
@@spacemanspiff2137 He means, that not all real numbers are algebraic, that is solutions to polynomials.
@@Brien831 Sure, but when we construct the reals from the rationals, we do it in a way that includes all non-algebraic numbers too.
We create an equivalence class where each real number is related to a Cauchy sequence of rational numbers. For example, pi can be related to the sequence (3, 3.1, 3.14, 3.141, …) where each member of the sequence is rational, but the limit is non-algebraic.
@@Brien831 Then he's confused as to what we mean. Constructing a number system is not the same as finding polynomials to compute those numbers. Constructed numbers are not the same thing as computable numbers.
Along with being clearly a fantastic mathematician, he is truly a philosopher.
Just loved that discussion. Hope you're back soon!
It is also interesting to notice that we it was important (and easier) to define the negatives using a ordered pair.
It is important to ppint out we can make an ordered set using the set without order, which is also a cool idea I had forgot and googled to recall.
Yes definitely! There many ways to do it but my favourite has to be Kuratowski's encoding of the ordered pair. And you can use that ordered pair to make ordered sets.
7:15 is a real "rest of the owl" type moment. Maybe the video would be a little hard to follow, but it would be nice to how you can go from there to the negative numbers to the rest. Also, why does it end at complex numbers? Are those 5 types ALL that there is?
No, it's definitely not ALL of it. You could go from the Complex set (2-dimensional numbers) to the Quaternions (4-dimensional numbers), or from numbers to matrix algebra. These are just the most common and useful examples. But you could invent your own fancy algebra with its sets and rules. If it is logically consistent, it's as valid as the Natural numbers.
I imagine it wasn't included because it wasn't really the point of the video and would've taken a while, but I can give it a go if you want:
Once you define the natural numbers, you can define addition on them. Then, integers can be defined as pairs of natural numbers (a, b) with the understanding that (a, b) is equivalent to (c, d) if a+d = b+c. (This is what was included in the video.) (a, b) is easily interpreted as "a minus b".
Once you define the integers, you can define multiplication on them. (The details of this aren't really important, but if you're curious I can explain it too.) Then, rational numbers are defined as pairs of INTEGERS (i, j) with the understanding that (i, j) is equivalent to (k, l) if i*l = j*k, and where neither j nor l are allowed to be the integer 0. Here, (i, j) can be interpreted as "i divided by j".
The reals are more difficult. First you need to define the concept of the equivalence of limits of infinite sequences of rational numbers - you can use an epsilon-delta style definition, where a sequence u_0, u_1, u_2, u_3... has the same limit as v_0, v_1, v_2, v_3... if u_0-v_0, u_1-v_1, u_2-v_2, u_3-v_3... etc. limits to the rational number 0.
Formally, it can be done in the following way: The above sequence limits to the rational number 0 if for every rational number e>0, there exists a natural number N such that for each natural number M>N, the rational number (u_M - v_M) * (u_M - v_M) < e (i.e. the square of the differences of the Mth terms of the sequences is less than e). Less formally, that means that 'if you go far enough along the sequences, the two sequences get arbitrarily close to one another and stay there'.
Now we're ready to define the reals. The reals can be defined as INFINITE SEQUENCES of RATIONAL numbers (x_0, x_1, x_2, x_3...) with the understanding that (x_0, x_1, x_2, x_3...) is equivalent to (y_0, y_1, y_2, y_3...) if the two sequences have the same limit.
Once you've got reals, complexes are easy. Just define the complex numbers as pairs of real numbers (x, y). Addition of complex numbers is defined as (x, y) + (z, w) = (x+z, y+w), and multiplication is (x, y) * (z, w) = (x*z - y*w, y*z + x*w).
If you have half an hour, check out Frederic Schuller's lecture 3: "Classification of sets" in his "Geometrical anatomy of theoretical physics" lecture series, starting from about 57 minutes onwards. He does this construction from naturals to reals, and goes through some examples of operations (like how addition/multiplication is defined in the set coding etc.)
