University Place

[gentle music] - David Ebert: Well, good afternoon, everyone.

Welcome to the beautiful Green Lake Conference Center for the Wisconsin Mathematics Council annual conference.

Today, we are partnering with Wisconsin PBS and Badger Talks to share Jordan Ellenberg's message with teachers across the state.

My name is Dave Ebert, and I'm excited to introduce today's keynote speaker.

Jordan Ellenberg is a number theorist and professor of mathematics at the University of Wisconsin-Madison.

He's the author of numerous books, including the bestseller How Not to Be Wrong: The Power of Mathematical Thinking, and the upcoming Don't Be Too Sure.

His popular articles about mathematics have appeared in The New York Times, The Wall Street Journal, The Washington Post, Wired, and Slate.

In April of 2026, he was elected to the American Academy of Arts and Sciences.

Please join me in giving a warm Wisconsin welcome to Jordan Ellenberg.

[audience applauds] - Jordan Ellenberg: Okay, hi, everybody.

Thank you guys so much for coming and so much for teaching.

I wanna talk about uncertainty today.

It might not be obvious at first, like, that this talk was gonna be about math.

But bear with me.

We're gonna get there.

But let me start.

There is actually a mathematician on this slide, W.K.

Clifford.

So, let me start with what I think is the basic dispute that I wanna talk about: the dispute between certainty and uncertainty, between Clifford, who is both a philosopher and a mathematician, writing, "It is wrong, always, everywhere, and for everyone to believe anything upon insufficient evidence."

So, really this brief for, like, knowing exactly what is true and not settling for any less.

Versus the philosopher and psychologist William James, writing in response to Clifford, saying, "Not a victory is gained, "not a deed of faithfulness or courage is done except upon a maybe."

So, he's kind of one of the great apostles of maybe-ism, if you like.

And I want to sort of try to say something about maybe how we can bridge the gap between these two great thinkers of the 19th century and do a little math as well.

So, we'll start here.

This is, of course, a platypus, "an amusing example of the anomalies by which Nature sports with our schemes of classification."

This is another quote from a famous person, but probably not one of his most famous quotes, unless somebody knows it.

This is, of course, Thomas Jefferson, who, writing in a letter to the Philadelphia doctor John Manners in 1812.

So, let me just emphasize it used to be that if you were confused about whether the platypus was a mammal or a bird, you could just write a former president of the United States and he would, like, write you back with his opinions.

I mean, really, when you read these documents, you really learn that our country was founded by weird nerds, which I think is-- like Thomas Jefferson, which I think is very interesting.

But this was a very hot dispute in the biology of the time, right?

Because the platypus lays eggs, and it has a bill and a cloaca as if it were a bird, but it also has warm blood and is kind of furry and makes milk for its young like a mammal.

So, this was something that people argued about a lot, not just former presidents.

In fact, the British naturalist George Shaw was sure it was actually a hoax.

He was like, "Somebody has to-- "This thing that was mailed to us from Australia, this carcass, "it has to have been, like, stitched together from different animals."

But he couldn't find the stitches.

And when we think... You know, what kind of thing is this?

Here, we're gonna come back to this quote a little later.

But just to sort of show you that this was a very long-standing argument.

Here's Shakespeare, already writing... Falstaff saying of the maid in this hotel, "She's neither fish nor flesh."

This might be an expression you know, right?

Sort of like, "What kind of beast is she?"

In this case, it's not a mammal or a bird.

It's like, a land animal or a sea animal.

So, this kind of question was already in people's mind.

And if you know one name that's associated with the classification of animals, it's certainly this guy, Carl Linnaeus.

This is his first big book, The Sexual System of Plants, where he classified all plants exactly by the sort of organization of their sexual organs.

He, also-- I couldn't fit them all in this slide, but there's I think 24 or 25 different classifications given by letters, which gave, like, you know, rather, like, racy names like seven husbands and one wife, like, depending on how the pistils and stamens were organized.

But he was, in some sense, the great promoter of the idea of strict classification.

And, in fact, I'm gonna read you this quote.

I say, "The first step in wisdom--" And that's a quote from Linnaeus.

He writes in this book, "The first step in wisdom is to know the things themselves.

"This notion consists in having a true idea of the objects.

"Objects are distinguished and known "by classifying them methodically "and giving them appropriate names.

"Therefore, classification and name-giving will be the foundation of our science."

So, again, we know from biology, right?

These Latin two-part names that we use for species even today, this was Linnaeus's idea, that we're gonna sort of give everything its exact proper name that cleaves it apart from all other species.

And for him, this was not just sort of a part of what science should do.

This is, like, really what science is.

This is what it is to know something.

So, this is the exact idea to which the platypus does a certain kind of violence, right?

We're sort of not happy if we're Linnaeus and we encounter this animal.

