The President's Frontier Award was established with a $2.5 million donation from trustee Louis J. Forster. Forster helped design the award to support exceptional scholars among the Johns Hopkins faculty who are on the cusp of transforming their fields. The award recognizes one person each year with $250,000 in funding for their work.
This PFA lecture features 2020 recipient Dr. Emily Riehl presenting on Category Theory. Thinking categorically can help serve as a guide post as you trek the grand scheme of (mathematical) things. As written in Quanta Magazine in a 2020 interview with Riehl, “category theory and its next-generation version, higher category theory, are central to many fields of math, from algebraic geometry to mathematical physics. In those areas, Riehl said, ‘I think it would be impossible to describe the kind of basic objects of study without categorical language.’”
Join us for this fantastic event honoring the brilliant work of Dr. Riehl, opened by Provost Sunil Kumar and Dr. John Toscano. Denis Wirtz, Vice Provost for Research, provides closing remarks. To learn more, visit jhu.edu/hopkinsathome
Transcript
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>>Welcome everyone and thank you for joining us for the sixth President's Frontier Award lecture.
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Before we go to the lecture itself, I'd like to say a little bit about the award. The President's Frontier Award was initially established with 1.25 million dollar donation
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from trustee Louis J Forster, who is now the Chair of the Board of Trustees, who is a
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1982 Arts and Sciences graduate, and a 1983 SAIS graduate. The President's Frontier Award
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has been awarded six times so far, and it will be expanded for another five years thanks to
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Mr. Forster's continued commitment with another 1.25 million dollar gift.
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The award is designed with the goal of supporting exceptional scholars among the
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Johns Hopkins faculty who are on the cusp of transforming their fields. So the aspiration
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isn't merely transforming their careers, but their fields themselves. And I'm delighted to
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say that there is no doubt in my mind that all six of the winners so far easily meet
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this very high bar. And so I could not think of a better award to celebrate the born innovation and
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pursuit of excellence that Hopkins is known for than the Presidential Frontier Award. As Provost
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I'm contractually forbidden from commenting on the actual work of somebody. And even though I
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know a little bit of Matt, to attempt to speak on category theory, even for people who know a
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lot of math is a dangerous proposition. However, I can't stop myself from saying this simply because
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it's, for me, it's one of the most beautiful parts of mathematics. Sometimes by asking the question,
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how would this problem be solved if it wasn't as special as it is? What is even more general?
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The answer becomes easier. In my own field, probability, a great breakthrough was when
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somebody realized the best way to think of a probability is simply the length of an intro.
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From that abstraction, suddenly a lot of ideas simplified. And category theory represents
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one of the logical extremes of that kind of reasoning and therefore is one of the most
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elegant ways of saying, if I free myself of special structure,
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will the more general I get make the problem easier and more elegant? And I think Emily
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represents the highest form of that art and so I'm delighted that she is the sixth President's
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Frontier Award lecture winner. So with that, let me turn to John Toscano, who is the Interim Dean
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of the Krieger School of Arts and Sciences to introduce our award winner. Thank you again.
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>>Great. Thanks, Sunil, appreciate that. When I, when I first learned last winter that Emily
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Riehl would receive this year's President's Frontier Award I was obviously delighted, but
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not at all surprised. We knew Emily was a star from the moment she arrived at Hopkins in 2015.
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Emily has been taking the mathematics world by storm, as her colleagues say. She studies category
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theory and you heard a bit about that from Sunil. Math Department Chair, David Savitt describes
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category theory as providing a framework in which outwardly different ideas can be viewed
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as examples of a single more general concept, which can then be specialized to new concept
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context to obtain fresh insight. Although Emily is less than 10 years out from earning her PhD,
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she's already widely recognized and admired for her ability, seemingly effortless to use
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this framework, to build bridges, to other disciplines, bringing her colleagues and students along with her. She's built a reputation for consolidating the approach of category theory,
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simplifying the literature in streamlining contact complex proofs. David also described
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Emily as a true force of nature. In her department for her ability to attract visitors and post-docs,
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offering sought after seminars and graduate classes, and pursuing a formidable research program, all at the same time. In addition, while writing books is rare for any mathematician,
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let alone one in her early stage of career, Emily's written three, including an introduction to category theory, a book that prepares graduate students for research
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and indeed has a fourth on the way. Emily's books, along with her numerous journal articles
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have earned her richly deserved praise. She's a groundbreaking mathematician who has already made
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monumentous and lasting changes to the field. She is also a highly collaborative convener and
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organizer in the academic community. For example, she hosts the end court end category cafe,
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which is a group blog on math, physics, and philosophy. She founded the Kan Extension Seminar,
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which is a graduate reading course in algebraic topology, and also organized a seminar program,
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Higher Categories and Categorification for the Mathematical Society Research Institute.
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She's doing no less than changing the culture of the field. Emily received her PhD in 2011
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from the University of Chicago and spent four years as an NSF and Benjamin Peirce
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Postdoctoral Fellow at Harvard before joining us here at Johns Hopkins where she was promoted
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to Associate Professor in 2019. She has been recognized with numerous prestigious awards,
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residencies, and invited lectures, mentored graduate students and post-docs and won awards
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for her undergraduate teaching. It's easy to see why the selection committee unanimously chose to
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recommend Emily to receive this year's President's Frontier Award. Emily, because this award was
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designed, as Sunil mentioned for exceptional scholars on the cusp of transforming their fields,
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it's tailor made for you. Congratulations. The virtual stage is now yours. >>Great. Thanks so
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much to Provost Sumar and Dean Toscano. That's really quite an introduction I'm very honored.