@@rtg_onefourtwoeightfiveseven Great explanation! A few points:
- Using the interpretations of the set-theoretical integer (a, b) as "a minus b", it is easy to see how we should define operations on them: from basic algebra we have (a-b) + (c-d) = (a+c) - (b+d) and (a-b) × (c-d) = (ac + bd) - (ad + bc), so we should define (a, b) + (c, d) := (a+c, b+d) and (a, b) × (c, d) := (ac+bd, ad+bc). Similarly the interpretation of the set-theoretical rational number (i, j) is "i divided by j" and we have i/j + k/l = (il + jk)/jl and i/j × k/l = (ik)/(jl), so we should define (i, j) + (k, l) := (il + jk, jl) and (i, j) × (k, l) := (ik, jl) (here all denominators are assumed to be non-zero). To show that these definitions of equivalence, addition, and multiplication actually work (because we could have chosen any old definitions, so we need to show that these definitions actually behave well), one needs to show that the notions of equivalence are actually equivalence relations and that the operations preserve these equivalence relations (e.g. for addition of integers we would have to show that if (a, b) ~ (a', b') and (c, d) ~ (c', d') then (a, b) + (c, d) ~ (a', b') + (c', d'), i.e. (a+c, b+d) ~ (a'+c', b'+d') - here I am using the notation ~ for equivalence whereas ≈ was used in the video) - then it makes sense to define the operations on the equivalence classes themselves (e.g. [a, b] + [c, d] := [a+c, b+d], where [a, b] is the equivalence class of (a, b) etc.). Also, just to fill in a detail, the addition and multiplication of two real numbers (defined as sequences of rational numbers) is just done componentwise - that is, (x_0, x_1, x_2, ...) + (y_0, y_1, y_2, ...) := (x_0+y_0, x_1+y_1, x_2+y_2, ...) and (x_0, x_1, x_2, ...) × (y_0, y_1, y_2, ...) := (x_0×y_0, x_1×y_1, x_2×y_2, ...). Also Alex missed a point that a real number isn't defined as (the equivalence class of) *any* infinite sequence of rational numbers; we have to restrict to just so-called Cauchy sequences of rational numbers (I won't give the definition here because this comment is already getting very long, but feel free to ask).
- Because Alex has explained the standard set-theoretical definitions of Z, Q, R, and C, we can now explicitly write down the embeddings of N into Z, Z to Q, Q into R, and R into C, and hence we can write down the composite embedding of N into C mentioned in the video :D Using the interpretations of (a, b), (i, j), (x_0, x_1, x_2, ...), and (x, y) it is again pretty clear what these should be: The natural number n embeds into Z as the equivalence class of (n, 0), denoted [n, 0]. The integer i embeds into Q as the equivalence class of (i, 1), [i, 1]. The rational number q embeds into R as the equivalence class of the constant sequence (q, q, q, ...) (which is a Cauchy sequence), [q, q, ,q, ...]. Lastly, the real number x embeds into C as the equivalence class of (x, 0), [x, 0] (the complex number (x, y) is of course to be interpreted as x+iy). Therefore the composite embedding of N into Q is n ↦ [n, 0] ↦ [[n, 0], 1] ↦ [[[n, 0], 1], [[n, 0], 1], [[n, 0], 1], ...] ↦ [[[[n, 0], 1], [[n, 0], 1] [[n, 0], 1], ...], 0]. That is, a complex number is an equivalence class of pairs of real numbers, which are equivalences class of Cauchy sequences of rational numbers, which are equivalence classes of pairs of integers, which are equivalence classes of pairs of natural numbers, which are sets - and since a pair is a certain kind of set* and equivalence classes are sets, this means that a real number is a set of sets of sets of sets of ... of sets of sets :D - I don't know how many layers deep this goes (and it is not at all significant), but one could count if one was feeling particularly masochistic xD
- Finally I will note that the steps from N to Z, from Z to Q, and from R to C (each being taking equivalence classes of pairs under some equivalence relation) are comparatively simpler than the step from Q to R. In particular, you cannot avoid the use of infinite sets in the step from Q to R, and this is reflected in the fact that N, Z, and Q are countably infinite but R is uncountably infinite. Therefore if one doesn't want to deal with uncountable infinities, one can bypass the real numbers and consider only complex numbers with rational coefficients (indeed, in practice it is quite rare to work with complex numbers with irrational coefficients anyway) - these are called "Gaussian rationals" and are denoted by Q[i]. Then the composite embedding N ↪Z↪Q↪Q[i] is given by n ↦ [n, 0] ↦ [[n, 0], 1] ↦ [[[n, 0], 1], 0] :) If you find complex numbers with rational coefficients too restrictive, then you could alternatively consider complex numbers with algebraic coefficients; the algebraic numbers A are a countably infinite set, and hence A[i] = {x+iy | x, y in A} is countable.
