What is a Number? - Numberphile

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

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!

Asaf is great, I hope we'll see more of him in the future! 😊

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 feel like this should have been your first video

This felt like the beginning of a 5-part series.

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.

Finally asking the real questions.

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 😅

"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!

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!

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.

I still remember my college roommate explaining this to me while I was high.

"beliefs lead you to being sure that you're right" What a wonderful quote. I love it

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 :)

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!

"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

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.

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.

We've been waiting for this video for ages