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So this award was announced in January and quite a lot has changed in the world since then. So I
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wanted to start by acknowledging that I certainly knew in January how lucky I was to, I mean,
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not only to have found a tenure track position, but to have found a tenure track position at Johns Hopkins. When I came here in 2015, I didn't really know my departmental colleagues very well,
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we'd only just met a few months before, but already, one of my colleagues, Steve Wilson
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threw a party to welcome me in the spring before I officially arrived. And put me up in his
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apartment when I was looking for a place to stay in Baltimore. He's the one who then nominated me
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for this award, you know, completely, without my knowledge, several years later. And I was humbled to even have been considered among the finalists much less to actually receive it. I know
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that a lot of people are really suffering right now. It's, it's really been a terrible year. And
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you know, there's something that feels a bit incongruous at times and sort of persisting with, you know, academic inquiry in the face of all of the present and urgent suffering. I do,
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I guess also think that there's something hopeful in a basic science research. I mean, certainly there was some hope that sustained Katalin Kariko and Andrew Weissmann in the 1990s when they
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were following up on this crazy idea of an MRNA based vaccine. And you know, it was years of
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sort of patient development of expertise in my colleagues Lauren Gardner and Ensheng Dong
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on the epidemiological modeling that allowed them to then so quickly stand up the Johns Hopkins coronavirus dashboard that's now been such as key policy tool used worldwide. And thinking towards
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the future, I'm hopeful that my colleagues at the Agora Institute and elsewhere around the University can help us dream up a better, maybe more functional political system and
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some policies that can sort of test or sort of, sort of survive the gauntlet that we have today.
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Anyway, I just wanted to say thanks to everyone at Johns Hopkins, I, I feel very fortunate to be at an institution that is you know, protecting us in many ways and allowing us to
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continue this very long ranging inquiry, even at such a turbulent time.
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What I'm going to invite you to explore with me today is the realm of abstract mathematics,
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which is where I like to spend my time. What I'm going to describe is a joint project with a longtime collaborator Dominic Verity. And even before I arrived at Johns Hopkins,
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we were working on redeveloping the foundations of infinite dimensional category theory. So these are
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infinite dimensional categories are homes for a sort of very complex objects that arise in mathematical physics or in mathematics in general. And
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we've been working on a sort of new approach to develop the basic theory to kind of rigorously prove all the theorems from, from the very beginning,
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so they can be used by other mathematicians. And specifically what I want to tell you about today is a new result that was proven in 2020, and very much with the help of the frontier
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award. And what it says is that there is a formal language that can be used to encode statements
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about infinity categories and every statement that is written in this language is then invariant under change of model. Okay. So what's going to be challenging about this is I realized that none of
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those words make any sense. So mathematicians have developed this very precise language. It sounds
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like English on the surface, but we've redefined all the words to mean something very particular. And it's, it's kind of a lingua franca around sort of spoken common among mathematicians all around
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the world and completely opaque to everybody else. So I'm going to explore all of the ideas
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of a formal language of models of invariants under change of model by starting with something that I
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hope is a bit more familiar to us which are the natural numbers. So when we think about
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the natural numbers, so, so what I'm referring to are that the counting numbers, I like to include zero, but if you'd rather not, it doesn't really make a big difference. So zero one,
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two, three, four, and the question is sort of what are they really? So there's some sense in
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which this is kind of a philosophical question, but so a way that a mathematician might approach
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defining the real numbers or sort of saying what they are is by describing some of their properties. So we know we can add numbers, we can multiply, we can, there's certain properties of
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that addition and multiplication. We can take any natural number and factor it uniquely as a product
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of primes. But the problem with the description or definition of the natural numbers along those
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lines would be that it would be very long. You know, we humans have been studying the national numbers for millennia and there's, there's quite a lot that we can say about them. So in the
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late 19th century Giuseppe Peano formulated some axioms, they're commonly referred to as the Peano
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postulates in a paper, erythematous, Arithmetices principia, nova methodo exposita. And so these
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are some true facts about the natural numbers and it's a, it's a sort of a shortlist of true facts and so I'll sort of say what they are. So one of them is that there's a natural number zero,
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and then every natural number has a successor that's also a natural number. The successor
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is meant to be like the next one in the sequence, so the successor of zero is one successor of one is two, the successor of two is three, and so on and so forth. So zero is not the
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successor of any natural number, meaning that this sequence doesn't wrap around to the beginning, it just kind of carries on. And moreover, no two natural numbers have the same successor. So every
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new natural number that you define as a successor of a previous one is a, is a distinct number. So
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we can generate infinitely many natural numbers in this way. And then the final axiom, this is the
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complicated one, it's also sort of where the real power lies. If you're familiar with the principle of mathematical induction, this is secretly justifying that. So it says that any set that
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contains the number zero and then is closed under the successor operation, meaning it contains the
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successor of any natural number, it contains that necessarily contains all of the natural numbers.