*the most common set-theoretical definition of ordered pairs is (X, Y) := {{X}, {X, Y}}
@@leogama3422 Or matrix algebra where each number is a matrix and each of those are quaternions.
"I don't want to have any concrete set of beliefs because beliefs lead you to being sure that you're right, and you can't really know." Amazing answer.
Mathematics without numbers is like having thoughts and ideas without a language. It can exist, but will be limited to very rudimentary concepts. Humanity went so far as a result of the ability to convey and pass-on complex ideas, and numbers are the language of maths.
Well, I would say that sets are the language of maths, there were no numbers in this video, only sets.
I agree. Numbers lead us and in almost mathematic theory, numbers lurk somewhere almost everytime. Even the greeks, for who geometry was considered as holy, couldn't escape from them.
This video could not have been more amazingly timed. I've been recently looking into Surreal numbers, and I've found myself wondering what numbers even are. Although I can't say I'm more enlightened right now after finishing the video :)
It really took more than a decade just to answer this on a channel literally named "numberphile" lol
I have read this guy's posts for years on Mathematics Stack Exchange. I never knew what he looked like, but I recognized his name immediately.
If I understand correctly, Abstract Algebra is the field of mathematics that considers properties of numbers, without really caring about precisely what a number is. For example, any set of objects that can "add", "multiply", "subtract", and "divide", can be called a *field* (assuming those operations are well-behaved in a really specific way). The set of real numbers are a field, but the set of integers is not because sometimes dividing two integers doesn't result in another integer. Matrices, similarly, are not fields because of how they multiply together, but they do fit a looser kind of structure called a "ring".
At least, that's my really vague understanding.
I've always wondered (and it may have been answered before), but what happens to the brown pieces of paper after every episode? Are they kept and archived, for example?
They're stored in a vault after being meticulously scanned and used to seed the most perfect random number generator ever devised. Probably.
They have given away some of their brown paper before to fans. Not sure what they usually do though.
I think some have been auctioned for charity
They all become part of the set that includes all brown pieces of paper, which in turn is part of the set that includes all sets of brown pieces of paper, which in turn is part of the set that includes all sets that includes all sets... and so on.
Some are auctioned on eBay, I think there are links in the channel homepage
Oooooooh, so this is what I mean when I say "how do you know it's two trees when you look at two trees, and not four halftrees?"!
It's the sets that you're arguing, not the number of them!
Brilliant!
How do you know it's "any number of trees" rather than "one thing where the mind extracts two trees (or four halftrees)." Basically, how do you go from one specific quantum configuration of the entire universe, to a bunch of subsets? It's theoretically possible because a specific configuration often factors neatly into subparts, much like most numbers factor into primes. But maybe the factorization is solely in the model, and has no counterpart "out there." Matrix ideas come to mind. Could just be one scene, and once the scene is gone (plug pulled), the trees are gone as well; the abstraction "2 trees" does not refer to anything. Well. This was a fun tangent. :)
@@matteyas No no, you're absolutely hitting it spot on! Counting is as much about defining what to count as it is about actually counting.
I like how numberphile has existed since 2011 and just now they made a video explaining what numbers are
This definition of N and Z is amazing, I’ve never seen something this abstract but I love it
The definition for Z is a "eqivalence relation" and the equivalence classes are used to define -1, -2, -3 and so on.
I liked Brady's answer at the beginning, where numbers are the abstraction of (physical) quantity. I'm in the "numbers exist" camp. :)
I think that's still a bit incomplete, I think it would be a bit more precise to say that "numbers are abstract concept for the abstract concept of quantity". While it is certainly helpful as a shortcut to be able to think about "3 mammoths", there isn't such a thing a mammoth. What is a mammoth, anyway. Where it starts and where it ends, does it include the air in its lungs and is 1 mammoth before shedding the winter fur the equal to the very same mammoth in the winter? We do understand what we mean when we say 1 piece of something, but I wouldn't say that this is a really a thing that universe reflects. And it doesn't get helpful when you get to quantum level. I'm obviously in the "numbers may not exist" camp :-)
I think numbers reflect quantities and properties of real things. You can't have pi apples, but if you _could_ have pi apples, the number pi would reflect that quantity of apples. In that sense, I think it makes sense to say that numbers are real. Whether numbers "exist", on the other hand, depends on what you mean by "exist" with regards to an abstract concept.