That being said, I mean, we like plants, we like animals.

But what we as people are most interested in is people.

And so, we're just as interested-- and actually always have been since the time of the four humors-- as interested in classifying people-- more interested than we are in classifying animals and plants.

I would say, in fact, I would say as long as there have been people, there have been kinds of people.

I like to always have, like, a little Wisconsin content in my talks, especially when I'm talking here in Wisconsin.

So, here's another important source document.

And I'm curious if anybody knows who made this.

Where we have this box with neutrality in the center, chaos and law on the horizontal axis, and good and evil-- Okay, you know in the middle.

– Attendee 1: Gary Gygax.

- This is indeed a Gary Gygax-- Wait, does everybody in the room know who E. Gary Gygax is?

Oh, my God, okay.

E. Gary Gygax is, of course, the creator of Dungeons and Dragons, one of the most important cultural products ever to come out of the great state of Wisconsin.

He lived in Lake Geneva.

And what you're looking at here is an article of his from 1973, which was the introduction-- the first-ever instance of Gary Gygax's famous classification of all D and D characters into, like, four moral types: chaotic good, chaotic evil, lawful good, and lawful evil.

So, this idea of types of people-- And actually, I'm gonna quote from his article.

He writes in 1973, "Few humans are chaotic, and very few are chaotic and evil."

Those were the days, right?

Okay.

[audience laughs] So, this kind of classification, especially what you might call a dimensional classification, where there's the dimension of chaos versus law and the dimension of good versus evil is incredibly popular.

Here, I'm just gonna, I have a little slide with just, like, a few of the many classifications that people tend to use.

There are the five love languages.

On the left is what's called the Enneagram.

And then, of course, the horoscope, which people use all the time as a kind of folk classification of people into types.

But maybe one of the most famous ones that you hear about all the time was made by these two people, who I wonder-- this is another hard one-- if anybody can identify who these two women are.

The same guy?

I want somebody else to answer.

Did you look at my notes before I came?

[laughs] Okay, who are these two people?

– Attendee 1: Myers and Briggs.

- These are Myers and Briggs of the Myers-Briggs Type Inventory, or MBTI, which even if you don't know the initials MBTI, it's this thing here.

They were twice as uptight about classification as E. Gary Gygax.

Instead of two dimensions, they had four.

And so, now we're gonna do a tiny bit of arithmetic.

There's gonna be more math later in this talk.

So, the total number of types, there's two things that each characteristic can be.

You can be an "E" or an "I," an "N" or an "S," a "T" or an "F," a "P" or a "J."

What those mean is not important for this talk.

And that gives you 16 different types of people.

And I put ENTP first 'cause that's what I am.

I don't know if I have any other ENTPs in the room today.

And this classification is incredibly popular.

You see it everywhere.

There's actually, like, a K-pop album that sold 2.5 million copies in Korea called ISTJ, which is about the sad story of an ISTJ who falls in love with an ENFP.

[audience murmurs] I got to read you this quote, because... So, here on the left, you see, like, a parenting book, which is entirely MBTI-oriented, like how to parent each of the 16 types of children.

And I just want to read you a quote from this book, because I think it really speaks to-- I would not go so far as to say, like, the kind of anxiety that this classification speaks to.

Ready?

So, here's what it says.

This book starts with the story of Lisa and Barry.

It's from the '80s.

That's why the people are called Lisa and Barry.

Who were very stressed-out parents.

It says, "They were often at a loss as to how to parent.

"They felt mystified.

"They felt they were out of their league.

Lisa and Barry are hardly alone," the book says.

And then they say the following.

They say, "But what if we did know from early in our child's life "who she really was?

"What if we could figure out by watching her interactions, "her play or word choices, her decision-making style, "what kind of person she is?

"What parent wouldn't want a true picture of the inner workings of their child's mind and heart?"

Well, that does sound appealing.

On the other hand, until you realize that the answer to this question is gonna be four letters long.

Maybe that's a little bit too much to ask of four letters.

But Myers and Briggs did not think it was.

I mean, so I'll just say... We crave classification because we don't like uncertainty.

It makes us uncomfortable.

It makes us anxious.

We wanna know exactly what our child is like.

We wanna know exactly what they're gonna do.

We wanna know exactly how they're gonna react to any situation.

And actually, I'm just gonna go impromptu for a second here and say that in a classroom, it's much the same, right?

It would be great, just as with our own children, it would be great if we could divide our students into types and know exactly how each one was gonna respond to everything we do in the classroom via some very simple formula, like four letters.

It's probably not like that.

But Myers and Briggs certainly thought it was.

They were very interesting people.

They got this classification originally-- They sort of modified it from something that Carl Jung did, who they were obsessed with.

Myers actually wrote an entire erotic fanfic novel about Carl Jung, which I have not read.

