 I'd like to just turn to introducing the panel members. We're very fortunate to have Almas Steingart and Stephanie Dick here, two of the rising stars in History of Science and Peter Gallison. Almas Steingart is a lecturer in the Department of the History of Science at Harvard. Her research focuses on 20th century mathematical thought, Steingart is interested in the ways in which mathematical ways of thinking impact a wide range of disciplines, including the natural and social sciences and even the humanities. She is currently completing her first manuscript, Pure Abstraction, Mathematical Thought and High Modernism, which tracks the proliferation of mid-century mathematical thought dedicated to abstract and axiomatic ways of reasoning. She has also written widely on topics such as mathematical visualization and computer graphics. Steingart earned her PhD from MIT in the program in history, anthropology, science, and technology and served as a junior fellow in the Harvard Society of Fellows. Stephanie Dick is an assistant professor of history and sociology and science at the University of Pennsylvania. Prior to that time, she was a junior fellow in the Harvard Society of Fellows. She completed her doctorate in 2015 at Harvard in the History of Science. She is a historian in mathematics and computing in the 20th century United States. In particular, her first project is a history of automated theorem proving that documents attempts to design and implement programs for mathematical theorem proving in the mid-20th century. Peter Galison is the Joseph Pellegrino University professor at Harvard. He received his 1983 PhD from Harvard in theoretical high-energy physics and in the history of science. In 1997, he was awarded the MacArthur Foundation Fellowship. In 1998, the Pfizer Award for Image and Logic. In 1999, the Max Planck and Humboldt Shifting Prize. And in 2018, the Pius Prize in the History of Physics. His other books include How Experiments End, Einstein's Clocks, Paul Coray's Maps, and Objectivity. Collaborating with Rob Moss, The Two-Directed Secrecy, which premiered at Sundance and then Containment in 2015 about the need to guard radioactive materials for the 10,000-year future. Galison partnered with South African artist William Kentridge on a multi-screen installation, The Refusal of Time, 2012. He's a co-founder in 2016 of the Black Hole Initiative, an interdisciplinary center, which is the focus for his current philosophical and filmmaking work. So I'd like to welcome the three panelists. And then I am going to just do a quick reading from Barry's book, Imagining Numbers. I don't know if this is the origin and the phrase, the longest conversation. I think that actually came from a friend of yours, didn't it? But in any case, Barry says, the great glory of mathematics is its durative nature, that it is one of humankind's longest conversations, that it never finishes by answering some questions and taking a bow. Rather, mathematics views its most cherished answers only as springboards to deeper questions. And I guess I would just start things off with the observation that if I can be simplistic and telegraphic, science grows through destruction and modification. Mathematics grows through accretion and reinterpretation. And so there's a difference between the history of mathematics and the history of science. Perhaps this is something we could explore. So we're going to have a conversation here. This is not the longest conversation. This is the shortest conversation. It goes until 2.15, and I am the time dragon. First, it's a great pleasure and an honor to be here as part of this group of distinguished scholars and artists who have been in long conversations for such a long time. And Barry might not know it, but as far as I'm concerned, I started my conversation with Barry before I ever met him. I was gifted a copy of his book, Imagining Numbers. And I really remember still kind of the vividness. I was surprised by the ability of Barry to bring together into a seamless conversation poetry, literature, history, and mathematics. So I just want to say, a quick word of what I find one of the most exceptional elements of Barry's writing about mathematics, which is, I think his ability not to fall into what has been, I think, the two most dominant approaches to writing about mathematics and its history. One of them I like to think, one approach I like to think frustrated isolationist. I think that people in that camp usually think about mathematics as something verified, treaties are kind of a topic that's just available to select few, and people write about mathematics in that camp, try to open it to the broader audience and to broader conversation, while all on maintaining that mathematics is somewhat unique and stands alone. On the other extreme, on the other side, I think there are those who can think about naive universalists that argue that mathematics is everywhere, right? This notion of mathematics, if we just open our eyes, if we just become, realize it, we will see mathematics everywhere, the world, the built environment around us. And I think what Barry does, which is remarkable, is actually being able to write in between this two extreme. He's able to write about mathematics and bring it into conversation in other intellectual pursuits, and he's able to do that without flattening mathematics into some sort of empty signifier, or some notion of mathematics existing there without humans being there to imagine it to the human. And the way he does it is exactly, the way he does it in this book, we're looking at ideas and topics such as imagination. In the beginning of the book, Barry has this moment that he asks the reader to come with him and think about the faculty of imagination. He kind of asks the reader to think about what it takes to imagine something. And I think once he does that, it's not surprised that he brings into conversation poetry and mathematics and literature. And it's really a testimony to his vision that he's able to do that. I just wanted to say that there's a historian of mathematics that had, somebody that has thought many, many hours, often many of them, would, Stephanie, I should say, about how to write a history of mathematics. And after I did surround mathematics, interested to a broad audience, Barry's ability to kind of write through cross-cutting themes and big questions will not forget the details of the mathematics. It's been inspirational until now and I'm sure it's gonna continue to be inspirational going forward. So I just want to say thank you for that. Well, I would also like to say it's just a profound and incredible honor and pleasure to be here. Barry has been one of the single most important interlocutors and mentors in the 10 years that I spent at Harvard University and his both encouragement and challenges and ideas and the way of approaching mathematical problem solving and question asking stays with me in ways that I won't be able to come close to articulating here today. So I also first encountered Barry through imagining numbers, but I got to meet the man alongside the book in a course that I was taking my very first year of graduate school that Peter was co-teaching with another Peter, a philosopher by the name of Peter Godfrey Smith. And the project of the seminar was to explore something called the distributed cognition hypothesis. And the distributed cognition hypothesis asked the question of whether or not it actually makes sense to think of cognition as being bounded by skin and skull or whether the ways that we think cognitive acts are in fact distributed sometimes among people, sometimes between minds and certain technologies or physical apparatuses. The course asked us to take seriously the idea that the ways we think our cognitive capabilities might actually be a function of social and material and technological contexts as opposed to just the uploading and