Three unicorns does not exist. But four apples do exist.
@@miroslavhoudek7085 I like your thinking, but it has led me to a problem. As with the mammoth who shed his winter fur, my son got a haircut and now ceases to exist. I was going to try and save him, but I couldn't get others to agree whether or not air in his lungs was an appropriate part of his definition. It's sad, but on the other hand, I need no longer save for his education. The fuzziness of an object's definition is no argument against that object's existence. Words, like numbers, colors, feelings, quantum-level objects and other existent things, are slippery - but they are also certainly fun to play with. There are many interesting ideas as to what numbers are, I am mostly saying that they are. I'm thinking there has never existed something concrete without something abstract. Particles, for example, were never without their lawful relations. Nor were they ever without number. One problem with existence is the presupposition that existence must be synonymous with physical existence. For me, it feels like an insistence that objects should not be allowed to cast shadows.
@@kazedcat You will need many more than three unicorns to adequately describe European literature! But, in the restricted sense of "flesh and blood" existence, I agree with you. :)
Yeah, we're going to need more Set Theory Videos. It's always been the field of Maths that irked me the most and this helped put that into words: it feels like you can just change up the rules on the fly, which by all accounts, REALLY doesn't feel like Maths. Though the way it's described here, it's more like changing languages rather than the rules. Regardless, we definitely need more.
This video helped me make sense of category theory for exactly the reason you described:
Category theory gives you tools to “change the rules” as it were, while still giving you confidence that either 1) you haven’t lost what you had before, or 2) what assumptions you need to give up in order for the rule-change to make sense, or 3) that the old rules were fundamentally unsound
Agnosticism as a function of not wanting to be mislead by your own beliefs is a fantastic way to approach life... Kudos to Asaf
8:40 is actually a very interesting way of thinking, more people should think like that
Once you believe that philosophy then you must be sure that you're right. And you're right back where you started from :-)
i agree
@@dannygrasse950 no
@@dannygrasse950 wow you completely blown out the philosophy in like five seconds how smart you are👏👏👏👏
@@dannygrasse950 agnosticists don't care if they're right though...
I feel like this should have been sorted out earlier in the life of this channel, pretty essential to a “Numberphile” to know what a number is.
An intuitive understanding is usually enough.
I don't need to know what water is to drink it. Plus, as he himself mentions, there's no true answer.
There has been several videos on it
Awesome. Thank you for sharing this conversation.
Congrats on 4 million subs!
Incredible, best number theory.
What’s an abstract thing? What’s a quantity?
Loved this video. Asaf making set theory interesting AF
Congrats on 4 million subscribers 👏 you deserve much more.
4,000,000 + i subscribers?
What I learned: the natural number 1 and the integer number 1 share the same symbol but are defined entirely different behind the scenes.
Actually, there are many different valid ways of defining the natural number 1 and the integer number 1. The natural number 1 has many entirely different ways of being defined. The way Peano defined the natural number 1 is entirely different from how it is defined in this video, and both definitions are valid.
The process of "lifting" from the primordial naturals to the complex numbers, simplified: the same "trick" used for the integers which are basically pairs of integers with the same difference is used here - rational numbers are just pairs of integers with the same ratios, with the plus operator changed to times in the equation. Real numbers are much, much more complex to construct, with multiple canonical definitions, but you could think of them as being defined as open intervals of rational numbers, bounded from a specific side; the boundary itself is something that might not be expressible as a rational number (i.e. all the numbers whose square is less than 2, certainly expressible only with the rationals, has a boundary of √2) and we need the uncountable reals. Constructing the complex numbers is a breeze compared to the prior infinite mess - you just make pairs of real numbers.
Of course, you can go much further, or take different directions along the original path. The construction of natural numbers, for example, eventually yields ℕ which is not in itself a number, but it still looks like one, since as a set it has the same structure as all the natural numbers themselves. So you could treat it as a sort of a "limit" of the increasing natural numbers, ω (as an ordinal number) or ℵ₀ (as a cardinal number) depending on whether you care about order or just the size. If you care about order, you could continue with the same construction ad infinitum, with ω + 1, ω + 2 etc. Eventually you find other limit ordinals lying beyond, and specific infinite sets of them have the same size and thus yield other, much bigger, cardinal numbers (actually only ℵ₁, ℵ₂ for all ordinal numbers if you accept the generalized continuum hypothesis).
