Episode 33: Daniel Sutherland discusses the philosophy of mathematics
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What is the central question about numbers that the episode begins with?
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Hello and welcome to Elucidations, a philosophy podcast recorded at the University of Chicago. I'm Matt Teichman. And I'm Jamie Edwards. With us today is Daniel Sutherland, Professor of Philosophy at the University of Illinois at Chicago. And he is here to talk to us about the philosophy of mathematics. Daniel Sutherland, welcome.
Thank you. Thanks for having me.
One of the perennial questions in the philosophy of mathematics is what is a number? Maybe we can just start there.
Well that's a very good question. Numbers appear to be abstract objects and by abstract object philosophers mean objects that don't seem to occupy a spatial or a temporal location, and also a kind of thing that doesn't enter in causal relations with other things. And in fact numbers are one of the prime candidates for Abstract objects should there be any. Now, various people have been quite puzzled about the role of abstract objects in general, whether there are any, there are people who've been opposed to admitting that there are abstract objects and have tried to explain away cases of apparent abstract objects, and that's the case in philosophy of mathematics as well. One of the troubling things about allowing for abstract objects is that it's hard to see how they fit into our conception.
Of the world as we understand it today, that is to say, through science, where we try to explain things through physical laws and causal interactions. And that has given a lot of people motivation to resist the idea of abstract objects altogether. Another more pressing problem within philosophy is to explain how it is that we could have knowledge of abstract objects. abstract objects and their properties if they really are abstract, if they don't occupy a spatial and temporal location, if they don't enter in causal relations, then how do we have knowledge of them? This issue has actually been tremendously formative to philosophy itself because it looks like typically the way you know something is to stand in some sort of causal relation to it.
So the paradigm case here would be um perceiving the things in the room that you're in right now. light strikes an object, it reflects off of it, it impinges on your eye, it causes neurons to fire. That's a more detailed scientific picture that we have now than was available at an earlier point of time. But still the idea is that you enter in some sort of causal relation to an object, and that's what justifies your knowledge that, say, for example, there's a cup on the table right now. Now. Now if we're talking about an abstract object, it is by definition something that doesn't enter into causal relations. So how can you have knowledge of it? Plato and other rationalists thought that we have some sort of perception of things that are abstract and they described in perceptual metaphors our knowledge of the forms, which you may have heard about, and also numbers.
Why do philosophers find abstract objects like numbers philosophically problematic?
These are ideal abstract objects that we don't under into causal relations with or least no direct causal relations with. So that raises a puzzle. What are you attributing to human beings a distinct mode of perception? And if so, how do you explain it? And that has been a big problem for those who believe that you have knowledge of abstract objects is giving an account of the epistemology, that is to say, the theory of knowledge, giving a theory of knowledge of these abstract objects. How is it that you can know things about them? This problem has perplexed people so much that some have looked for alternative views about what numbers are. They have tried to locate numbers in the mind. or perhaps in the physical world.
One way to do this, and this is uh a simplified way of thinking about it, you might think that numbers are in the world in the sense that the number two is really summarizing or somehow an indirect way of thinking about two things, two things that you experience in the world.
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Chapters
8 chapters
1
What is the central question about numbers that the episode begins with?
0:00–3:49
2
Why do philosophers find abstract objects like numbers philosophically problematic?
3:49–6:55
3
How can we have knowledge of abstract objects that have no spatial or causal ties?
6:55–10:12
4
What are the three main philosophical responses to “What is a number?”
10:12–14:15
5
How does the view that numbers are concrete, world‑based entities try to solve the epistemology problem?
14:15–17:33
6
What is logicism and how does it attempt to reduce mathematics to logic?
17:33–21:49
7
Why do Kantian and intuition‑based approaches argue that mathematical knowledge is special?
21:49–25:53
8
What unresolved issues remain about the nature and cognition of numbers?
25:53–30:57
Speakers
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