downloading of things into the mind and out of it. And we read Barry's book in the context of that seminar and it was such a perfect fit with our set of questions because in it Barry proposes this notion of the collective imagination in mathematics where not just the ideas that are important or significant for mathematical work but the very conditions of possibility for certain forms of imagination and with it intuition in mathematical work are a function of what communities or histories do together and are in no simple way just the sum of the parts of a community but that intuition might be an emergent property both of communities or of histories and I thought that was such a profound idea and I am quite convinced that 20th century knowledge perhaps in particular demands of us that we think about cognition not as seated in the heads of individual people but as something that is made possible by practices, by methods, by the visions and the collective work of communities over time and in the book project I'm working on that is perhaps the single most primary question I'm interested in how new forms of intuition and imagination about mathematical work emerges a function of interaction with computing machinery and I was profoundly honored to have Barry sit on my dissertation committee and constantly be pushing me to ask questions around that theme and it would be impossible to overstate how much every meeting with him redirected and reshaped the conversations I was having with myself precisely because every meeting with him was a conversation about history, about poetry, about literature, about intuition, about big data, about machine learning, about statistics, about everything I've never known a person who is so able and willing and interested to talk even with young graduate students about anything under the sun that helps them to pull their thoughts together so it's really a pleasure to be here and be a part of this conversation and to be a part of conversation with Barry. Such a pleasure to see how Barry's influence has continued across the years to my favorite historians of mathematics. Barry had a huge impact on me as when I first came to Harvard as a undergraduate I had spent a year at a cochlear technique in France before I went to college and I studied, I was working in a physics lab but I was studying with a mathematician, a great mathematician, Lauren Schwartz and who taught a course on distributions and convolutions which I enjoyed but I made mistakes right from the beginning. The first mistake, well, first of all I didn't look like the other students I had hair down to my shoulders and a headband, they were in the military so the students were all properly attired in military form and so I raised my hand, asked questions and people explained to me, you don't ask questions if they go right away. So I said, okay, well then obviously the next thing I'll do is I'll go ask questions to him in his office and then someone explained to me, you don't go to a military school, it's not allowed. So he was actually very nice to me when I went I had a very nice conversation with him but I learned the lesson so I didn't think there was tabo-platique, the sections and so I went to the section, you know, raised my hand, I have a question and someone said to me, no, you don't ask questions in his office, I think. So I said, so where do you ask questions? And I said, you don't. So at the tabo-platique there was a guy, sort of a bureaucratic queen who sat there with his pencil like this and you know, he would say, garnison au tableau, résume la question suivante and he would write on the board and so you answered his questions but he didn't. So when I got to Harvard I was a little math traumatized but there was a course that swept people up called Math 55 and I took that. It was very fun, lots of interesting people who've gone on to do interesting things like Steve Gates and we had a great time and then I went to see what to do next and that's when I met Barry and he said, well, come to my course on complex analysis. So this was 43 years ago and I went to his course and I loved it. I mean, I think when we talk about the longest conversation of mathematics and narrative in science in a way Barry combines narration and storytelling with a conversation. It's not just performative where you ignore it. We were talking to these as both Stephanie and Alma have said, he really engages and colleagues profit from this every week at the Society of Fellows when we have something we wanna know about mathematics or a mathematician's work. Barry will momentarily look up as if to begin a story and then he tells a story about the mathematics that is completely riveted to everybody whether they're from English or history or art history or the other sciences, biology, physics. So that continues and I love the sort of, in these stories are the motivation and the appeal of mathematical reason a sense of why you would want to reason this way and they're not so crystalline formed that you feel shut out from them. You will address to you whether you're a student in a graduate course on complex analysis or a colleague asking about something that's going on a response that shows you why he cares about it or why we should care about it or how we can join him in caring about them. And I think that both Stephanie and Alma have mentioned this but the diversity of interests that I see in Barry the way he thinks about things is not just a collection of isolated bits they're actually for him very organically connected and the interest in narration and the narrative structure of mathematics the narrative quality of how he teaches the way he's interested in the narrative structure of literary or philosophical arguments it's really of a piece it's telling a motivated story that's propelled through ideas to the next thing and conveying that sense of an impelled movement is I think an enormous privilege for all of us who have been in a conversation with Barry over these years. So I have a question for you both. Watch a conversation. How do you, I mean in a way both of you have been interested in mathematics not only as a conversation among mathematicians but between and among mathematicians and others whether they're computer scientists or formal proofs or machine learning or collectivities of people and both of you are interested in how the collective accomplishment works. How do you see the history of math as a conversation with both in, you know, as a hub with spokes as well as its own intrinsic historical conversation in your own work, I mean, start there. I mean, I'm happy to say I think this is kind of one of the most interesting question that for a story in mathematics is exactly how to be able to write in student mathematics that sees mathematics in conversation without a field. I can say that for me part of my interest right now lately have been a lot about the kind of rise of axiomatic thinking. And what's interesting, if you, what I realize if you look at the history of mathematics not, and I think Mark actually already noted that in the Marks that you read by Mark. If you think about mathematics not just a collection of theorems but as thought processes, as practices. It allows you to tell a much richer history of mathematics. And if for example, in my work I look at the use of the kind of rise of axiomatic thinking. And if you just turn to that you realize that the same modes of reasoning the way of thinking mathematically specifically axiomatic thinking was then taken out in the 1950s. You can find both social scientists, psychologists calling for incorporating axiomatic thinking into psychology. Or you can find actually architects saying we need to take axiomatic thinking and be influenced by one of mathematics. So there's a way in which it's all about how do you think about mathematics itself and how do you approach the field itself. As not just a collection of