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Okay. So these are some facts that mathematicians accept about the natural numbers. We sort of believe in our bones that these are true. I've reformulated these postulates in, so formal
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languages are going to be a theme later on, so I've rewritten five axioms, zero is a natural number, every natural number has a successor and so on in this formal language. So there's a
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formal syntax that mathematicians can use to write sentences. And then you can ask a computer,
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whether the sentence is a logically well-formed mathematical sentence or not. So that's the,
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this idea of a formal language. This is sort of how I would teach a computer about the Peano
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postulates and the computer would say, okay, these sentences make sense at least. They're logically reasonable. So another mathematician, Dedekind, was studying the, or formulated a very similar
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series of axioms at around the same time in a paper, Was sind und was sollen die Zahlen. So
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what numbers are and what they should be. Which has this starts with this very beautiful quote
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in German, so in science, nothing capable of proof should be accepted without proof. And what's so exciting about the Peano postulates, what's , why we can think of the natural numbers
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as being defined somehow by these Peano postulates is all of the other properties that we know about them follow from these axioms. So using just these sort of five simple statements about the natural
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numbers, it's possible to define addition, and then prove that addition is communitive,
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that A plus B is equal to B plus A, and to prove that it's associative, that you can add any finite
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collection of natural numbers together in any order and the result will always be the same. You can similarly define multiplication, prove the distributive property of multiplication over
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addition. And then prove more sophisticated statements in number theory, starting from
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what's called the fundamental theorem of arithmetic, which is the property that any natural number can be factored uniquely as a product of primes. So there's a sense in which
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if we're asking what the natural numbers are the, the Peano postulates give an answer to that somehow. If it's a list that we can write down and
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not too much time that somehow generates all of this complex mathematical structure. Number theory
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is very much still an active research area today. Okay. But mathematicians don't like sort of,
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you know, founding every area of mathematics on its own set of axioms. There's, there is a
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desire at around the same time around the turn of the 20th century to come up with a common foundation for all mathematics. So a list of axioms, like these Peano axioms that
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characterize the national numbers, but that we can use to talk about all kinds of mathematical structures. So not just numbers, but geometric structures, analytics structures as well. And
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the accepted common foundation of mathematics that we implicitly refer to today is something called
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set theory. And so there's a secondary question, which is, can we reason about the natural numbers in set theory? Can we use some, maybe more primitive axioms, axioms about sets
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to begin to talk about the natural numbers? And so there, it is possible to construct the natural
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numbers formally in the system of set theory. And in fact, there are different ways to give a construction. So, so set theory includes, so what we're trying to do is we're trying to
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model that's, that's the word the mathematician would use, to sort of model or define, introduce
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the natural number to zero, one, two, three, and so on as sets. Because for mathematicians, somehow sets are more primitive than numbers. So, so how do we do this? So one of the axioms
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of set theory tells us that there was an empty set. So this is the, it's written as a zero with a line through it. So we can associate the natural number zero with the empty set.
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So then in set theory, you're allowed to take a collection of sets and form a new set with those
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sets as elements. This is one of the things I don't actually like about the axioms of set theory, the elements of sets in formal set theory are themselves sets, which is very confusing, but
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this is just how the world works according to the axioms of set theory. So once we have the
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empty set as a set, we can use that as an element and form another set. This is the set containing the empty set, and that's how we're going to think of the number one, as a set. And so now,
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if we wanted to find the number two using Von Neumann's construction, what we do is we take the two sets that we've mentioned so far, there's the empty set that doesn't have any elements,
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and then there's the set that has the empty set as its element. And I can form now a two element set,
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including the empty set and the set containing the empty set, and that's the number two. And then we define the number three using the three sets that I've just mentioned. So we have
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the empty set, the set containing the empty set and the set who has two elements, one of which is the empty set and the other is the set containing the empty set. That's the definition of three
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in Von Neumann's natural numbers and four is then the set that contains the empty set, the set of the empty set, the set of the empty set, plus the set of the empty set, and then the set
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that has all of those three sets as its element. Okay, I'm going to stop there or I'm going to
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run out of time. But in parallel, Zermelo proposed a different construction. It starts
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out the same we're identifying the empty set with the number zero, and we're identifying the set, containing the empty set for the number one. And then to form the number two what I'm going to
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do is build a set, which has a single element and that element is the set that corresponds to one. So this is the set of the set of the empty set. And then three is
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the set of the set of a set of the empty set and four is the set of the set of the set of the set of the empty set, and so on and so forth. Okay, so these are complicated constructions,
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but they're valid within the rules of set theory with the accepted axioms of set theory. And when I collect all these sets together, I have two different models of the natural members.
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I have defined that the set of natural numbers in these two different ways, and you can check that they satisfy the Peano postulates. So this gives us a construction of the natural numbers
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in set theory. Okay. But what we need to observe about these constructions is they're not the
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same. They're, the reason is the sort of way the successors formed is different in the
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two different constructions. With the Von Neumann construction of the naturals and this are Zermelo construction that natural members, and they're not literally equal. Which seems weird because
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we think of the natural number as being some sort of unified concept. And now we have two
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different models of this one thing. So that's okay because mathematicians have a more refined
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notion of equality and Dedekind proved in this paper...that something that's now referred to as
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Dedekind's Categoricity Theorem. It says that if you have any triple given by a set, and so
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that's like the set of all the natural numbers, an element, zero and a successor function, then as
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long as that data satisfies the Peano postulates, so the five axioms that I mentioned previously,
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then those triples are isomorphic. So I'll, I'll say more later on about what I mean by isomorphic,
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but its etymological roots are same shape. So these are the same, in some sense, not equal,
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not literally equal, it's a more sophisticated sense of equality. And a corollary is that I mean,
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from the point of view of mathematics, having two objects that are isomorphic is awesome because it means that all kind of meaningful properties of one also hold true of the other. So
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a consequence of this isomorphism between these two constructions of the natural numbers is that
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all number theoretic properties of the Zermelo naturals hold equally for the Von Neumann naturals. So each is kind of, can be used to prove a theorem and then if that theorem will be true
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of the other model as well. Okay. But there's one subtlety here that was pointed out by Paul
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Benacerraf in 1965 paper, what numbers could not be, sort of referencing to Dedekind's, what
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the numbers should be. And he asks a question, is three an element of 17? So this is a really
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wild question. I mean, on the one sense it's like total nonsense. I mean, it doesn't make any sense for a number to be an element of another number. I mean, that's just not what numbers are about.
