Benoit Mandelbrot, Beauty and Finance

Is there anything that shouts “mathematical beauty!” quite like fractals? Fractals are a core pillar of how mathematicians and educators sell math to the public. That makes sense — they really are beautiful in a way that is both visual and intellectually pleasing.

In fact, just a few hours ago I showed this image to my 4th Graders, and I got exactly the “ooohs” and “ahhs” I was hoping for.

sierp-det.GIF

The name associated most strongly with fractals is Benoit Mandelbrot’s. Mathematician, visionary, Jewish WW2 refugee, early adopter of computational visualizations, really a very cool person, and discoverer of the Mandelbrot Set. He seems likely to join that canon of Great Mathematicians. (There is such a canon, right?)

Plus, there is an awesome song about him:

And it was only since I’ve started reading more about finance that I realized there was a part of the story that wasn’t being told. Because while there is no denying the beauty of fractals, Mandelbrot’s study of fractals is caught up in the “wild randomness” that he sees in financial markets.

Just to be clear, I’m not saying that this is some sort of big secret. Mandelbrot wrote and spoke frequently about this, and I came across this nearly immediately while studying finance. Because within finance, Mandelbrot emerges as a critic of the sort of quantitative financial engineering that I’m trying my best to understand.

(Shoutout, Nassim Nicholas Taleb.)

The sort of financial models I read about in textbooks all represent the movement of a stock as a kind of random walk, weighted by certain probabilities. So IBM may be more likely to go up or go down, but essentially its price at a given time has to do with how the stock travels through this field of uncertainty — and it is moving randomly. Maybe its price is more likely to rise than fall, but still it’s governed by the same laws as coin flips.

In other words, the value of IBM follows a normal distribution, i.e. the conventional bell curve.

That’s how most quants (apparently) think about finance, but Mandelbrot thought this was a big mistake. He thought the markets were governed by wild randomness, a propensity to run towards the extremes. Here is a piece where Mandelbrot (co-written with Taleb) describes the difference between his model and the conventional one:

These two models correspond to two mutually exclusive types of randomness: mild or Gaussian on the one hand, and wild, fractal or “scalable power laws” on the other. Measurements that exhibit mild randomness are suitable for treatment by the bell curve or Gaussian models, whereas those that are susceptible to wild randomness can only be expressed accurately using a fractal scale.

I’m not quite ready to try to describe what he means by “fractal scale.”

Does it mean anything that the financial side of Mandelbrot’s work is less often shared with students and the public? Maybe not. It certainly doesn’t exhibit the same gut-punch beauty as his images, and finance is probably not the quickest way to gain an appreciation for fractals.

And yet…so often, mathematicians and educators are eager to emphasize the beauty of mathematics. And I agree — math can be beautiful! But it doesn’t seem quite right to say that mathematicians simply chase beauty. Mandelbrot wouldn’t have studied fractals if he wasn’t trying to model financial data, and the story of how math at all levels is entangled with finance is deeply undertold.

How much real-world complexity can we tolerate in math class?

I saw this, and I have a reaction:

https://twitter.com/jrosenberg6432/status/1110333373149601792

I think of this as a matter of the complexity that we are exposing students to. Part of what makes these math modeling experiences engaging is that they are relatively simple. Not to say of course that these problems are easy for students. They aren’t. But when they are engaging it’s because the context is new, but the structure is discernible to students. That’s what engagement is: that feeling of novelty, along with the feeling of I-can-do-this.

And the reason why students can do this is typically because we’ve chosen a dataset that matches either a proportional, linear, exponential or quadratic function, or a pair of these functions. (True: it’s hard to choose the appropriate variables, and if you choose the wrong variables you won’t have very much fun. For that reason teachers usually make the smart choice to help students choose useful variables before they have time to tackle some task.)

Anyway I don’t want the point of this to be that the modeling experiences we create in math class are easy or pointless. That’s not what I’m trying to say. What I am trying to say is that part of what makes them engaging is that they are relatively simple modeling problems. And what makes them relatively simple is what makes them mathematical modeling problems as opposed to scientific ones. What I mean is, students don’t know the context in depth, we are telling them that they know enough to make predictions — they are relatively protected from the complexity.

What makes scientific modeling a different kind of engaging is that it’s in a way more serious contextual work. You learn to see new variables and new factors, and then you’re trying to coordinate them into newer, more powerful models. But there is a sense in which this is more serious work than mathematical modeling, which (by dint of being part of math class) engages less with the context.

