Numbers | Send Me To Sleep
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What foundational question about numbers does Andrew explore?
So, in 1910... Prominent mathematicians and philosophers Bertrand Russell and Alfred North Whitehead published a book. And that book was called Principia Mathematica. And this book was their attempt to answer a seemingly innocuous question... Why is it that 1 plus 1 equals 2? And to hear that question for someone with a very base level understanding of mathematics... might sound quite silly it may even sound too obvious to be asking seriously especially as a prominent mathematician but in asking this question and seriously attempting to answer it russell and whitehead are tapping into something about not just numbers but mathematics that is quite mysterious and that is the fact that in actuality we very much take these concepts such as one or two or three
or even add or equals, very much for granted. I think when we all first begin to learn maths, Well, one plus one equaling two is perhaps one of the first things that we learn. It is foundational to the idea of mathematics. And further still, I think that there might be something rather intuitive about the idea to people. It's easy to picture taking one thing adding another thing, and calling those two things combined, too.
How did Russell and Whitehead attempt to define the concept of 'two'?
But, Russell and Whitehead realized that it was actually a lot more difficult to formally, truly, and accurately describe What it meant for something to be one. What was this concept of oneness or two-ness for that matter? You know, I could say that I have two hands. But what is it about two? These hands in front of me that give them their two-ness. And... In part, I know that Russell and Whitehead attempted to answer this question, partially using something called set theory, which is a very complex field of mathematics and I won't pretend to know very much about it at all. But, to my understanding... they were able to say something roughly like this, we could perhaps, when, for example, discussing the idea of two, use a metaphor, an analogy,
in collecting items of two so as i've already said i could say that i have two hands And that's all well and good, but that doesn't really give us much of an idea of really what two is. But perhaps if I then collected two apples and put them by the side of my two hands, and we compared... the similarities between these two examples of two. And then say, perhaps, I were to go and get two pencils and put them by the side of my two hands and the two apples.
What challenges did Russell and Whitehead face in their mathematical endeavors?
We were to get two ducks, two tables, two chairs, two stars, two planets, and put them all next to each other equally, and ask ourselves, what is it that these sets of two things all share in common? And if we extended that out to all possible examples of what two things might be, and we tried to understand completely what it was that those sets of two things all shared, then we could call that two-ness. We could collect all of those things, all of those pairs, into a set. And we would call that set the number two. And this is, possibly to some listening now, An overcomplicated way of describing two. Like I said before, it's something that feels very intuitive if we see two things. Well, of course, that's two.
But... That is actually no definition of two at all. Just because we can intuit something doesn't mean that it is necessarily logical or explained in of itself. And so this was one of the techniques that Russell and Whitehead attempted to use to to describe what numbers were, as well as many other techniques in formal logic. And that really was what the Principa Mathematica was, or at least attempted to be. It was primarily a work of philosophy. There was very little of what we might traditionally describe as mathematics in the book, despite its incredible length and detail and complexity. Because you have to understand that These people were trying to foundationally write the rulebook, as it were, for mathematics itself.
And they were trying to prove that mathematics as a system was consistent in of itself with And it took them some years to produce the Principa Mathematica. I forget how many, but a lot of strain and effort went into it.
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Chapters
8 chapters
1
What foundational question about numbers does Andrew explore?
0:00–3:09
2
How did Russell and Whitehead attempt to define the concept of 'two'?
3:09–5:58
3
What challenges did Russell and Whitehead face in their mathematical endeavors?
5:58–11:34
4
What is Godel's incompleteness theorem and its implications?
11:34–20:38
5
How do different types of infinities compare?
20:38–26:16
6
What are prime numbers and why are they significant?
26:16–33:20
7
How do personal experiences shape our relationship with numbers?
33:20–38:19
8
What unique perspectives on numbers does Andrew share at the end?
38:19–1:02:44