theories that are timeless, but as modes of reasoning as a way of thinking that are very much embedded in the world. But to me it being a way of kind of getting at this question, exactly this question. Yeah, it's a great question. I think there's a sort of tragic way to frame most of the work that I am doing which is as being about the sort of breakdown or failure of communication between mathematicians and a different community of practitioners. So the people that I study are really keen on taking advantage of the profound power and speed and efficacy at working with formal systems that modern digital computers offer in order to try to introduce them to the work of mathematical research or to the act of mathematical theorem proving. And they can be quite good at it. Computer proofs are computers that made possible some very challenging and powerful mechanisms for theorem proving. And yet often mathematicians have not been very interested in this work. In 1983 the American Mathematical Society tried really hard to get working research mathematicians in conversation with people doing automated theorem proving work. And it didn't really go anywhere. And one of the reasons for that is that mathematicians are interested in so much more than whether or not a given statement is true. They care a great deal more about the kinds of insights that a mathematical proof can provide but only if you can read it. Only if you can understand it. Only if it is surveyable. And now proofs do a lot more than just certify the truth of something for mathematicians. And so a lot of the automated theorem proving work especially in the second half of the 20th century that's been done sort of failed to start an engaging conversation between mathematicians and computer scientists. And part of what I've been trying to explain and explore is what is it that mathematicians want from a proof? And the answer is all kinds of things. Proofs offer up new methods for thinking about objects or behaviors. They give new insights into the relationships between different fields. They offer new possibilities or analogies between kinds of problem solving or kinds of artifacts or objects in mathematics. And so in a certain sense, part of what this project has been about and one of the questions Barry has been so helpful in his approach to the history of mathematics has been so enlightening about is that mathematics as Mark said is not just about theorem proving and it's not just about collecting theorems. It's also about all the forms of intuition and imagination and practice that come along with exploring mathematical domains. So that's an anti-answer to your question. But I have found that looking at the ways that mathematics interfaces with other disciplines that might seek to make contributions to it, sometimes they don't actually have on offer what mathematicians are looking for which is an interesting way to think about the problem. Do you think we'll ever get there? Maybe, I'm a historian. Keep. Yeah, yeah. I'm a crystal ball. I wish I did, I wish I did. Interesting things are happening now but the story of the 20th century is almost the story of sort of misinterests as I think. Well both of you mentioned imagination and it's in the title of Barry's book. How do you think about imagination within the historical frame of things that you study on? I have a good story about Barry which is about imagination. So a few years ago I became interested in the way in which, it's a question about imagination in mathematics and I became interested in the way in which mathematician imagined higher dimensional spaces. It was something that was fascinating to me. And I ended up writing a paper about the way some mathematicians ended up using computer graphics in order to imagine higher dimensional space. And I had Barry read the paper and after Barry read the paper he got back to me with comments and one of his comments he said, I have an idea for you, I have a suggestion and I think what you need to do is you need to try and imagine one of something in a higher dimensional space that doesn't come easily to you and then you need to describe it. Now what Barry didn't realize is that the reason I became so interested in that is because I utterly failed in trying to imagine higher dimensional space. I had no ability to do that, right? But I think this is kind of, but I still took Barry's idea, the idea that I need to try and imagine a higher dimensional space and then what he asked is for me to be able to write and describe another texture of what went in my mind. What's the kind of texture of this imagination? And I failed in this case when it comes to trying to imagine higher dimensional space. But I think that as when I write about mathematics and when I try to describe some sort of mathematical problems in my own writing, I do take this advice about imagining for myself and trying to imagine as a way to start a process of writing itself. And so it's both, what I like about it is both the historical question, how in different times mathematician went about imagining and here comes, I look at computer graphics. So how did it change? How, what's the possibilities of imagining geometry changed with the introduction of new technology? But also as a way of thinking about writing about the history of mathematics. So both it's an historical question, but it's also, I took, I took, that's what I took Barry to say. I hope that was part of what you were saying at the time. As something that I should think about for writing about the, kind of for myself writing about history of mathematics. But I think it's, you have to think about this question of imagining, not just, I think geometry allows it more easily to this question of imagining, but if you take Barry's work, just imagining the square root of, then I don't mind if it didn't require sort of imagining itself. So it's always there in the sense of it. What do you think, Stephanie? Do you think that as imagination changed, it's how we understand it or how it's understood in the community of mathematicians, applied mathematicians over the period you've looked at in the last 100, 120 years? Yeah, certainly. I mean, there are sort of two camps of people working with computers on this question of intuition or imagination or the creativity that is required to do especially higher mathematical problem solving or theorem proving. There's the one camp who says, oh, we can automate that. Everything mathematicians do, all their short hands, all their exploratory techniques, we can represent them as heuristics that can in turn be automated. There's a camp who truly believes that what might feel esoteric or irreducible or unformalizable even to mathematical practitioners could with the right knowledge engineer be elicited, formalized and automated, if not now, then in the near future. And there's another camp of people who hold imagination, creativity, intuition, certain human mathematical faculties, they hold them apart and say, well, whatever computers are gonna offer to mathematics, it can't be those things. They are unautomatable. They are uniquely human in some way. And what I've been really interested in is what happens to the idea of what imagination or intuition, they're not the same thing that they often get grouped together by these people. What happens to them when they're actually making these computer programs? And what struck me is really interesting is that even for those who wanted to cordon off imagination or imaginative acts or intuitions as something uniquely human, they still get translated in a certain way into a language that the computer can understand. So for example, there's one group I've been very interested in studying who wanted to build collaborative fear-improving programs that would take human intuitions as inputs to guide them on their nearly deductive and nearly mechanical search for proof within some formal space. But