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I mean, it's, you know, there's, there's sort of no way to answer that question because it doesn't
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make any sense. But on the other hand, if we're thinking about these numbers as something that are constructed formally in set theory, this question makes perfect sense. It's something
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that you could write in the formal language, this language that a computer would recognize and say, yes, this is a mathematically valid question. You can write this question in the formal language of
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set theory. And moreover the answer to this question depends on which model you're using.
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So the way that the Von Neumann natural numbers are constructed, the set 17 contains 17 elements
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and one of those is the element three. So three is in fact, an element of 17. Whereas in the Zermelo construction of the natural numbers, the set 17 contained a single element,
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namely the element 16. So three would not be an element there. So it's true for the Von Neumann natural numbers, but false for the Zermelo natural numbers, but also totally nonsense. So this is
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a question that we're going to have to leave open, which will I'll address again at the end. The question is can we sort of restrict our formal language? So there's a common formal language,
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first-order logic that mathematicians use to write very rigorous sentences that allows us in
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the case of the natural numbers to ask nonsense questions like is three an element of 17. So a question that I'll leave open for now is whether we can restrict that language so that it only
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includes meaningful statements, mathematically meaningful statements about the natural numbers.
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So just to summarize the story so far, that if we're reading about the natural numbers in the classical foundations of mathematics, we need a model. So something like this set of Von Neumann
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natural numbers or Zermelo natural numbers that are built within the axioms of set theory.
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So the problem with these models is they're not unique. We have two different constructions and in, you know, in fact there are many, many more. But there's a sense in which they are
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the same. So mathematicians have a more sophisticated notion of sameness, not equality, but something called isomorphism that I'll tell you more about very soon.
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But a problem remains, is there a way to restrict the formal language of mathematics include these nonsense statements, like is three, an element of, of 17? And the idea that's going to appear
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in the solution is that we need some sort of more precise taxonomy of mathematical objects so that
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we don't ask whether X is equal to Y, if X is a natural number and why is a triangle? I mean,
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that's, that's kind of the problem with using set theory as a foundation for mathematics is that everything in set theory is a set. That's sort of the only sort of stuff we have to build more
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sophisticated mathematical objects. So you could absolutely ask, is this number the same as this triangle, even though that doesn't make sense to sort of intuitive humans. And the, what we're
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going to bring in to help us solve this problem to kind of come up with this taxonomy of mathematical
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objects is something called category theory, which is the area of mathematics where I work. Okay. One
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of the things that we can do in category theory is we can define what an isomorphism actually means. So I'm going to give you the definition of this idea of seeing shape of mathematical objects.
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So it's something you can define in any category. So I have to tell you, firstly, what a category is. So this is a mathematical definition. It's kind of bread and butter for our math majors, but
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maybe less familiar for the rest of us. So what is a category? It's something that consists of some
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collection of objects. You can think of these as being sets if you'd like, or as being some sort of data types. So they're objects of some specified kind, but they really can be whatever you want.
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And then we also specify some arrows between the objects. So an arrow should have a name telling you sort of who it is. And it also has a specified source object and a target object. So
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we're drawing an arrow is pointing from one thing to another. And this you can think of as like a function or maybe it's a algorithm that converts a thing of type X into a thing of type Y. And then
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on top of that, we have two additional structures. So each object necessarily has an identity arrow,
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which you can think of as the process that does nothing, that sort of leaves the state unchanged. And then whenever I have an arrow from X to Y, like F, and an arrow from Y to Z, like G,
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there needs to be a specified composite arrow, which you can think of as, you know, running two processes in sequence or composing a function with another. And this would be an arrow from X to Z.
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Okay. So this is a very, very abstract definition as a Provost Sunil said in the introduction.
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This is all sorts of mathematical objects assemble into categories. They're kind of all over the place. And what's cool about them is you can define a notion of isomorphism at this
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level of generality. So between two objects in any category, no matter what that category is. So what
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is an isomorphism? So you, the data consists of two different objects in a common category. So A and B have to be in the same category. And then an isomorphism consists of an arrow,
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one pointing from A to B and one pointing from B to A, so that when I compose the arrows, if I
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compose them one way around, I get an endo arrow of A, and that needs to be this identity arrow.
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And similarly, if I compose them the other way around, I get an endo arrow of B and that needs to be the identity arrow again. So the idea of an isomorphism is F gives us some
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way of translating from A to B and G gives us some way of translating from B back to A, and
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this condition about the composition says that no information is lost in those translations. You can pass from A to B and then back to A, and you've come back to exactly where you started,
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right? So again, what's, what's powerful about this definition of isomorphism is it can now be deployed all across mathematics for any type of mathematical object, whatever you want. You can
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ask are two objects of that type isomorphic or not? Are they same in this sense? And so
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in particular, we've just defined this notion of category. So categories are now themselves
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mathematical objects, and we can ask, what is an isomorphism then between categories? What does it mean for two categories to be isomorphic? And so there's an answer for that. But before we think
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about it, let's think about an example. So there's a category of matrices defined as follows. So it's
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objects are just natural numbers, so like zero, one, two, three, four. So we have natural numbers,
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that's the objects. And then an arrow from the number N to the number of M is an M by N
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matrix. So an arrow from three to four would be a four by three matrix of real numbers. So it's
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got 12 different real numbers in an array. And then we have a composition and identity.
