What is a Number? – Numberphile

Featuring Asaf Karagila.
More links & stuff in full description below ↓↓↓

Asaf is a UKRI Future Leaders Fellow. Asaf’s blog – http://karagila.org

Asaf’s Twitter – https://twitter.com/AsafKaragila

Numberphile podcast featuring Asaf – https://youtu.be/b6GLCTh5ARI

All the Numbers with Matt Parker – https://youtu.be/5TkIe60y2GI

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Comment (360)

  1. Can someone explain to me why the information at 3:46 is correct? We were given the relation n+1 = n U {n}. That implies 3 = 2 U {2} = {0, {0}} U {{0, {0}}} = {0, {0}, {0, {0}}}. This is not what is listed for 3. What am I misunderstanding?

  2. I really enjoyed this video and this speaker, but I was stressing so hard about the marker when he was explaining things with it uncapped, it is going to dry up with all that waving hands!
    Excellent explanation and infectious enthusiasm.

  3. I think the question of "were numbers necessary" is redundant. Whatever path we took would result in us thinking of that path the same we think of numbers.

  4. If you take the low-tech approach you could argue the original purpose of numbers is counting stuff. In that sense numbers are part of the toolbox humans use to describe the world around them: There are three red apples on the chair. So do numbers exist? They exists in the same way "Red" exists. It's a characteristic of the part of the world we are describing.

  5. Really enjoyed this. It's similar to what I was thinking in my own videos. Number theory derives from set theory. He's saying "mammoths and cavemen" but I used "lions and hyenas" in my counting videos as set theory examples. Very cool stuff. Loved this vid. Keep up the great work!

  6. Isn’t 0 technically not a quantity? Not trying to be a contrarian, I just noticed him wrote down ‘0’ as an example of a number and I’n wondering what makes it so.

  7. I wish he would have gone deeper in that idea at the very beginning.
    How did we go from seeing 2 rocks, 2 hands, 2 seeds and from that we figured out “oh that’s the same “2” every time. I have the same number of things.
    What’s the process that allowed us to figure that out ?
    I remember a quote from a mathematician or a philosopher that said someThing like “it must have taken us a considerable amount of time to understand that 2 rocks, 2 seeds are instances of the same underlying idea” but can’t remember who

  8. I cannot jive with zero being a number… i was always told it was just a place holder.. a symbol for the lack of anything, a number should effect another number or be able to be manipulated by another number .. zero is just zero

  9. I have a huge problem with the notion of set, in the wake of Cantor, whereby even uncountable concepts can be represented by a set. I can understand that within a formal system, such a thing can be reified, but I reject being called an "intuitionnist" by formalists, as if that means being irrational. Maths either deal with the real world, or it deals with itself as a formal language. But that formalism leads to tautologies and in no way does it tells us anything useful about the real world. Countable infinite sets such as N make sense to our intuition, because they embody the fundamental notion of cardinality (and factor). I would claim that Pi and e, aren't as much numbers as they are constants. The set or Rationals makes little sense to me ; I don't see why, for instance, it would not be bound between zero and one, and numbers like Pi would mean the natural number 3, plus the irrational decimal expansion. Logicians tried to make sense of the world using set abstractions, but they have largely failed to make a system that can be used to determine truth on the absence of semantics. Structure itself is misleading and becomes rapidly equivocal. It shouldn't surprise anyone that when pushed to its logical extents, formal mathematical langage was shown to be incomplete by Gödel. The paradox is the horizon of the field of logic : it is the virtual boundary that arises at its perceptual limit. The horizon is not a property of the ocean, it's a property of the observer. The same is true of formal language : it always leads to virtual, undecidable boundaries, not because these boundaries are intrinsic to the ontology of language (and therefore knowledge) but because the operational aspect of language is limited by its users. Therefore one can always craft more language to palliatively cover new concepts, the same way one can advance on a ship over the ocean ; but the horizon keeps moving. And no, there is no land in sight ; we're talking of the infinite ocean or existence there.

