Episode 66: Haim Gaifman discusses mathematical reasoning
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What is the main topic discussed in this episode?
Hey, Matt Techman here from Elucidations. Before we get going today, I just thought I'd ask if you're a fan of the show, to maybe go to our iTunes page and leave a rating and a review, and that way more people can discover it. Alright, thanks.
Hello, and welcome to Elucidations, a philosophy podcast ordinarily recorded at the University of Chicago, but which today is being brought to you from New York. With me is Chaim Gaifman, Professor of Philosophy at Columbia University, and Emeritus Professor of Mathematics and Roosevelt Chair of Philosophy and the History and Philosophy of Science at the Hebrew University of Jerusalem. And he is here to talk with us about mathematical reasoning. Chaim Gaifman. Welcome.
Thank you. So a big
question that philosophers have asked for pretty long time about mathematics is are numbers real entities independent of what people say and think about them? Or are they just sort of cultural constructs?
What is the central philosophical question about the reality of numbers?
You know, you might imagine, I don't know, maybe if a second asteroid had hit and, you know, killed off all the proto Homo sapiens, you know, human civilization had never started, would it still be true that two plus three equals five and everything else that's true about all the numbers? Or are all the facts about mathematics just determined by kind of stipulations we make and definitions we make? So are numbers just kind of social constructs?
Okay, I think the the question is a tricky one because if you push people to say what are numbers, they find themselves a little bit confused. The numbers are not things that you can hold in your hand. So uh let's go on with this game and assume that there have been no humans or the earth was destroyed. Or something like that, or let's assume a stage in which they are no longer humans. Now people find it very easy to imagine that there will be rocks and stars and galaxies and so on. There is no problem. But they should pay a little bit attention. to the kind of scenario that they themselves assume. Because suddenly there would be, say, eight or nine planets, depending how you classify them. And eight planets, this means you can arrange them in groups of four and four, or four groups of two, and so on.
And the galaxies will have a certain distance from each other, and the distance is measured by using numbers and certain dimensions in physics and so on. So all the apparatus that you are assuming in describing, so to speak, an objective world without humans still is meaningless without using mathematical concepts.
How does Gaifman argue that mathematics describes an objective reality even without humans?
I can ask you, would there be three rocks there, or would there be nine rocks there? And if there are nine rocks there, could it be arranged in three groups of three and so on? And you are using mathematics all throughout this description. So to say that there would be no mathematics really makes no sense. You really cannot even conceptualize what the world would be there. Either you are completely uh skeptical about existing of something without humans, and this is one position, but if you think that there is a physical world to exist that can exist without humans, you must give the same status to mathematics. The fact that you don't have a little Rocks or little pebbles which you call number one, two, and three and four makes no difference.
Obviously, if we are going to imagine what the numbers could be, you can posit abstract entities to play the role of numbers, just like you posit an abstract entity which is called the equator. to give you a description of the earth. I mean when you come to the equator the you won't find any line in the sand and nothing to signify this. But still the equator as an abstract object, which is very useful for describing the physical world, has a certain status of an objective entity and the same thing is about mathematics. Let me give you another example which shows that what should interest us is not the existence or non existence of numbers, but questions about mathematical truth. Okay? So for example, suppose you consider an hypothesis in number theory.
The hypothesis is that there are infinitely many twin primes.
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Chapters
8 chapters
1
What is the main topic discussed in this episode?
0:00–1:20
2
What is the central philosophical question about the reality of numbers?
1:20–3:16
3
How does Gaifman argue that mathematics describes an objective reality even without humans?
3:16–6:50
4
Why does the twin‑prime conjecture illustrate the existence of objective mathematical facts?
6:50–11:20
5
What reasoning solves the classic domino‑tiling (chessboard) puzzle?
11:20–13:52
6
How does the symmetric‑strategy coin‑placement game guarantee a win for the first player?
13:52–18:25
7
What does the discussion reveal about mathematics as an accumulative discipline versus philosophy?
18:25–22:33
8
Why are abstract mathematical structures considered real rather than mere social constructs?
22:33–40:44
Speakers
1 identifiedMore from Elucidations
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