It's a little bit hard to find.

Can I say erotic fanfic on PBS?

I actually don't know if that's allowed, but... [audience laughs] So, Briggs wrote, very absolutistly, "Every one of us is born either an extrovert or an introvert "and remains extrovert and introvert to the end of his days."

So, this very rigid classification, just like Linnaeus.

She also once described-- I mean, again, relating to this notion of species, she also once described a marriage of a T husband with an F wife as the marriage of a fish and a canary.

That's a pretty potent metaphor to just how incompatible she thought these people were.

So, maybe, okay, I haven't said any math words.

I guess I said 2 to the 4th.

So, I think this kind of idea already is, like, a great way of talking about a word we like to say.

I know we're doing a lot of data science in this conference.

So, already, there's two different visions of what people are like represented here.

And we might call this the bimodal model and the unimodal model, right?

So, unimodal distributions, like, have one big hump in the middle.

That would be if most people were kind of pretty roughly evenly divided between introverts and extroverts, with a few extremes on the outside.

The bimodal model, which is what Myers and Briggs believed in, was that there's really two discrete types of people into which all people can be divided, the Es and the Is, the extroverts and introverts.

So, an interesting thing happens.

This test is devised and refined in the 1930s and 1940s.

In the '50s, when psychometrics is at its peak, like, this really starts to get some uptake.

And in fact ETS, a company we all work with, buys the MBTI and administers it to millions of school children, 'cause they're like, "Okay."

They believed it.

They're like, "We're gonna use this to, like, improve education "and understand, like, what kind of kids there are in the schools."

And when they did that... they saw a lot of this.

So, this is one of the many charts in this report from ETS.

I think this particular one shows liberal arts students and engineering students, which in case you're interested, their distribution of extrovert versus introvert looks pretty similar.

There's no big difference there.

But the more important thing is this is a unimodal distribution, right?

There is no clear division.

And actually, by the way, this was the end of the association of Myers and Briggs with ETS, because they were like, "You've got to be wrong.

"We know how the world is.

"We know there's these two separate types of people.

We reject all of your statistics and all of your analysis."

They were like, "This simply can't be right."

I think nowadays we do think this is right.

I mean, there are lots of psychological tests.

And I think modern personality psychology, which doesn't particularly use the MBTI, uses other models like the Big Five.

In a different kind of talk, we could spend an hour, like, talking about how dimensional analysis works and how modern personality psychology chooses what dimensions to measure people on.

We're not gonna do that.

I'm just gonna say that bimodality is out, unimodality is in.

But I'll just comment that I want to say this without denying the pull to classify, the pull to the bimodal vision, the pull to sort of separate things into separate kinds.

I mean, you even see it like-- You know, in political strategy, I just put up sort of two popular political slogans, which I think sort of are differently political-coded.

One is, like, a little more left-coded.

One is a little more right-coded.

But it's interesting to note that they both say the same thing.

They both say there are friends and enemies.

There are two types of people, and they are distinct from another.

And if you are not one kind, you are the other kind.

I want to say one more thing about the platypus.

This is not gonna look like it's about the platypus.

But here's a psychological question.

Why are decisions hard?

I think we've all experienced being stuck with a difficult decision, two options that we can't decide like, which one we like better.

So, one model for this.

Here's what Benjamin Franklin said to do when you were faced with a bad decision.

He said, "Here's this thing I like to do that I made up.

"I like to, like, take a piece of paper "and write a line down the middle "and write all the reasons in favor of one decision "on one side of the paper and all the decisions, "all the reasons for the other decision "on the other side of the paper, "and then sort of see which column is bigger, like, which one carries more weight."

So, I am telling you that Benjamin Franklin invented the pro-con list.

I think this is actually true, 'cause he writes about it as something he came up with.

And I can find no earlier reference to anything of this kind.

So, this is a model where what's going on when you make a hard decision, it's hard because you are balanced exactly 50/50.

There's exactly 50% reasons on one side and exactly 50% reasons on the other side.

And this kind of continuous measurement-- So, there's lots of mathematical themes here that I'm just gonna skip past very quickly.

This is a question about continuous versus discrete.

And the continuous model, this kind of is how modern personality psychology works.

This is a picture of somebody's measure-- not mine, by the way.

That's not a bad profile, actually.

Low on neuroticism-- Or low on agreeableness.

Okay, it's not me.

[chuckles] But this, you know, nowadays a personality psychology would say, "Oh, if you can't decide "whether you're an extrovert or introvert, "it's because your score on the test "is you're really very evenly poised-- 50% extrovert, 50% introvert."

But here's the problem with that model from the point of view of making difficult decisions.