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And then there's a similar category of vector spaces. So the objects here are written sort
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of RK, RM, RN. These are RN is an N dimensional Euclidean space. So this is the collection of all
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points in N dimensional space. So if I have an ordered list of N real numbers, that defines a point in RN. So these are sort of much bigger somehow as mathematical objects than just
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the numbers themselves. So these are the objects now, these Euclidean spaces, you can think of a point, a line, a plane, these are examples of these Euclidean spaces. And an arrow
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now from RN to RM, from N dimensional space to M dimensional space is a function that takes an
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N dimensional point to an M dimensional point and then satisfies this property of linearity which is
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probably unfamiliar to many of you, but most of our students at Johns Hopkins learn about linear transformations at some point in a linear algebra course. Okay. So it turns out,
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so it makes sense to ask whether these two categories are isomorphic because they are both objects of a category of categories. They're not in fact isomorphic. However, they are same in a
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more refined sense of sameness, something called equivalents. And a consequences of this is that
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all category theoretic properties of the category of matrices also hold for the category of vector spaces. So a way I like to think about this equivalence is we actually teach two different
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linear algebra courses at Johns Hopkins. In one of the courses we spend most of the time in the
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category of matrices learning about a row echelon form and matrix operations and so on. And in the
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other course, we spend most of our time in the category of vector spaces. I think you may be
30:15
more abstractly about inner products and so on and so forth, but somehow those are the same course. So both the matrix course and the vector space based course are teaching the same subject
30:23
of linear algebra. And the justification for that is that this category of matrices is equivalent, so the same in the sense of categories to the category of vector spaces.
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But again, these categories are not isomorphic, so they don't satisfy this reduced property of sameness. And that means you can ask questions that are kind of nonsense from the point of
30:46
view of category theory, but which will have different answers depending on which category you're thinking about. So here you can ask whether your category has countably infinitely, many
30:55
objects. Something that's a bit wild is that's actually the smallest notion of infinity, that's countable infinity. There are other larger notions of infinity. And in this case, the category of
31:04
matrices does have countably infinitely, many objects where the category of vector spaces has many, many, many more objects than that. So somehow these categories can be equivalent
31:13
with having different numbers of objects, which is a bit of a strange thing to think about.
31:21
Okay. So I said that the categories of vector spaces and matrices are equivalent, not isomorphic, and what does that actually mean? Well, like isomorphism was a concept
31:30
you can define in any category, equivalence is a concept you can define in a 2-category. So really
31:36
categories themselves live as objects in some two dimensional category, and that's where this
31:41
notion of equivalence can be understood. But if I defined a 2-category for you, then we could
31:46
ask the same question, what is the notion of, what's the correct notion of equivalence between 2-categories? And that's something that you can define, but you need a third dimension. You
31:55
define that enemy three-dimensional category. And three-dimensional categories also have a notion of equivalence that's correctly defined in a four-dimensional category and four-dimensional
32:03
categories, have a notion of equivalence to find in a five-dimensional category, five categories of notion equivalents in a six-dimensional category. And you can see this is never going to stop until
32:11
we get to infinity. So these are actually the sort of categories that I spend most of my time with,
32:17
infinite dimensional categories. And let me introduce them to you now. So an infinity
32:23
category has, like an ordinary category, it has some objects and some, one arrows between them.
32:30
I say one, cause I'm thinking of an arrow as being something that lives in dimension one. I mean, if you like, the picture of an arrow is a sort of one dimensional path from X to Y. But now we
32:40
have an addition, these two-dimensional arrows. So one arrow had a source object and a target object,
32:45
a two-dimensional arrow, alpha here, has a source one arrow, which is F and a target one arrow,
32:51
which is G, and there's a requirement that F and G need to be parallel arrows. So they have a common source object X and a target object Y. And then we have three-dimensional arrows between parallel
33:02
two-dimensional arrows. So here are alpha and beta are two dimensional arrows from F to G. And so we can have...as a three-dimensional arrow from alpha to beta. And think of it as inhabiting
33:14
this sort of three-dimensional ball. So X and Y are sort of objects here and F and G define
33:19
paths between them and alpha and beta inhabit sort of the front and back faces of the ball
33:24
and the interior is really where, so it's somehow a three-dimensional object and this continues in
33:30
all higher dimensions. So the pictures are sort of harder to draw. And like an ordinary category,
33:36
these infinite dimensional categories need to have identities and composition, but it's, it's a little bit more subtle. Now these are, composition is, is not an operation. I don't
33:46
have a well-defined composite of any two arrows. Instead, it's something that's witnessed. So
33:52
for any arrow from X to Y and arrow from Y to Z, you can recognize when an arrow from X to Z
33:57
is a composite of those arrows, but it's not a unique property anymore. And the reason that
34:05
this definition is as complicated it is, is it's reflecting what we see in the examples
34:10
when we meet in the real world. This is actually the point in the talk that comes the closest to the real world, so let me take you there. So any space, so space is something that a term that a
34:21
mathematician would recognize, so any space has an associated, infinite dimensional category called
34:26
the fundamental infinity groupoid of that space. So let me describe a space for you so we can have
34:32
a common visual metaphor. Let's say you're baking donuts, but you're doing it badly. So you've,
34:38