  10. I wonder how other necessary operations would be defined with this representation, such as division.
    Feels like this isn’t the most natural form of a number for that.

  11. I appreciate this very complimented system of working out what a number is, but maybe its me coming from a non-maths/more language background, but I feel like numbers aren't this complicated, its the abstract representation of an amount, but its a language, and there are numerous types of 'number'. Absolute amounts like people use in maths, but the examples given like Binary and images, are made up of numbers, but not numbers themselves. A number is a way of counting units, or some constant as a reference like amount of atoms in something. Once you start abstracting like 1/2, then 1/2 isn't a number, its half of that absolute singular number. 1 is a constant as a number, like having 1 apple, if you have half, that nots a number, but half your apple, half of that 1. But of course, me not having the PHD here and years of thought, may have oversimplified and missed the entire point of the video, but honestly once the x+y came out, I got overwhelmed.

  12. A c'mon there IS an answer to the question! Number is a quantitative measure that has an actionable utility in a certain context. If you can use it in a setting, and if you decide on the definition of what "1"s are and what nothings are (according to the type of action you want to take), what you do is automatically using numbers. That's it.

  13. But to recognize "a collection of mammoths", you first need to define the criteria for fitting a "collection". You need to recognize that a) this is a mammoth and b) that one over there is also a mammoth (and thus belongs in the same set).

    Now replace the word "mammoth" with the word "number". Sure, all numbers belong in a set, but how do we recognize that something is a number in the first place?

  14. 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.

  15. 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.

  16. 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.

  17. 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

  18. 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.

  19. 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.

  20. 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.

  21. I think sets and numbers seem pretty simple when you look at them from 30,000 ft above. They’re just the best tools for the job at describing a quantity and a collections of things in an abstract way that I’d assume mirrors the way we remember those things in our brain. A differently constructed brain may recall and think about things in a fundamentally different way and could possibly perceive everything on a fundamentally different way which would lead to different things than numbers, I think.

  22. In defense of the humanity Electrical Engineers, they actually use 'i' to denote current (albeit in the wrong direction of current flow), and since 'i' was already taken, they use 'j' to denote complex numbers. Electrical Engineers use two eyes but not two 'i's.

  23. As long as there are beings who wish to describe their environment, numbers will inevitably be discovered. Math then becomes a utility for us to describe that detail which words cannot capture accurately. I think the essence of numbers are discovered and not invented because, well probably that invention is human concept designed to fulfil our egos. In reality, everything we've "invented" had already existed in the future, we had just discovered the processes required to get there.

  24. 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 are 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.

    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 surreal. 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, as the second component of them is not set-sized (transfinite surreals or infinitesimal surreals are already proper class-sized).

    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.

  25. I disagree with what is given as the natural numbers.
    My understanding is that the natural numbers are the counting numbers, which start with 1. You don't' start counting anything with zero.
    The Whole Numbers are the natural numbers with the number zero included.
    The integers, taking things further, are the whole numbers with the (additive) opposites included.
    Has something changed?

  26. "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

  27. So your answer was YES numbers are a human creation- because they always need a context- and only a sentient being can GIVE them context. Its the same for words/language too obviously.
    The things themselves exist and in a given quantity without any context still, but there would be no meaning or way to talk about them and their quantity till the context was established.

    So obviously aliens, or smart animals that can properly count (is there any?) could also give some context to them too.

  28. 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.

  29. "i" isn't a number. "i" is an operator, like the negative sign "-". Like the negative sign is not a number, but an operator to tell you which direction the number should be applied to, so is "i" an operator to tell you that the number applies to an orthogonal number line. 1i is a number, and -1 is a number but for some reason, our convention allows us to drop the 1 in 1i, which makes "i" sort of count as a number, but we can't do the same with "-", because no one would recognize "-" by itself as meaning "-1".

    So I think it is important to always distinguish the pure concept of "i" as an operator and "1i" as a number in your mind, just like you would distinguish "-" as an operator and "-1" as a number.

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