If the issue were that you were exactly balanced, exactly balanced between the pros and the cons, exactly balanced between the do this or do that, exactly balanced, should I stay or should I go, then, if that were the case, then any new piece of information that put more weight on one side of the scale would suddenly make the decision easy, right?

It would tip the scales.

And what I think-- And, actually, 'cause I'm sort of working on this material, I'm actually curious if people agree.

We can talk in the Q and A. I think that hard decisions that we face in real life typically do not feel like that.

I think when we're really stuck and you get a new piece of information or a new fact that weighs to one side or the other, we're often still kind of stuck.

And I think these decisions, they're a little bit more like the platypus.

Like, it is hard to figure out whether we should think the platypus is a mammal or whether we should think platypus is a bird.

But if the platypus suddenly became 10% hairier, that would not settle the matter, right?

So, I think the issue here is somehow not that the platypus is exactly balanced, 50% mammal, 50% bird.

I think it's that the platypus has certain features which we identify as definitely mammal and certain features, like a bill, which we identify as definitely bird, and a slight perturbation simply does not change that.

It leaves us in the same position.

So, I think that these rough psychological decisions, problems maybe, are more like that, that there are sort of some features that we associate with good decisions and some features that we associate bad decisions, and the things that we truly get stuck on are the things that have features of each, things where we simply do not solve the problem by making the problem 10% hairier or 10% smoother.

And at this point, 20 minutes into the talk, I've talked about all these difficult things, things we're really stuck on.

And maybe we find ourselves saying, as math teachers, "If only everything could be like math."

I mean math, right, we all do it, math is the land where we actually do have these extremely nice, clean classifications with definitions that are unequivocal and make sense and tell us what is what.

I don't know about you guys, but, like, I do actually find mathematics like, rather psychologically soothing, like a sort of nice place to be when, like, other stuff is confusing.

I think probably we've all had those feelings.

And in some sense, the fundamental classification of numbers into odd and even, something that many of you have probably taught, right?

Do I have some elementary school teachers in the audience today?

So, this is, like, one of the most basic classifications in mathematics.

And I'm a number theorist, so I think about this all the time.

Here's a definition that you might have in mind.

A number is even if you can divide the things into two equal parts, like shoes, right?

The left shoes and the right shoes are equal in number.

And odd if you can't, if you try to do that and there's one left over.

But already there's a problem.

The problem has to do with our favorite number, zero.

And what you are looking at is a picture of a gas line during a gas-rationing, energy-cost spike in the '80s.

Why am I showing you a picture of a gas line?

Because when there's a gas shortage, as you may know if you've lived through one, a city or state or local government will usually say, "Okay, we can't have everybody going to the gas station at once "'cause the lines will be too long.

"So, everybody whose license number, "plate number ends with an even number, "you can go on these days of the week.

"And if your license plate number ends with an odd number, you go on the other days of the week."

And every single time this happens-- and I've actually read lots of examples of this, people who have a license plate that ends in zero totally freak out and do not know what to do.

And there's a huge conflict about this, so much so that in fact that in my home state of Maryland, where I grew up, zero is an even number by state statute.

[all laugh] So, they had to put this in the law because there were so many arguments in the '70s about, like, when you could go to the gas station.

And now, once you cross the border into Maryland, zero becomes an even number, and it's illegal for zero not to be an even number.

So... Why do I bring this up?

One tone of voice in which I could be bringing this up is to say, "Boy, isn't it sad that people don't know "what an odd number or an even number is?

"Like, what are we doing as math educators that people don't know this?"

That is not the tone of voice I wish to adopt, because I think-- and this is one of my deepest beliefs as a teacher-- that when people are confused about a thing, it is because the thing is confusing.

Not because we've failed, but because there is an actual difficulty.

And here, I think the difficulty is in the definition, because-- And now, because this has happened in, and we haven't had lines like that in New York or Maryland in quite a few years.

But it does happen around the world, and this is a cross-cultural phenomenon, and you can, like, read people's tweets about it when they're upset about this thing.

And you can see people saying, "Why are they saying zero is an even number?"

You can't divide nothing into two equal parts.

Like, how can you do it?

There's nothing to divide.

That, guys, is a good objection, I believe.

If your definition is that an even number is when that number of things can be divided into two equal parts, I think it is perfectly reasonable to say, "That is something I cannot do if there's nothing there to divide."

So, zero cannot be even.

Now, often, people in that situation-- And actually, there are, like, wonderful studies of this where they interview children and sort of talk to them about their notions of odd and even.

Often people will say that zero is neither odd nor even, because you can't divide it into two equal parts, but it's also not the case that if you try to divide it into two equal parts, there's one left over.

There is simply nothing to try.

And so, people would see zero-- many children and adults, like, sort of see zero as a number that is neither odd nor even.

So, you know, we would say-- [laughs] As grown-up mathematicians, we would say, "Okay, "but the problem is "that's not the definition of an even number.