you've got all your donuts and you put them in a tray, a baker's dozen donuts, but then you put them too close and so when you put it in the oven, I realize this isn't how you normally bake donuts,
34:47
but let's just do this anyway, you put it in the oven and they all kind of run together. They all sort of stick together. And the space I actually want to imagine is the glaze on this sort of sheet
34:57
of congealed together donuts in the oven. You can imagine you're an ant sort of walking around this
35:04
space. This, a mathematician would call this the sort of oriented surface of genus g or genus 13,
35:11
where 13 refers to the baker's dozen donuts that you congeal altogether. Okay. So what's
35:17
the infinity category here? So the objects of the infinity category are the points, the positions,
35:22
the ant could be on walking around the glaze of this congealed together sheet of donuts. The one -dimensional arrows are the paths that an ant could take between two different points. So
35:32
the two-dimensional arrows are something called homotopies. These are notions of paths between paths. I don't know how to describe this in terms of ants, but it's like a continuous deformation
35:40
from one path to another. And the reason I like this space is you see that some paths are sort
35:47
of deformable and continuously deformable and others aren't. You have to respect the holes in,
35:53
you know, an ant can't sort of teleport across the hole in a donut. It's got to kind of walk around sort of one side or the other. And in a way, these continuous paths, there are sort of paths
36:03
between paths, between paths, between paths in all dimensions and that's the notion of an infinity category here. So these are the sort of infinity, these are the sorts of categories that might
36:13
collaborate in Dom and I's study. And again, if we want to reason about them rigorously using the
36:19
common foundations of mathematics, sort of this lingua franca that all mathematicians use, we need models. So we need a way to construct instances of these, construct sort of rigorous mathematical
36:32
versions of these infinite dimensional categories within set theory. Now, these are sort of much, much, much, much bigger objects. So they live in a much, much larger set theoretical universe.
36:42
And what's distinctive about the way that Dom and I think about them is we're sort of zoomed out
36:49
an extra category level. So, so it's kind of classical at this point, mathematicians like
36:56
to think about the infinity category of spaces or the infinity category of spectra, or there's an
37:02
infinity category of homotopic coherent diagrams, which was first imagined here at Johns Hopkins by
37:08
Michael Boardman. So these are sort of specific instances of infinity categories. And what Dom and I think about instead is something that we call an infinity cosmos. It's the universe,
37:17
the infinite dimensional categorical universe in which these infinity categories live as
37:23
object. So, I'm drawing here in orange are these specific infinity categories,
37:29
which are themselves, you know, very, very large mathematical objects. And we're thinking about them as themselves being objects in a much larger thing, the infinity cosmos of quasi categories,
37:39
which is one of the models of infinity categories or the infinity cosmos of Segal categories. So
37:45
the right visual metaphor is the sort of Men in Black outro, which takes the power of 10
37:51
video where you, you know, start with a couple in Millennium Park in Chicago and then sort of zoom out. And then eventually you see the United States and then you see the Earth and that you see
38:00
the solar system, and then you see the Milky Way. And then at the very end, you, you see that, that whole thing lives inside a marble that some alien is playing with. So these infinity cosmos or
38:10
the marbles. So they're, they're small somehow because we have different models that we're thinking about at once, and we're sort of juggling between them, but they're also very big because
38:17
they contain the whole universe within them. Okay. So, what Dom and I have done is that we have
38:25
proven that the theory of infinity categories, so theorems that mathematicians want to use about infinity categories can be developed in any model and are independent of the features of the model.
38:35
And the way we use that is we've observed that when you zoom way, way out, and you're thinking
38:40
at the level of an infinity cosmos, all of the models kind of look the same. So there are common
38:46
properties of the infinity cosmos of infinity categories in one model and the infinity cosmos
38:51
of infinity categories in another model. And if we prove a theorem starting from those common properties, then it applies, it's one of these very abstract results that then applies in all
38:59
sorts of specific contexts. Usefully though, our theorem is kind of general enough to
39:07
also encompass analytically proven theorems. I do refer to the coordinates of a specific
39:12
model. But, we still have this problem of these sort of evil statements that are not variant under
39:19
equivalence between infinity categories or under change of models. So these sort of questions that
39:25
somehow are beside the point that are asking about features of the model, as opposed to features of the infinity category. And there's a question about how to exclude statements like that.
39:36
And so, we are developing a formal language for infinity categories that are based on a
39:43
formal language for category theory developed by Michael Makkai. And the idea is he's looking,
39:51
using a sort of familiar first-order logic, so the familiar formal language of mathematics,
39:56
but then imposing some additional restrictions. So one of the restrictions is that equality cannot be
40:03
sort of used everywhere in a formal sentence in the language of categories. It's permissible to
40:09
ask whether two parallel arrows are equal, but it's not permissible to ask whether two objects are equal. And then there are also restrictions involving the quantifiers. So this upside down A
40:18
is, reads to a mathematician as for all, and the backwards E reads to mathematicians as there
40:24
exists, and there are also restrictions there. So Dominic and I have developed an analogous
40:31
signature. So these are describing the structure of the variables in a formal language for infinity
40:36
categories. You can see it's considerably more complicated than the signature for ordinary categories, but infinity categories are somehow infinitely more complicated than
40:45
ordinary categories, and the fact that this is sort of finitely more complicated, is I think a
40:51
better than one could have hoped. Okay. So there's considerably more to say about that, but I think
40:58
at this point, I'll leave that to the references. So Dom and I are very close to the end of a book,
41:05
Elements of Infinity Category Theory, that I hope will appear in 2021. It's also currently available
41:10
on the web. I should say this, the intended audience for this is sort of graduate level
41:16
mathematicians, but there's some pretty pictures and maybe you want to check it out. I also just want to say thank you to everyone who's helped me here in so many different ways. So.