If you stick with the finite natural numbers instead, you could take a different turn at the real numbers, if you realize the boundary I was talking about requires a parameter - a metric (a way to assign distance between two numbers). The usual way yields the real numbers, but there is a special kind of metric that yields the 𝑝-adic numbers for every prime number 𝑝. This is an infinite family of number sets, each with its own unique elements, without order but infinitely cycled and fractal. It's beautiful and some people say it may actually describe reality at the quantum level better than the real numbers (which work on human scales quite well). This discord is ultimately solved however, as the complex numbers can (as a field) be reached from both the real number and any 𝑝-adic number system, so the families eventually reunite.
The complex numbers, as pairs, also need a single parameter, in the form of what that 𝑖 is equal to when squared. Only three values really turn out to form unique sets: -1 gives you the usual complex complex, 0 gives you the dual numbers (with an infinitesimal second component), and 1 gives you the split-complex/hyperbolic numbers (these are actually just pairs of numbers, but rotated 45 °). Only the complex numbers behave properly, as one would expect from numbers (i.e. they form a field).
Beyond the complex numbers, you can find many more wild lands full of wonders and mythical creatures. You could get inspired by the dimensionality of complex numbers and continue with that, combining different complex, dual or hyperbolic components into bicomplex numbers and beyond, or you could take the idea of the square root of -1 itself further and make up the quaternions, octonions and others. Even matrices turn out to be numbers, at least somewhat.
Other possibility is to tackle the infinite again and you find the hyperreal numbers, essentially defined as sequences but with specific sets of indices regarded as "important", through an object known as an ultrafilter, something so detailed and intricate that it cannot be constructed through simple expressions, only proven to exist. If you get scared by this object, turn up to the surreals which are built up of a hierarchy where the newer surreals are built up from pairs of specific sets of older surreals. This way, you contain the hyperreals, but also all the ordinals and cardinals from above, finding something that is simply too big to actually be a set itself. It is called a proper class which is an object that doesn't exist in a set theory, yet you could describe all its set-sized parts and thus prove theorems about it, in a sense. The surreal numbers, despite not being a set, have all the algebraic properties of the rational and real numbers, i.e. they form a field (the largest field that there could be).
Beyond the surreal numbers you have to abandon the constructions and accept the symbolic. The surreal numbers essentially give rise to specific "meeting points" between sets, thus for example you could define the number ε (the simplest infinitesimal number) as the meeting point between 0 and all the real numbers larger than 0. Even with its tremendous size, the surreal numbers still contain "gaps" so to speak, for example all the finite surreal numbers (which could be represented by the reals) and any set of transfinite surreals will always have a gap that itself is not in the set. You could call that gap "∞", the point between the finite and transfinite. Of course there are other gaps, like 1/∞ is the point between 0 and the positive infinitesimals. These gaps are not surreal numbers, as the second component of them is not set-sized (transfinite surreals or infinitesimal surreals are already proper class-sized; any interval on surreal number is).
To even consider these numbers with classical set theories, it is required to use an approach similar to calculus, where instead of using the hyperreal numbers (as you could), you simply define everything in terms of approaching to an arbitrarily close distance. You approach these concepts and thus you can talk about them without ever being able to construct them in a standard set theory.
And by the way, a pair (a, b) is just {a, {a, b}}. So yeah, everything comes from a set.
"Boundaries/limits of infinities..." Set theory is so sad, that it's better to take as an absurd joke. Well, as Wittgenstein called it. :)
@@santerisatama5409 No, the issue lies with your (and Wittgenstein's) abilities.
@@santerisatama5409 Set theory and the notion of infinite cardinals, deliver impressive results completely coherent with finite theory. I see no reason to abandon a most interesting concept because some dislike the notion of it. Finitism is interesting in certain settings but formalism is simply that much more interesting. Do you even understand set theory? How to determine, if two sets are the same size? Or are you just a misguided philosopher overstepping their bounds?
@@noonehere0987 Non mathematicians arguing against axioms and concepts with philosophy is always such a joy.
@@Brien831 Cantor's notion of "infinity" boils down to oxymoron of "finite infinity". If you want to do paraconsistent Cantorian joke math, fine, but don't call it consistent.