in order to make human intuitions useful to a machine, they turn them into a waiting mechanism. So at different runs during the computer program, you input a waiting template that says something like, short clauses are preferable to longer ones or multiplication is more important than division or this operation should be executed before this operation. So they take this beautiful, sort of folk-psychological Jacques Audemars inspired notion of human intuition and they turn it into a waiting mechanism that gets input by a punch card to a computer program. So the history of intuition is tied up with surprising practices or constraints or ideas sometimes. And I have no idea how to write the history of subjective experience. Peter has a book on objectivity. We are all awaiting the book on subjectivity. Alma and I have been talking about this for 10 years. What it would look like actually to try to write a history of human intuition and Barry's book is probably the closest thing we have to it. And so most of what I have to say on the subject is about the way that the people I study try to define and then work with different formulations of intuitive or imaginative ideas in mathematics which ends up looking quite surprising sometimes. You know, it's interesting imagination has had such a ambivalent relation to the arts, the sciences, mathematics. I mean it's, philosophically, for a long part of its history, imagination was considered to be a bad thing. And either bad because it was a mere recombination of things as opposed to a deeper working out of structures that underlay them. Certainly in literary and literary philosophical circles in the 19th century there was a fairly common view. But it was also something that psychologists thought of as being a valuable and measurable property. I mean before Rorschach when people looked at ink plot tests you had specific, there were specific tests that you had for each of the faculties. There was a test for how quickly you could calculate and how many numbers you could remember. Could you remember? Remember one, two, seven, four, three, can you imagine? One, two, seven, four, three, five, seven, six. And so on and then it put each of the, in the faculty conception of mind each of these had a specific test. The test for the specific faculty of imagination was how many images, how many things you could see in an ink plot. So you would hand somebody an ink plot and there would be a bell and you would say, cow, light, bottle, table. And then you would have to show where you saw those and the more, the merrier. So when Rorschach began his work in the early 20th century he found that people would say, oh, you know, this is a test of the imagination. And for Rorschach it wasn't at all, it didn't care at all about the imagination. He didn't believe in the faculty concept of mind to begin with. He said this was a test of perception. What are the characteristic modes by which we come to the world? Do we emphasize blank spaces? Do we look at shades of color? Do we emphasize color above form or form above color? And he used that to try to give a kind of set of coefficients, so to speak, of our perceptual tuning, what we brought in and what we left out in our encounter with the world. And, but he said, because everyone believes that it's a test of the imagination, it's a useful fiction to let people come in. So people were taught to not contradict the patient or the examinee when they came in and said, is this a test of the imagination? You're supposed to say, mm-hmm. But it had nothing to do with that. And so it's been, he devalued that, I mean, maybe. So I think that it's interesting to think to what extent imagination is tied to images, to what extent is it a poor cousin, to real poetical or literary or philosophical creativity, to what extent is it the triumphant characteristic of the faculty of being able to see to what extent does it have nothing to do like the square root of minus 15 with sight or imagined or recalled or recombined sight at all. And in that sense, I think it's a dynamical category that like intuition, there's nothing intuitive about intuition, right? It's not, there's nothing simple about simplicity and there's not the imagination as it's shifting set of allegiances, friends and enemies. I think there's also a gender story to imagination. I mean, if you look, wait, if you think about Eda Lovelace, who's a good example, people like to think about her, right? She was, at her time, she was told that she would, she would never be successful because she was gonna let her imagination run free and she was too imaginative. But that was kind of what she was told, this is why she had to, she had to study mathematics as a way of, the idea was that she was studying mathematics as a way of kind of teeming down her imagination to correct. So there's also kind of very much a gender component to how we think about this idea of imagination and logic, which is part of this history when we think about it as, like you said, as a shifting, historically shifting category. And maybe the same could be said for all of these hierarchies of human faculties we would like to build up. We have a colleague, Lorraine Dasten, who has this wonderful article about the act of calculating large numbers in the enlightenment and there was once a time when the ability to do large calculations or to work in your mind with big numbers was associated with virtuosic mathematical genius. But as soon as there were machines that came or were invented that could take on the act of working with large numbers, that kind of mental work became merely mechanical and mathematical genius and virtuosic mathematical thinking became associated with other capacities quite independent of sort of arithmetic or working with large numbers. But it was around that exact same time that human calculators, primarily women, started taking on low paid, low skilled labor as calculators, as calculating machines or human computers. So there's always a story being told about whose capacities and whose faculties are being valued, whose are being devalued and how the work of mathematical labor gets distributed across different communities of people and people in technologies. The same is true actually in automated theorem proving the anthropomorphic language that the people I study use to talk about the kind of work they think a computer will be able to do is incredibly telling. So some people, those same people who think all of your faculties are at least in principle automatable, talk about computers as being colleagues or mentors or coworkers. On the other end of the spectrum though are those who talk about the work of computers as being assistants or servants or slaves. And by using anthropomorphic language of this kind they reinforce ideas about what kind of mathematical labor is valuable, what kind of mathematical labor is not valuable and they redraw these lines between what is uniquely human and impressive and what is merely mechanical and therefore a woman could do it or a machine could do it or, you know. So it's a fraught history, this imagination or creativity history maybe. So I remember in the chess playing community a few years ago there was this term chimera and the idea was that the chess player would become sort of amalgamated in some sense with a chess engine. Could this, I mean do you see much of this in the world of mathematics over the last few years in terms of automated field proof? I'm not sure, I need to know a bit more about what they have in mind I think. I think the idea is that the machine and the human are sort of greater than the sum of the parts. There's a sort of synergy here. Yeah, I think some people would want to say that that there will be modes of reasoning or forms of problem solving that are emergent properties of different relationships between the people and the technology but I still don't quite know what it looks