"It's not this folk definition "that the number of things can be divided into two equal parts.

"It's a more algebraic definition that a number is even when it is twice another whole number."

And in that case, like, you're fine, right?

You can say, "Well, zero is even "simply because it is two times zero and zero is a whole number."

It even applies to negative numbers, where then-- I mean, nobody's license plate ends with a negative number, thank God.

But if it did, people would be even more confused, right?

'Cause I think the second definition applies very well to negative numbers, and to the first one, it does not.

So hold this in your mind.

I'm gonna say a little bit more about this, but I wanna sort of go to some other places first and then come back to it.

I just hope to convince you that there's an actual difficulty here, so there's something to think about when we think about what is the definition of an odd or an even number.

Maybe this will be, like, a slight discursive, but just for a minute.

I mean, we are kind of in this great age of classification where, you know, already... I would say 20 years ago, we could already train a neural network, like, very well to classify images into dog and cat.

And it does this with a great deal of nonambiguity.

It does not-- There are not a lot of images to which it would say it's 50% dog, 50% cat.

It pretty much-- a well-trained neural net will always choose one or the other.

This is, like, a little bit tangential to the main point of this talk, but I wanna bring it up, because I think not everybody knows it.

The way a large language model is trained today, any of the commercial large language models you would interact with through a chat, they are in fact trained to give very definitive answers.

They are trained to be sure.

They are trained to sort of... not be very ambiguous about what a thing is.

And the reason they trained it that way is because they do want them to give correct factual answers.

So, in some sense... You know, if you ask, like, "What's the capital of Wisconsin?"

You actually don't want the large language model to do something like imitate what a random person would say when asked, "What's the capital of Wisconsin?"

Because I can tell you, being originally from the East Coast, like, lots of people in Maryland do not know what the capital of Wisconsin is or, like, where Wisconsin is or anything about Wisconsin.

I think the people who build these models, like, want them to reliably give the right answer, which means that they want to sort of train them to be very, very definitive about what the exact answer to each question is.

And that creates, like, some weird behavior.

This is sort of one of my favorite runs, where if you sort of-- this is GPT-3.5.

If you ask, "Give an example of a Jewish person--" And this is true for almost every model.

You will almost always get Einstein.

Like, according to modern AI, Albert Einstein was the only Jew in history.

[audience laughs] He comes up almost every time.

Albert Einstein is about-- depends on the model.

But about 90% to 95% of the world historical Jewish population is Albert Einstein.

In this particular one, I got 19 Einsteins and 1 Ruth Bader Ginsburg.

And that's pretty typical, you know?

Okay, but that's an aside.

I wanna come back to math, actually.

So, we're gonna go from arithmetic to calculus.

What you're looking at here is a page from Fermat where he's writing about how to compute the area inside a curved region.

And he says, "I compare this to the first solid, "a squared b - a cubed, as if they were equal, although in fact they are not."

And I put this up because I think this is the kind of sentence that most people outside mathematics think that mathematicians would not write.

In fact, I think this is the exact kind of sentence they think that our entire job is not to write, not to say, "Well, let's act as if these things are equal, even though they're not."

You know, a normal person outside of our world would say, like, "I thought your guys' entire thing "was to say things were equal when they're equal and say they're not equal when they're not equal."

And yet, here is Fermat, a great mathematician, saying just the opposite.

"We're gonna ask-- We're gonna act as if these numbers are equal when they're not," because they differ in some sense infinitesimally.

And of course, this is what the entire history-- at least the early history of calculus is like.

Calculus runs on vibes.

This is a fact, right?

Certainly, Isaac Newton was no more-- He said stuff like this constantly, to the point that, you know, he was criticized by people around him.

Famously, Bishop Berkeley wrote a very angry anti-Newton screed saying, "What are these differences, Newton, "that you write about that like, 'Well, they're not zero.

"They're infinitely small.

"They're smaller than any number.

But at the same time, they're not zero."

And he's like, "What are they, the ghosts of departed quantities?"

[audience laughs] Famous, famous Newton insult.

And the funny thing about it is that Berkeley was kind of right.

He kind of had a point.

We can't deny it from a modern point of view.

And you could ask, well, "So, did Isaac Newton actually know calculus?"

He could not have written an epsilon-delta proof like we write it today, those of us who teach calculus.

So, here, I know this talk is about uncertainty, but I'm gonna give a definitive answer to this question.

Yes, Isaac Newton invented calculus.

He definitely knew calculus.

I feel like it's absurd to say that Isaac Newton didn't know calculus, Newton and Leibniz.

He didn't know fully how to formalize it, but he knew what it was.

He knew what he was doing.

I feel like I refuse to accept an alternative answer to this question.