Numbers are representations of what happens in space-time.
absolutely amazing guest
Next time on Numberphile: What is maths?
I'm almost sure if you ask 10 mathematicians you will get as many answers,,,
"How does π taste like?"
"Are mathematicians lizard people?"
"Where is this bar in which mathematicians use to make such strange orderings?"
"Why doesn't the bar's owner forbid mathematicians from entering once and for all?"
So many questions...
What is maths?
Baby don't count me...
Don't count me no more...
The plural of math. Next question.
8:41 "I don't want to have any set of concrete beliefs because beliefs lead you to being sure that you're right, and you can't really know". Great quote. I wish more people thought like this about more than just maths.
Unfortunately agnosticism requires an above average understanding of philosophy. But I go one step further as I'm an ignostic
A terrible way to go about life; keep it to the very abstract or you become miserable and misery-inducing. "I can't be sure if this book is fantastic" "I can't be sure he loves me" "I can't be sure you exist and genuinely feel pain". It's easy to run from the world, to try to live without "a set of concrete beliefs", but it's a baseless belief in its own right.
I can't believe you would say such a thing. Ah well, can't let it bother me. :D
I love this channel so much!
Numbers are not a representation of quantity but of order. The main problem with defining numbers from the natural set is that you are using the last codification of the chain of order, so you are not trully defining anything.
Numbers could be used either for order or for quantity, could they not?
The numbers themselves must be more abstract and flexible than whatever they are used for.
What does 1+2i order exactly?
And yes, random CZcams person, the legendary Von Neumann wasn't "truly" defining anything. He was the one confused. Not you. XD
"What is a Number?"
Something I have difficulty getting from the ladies
I remember first learning about how the naturals, integers, rationals, and reals could be constructed from sets and equivalence relations, and it felt like a curtain was being drawn back. It made so many things make sense that I had previously just been told were true - why infinity isn't usually considered to be a number, why certain definitions of numbers make more sense than others, etc. Also, stuff like why limits of infinite sequences are considered so important in analysis.
Luckily, it's been shown that the surreals are consistent if and only if the reals are consistent, so infinity not being considered a number is outdated, though still taught given the prominence of standard analysis.
This was one of the most "mindblowing" eps that i ever seen.
This is a great mindset. Don't worry too much about what numbers "are." Focus on what you want to do with them.
42 is not a number. It's the meaning of the universe.
Its the answer, not the meaning. You still need to know what the answer means.
The only thing I would correct is that in Electrical Engineering, we use j to represent sqrt(-1) because we use 'i' to represent the quantity of rate of change of electrical charge per unit time as current.
X is way more abused in notation than any other letter.
I’d enjoy hearing a more detailed discussion of how, specifically, the uncountable sets are built from the countable sets. This would be the jump from the rationals to the reals.
Somehow it feels the numbers and the operations are just profound!
Number theory deals with whole numbers whereas function theory sees numbers as always being complex numbers. So yes, it very much depends on the context but the modern view in mathematics I'd say is that without any further context, a number is a complex number.
You said complex number, using the word number to define number…
What about quaternions in that case?
@@jacobgoldman5780 I didn't _define_ anything. Also, what is it that you're saying?
I wish it were the case that without any further context, a number is a complex number - during my undergrad all of analysis was done with real numbers :/ (which is a damn shame because complex analysis is much more elegant imo).
I think the reason a lot of people take "number" to mean "real number" is because they're so used to thinking of the numbers as existing along a line (rather than e.g. in a plane), when in fact they almost never need to use this linear property unless they are reasoning with inequalities
@@schweinmachtbree1013 weird, all my analysis was complex analysis and I'm just a computer scientist.
I like to say that "Mathmatics" are an evolutionary growth of symbols based on our observation of space around us. Let's take the example of colors, they too are a representation of quantization. In some cultures blue has 50 different conceptuals as they are named after the space around them (ie., blue sky, blue ocean, blue flower, etc..) Mathmatics, like colors are a conceptual of what we seek viewing the space around us no matter how large, or small. The growth of those concepts are a collective evolution we, as an intellegent being, symbolized to understand that space. Some would argue the outcome of that space has always been there. Understand, the color of the sky was always there, but we gave it the language/symbol of that color over time.
mathematics is not empirical. Mathematics is in most cases, completely unrelated to observation. There are whole fields in mathematics, that have nothing to do with observation. Finite Galois theory and a lot of abstract algebra in general. Mathematics creates logical models, based on a set of certain axioms. Some times, such a model might match nature or the world in some way, but that is only a subset of mathematics as a whole.