like in practice or what they think it would look like in practice. Yes, definitely. That they will be more than mere extensions of our faculties but rather create new forms of intuiting or thinking is something that lots of people are at least in principle excited about. What, chess players? What human machine? Human machines. Small animations. Right, I mean the history of, I think we talked about it a lot in the past but the history of human machine when it comes for example, the chess is a good example, right? So chess used to be, when you think about chess playing, it was considered to be that's when machines would become intelligent, when machines would be able to win chess game and of course once that happened, kind of the bar was raised, that no longer was considered to be the ultimate test case so then Game Go became the new kind of, you know, the way we are gonna measure and then when we got there, so there's a way in which this is a constantly, again, it's a constantly moving construct of how we're gonna think about what the concepts of intelligence and this sort of kind of what faculties and machines and humans can develop together. And of course with the game, with the recent one with Game Go, part of their argument was that it wasn't pretty intelligent because it didn't mimic what how a human would go about reasoning, how a human would go about reasoning about the game of Go, that was kind of one of the main things that people were left with the game. So I think there's a way which as the sort of human machine boundaries are gonna continually being drawn and redrawn, it's a question about the faculties of reasoning, faculties of nutrition, I think that's what Stephanie's book shows that for the earliest period exactly now. But this is, if what you show for the very first, most likely gonna continue to play out and continues to play out today as well. Yeah, it reminds me of a debate that happened in physics where the between CERN and Berkeley in the 60s and the CERN folks, there were all these millions of bubble chamber pictures coming out of the apparatus, literally millions, tens of millions sometimes. And so people would sit there at scanning tables and code these images into a form that the computer could process. And the Europeans said, this is terrible, this is a kind of Dickensian workshop where all these people are consigned to doing this routine work. How can we avoid that? We have to solve the pattern recognition problem in general and liberate all these people from their light tables and projectors and computers. And Albert said, no, no, no, here's the problem. People are good at some things and they're not good at some things. They're good at sort of seeing an emergent pattern or something unusual that doesn't fit a pattern and they're terrible at routine tasks and calculation and not making arithmetical errors and transcribing. So get the machines to do the things that are good for machines to do and then leave the scanners to do the things that they're especially good at. And so they started to train the scanners to be in a way like a doctor who studies routine pictures of skulls until you see one that you say that's not normal, that's pathological, that's a tumor. And only in the case of bubble chamber and other images it was, that's not normal, that's a discovery. So the valuation was very different from being catastrophic to being good, but it was somehow distinguishing between the routine and the exceptional in images. And, but as of course these techniques have changed in different areas, classifying stars with something computers couldn't do and by their spectra and now computers can do that pretty well. So pattern recognition is itself constantly changing and so the separation line between the things that computers do and the place where we humans are useful is a moving point of separation. So the chimera in a sense is a amalgamation, a side work of changing partition. I don't know how chess players think about this. Well, I think they do, you know. I think, now I think it may have evolved beyond that to the point where, you know, I think many chess players now would say that their training as a chess player is absolutely dependent upon their computer partner. Will that ever become the case with mathematicians? I have no idea. Would the chess players don't feel like they're done for? No, I don't think so. I mean, there is a question. I mean, I think actually, I talked about it with Barry, when computers were introduced originally, most mathematicians were very much against it. They were using computers in mathematics and I think today, I think that most mathematicians end up using computers in the work, very much about it, so I know I'm right. But I think there's also a certain kind of what's the community standards are changing as well. And part of it is exactly figuring out to what degree of computer, to what degree and to what end computers are being used. It's not to a place, but it's, you know, we talked about the use of experimental mathematics or the way of kind of bringing out conjectures, it's a topological route. So it means a self-heavism truth and then the question of intuition and imagination comes very easily, but with Hilbert, it's exactly the point that Axiom is arbitrary, right? If you follow Hilbert's model for geometry, by definition, Axiom has to be arbitrary. It's no longer something that's self-evident truth. And I think though, you can kind of tell the story and the story is partly being told as the way in which intuition moves out of mathematics, that with this rise of formal systems, the intuition kind of falls out of fashion. I think this is a, I don't want to say it wrong, but it's a one way of telling a story or a simplistic way of telling a story because even if you take Robert Key, which we're considered to be kind of the best example of taking the modern axiomatic in the kind of formal work of Hilbert, they still talk about intuition because you, as they talk about the kind of mathematician, you have intuition to the kind of objects that you kind of build intuition during your studies for the object that you are studying as yourself. So what you mean by the word intuition ends up changing in this new, it's no longer a man, this kind of self-evident truth is what's intuition. But I think that I even, you talked about it as a mathematician, if you're working on a certain problem, part of working on a problem is building a certain kind of intuition for the material that you're studying. So part of it is exactly trying to be careful of what we mean when we talk about the word intuition, how we use it, and to say that when we use it, when we talk about axiomatic in kind of a Klingian way, when we talk about axiomatic in a Hilbert way, which we actually maybe mean two different things in part, and the role imagination itself kind of moves. I mean, the work, if you think about the work of Merthens that kind of wrote about the changing mathematics at the end of the 19th and early 20th century, for him the big divide was the largest logic and intuition, that used to be the big divide, I think, the way to tell kind of the break in the history of mathematics. I think that a more interesting break that happens in the 50s, if you look at the way mathematician talked about two kinds of mathematical lines. It's about the theory builder and the problem solver. So you take Eredoche as being the problem solver, and then you have people that are theory builders. It's a different way of thinking about intuition, of really thinking about kind of faculties of mind that's coming into mathematical, compartmental mathematical, practicing mathematical creation. Do you think that that's coincident with sort of algebraic and geometrical, or do you think that that's a separate? I think that might be just another dichotomy to add to this list that Alma is constructing of moments in history of