I’m not trying to hate on mathematical modeling here, but I am trying to articulate something that I’m moving towards. I really do think kids deserve a chance to connect with applied math problems, “real world” math. At the same time, I’ve been frustrated with what passes for “real world” in math because it doesn’t take the context seriously. I have a great respect for complexity and people who study it. I’d hate for students to get the message that math can just march in and math all over the place and solve everything. You have to have some humility and learn about the world before diving in with an equation!

One question I’m asking myself these days is whether there is a way to take the contexts more seriously in math class. Is there a way I could bring more applied math into my classes without getting lost in the complexity, or ignoring it entirely?

Trying to write arbitrage puzzles

I’ve been trying to write little arbitrage puzzles. I’m hoping to make them accessible to kids. Here’s one I’m working on now.

It might rain on April 15th. Then again, it might not.

Your first friend agrees to make a bet. If it rains, you get $10. If it doesn’t, you have to pay your friend $5.

You have a second friend who wants the opposite bet. If it rains, you pay your friend $5. If it doesn’t, they’ll pay you $10.

What do you do, and why?

My take on a solution in the comments.

Is this OK?

Is it OK to abstract away all the history (and suicide) for the sake of a good puzzle?

Forget for a second whether or not this particular video is ok. (I wouldn’t show it to my students; you never knows what someone is dealing with.) But doesn’t this say something about mathematical culture? What are we training mathematical people to ignore?

Axiomatization of “Story”

Axiom 1: At least one story exists.

Axiom 2: There exists an “empty story,” i.e. a story where nothing happens.

Axiom 3: Two stories are the same if and only if they contain the same events in the same order.

Axiom 4: If X is a story and Y is a story, “X then Y” is also a story.

Axiom 5: For every story X, there exists a story Y that contains story X. In this case we say that Y is a telling of X.

Axiom 6: A story exists that contains the empty story, the story containing the empty story, the story containing that story, the one containing that, etc. forever. This is called the neverending story.

Since there is a story where nothing happens, there is also a story that is a telling of that empty story. As a result, the story where nothing happens and then someone tells a story about nothing happening is also a story. This can keep on going indefinitely, populating the entire universe of stories with retellings about nothing.

Geometry and theology

Boiled the elements down into the axioms
Mistook a fax for remedial tediums
It seems Ezekiel speaks to some
My mind was hazy and numb
And left hand gripped a clump of palladium
Saw the beast with the wings and the talons
The simple answer but it felt out of balance
Bad news like a blue screen of death
Besides the point, but which hue seems best?

I’ll keep conducting these autistic symphonies
These sentences have sentenced me
Like I didn’t have the sense to sense the mere
The presence grows weird
Doesn’t make sense but I don’t fear, not a damn thing
I live like a man who’s already dead
Like I had a motorcycle but my name is Zed
(I’ll be missed), said another clumsy alchemist
Like he just learned predicate calculus
The existential quantifier, a backwards EX
My rap career is a cataloging of defects
Copy edited by Ryan Seacrest
Like he must be new to this
Milo doesn’t exude hubris, chuuch
Like he must be new to this
Milo doesn’t exude hubris

Modernism in Mathematics

Jeremy Gray makes the case (in here) that modernism applies to mathematics. His modernism consists largely of a move away from representations and towards formal approaches.

So on Lebesgue’s theory of the integral in 1903:

“The axioms specify what the integral is intended to do. They do not start from an idea that the integral is about, say area, or any other primitive concept. It is necessary to show that there is a model of these axioms, but once that is done it is at least possible to prove properties of the integral directly from the axioms and without reference to any model of them. The axioms are sometimes said to define their object implicitly, or to create it. There is no reference to a primitive concept available via abstraction from the natural world.”

And on Kronecker and Riemann:

“Neither man suggested that objects cannot be studied via their representations, but both believed that one must be vigilant to ensure that one establishes properties of the objects themselves and not the properties of merely this or that representation, and to this end it was best to avoid explicit representations whenever possible.”

I didn’t know about the Hausdorff paradox, which feels a lot like Godel. Gray’s summary: “on any plausible definition of the measure of a set there must be non-measurable sets.”

Borel ended up critiquing the use of the axiom of choice to call the paradox into question, but this was another step (apparently) in pushing people to accept that definitions of area are inherently imperfect — pushing us further away from meaning and belief in the representations.