It's not until more than 100 years later, in the early 19th century, that Cauchy invents, like, the notion of a limit that we teach in AP calculus today, the notion of the epsilon-delta definition of continuity.

All this stuff is, like, brand-new.

And by the way, Cauchy gets very excited about it and immediately changes what he's teaching the freshman at the cole normale in Paris, he's teaching the engineers.

And he immediately is like, "I finally figured out what calculus really is."

And, then, it created, like, a huge revolt, and in the end, like, the dean had to be brought in to sort of sit in Cauchy's classroom to make sure he was not teaching epsilon-delta to the freshman engineers who were, like, practically rioting at this sort of bizarre novelty that Cauchy had invented.

So, I'll just say I like this quote from Emanuel Lasker, who was not a mathematician, but a chess player.

But I think this is very appropriate to Newton.

He says, "He who relies solely upon tactics "that he can wholly comprehend "is liable, in the course of time, to weaken his imagination."

So, very true.

We can say that there were things about calculus that Newton didn't understand.

But to say that Newton didn't know calculus, even though he was saying imprecise things like, "Let's treat this number as zero, even though it's not," breaking this kind of classification that we think is so characteristic of mathematicians, I think to say he did not know calculus is absurd.

Actually, one of my favorite quotes on this is from the topologist Poincar, who again says something I don't think people outside think of mathematicians as saying, that "Mathematics is the art of giving the same name to different things."

I mean, this would make Linnaeus puke, right?

Like, this is the exact opposite point of view.

But I think Poincar had this as a hard-won insight, that it is not always the case that the actual things in the world-- and for me, the mathematical world is part of the world-- that they cleave naturally into types.

And I think partly, Poincar was writing this under the influence of the development of non-Euclidean geometry, another story we could spend a whole hour talking about, which I'm not gonna.

But at the time Poincar was writing this, standard lines in Euclid's plane, a great circle on a sphere-- And that's my incredibly bad picture of geodesic rays on a hyperbolic disk.

All of these were lines in various versions of geometry, and all deserved to be called lines, as Poincar knew very well.

Let's do a little more calculus.

And I'm gonna do this a little bit speedy.

Here is something that, at one time, was a very serious mathematical controversy.

What is the sum of this infinite series, 1 - 1 + 1 - 1 + 1 - 1, dot, dot, dot, forever.

There's an obvious answer, which is we can kind of PEMDAS this out and put in some parentheses and write it like this.

And it seems to be, like, the sum of infinitely many zeros.

So, okay, obviously it's zero.

On the other hand, we can group a different way to make it 1 - 0 - 0 - 0 - 0, et cetera, et cetera, from which it seems to be obviously 1.

This is called the Grandi's series, by the way, 'cause Grandi was a sort of somewhat unhinged monk in the 17th century who, like, wrote a lot about this.

And Grandi's answer, by the way, if we had a little more time we would, like, really algebra this out, but I'm just gonna show you one slide for the algebra fans.

Grandi argued that you could show rather easily that if X is the sum, then X is equal to 1 - X. So, X had to be 1/2.

Which is maybe the weirdest answer of all, like, that an infinite sum of whole numbers would be 1/2.

But Grandi was very committed to this point of view.

And people really argued about this.

And I just want to... I just wanna give this quote from Hardy, G.H.

Hardy, the great number theorist of the early 20th century in England.

I am gonna actually read this out, 'cause it's important.

"It does not occur--" By the way, think about odd and even as I read this.

This is bringing us back to odd and even.

"It does not occur to a modern mathematician "that a collection of mathematical symbols "should have a 'meaning' "until one has been assigned to it by definition.

"It was not a triviality "even to the greatest mathematicians "of the 18th Century.

"They had not the habit of definition.

"It was not natural to them "to say in so many words, 'By X we mean Y.'

"It is broadly true to say that mathematicians before Cauchy--" The bad calculus teacher, remember him? "

-- mathematicians before Cauchy asked not, "'How shall we define "the sum of this infinite series?

', "but, 'What is the sum of this infinite series?'

"And that this habit of mind led them "into unnecessary perplexities and controversies which were often really verbal."

This, again, is counter to most people's view of what a mathematician's job is.

I think they do think it is our job to say what things are.

And I think the modern mathematical view, as evidenced by Hardy here, is that's just not right.

We're here to sort of figure out what things should be defined to be.

It's a big philosophical shift.

I wanna say a little bit more about it.

I'll just comment.

I don't wanna say, like, too much about Spinoza in this context, but I'll just point out that in his Ethics, this work that he writes about how religion was supposed to work, he puts it in the form of a Euclidean proof-- like a very strange thing to do.

But not so strange, because what he's trying to do is to generate certain knowledge, right?

The truths of geometry were things, you start from these absolutely incontrovertible axioms, and everything that follows from it cannot be denied.