mathematics where mathematical communities are defining themselves in opposition to one another. And in the 19th century, there's an opposition between those who believe that the correct set of tools and mode of thinking with which to do geometry is diagrammatic, or it's tied to physical models. It's about that kind of knowledge and synthetic intuition, and those who believe that the new algebra is the right way to solve geometric problems. I think I wouldn't want to reduce that dichotomy to the others, but we could add it to the list. We could add it to the list. What strikes me in some of those early 20th century discussions is that both the anti-imagistic and the imagistic schools, they both recognize the extraordinary power that these images afford. But the anti-imagistic people sometimes think that it's too powerful, that it leads us astray, like the iconophilic dangers of the icon in theological thought. That it's supposed to be a step to something else, a deeper understanding, perhaps, or a deeper knowledge of God in the theological domain. The danger is that it ends up being an object in its own right, and that Joseph Kernar, who I see back there, has taught me long ago, is that iconoclasts are always iconoclastic about something. They're not iconoclastic in general. And so, his witch images, they want to break. And I think in science and mathematics, that's true, too. It's like, what are you worried about? And like in Poincare, it makes a distinction between people who think in an algebraic formal way versus those in a more geometrical, intuitive way. He's not just, he has something specific in mind. And he thinks that his worry is that people give up too easily on the power that the images hold. My feeling about Poincare is he often thinks it's like a carpenter saying, hey, I could build a house without a hammer. And Poincare is, why would you do that? I mean, that's just crazy talk. I mean, of course you could use a rock to knock your nails in, but a hammer is specially made for just that job. Why would you give it up? And when he talks about geometrical intuition or about the ether in physics, it's not because he's not, he isn't. And he doesn't think that there's an ether blowing in the, by 1904 or 2005, he's quite clear on that, but he thinks it leads to all sorts of things. Maxwell discovered the displacement current in electric waves because he was reasoning about something he thought was the ether. And then he goes on and he has another formulation where he doesn't use that at all, but it's a helpful way of moving. The characters say it's too helpful. It will blind you to relations and structures if you're too fixed on it. Like, you think you know parallel lines are defined by railroad tracks? Well, you're gonna miss out on a lot of the mathematical adventure that awaits. I mean, Gilbert is great for that, because on the one hand, he's celebrated as exactly this kind of formula. On the other hand, he has geometry and the imagination, which is full of images. And it is, it's not, absolutely acknowledged that, like you said, that there is a place for images. But for him, for Hilbert, there's no place in the foundation of mathematics when you go to put mathematics on a certain base, then there's no place for images. And then, at that place, it's logic enough to form a system and not images. So, you know, the flip side of rigor, in some ways, is well-definedness. They're not exactly the same thing. And a lot of times, the discussions, at least at the boundary with physics, when mathematicians are really annoyed at physicists, it's not even mainly, I think, about rigor. It's that they say, although, of course, we all say, I don't know what you're talking about. We say that all the time. But I think, for mathematicians, it's often a case that they really look at the physicists and say, what are you talking about? And what role does, you know, historically do you see in, as you look in your different but related domains in the history of mathematics, what role does well-definedness play as a historical concept? I mean, the advent of automation has sort of reanimated a quite old debate between those who believe that a proof is something that convinces mathematicians that something is the case and those that believe that a proof is a fully formalized and complete step-by-step demonstration that something is the case within a sort of formal system. And computers are actually quite good at formalization. They're good at well-definedness. They're good at following the rules in a sequential order. They can only do sort of complete and fully formalized mathematical work, which it turns out human mathematicians are not very well equipped, nor particularly interested in doing. But then there are those, as I've already discussed a little bit, we think that proofs are about more than a fully formalized and complete enumeration of the deductive steps that establish that something is the case in mathematics. And that's a story that's been written about quite a lot. But what actually interested me more when I looked at this history was what actually happened to what formalization meant. It turned out that whole new ways of formalizing were also being developed by the people who wanted to introduce computers to the work of theorem proving. And some of the new ways of formalizing, the new formal systems that were designed to help computers work in mathematics were quite different and precisely this way in that they removed human psychology from the equation. So there was famously a classicist by the name of John Allen Robinson, who working in 1965 said, all of our logical formalisms until now have been oriented around trying to have a single step of deduction or inference be clear to a human mind that how you get from one step to the next step is something that we can understand and follow. We have adaptive formalism to human psychology, but maybe now that we have these extremely powerful logical machines on the table, we might develop whole new logics built with different formal tools that are not so accommodated to what is clear to us in a single deductive step. So in that moment in the middle of the 1960s, there emerged this field of study called computer-oriented logics, which were about building perfectly logically sound and very powerful formal systems but that displaced human understanding from the equation of what formalization looked like. And so this was a moment again where two different ideas about what ought to count as a proof for what good mathematical demonstrations looked like are on the table like they have been for so many other moments in history, but at the heart of this conversation was a question of what does it even mean to work with a formal system? What should a formal system look like and need it be bound to us in some way? And sometimes the answer was no, but often the demonstrations produced by those computer systems, they were fully formalized, they were sometimes very powerful, but they produced these demonstrations that aren't just unintelligible because they're too long, but because they are built through the execution of formal steps that are not so easy for us to identify as a reasonable, logical step. So what formalization means and how it is done and how it relates to our understanding is also one of the things that's on the table in the conversation about automation. This is sort of the black box problem of AI, isn't it? It is exactly that, yes. Yeah. The AI does not know how to explain what it just did. That's right, nor is it usually asked to. Right, right. I think the union wants to ask. The union wants to ask, so do I. Yeah. No, I mean, I think it's a question of definition of the world of finance, exactly this. Where you see it is exactly in this boundary work, it's when the definition of this is come to that, I need to agree. So there's a way in which I was thinking, can we think about a case