41:27
>>Fantastic, Dr. Riehl, amazing lecture. If that's okay with you, I'd like to ask you
41:36
questions that were posed through the website or through chat. And the first question is from Dr.
41:45
Sonja Armstrong, who's asking, why do you include zero in the set of natural numbers? >>It's not,
41:53
right, this is always a point of controversy. So the set of natural numbers with zero is isomorphic
42:00
to the set of natural numbers without zero. So from that notion of sameness, it doesn't actually
42:05
matter whether you include zero or not. So it's really kind of a matter of personal preference.
42:13
I guess I like to have zero there because it's nice when you're adding numbers to have a number you can add that doesn't change anything, but if you'd like to take it out I won't object too much
42:25
>>Also from Dr. Armstrong, are you redefining the fundamental theorem of arithmetic since the
42:32
theorem refers to prime and composite numbers? How you do, do you define prime and composites?
42:39
>>Right. So, great. So the fundamental theorem of arithmetic is the result about the natural numbers
42:46
that says that any natural number can be expressed uniquely as a product of primes. And I guess here,
42:53
I should maybe say any non-zero natural numbers. So that might be related to the question about zeros, but so it is certainly possible to define a prime number in the sort of formal language for
43:06
mathematical, the formal language for natural numbers that we've been discussing. So a prime is
43:12
a number that has no proper factors. So if your number P is equal to a product, A times B
43:20
of other natural numbers, then either A is equal to one or B is equal to one. And that is certainly
43:26
a sentence that you can write down. It's negation then sort of the formal compliment of that
43:32
sentence defines the composite numbers. So those are certainly things that you can start to prove
43:37
theorems about beginning with the Peano axioms. >>Thank you. What do you see as the advantages
43:47
of category theory over set theory as a foundation for mathematics? >>Right. So
43:56
I think it's useful to have a foundation for mathematics that's a little closer to mathematical
44:02
practice. So sort of famously when mathematicians in the early 20th century, we're trying to
44:11
write very, very formal versions of proofs of, you know, sort of basic results. Two plus two equals
44:16
four, that all mathematicians accept are true. It, it took, you know, hundreds of pages. This is,
44:22
this is just really, really, really difficult, and it's also kind of completely unreadable.
44:28
So more recently as a mathematics gets very, very complicated, there has been
44:35
increased interest in using computer proof assistance. So what a computer is doing actually is it's, it's not at this stage helping you come up with new ideas or coming up with new theorems,
44:45
but it's checking that the proof that you've claimed you've written is actually correct. So,
44:53
so the idea is that you know, this is very cutting edge today in, in, in limited use, but the idea is
44:58
that in the future that a mathematician might be able to formally verify all of the little lemmas,
45:05
all of the building block results in their papers. And so when you write a paper and share it with
45:10
other mathematicians, you could also share this computer file that checks that everything that you've claimed is true is in fact, correct. So there's a sense in which this should be
45:20
possible because there are these formal languages that you can teach a computer to recognize whether a statement is valid and you can also teach a computer, the basic rules of inference
45:31
in mathematical proof. So that if A is true and A implies B is true, then B as true, that's called
45:37
modus ponens, it's one of the logical principles. That's kind of the building block of mathematical proofs. But for this to be practicable, so for a mathematician who has, you know,
45:47
is also teaching and is also advising students and, you know, you know, has limited time to
45:54
be able to write every single step of every single logical argument in a computer, you
46:00
really need the formal foundations to be much closer to mathematical practice. You need this sort of sentences. You would teach a computer to be close to the way that you're
46:07
thinking about these objects already. And set theory just really is not that. So one of the
46:13
most basic concepts in mathematics is the notion of a function between sets and the way that functions are thought about or formalized in set theory, is somewhat complicated. It's a formalized
46:23
as a graph or as it's relation. And you know, it's just, it's just not the way they're mathematicians
46:29
tend to think about functions. So there are other formal systems that are more categorical
46:35
such as type theory, for instance, that I think are closer to mathematical practice. And this is one of the things that I'm very interested in as well. >>A couple of questions about your career,
46:47
your scholarship, you as a person...of course one is from Andrea was asking,
47:00
how did you, Dr. Riehl, find your way into this field of mathematics? >>I mean,
47:11
it's kind of a very human story. I, you know, I loved mathematics as a child and loved it even
47:17
more when I started to learn about proof-based mathematics and, you know, kind of how creative it is and how you know, how much writing is involved. And before I started my PhD, I went to
47:28
Cambridge, England for a year and took a category theory course there. And it was kind of love at
47:33
first sight. It felt to me like, this is the style of proof that I liked the best. This is the way I like to think about mathematical objects. And so that was that. >>Terrific.
47:45
Also these question, right, from...what is a common misconception people have about your
47:52
research? Either experts or non-experts. >>So there's this there's this sort of
47:59
classical phrase, abstract nonsense to describe proofs and category theory. And
48:05
some people view it as a pejorative, other people embrace it. So I guess you know, when the first
48:12
paper in category theory was written in the 1940s, it was simultaneously viewed as very profound
48:19
and completely meaningless. And I, I think I think that could still apply. I don't know.
48:28
>>So I think we all have a same question and that's okay too, you know but from Jenny,
48:35
how does one apply category three through so-called real life problems?
