174 | Tai-Danae Bradley on Algebra, Topology, Language, and Entropy

Sean Carroll's Mindscape: Science, Society, Philosophy, Culture, Arts, and Ideas

Sean Carroll | Wondery

Society & Culture, Physics, Philosophy, Science, Ideas, Society

4.8 • 4.4K Ratings

🗓️ 22 November 2021

⏱️ 82 minutes

🔗️ Recording | iTunes | RSS

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Summary

Mathematics is often thought of as the pinnacle of crisp precision: the square of the hypotenuse of a right triangle isn’t “roughly” the sum of the squares of the other two sides, it’s exactly that. But we live in a world of messy imprecision, and increasingly we need sophisticated techniques to quantify and deal with approximate statistical relations rather than perfect ones. Modern mathematicians have noticed, and are taking up the challenge. Tai-Danae Bradley is a mathematician who employs very high-level ideas — category theory, topology, quantum probability theory — to analyze real-world phenomena like the structure of natural-language speech. We explore a number of cool ideas and what kinds of places they are leading us to.

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Tai-Danae Bradley received her Ph.D. in mathematics from the CUNY Graduate Center. She is currently a research mathematician at Alphabet, visiting research professor of mathematics at The Master’s University, and executive director of the Math3ma Institute. She hosts an explanatory mathematics blog, Math3ma. She is the co-author of the graduate-level textbook Topology: A Categorical Approach.

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Transcript

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0:00.0	Hello everyone and welcome to the Mindscape Podcast. I'm your host Sean Carroll.
0:03.9	A lot of people, I think, when they think about math, when they hear the words math and doing math,
0:09.5	they think of what a math mathematician would call calculating, right? Performing some calculation,
0:14.1	whether it's something simple, like calculating the tip on a restaurant bill,
0:18.4	something maybe more complicated, like calculating the orbits of planets and stars and galaxies and so
0:24.2	forth. But that's actually very, very little of what professional mathematicians do. We use math
0:29.6	in that way. But professional mathematicians are engineers of concepts. They invent new kinds
0:35.8	of concepts. They put them together to prove theorems and they draw connections between concepts
0:41.2	that you might not have known about before. So today's guest, Tide and A Bradley, works at a very
0:46.3	interesting intersection of algebra and information. Algebra being in the mathematician sense,
0:54.4	a kind of study of structures in mathematical objects. So ways you can multiply things together,
1:02.0	ways you can combine things and also ways you can take things apart that relate to each other and
1:06.5	then build them back into the original whole. And then information in the sense of what relates to
1:12.4	what, what do you know about one thing by learning about another thing? So here's an example that
1:18.4	you might not think of as originally mathematics, but that is absolutely closely related to this
1:24.2	stew of algebra and information, language. The language you're hearing right now, the set of
1:30.2	words that you're listening to, has a structure in it, right? Because when certain words pop up,
1:35.6	other words are more likely to pop up right next to them. And of course, you can take a completely
1:41.2	mindless approach to understanding this. You can just dump it all on a computer, dump a huge text
1:46.1	in a computer and ask, what are the statistics of this? What words appear next to what other words?
1:52.0	But you can also think like a mathematician. You can say, well, why is it like that? What are the
1:57.2	structures underlying this kind of thing? And it's kind of an interesting mathematical move to
	...

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