of world of finance that stays within them? So when you don't talk among across communities, I think this question of standards, maybe it's not quite this question of like, well, the question of standards are still within specialties, within kind of communities of specialist within thematics. I mean, I've done a little bit of work on the classification of financial groups and within the community it was very much agreed what counts as standard, but it doesn't mean that people outside core group of mathematician has already agreed. So it's exactly in these places that the kind of boundaries between different communities that you see that, but I don't have other good example within the kind of mathematics when you see that. I don't know physics and mathematics. I'd like to go back to Mark's statement here. He says, the history of math is not just the story of what was proved and how, but also includes the question what was believed and why. And it seems to me that we've been talking a great deal about intuition. If I'm working on a problem, I develop a great deal of intuition about that problem. I begin to somehow believe that this is true, but that motivates me to try to find a proof, perhaps without that belief which follows intuition that ever happens. Computers don't believe things. They just do stuff. It's a powerful thing. Barry shared an anecdote with me about communities of mathematicians who believe something is true in the face of all empirical evidence to the contrary. And that's a really powerful thing that speaks to the degree to which even when mathematicians make use of what looked like empirical practices, like looking, using computer generated data about what happens way out there in the number line or experimenting with cases on the page, or there are lots of empirical practices that go into mathematical problem solving or the cultivation of mathematical knowledge, but it is not reducible to that, it seems. If you can believe something contrary to all of the evidence, that has to be a big part of how paths are drawn through mathematical knowledge production, I guess, making. I mean, there's also the point of you have to believe something in order to go about trying to prove it, to begin with. Like, I mean, it's kind of the first act, it seems, this place of, like you said, right? It's the place of, in order to begin it, and choose a topic and go after it, you have to have some sort of belief as possible. As a human being. As a human being. Yes, I do. Unless you're advised, I've told you to do that. If you're advised, you're human. I mean, with AI, it's interesting because recently I've been looking at the way that image making takes place when you have sparse data, you know, from astronomy or from other domains. And you have to assume something to begin to form the image if the image is sparse. It's like, what kind of line are you gonna draw? You have a bunch of points. You wanna draw. You say, what's the best line for it? Well, you tell me it's straight, I can find which straight linear feature goes through it. Or if it's a circle, I can find the best circle, but you can't just say, what's the best without any kind of prior. And so that's been something that's been much debated. But then it worries the people in the community of astronomers because it feels, as they say, subjective. And what are you gonna put in? Do you think it's a prior assumption that it's maximum entropy, minimum information? Do you think it's gonna be another set of functions that something circular symmetric, or do you have to put in something? And so they say, well, let's see if we can get AI to look at these putative galaxies or classify galaxies or do something else. And so they say, this is right. This is gonna get rid of our subjectivities. Then the computer does it. And then they say, but how did the, what's the computer done? Well, it won't tell you. So I mean, if you're Netflix or Google or, it's fine, Amazon doesn't care whether they can interrogate their machine and find out why if you like this, you should buy that as long as you buy it. And so that may not be satisfactory to the scientists confronted. So I looked up the papers by the machine learning community and they say these amazing things. They say, at this point, the machine began to hallucinate. It's really, it's enhancing some zebrafish and it's put stripes where there are no stripes on the zebrafish. So, or another group said, well, we don't know why it did this. The machine seems to be learning to do some wrong thing. And so what's fascinating to me is that in this huge community, billions of dollars have been spent on machine vision for all sorts of reasons, no. And is that it's taken, moving to AI, at least in some of these scientific domains, was out of a fear of subjectivity. And then what they've done is they've gotten a machine that they ascribe subjectivity to. And the machine won't answer any questions. So it's like worse than having it, one of your colleagues would say, well, I think the prior should be maximum entropy because I really think we should assume the minimum about the world around us. And you can have an argument with your colleague. The computer's not gonna say anything. So, but I thought that was very interesting and it seems to occur over and over again because machine vision is in some driving cars and classification and face recognition, all sorts of things. Well, I believe that there's legislation before the European Union that wants to establish a criterion for explanatory behavior, is that right? Yeah, the question is the question of interpretability is the technical term. So most models in machine learning, they operate as a function of some crazy huge number of parameters, but it's impossible to unpack how any given parameter contributes to the behavior of the overall system or model. So interpretable, you can force a model to be interpretable so that we can look at the relationships between parameters and the overall behavior of the model that comes out, but you forfeit a little bit of predictive accuracy when you do that. And so the question is whether we should forfeit predictive accuracy and have interpretable models and the answer tends to be depends on what the problem domain is. So if it's criminal risk assessment scores being used in courts to predict whether or not a given defendant will re-offend or commit a violent crime or show up for their bail here or for their trial, the answer might be yes, because they have a right to know their accuser. And if this number is being generated by a non-interpretable machine learning system, perhaps that right has been violated in some way. The other answer, if you're trying to build the safest bridge might be we don't care about interpretability, we care about predictive accuracy and these machine learning models happen to be the best models we have ever built in terms of predictive accuracy. And so it raises a host of really important ethical questions, but also epistemological questions about whether or not or what kind of knowledge or explanation we want in different domains. There was a, Barry and I a couple of years ago read a series of articles about this exact question, including sort of one sided debate between Peter Norvig who is the head of research at Google and Noam Chomsky who needs no introduction. And Noam Chomsky apparently had said something at the 150th anniversary celebration of MIT to the tune of, I don't actually care if you can predict with perfect accuracy when and where the bees will migrate. I wanna know why bees migrate. So wanting a different kind of explanation of phenomenon, on the other hand, we don't know how to give mechanical explanations in a lot of problem domains and having the ability to predict well serves us in a number of cases. And so they had a sort of conversation about what kind of knowledge this statistical prediction was that there were more