48:41
>>So actually applied category theory is a new and growing field within category theory. And they are
48:49
thinking about modeling sort of compositional frameworks, whether these are like Petri nets or
48:56
you know, some sort of computer processes where you can run things in parallel, or you can run them in sequence, or you can do some sort of combination of them
49:03
or quantum systems. It's an extremely active research area. One of the applications is also
49:10
if say you've written a computer program, and you want to verify that it functions as intended, you know, that its implementation matches with its intended design. So that can be expressed as a
49:21
functor. In other words, an arrow in the language of category theory. There are also applications to mathematical physics of the kind of infinite dimensional categories that I was mentioning. So
49:34
there's a notion of a quantum field theory which what it means mathematically is it's a functor, so
49:40
again, an arrow between these infinite dimensional categories. So, so yeah, the applications are
49:47
there though a lot of mathematicians are motivated primarily by kind of aesthetics and, you know,
49:52
you want to know the answer because you want to know the answer and that would describe myself.
50:00
>>Thank you from...do you think this also....all of this could someday be considered basic enough in ordinary syllabus for high school students or
50:11
this is necessary necessarily advanced and only of interest to specialists? >>Great. That's a
50:17
good question. I don't anticipate teaching this to high school students ever, but I do dream about
50:22
teaching it to mathematics undergraduates. So the things that we teach our undergraduates today
50:28
are were so cutting edge, you know, a hundred years ago, 150 years ago, these were, you know,
50:34
were to blown the minds of mathematicians just a few centuries or a few decades before. So,
50:40
so our undergraduates today are handling extremely sophisticated mathematical concepts. This work
50:45
that I was talking about today is not well enough understood yet, it's complicated and challenging
50:51
to explain to a graduate student, but I hope that in a hundred years' time, we understand things
50:57
in you know, we, we really we understand to the point of being able to simplify and, you know,
51:02
perhaps have improved our foundation system, which I think at present is really holding us back and our teaching this sort of things to advanced undergraduates yeah. >>From...why or
51:13
how is functional programming related to category theory? >>Right. So I'm not a computer scientist,
51:23
but there is kind of sophisticated operations that you can have on data structures that can be
51:32
expressed in a categorical way. So a program you know, has, the way a program runs might be
51:43
affected by that current state, a machine is in and might leave that state that the machine is in
51:48
altered. So they, we have these sort of interactive input output and side effects and if you're understanding programs in that way, then understanding what it means to
51:57
combine programs to compose programs is quite complicated. But the language of category theory
52:02
can express that. So the people who designed programming languages like Haskell are aware of
52:09
category theory and are implementing categorical concepts in that programming language and the way
52:16
that people study programming languages can be informed by categorical thinking.
52:25
>>I guess I have a question also about what seems to be your acute awareness of the historical context. So who have been contributors
52:35
to your field. And I was wondering, where was that happy balance between knowing enough of
52:42
what's been done, but maybe at some point may diminish or...the original because you think
52:49
maybe not, maybe at this point you see a lot of open space in front of you or and versus almost
52:56
being happily innocent or ignorant of all those previous advances as,
53:02
as a way to be feeling liberated in your, in your pursuits. I don't know if you think about that.
53:10
>>I do. I don't have an answer for that. I mean, one, one thing that we're, we're lucky in mathematics is that we're, we're really in a very long conversation in a way that our colleagues in
53:19
the sciences aren't. You know, we Euclid has this beautiful proof that there are infinitely many primes that's over 2000 years old and it's still as true today as it was then, and as useful to us
53:29
as it's always been. So I think you can't, I mean, you can't ignore the history. But
53:36
you're right that just because somebody thought about something in one way 50 years ago, doesn't
53:42
necessarily mean that's the most productive way to think about it today. And I think that's one
53:48
of the exciting things about doing research in a teaching institution is, you know,
53:54
there are new people coming through all the time with new ideas and new interests and already
54:00
you know, infinite dimensional category theory is more popular among kind of younger people in my field than older people in my field. And so you know, interests shift and times, and there are,
54:10
my students are more interested in type theory. And I think you know, being part of this structure
54:17
where we can sort of challenge these hierarchies and have sort of established senior people and
54:24
ambitious junior people together in the same discussion, I think is really helpful in driving the science forward. >>I think it was said before, I think by, by John, that
54:37
you've made extra efforts, especially for, to really make your field more accessible. I think
54:43
this lecture, you just give epitomizes this, this, this, you know, you, you could be still trying to
54:48
impress and make contribution when vis-a-vis your peers, but you you've seen more than that. An
54:55
opportunity really to bring in kind of the next generation of mathematicians into the field and
55:01
your, your texts I think have helped tremendously. They're super highly cited where,
55:07
you know...especially in mathematics. So Emily, if I may, if I may I'd like to thank you for a
55:16
wonderful lecture and thank all attendees who gave great feedback, great questions, I think.
55:25
I hope also that you've inspired many to, to talk about math as a really a wonderful pursuit
55:35
and academic and, and really almost, I think you touched on, on philosophy
55:40
to some extent as well. So thank you again. And I like to mention that
55:48
the recipient of our next Presidential Frontier Award will be announced early next year. So
55:55
checkout who have been our past recipients besides Professor Riehl. It's quite a cohort. They are
56:03
really creating a huge and wonderful solid foundation to for students and peers at Hopkins.
56:12
So again, thank you so very much Professor Riehl for a wonderful lecture and
56:18
one congratulations again for receiving the Presidential Frontier Award. >>Thanks so much.