@@Brien831 Hmm, interesting. ;O)-
I'd argue that mathematics, in some sense, exists in a way that's detached from other forms of existence. That way the question of discovery vs. invention isn't exactly meaningful. The color of the sky has meaning in a universe with photons that can have differing wavelengths, but mathematics has no such dependencies on reality or some particular flavour of it.
According to intuitionist philosophy, mathematical languages ("symbols") are secondary to the primary intuitive/idealist ontology of mathematics, which is evolutionary and can't be exhausted by any mathematical language.
Undecidability of Halting Problem is not a spatial concept, it's temporal. Together with Curry-Howard correspondence it follows that proof-events are not eternal, but merely durations under Halting Problem.
S and K combinators of Schönfinkel's Combinatory Logic are not "growth" of symbols, they are a minimum of symbols for a Turing Machine. Nothing to do with "space", all to do with relations.
@@Brien831 Intuition is empirical. Intuition is non-verbal experience of the whole, and mathematical art is about translating intuition into mathematical languages.
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This should have been the very first video of the channel.
I was asked the question "what is a number?" by a professor. The answer we came up with was "anything you can add, subtract, multiply, or divide with."
That's not a bad answer really (not quite precise enough but in the right vein), they are a means to an end.
@@DavidBeaumont it's not precise, true, but it gets to the heart of it I think. A number as a quantity fails in so many regards. But numbers as objects that we have an arithmetic for works pretty well.
Informally, I usually think of, e.g. 3, as the unique property that all collections of exactly three things have in common (ok, that no collection with "fewer" things also has) , which sounds circular, but is cleared up by such a recursive definition. I've always loved set theory, and disliked the seeming feud between set theory and type theory.
Couldn't you just say that a number is an isomorphism class of sets?
@@ruinenlust_ Exactly, since two sets are isomorphic iff they have the same cardinality, but I was just trying to convey the idea a bit more informally =)
That works great until you get to other numbers that don't deal with quantity, like complex numbers.
What about vectors or matrices? I think of those as multidimensional numbers.
And quaternions, and tensors etc, but Asaf chose to stop at the complex numbers for brevity.
Actually complex numbers and quaternions are multi-dimensional numbers, not vectors. Numberphile has a video on quaternions. Vectors are "things" with a magnitude and a direction, matrices are series of vectors (with the same number of components).
@Gábor Králik
Quaternions do not form a field.
@@edomeindertsma6669 well, naturals don't, either
@@gabor6259 That's a dramatic oversimplification and misses the point of the video. Just like numbers, it entirely depends on your 'encoding', and neither encoding is generally more valid than the other. Complex numbers and quaternions could be considered numbers in their own right. But they can also be considered vectors, or subalgebras of the Clifford algebra in 2 or 3 dimensions, or a whole host of other encodings.
The same is true for vectors and matrices. Are vectors collections of numbers or things with magnitude and direction? Are matrices linear transformations of vectors or are they 2d arrays of numbers or are they series of vectors? There's a reason mathematicians abstract away from all those representations and simply define vectors as 'elements of a vector space', etc. 3Blue1Brown has a great video on it.
I loved the Parker-animations in this video!
For the definition of integers as pairs, you can think of (x,y) as x representing how much money you have in hand, and y how much debt you have.
So if you have (0,1) and (3,4), those represent the same thing because once you settle your debts, you'll have the same net worth either way, which is "1 dollar in the hole", and so that represents -1. (0,0), (5,5), (17,17) all represent the same thing of "zero total money" once you settle your debts.
In electrical engineering √-1 = j :)
A miserable little pile of digits. But enough talk, have at you!
I knew I wasn't the only one thinking this.
This channel is very charming, I think I’ll stick around.
I can't believe I got what a number is. 🤯 Not completly, but at least the basics.
Thanks for that.
That’s Numberwang.
You should have Alon Amit on from Quora! He has an amazing answer on what a "number" is, his basic conceit is that there is no obvious reason why we think of "1" as a number but a square matrix or a polynomial as not a number.