than one kind of it and that it has different stakes in different parts of society is obviously true, I guess. Are there cases in mathematics where AI enters an, I mean, if you have a computer doing tasks where it can be interrogated then there's a, you can understand historically and we know a lot about the history of that as to why people might not like it but it seems like something you could get used to but when it gets to be hidden under back propagating neural networks it might get a little more problematic. I mean, in the domain that you spoke about the additional features that some of these predictive features become a proxy for race and that's really, you know, like where do you live? And so one of the things that people have really objected to in automatic sentencing guidelines is if it's a proxy for race it's like the worst kind of bigoted judge without recourse. But in mathematics are there uses of AI and neural network? Deep learning? I don't think I'm still lying. There are a lot of other positions here. You know, this might be a good time to shift over to the question session. I would just like to read a short passage from various book imagining numbers before we do that. He says, even when we have, even when we have palpable historical evidence, it may be hard to weigh the importance of that evidence correctly for the most minute shifts of thought or change in notation. The appearance or disappearance of what would seem to be a harmless metaphor can I signal the evolutionary beginning of the new species of idea? At least from the history of science, I would say pedagogy is the important thing. That's part of how you learn to imagine is because you're a student in a certain kind of school and you learn from people around you. The other thing is that there's one historian of mathematics down in Kansas that said, and I think he's probably right about it, that mathematicians are, in fact, the group that comes from sciences that do most of the work in face-to-face interactions and most of the other sciences, that you actually only learn something in mathematics if you talk to people. That you almost never learn or what you can learn from just picking a published paper without ever talking to people in that field is minimal. The degree to which mathematicians, in fact, conversation is done in conversation is a huge part of this communal imagine. Doesn't that make the inner circle, outer circle problem very serious? Yes. Maybe technology mitigates that to some degree. I don't think so. I think it actually reifies the problem. So it was the development of this sort of early expert system called maxima that was supposed to do for mathematicians or take over for mathematicians all of their sort of drudgery of manipulating symbols on the page, do it for them so that they could formulate more conjectures, have more creativity, work in problem domains in which they were not expertly trained. It was just such a textbook story from a history of science perspective where nobody knew how to use this system. More than that, nobody knew at what point in mathematical problem solving this system might be valuable. And at the end of the day, they had to get everybody together at users conferences to teach people how to use it face to face. It's kind of technical knowledge is not outside of the story that Alma just told. It's actually very much a part of it in which access to technologies, knowledge of how to use them, knowledge of when they are useful, and then the ability to incorporate them into your problem solving practice requires you have access again to training to the technologies themselves and to the community that makes them meaningful. Yeah, that's a great question. I'm actually writing a book chapter about Maxima as we speak, so I'm serious about talking to you after it's over. I'm not saying this room for users. I've been. Excellent. Right, so the reason that we have not relegated all of these mathematical practices to the merely mechanical because we have built computer systems that can do them is because for the mathematical community, they still place value on the kind of knowledge and capability that comes along with learning how to do them. That could change. I don't know when or why, but the designation that something is merely mechanical doesn't just come along with having a machine that does it. It comes along with a whole set of valuational practices about what kind of knowledge and practice and training one ought to have in order to be a mathematician. We don't really learn to sort of do long division in the same way that we once did. So that's a designation of merely mechanical that's been more sort of comprehensive in its vision. But we're still in the very early history of what's happening with computers and mathematics, I think. So that may yet come or not, depending on how those valuations continue or change. In another field, you know there's a very striking example for medical students anatomy was the central course was the organizing core of knowledge. And now almost every medical school in the United States at least doesn't teach anatomy anymore. If you do it, you do parts of it on the computer in simulated form. And something that was thought to be central is now barely peripheral. One of the best examples of that is the new math, right? There was in a sense as an attempt, and there's a book written by Christopher Phillips about the new math in the United States, but an attempt to teach school mathematics, which will be somehow closer to track how, you know, what mathematics actually is, and it failed terribly, failed terribly so. But I think that the place of teaching mathematics in school is something that's been sourced for debate and will continue to be a source of debate. There's always going to be an argument of what's the right way to teach mathematics for school majors. Part of the point that Chris Phillips makes in that book is that mathematics, as you say, mathematics education into primary, secondary, and high school isn't necessarily meant to train you to be a mathematician. It's supposed to train you how to think properly. And so it's a very political debate how mathematics should or would be taught, especially during the Cold War, because it was tied up with this, not with being a good mathematician, but with being a good American, thinks and reasons properly, right? So you're a mathematical linguist. But it was in the Cold War that we had to do math, and it's now after the Cold War that we have a much more formalistic approach. So I don't think the connections to the Cold War actually work. Well, that's how it looked in that moment, but that this is a political conversation in other contexts in different ways, of course, yeah. There is this question of each person, it's probably, some people learn better in place-to-face interaction. I think that what the literature has shown, however, is that you have to have some place-to-face interaction. There is a certain kind of learning, and it's not just learning, it's learning what it means to be a mathematician, how to go about reasoning about mathematics, how to explain it, a new mathematical discovery. This is part of what, and you're probably gonna do it in the next few years in graduate school. But I think that what you see in the literature is that almost, the example of the example of people, that it's only when they become part of a certain kind of community, they're for expertise, is that when they really become, when they come to place-to-face interaction in part of the community, is that they really kind of become an expert himself, part of it. I don't know that there is, like I said, a certain kind of formula. You're a mathematician, so you want a formula, but I don't know that there is a certain kind of formula to exactly how to divide your time. I mean, people in the audience are probably answering that one. Join me in thanking me.