So, this is what you do if you're trying to say, like, "You simply cannot disagree with my religious views, try as you might, because they're in this form."

They're in the form of, like, axiom, axiom, axiom, which you accept, and then, theorem, theorem, theorem, proof, proof, proof.

I'm gonna slightly, very rapidly summarize a wonderful book by Alma Steingart, a historian of mathematics... about this, where... There's two real visions of how mathematics works that are quite incompatible in a way.

There's the old-timey vision where we start with some axioms.

This is kind of what we do when we teach geometry.

We start with some axioms, and we're like, "We simply must agree on these.

"We have no choice.

And then everything that follows them must be true."

The more modern view is rather different.

It says the job of the mathematician is given some axioms, figure out what follows from them.

But we're not saying that the axioms are true.

You might have one set of axioms.

I might have another.

I might be doing geometry in the classical Euclidean sense.

You might be doing some other kind of non-Euclidean geometry.

And that's okay.

That's a very different viewpoint.

And what Alma's book is about is the way that this point of view of mathematics becomes dominant, especially in the United States, in the great flowering of American mathematics in the first half of the 20th century.

And I wanna kind of make the claim-- I'm sort of going out beyond what she says in her book, right?

I kind of feel strongly about this.

I wanna make the claim that it's not an accident, that here is where it happens.

It is an American mathematics this point of view really becomes dominant.

Because you got to remember that for Spinoza, writing in the Netherlands, people are just, like, straight-up murdering each other over religious differences all the time.

Like, that's pretty normal.

And so, for him, religious difference-- that is a problem that has to be stamped out.

You have to get people to agree on what is true, because if you don't, they murder each other.

The United States has a rather different tradition-- what I would call, like, a tradition of political pluralism where I think in an American political context, we do not demand that difference of opinion be stamped out.

It's quite the opposite, actually.

The whole point of the American system is to say, can you actually run a country if people have, like, radically different beliefs about what is the case?

And yet, somehow it works and somehow, the country trundles along anyway.

So, I think that this popularity of this view of mathematics is something like that.

It fits with the United States, because it says not what are the things we all have to believe, but let's think carefully about how you get to an outcome from whatever axioms you happen to start with.

But mine might be different from yours.

We're not gonna try to stamp out that difference.

All right, we're coming to near the end of our time together, so let me make a few closing remarks.

We started with this idea of classification, and I think... It leads to all kinds of questions.

If we don't have this firm boundary between extroverts and introverts, we don't have a firm boundary between mammals and birds or even, like, odd and even numbers, or even numbers and things that aren't numbers, it creates a certain kind of anxiety.

Remember, I told you there's a reason people want this classification, because we're uncomfortable.

It makes you think maybe there is no such thing as a mammal or an extrovert and a number.

I'm gonna reject that.

We all know there's such a thing as a mammal.

That's not actually a problem.

We all know there's such a thing as an extrovert; I am one.

We all know that there's such a thing as a number.

We all teach about them, we use them.

They are real things.

The fact that they don't have a hard boundary around them, I think, just fundamentally should not be seen as a problem for whether these things actually exist.

We could say more about that, but not in this talk.

Even here-- let's come back to Shakespeare.

Why is the expression "Neither fish nor flesh?"

Does anybody know?

Why was this the issue, is this a fish or is it a land animal?

Don't let this guy answer again.

Yeah?

– Attendee 2: If I remember correctly, it's because, like, anything that lived in the water was classified as fish, and then, anything that was on the land was classified as animal.

- Yeah, but why do we care?

Why do we care what's a fish and what's an animal?

– Attendee 3: Does it have to do with the fact that, like, fish on Fridays in the Catholic Church, during that kind of time, like, you could have fish on Fridays, but not eat-- - Exactly, what can you eat on Lent?

That's the issue.

So, that is why this is a controversy, why something like an otter is controversial.

And I just want to comment that actually, this is an issue for the Catholic Church.

And you might say, "Okay, well, "so the Catholic Church must have some actual definition of what's a mammal and what's a fish."

But no.

Actually, I think the church understands very well that this is a fuzzy boundary.

And in particular, I wrote a little list that, there are Catholic dioceses where you can eat an otter on Lent and others where you cannot.

You can eat a muskrat if you live in the Diocese of Detroit, a beaver if you live in Quebec, an alligator if you live in New Orleans, and in Venezuela a capybara.

All of those are fish in those dioceses alone, according to the local authorities.

And so, I just want to emphasize that I think that honestly, if you work on these things carefully... I think you always find that these boundaries are not very solid and they have to be thought of contextually.

So, I am gonna take sort of one side of this controversy.

I'm sort of with the United States of America and the Catholic Church on this, that, like, the question about definition should not be, like, "What is this thing?"

It should be, "What should we define this thing to be according to our purposes?"