He's a cool guy, but he hates numberphile.
I would love to see Alon Amit on numberphile.
Numbers are the friends we made along the way
The intersection of mathematics and philosophy is an interesting set.
8:45 "Beliefs lead you to being sure you're right." It's a nice pov. But it is, itself, a belief. If this bloke had studied some set theory he's know all about these tangles you can get yourself into.
He did however mention that he didn't have a *concrete* set of beliefs, which makes his belief more open to other interpretations. I think his main point in that statement is although you think something might be a certain way, you don't neccessarily need to throw away other possibilities. He probably doesn't like the idea of "beliefs" because they are one-sided, whereas numbers as an idea are very much open to interpretation, in fact, defined as an interpretation of something.
Belief isn't by definition blind trust, I'll give that, but saying you believe in something already puts you on one side of a question.
And, what's a point in geometry?
0
It depends on your axiomatic system, i.e. what you choose to be your foundation of math. If you use ZFC or a similar axiomatic system, then a point can just be an ordered pair of numbers, like (1,2) for a point in the Euclidean plane. (Note that ordered pairs (like everything else in ZFC) are constructed using sets, so in ZFC a point is just a special kind of set.) But if you want to build math in a way that's tailored to geometry like the ancient Greeks did, you would use a different axiomatic system in which a point is a primitive object, meaning it has no definition (but can still be _described_ by the axioms).
An axiomatic definition of a concept. A point is a point, and it is always true. Just like ZF treats {} as a nullset that is axiomatically always true.
2^i
Aren't you defining here the point by the point itself?
A number is a mind operator that helps us reach specific future events
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When one says "a set", he means "one set". So basically one uses the concept of 1 before one has even "defined what 1 is". It is possible to consider the natural numbers as a starting point for the development of a description. The fancy pantsy way of introducing sets first is fine so long as the emerging constructions are consistent and useful, but that is something that shouldn't be taken in a religious way.
Are you sure that using “a” in that was used the concept “1”?
What if “a” is prior to “1”, and “1”just happens to in some ways/contexts be (sorta) equivalent to it?
@@drdca8263 we can do interesting things by playing with words, but it is perfectly possible to consider you have nothing, then something with no information beside being something, you get the natural numbers in an unbiased way and you can use that as one of your starting points for building descriptions. Mathematicians are fine with postulating as axioms false statements are right and are still able to build solid constructions. I don't see their problem with a set not being 1 set because you haven't defined 1 yet. One can argue, we can describe our universe either way, and you're even the one who is doing acrobatics, why block my approach? That's the definition of religious.
10:07 *"Could mathematics […] exist without numbers?"*
The greeks have been doing mathematics with geometry. So yes, mathematical thought can exist without numbers.
So do logic and a set theory, don't they? However, that was, I believe, the Hilbert's idea to represent mathematics in a unified way with the help of numbers and arithmetic.
Wrong, they did mathematics without algebra, but they had numbers and sets.
@@Eagle3302PL OP said that the Greeks carried out mathematical thought without number, not that they did mathematics without numbers. The Greeks did geometry without numbers (whereas today we usually do use numbers by coordinatizing the plane/space), and hence carried out mathematical thought without numbers.
Similarly these days we have branches of maths with no numbers. Algebra comes pretty close sometimes, but if you have an abstract notion of addition or multiplication then I'd say that you have numbers, because (for multiplication, for example) you have , a, a*a, a*a*a, ..., which are denoted by a^1, a^2, a^3, ... and (usually) satisfy the usual properties of arithmetic, e.g. a^{n+m} = a^n * a^m. Therefore a better example is something like category theory, which is super abstract and doesn't use any numbers at all (at least that I can think of off the top of my head), but rather draws heavily on pictures and diagrams.
@@schweinmachtbree1013 They still used numbers for counting lengths/distances.
@@Eagle3302PL yes, but the point is that the Greeks carried out mathematical thought without numbers - nobody ever said that they carried out all mathematical thought without numbers - obviously they did arithmetic with numbers. I am talking about the question "Can mathematics exist without numbers?", not "Can all mathematics exist without numbers".
I like the approach to the "Is 0 a natural number?" controversy. Essentially, "Yes. Get over it. Moving on..."
love how nervous he sounded when he said "I just thought of that" when answering "what is a number?"
Is 57 a number?
No.
Obviously not