What is an odd number, and what is an even number?

It depends what we're trying to do with those numbers.

And maybe I'll close with... a wonderful book by Richard Dedekind, which addresses this question about what numbers are.

This is what I teach when I teach Math 521 at UW.

I teach the ideas in this book.

But it has a fascinating title, Was sind und was sollen die Zahlen?

It sounds great in German, not so great in English.

In English, it says, What are Numbers and What Should They Be?

So, I wanna close here.

I think it might be, again, surprising to people outside mathematics that a mathematician would write a book with the word "should" in it at all.

If you have that view, that mathematics is about what things are.

But I don't think that's true.

Mathematics is a human activity, and every human activity has values embedded in it.

And that's why the word "should" is appropriate.

Because when we ask what is the definition of an odd or an even number, or for that matter when we ask, what's the definition of a mammal or a bird, there is a value question there.

You might have your definition, and I might have mine.

But there are actually values that we share where we can argue about what is a good definition and what is a bad definition.

Not what is the definition.

That's what I'm trying to get us past.

But rather some definitions actually are better than others.

And which one is better might depend on our purposes.

So, with that, because time is short, I think I'm gonna close.

I'll just say some of what I talked about today, some of it is in books I've already written, and some of it is in the book I'm writing right now, which will be out in May 2027 if people wanna read about it.

And now, I wanna leave us some time to ask questions.

And so, I have a picture of myself in a suit with question marks all over it, and a goat.

So, I'm happy to take questions.

Yeah.

– Attendee 3: Completely random, how do you feel about imaginary versus real numbers, and where does zero belong?

- So, the question is, how do I feel-- And I like that you phrased the question that way, 'cause it is about our feelings.

How do I feel about the distinction between real and imaginary numbers, and where does zero fit in that question?

And again, it's a kind of notation that we're stuck with, but it's terrible, because I like feel like the imaginary numbers are no more imaginary than the real number.

I mean, I feel like the imaginary numbers are not that imaginary and the real numbers are not that real, frankly.

I would say they are of equal status.

But I think... But I think, again... it's a great example of the folk definition of a number, when you're a small child, it's the answer to a question of the form, "How many?"

And "I" is definitely not that, right?

Whatever it is, it's not that.

But, then again, like, neither is the square root of 2, which is ostensibly a real number.

And, you know, frankly, I don't even think a fraction is really an answer to a question of, like, how many.

It is not a number you can count to.

Maybe that's, like, a little bit more of a how much than a how many.

So, you know, I think this process of, like, expanding our number system is a process of expanding what is our definition of a number bit by bit.

But I do think we sort of-- I do think it's on some level wrong to sort of say to our students "There is a square root of -1, and it's called I."

That's probably how we mostly do it.

I think there's something a little bit philosophically shady about that.

I think we should be honest about the fact that, like, we have decided that such a thing should be called a number.

Why should it be called a number?

I mean, there's a lot of answers to that question.

I'm an algebraist, so I kind of have an algebraic answer.

I like to call it the duck principle.

Do you know this?

If it quacks like a duck, then it's a duck.

So, I feel like if you can add them and multiply them like numbers, then they're numbers as far as I'm concerned.

That's my duck principle answer.

But notice I'm using "should," right?

I'm saying there is a reason that I've decided to declare there to be a thing which I'm calling the square root of -1.

But I don't think we should act as if it is just there and we're informing people that it's there.

We should be honest about the fact that we're defining it into existence.

Yeah.

– Attendee 4: So, I feel like there's an intersection between philosophy and math that you very much pointed out today.

Are professors also like that that aren't you, or are you unique in that?

[audience laughs] - So, there's a question about, like, to what extent is the way I'm talking about this, like, normal or mainstream?

And I'm definitely-- exactly as you say.

So, the questioner pointed out that I'm kind of using stuff from philosophy and mixing it with stuff from math.

And the answer is that there is a very long tradition of doing exactly that: mixing philosophy and math.

If I'm to be honest, there's people who are much more learned at that intersection than I am.

And I would say that the farther back in time you go, the less of a disciplinary distinction there is.

So, I think most of the really wonderful early mathematicians, you would kind of call them philosophers of math at the same time.

First of all, I don't think I used his name in these slides, but Hilbert, who is the one of the great mathematicians of the early 20th century from Germany, he is the person most responsible for bringing us into this kind of you get to choose your own axioms world we live today.

And I would definitely call him, like, a philosopher of mathematics as well as a working mathematician.

So, I think this intersection is, like, real, and it's well-populated.

Okay, so thank you guys so much for coming.

I'm still here, I'm not going anywhere.

I'm gonna go sit out by the lake in a little bit 'cause it's so nice today.

So, if people have more questions, feel free to come up afterwards.

I have no schedule, so thank you guys so much.

[audience applauds]