Another interesting point from Gray: you know that thing about the unreasonable effectiveness of math? That wouldn’t have made any sense in the 19th or 18th centuries because math was coextensive with science. Like, there’s nothing surprising at all about the connection between math and the world back then, because math was an attempt to describe the world.

I’m interesting to read more, but I’m feeling as if a question has been answered. Whether we call it modernism or not, this is the time in the history of math when the connection between mathematics and the empirical world was made problematic. If we’re looking for the origins of the idea that math is “useless,” it’s going to be in this movement in mathematics between 1880 and 1920.

Things that I’d like to read: on modernity and mathematics

The world has changed immensely over the past several hundred years. Mathematics has too. Are these changes all related?

Plato’s Ghost looks like a good place to start.

Plato’s Ghost is the first book to examine the development of mathematics from 1880 to 1920 as a modernist transformation similar to those in art, literature, and music. Jeremy Gray traces the growth of mathematical modernism from its roots in problem solving and theory to its interactions with physics, philosophy, theology, psychology, and ideas about real and artificial languages. He shows how mathematics was popularized, and explains how mathematical modernism not only gave expression to the work of mathematicians and the professional image they sought to create for themselves, but how modernism also introduced deeper and ultimately unanswerable questions.

Building on Gray’s work is this presentation by Susumu Hayashi, which introduces (to me at least) the notion of “mathematical secularization.”

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I also came across The Great Rift.

In their search for truth, contemporary religious believers and modern scientific investigators hold many values in common. But in their approaches, they express two fundamentally different conceptions of how to understand and represent the world. Michael E. Hobart looks for the origin of this difference in the work of Renaissance thinkers who invented a revolutionary mathematical system—relational numeracy. By creating meaning through numbers and abstract symbols rather than words, relational numeracy allowed inquisitive minds to vault beyond the constraints of language and explore the natural world with a fresh interpretive vision.

The focus is on early modernity and the shift to algebra, which is an earlier phenomenon than modernism. But maybe it’s part of the same story?

Also in the category of “is this related? maybe??,” there is a working group of philosophers that call themselves the Mathematics, Mysticism and Secularization working group.

There are all these -isms that I learned about in philosophy of math: empiricism, logicism, formalism, fictionalism (wiki). That’s part of this story too.

What I’m attracted to is the idea that math is as much a part of culture as anything else. Over the last few centuries Western society has gotten less and less comfortable with the abstract, invisible realm of religion and spirits. Wouldn’t that have an impact on how that culture thinks about that other invisible, abstract realm of mathematics?

People used to think of x^2 as referring just to a square’s area, but then it was emptied of that meaning. Is the break of algebra from geometry something like the break of philosophy from theology?

People used to think that mathematics was a search for ultimate truths, not just conditional ones. Are we living in a mathematically relativistic world?

Mathematicians sometimes talk — with pride! — of the uselessness of their work. Is that the end result of the sorts of processes described by these authors?

I have no clue, and I have no idea when I’ll be able to read those books. But the questions seem interesting and confusing.

Continuous induction is a thing!

Great question:

Great answer: Yes!

From the paper:

Consider “conventional” mathematical induction. To use it, one thinks in terms of predicates — i.e., statements P(n) indexed by the natural numbers — but the cleanest statement is in terms of subsets of \mathbb{N}. The same goes for real induction.

I think of induction as being about dominoes falling, but the focus on sets is a different way of talking about induction — even for the natural numbers.

To be OK with this paper’s explanation of continuous induction, we have to be OK with the idea that its sets that the inductive property — it’s not a process metaphor, it’s a property metaphor, or something like that. Maybe a useful metaphor is flammability, rather than dominoes. Some sets are made of stone. Others you set a spark and the whole thing goes up in flames. Those are the inductive sets.

Here’s what it means for a set of real numbers to be inductive:

Screenshot 2019-01-27 at 9.55.55 PM.png

The three conditions say what it means to be inductive, and the theorem states that if a set is inductive then it is fire: something that’s true for part of that inductive set has to be true for the whole thing.

(Something I’m wondering about real induction: can’t the spark can be planted either at the front or back of the set? I think with slightly different conditions for inductivity you could start this at b.)

The paper goes on to use this continuous induction to prove some of the classic theorems of calculus and analysis, including the Intermediate Value Theorem. Here’s my attempt to restate the proof in the paper.

So, I’m going to tell you three things about a graph. First, it starts like this and ends at 10:

Screenshot 2019-01-27 at 10.17.09 PM.png

Second, the function is continuous.

Third, the function does not ever cross zero.

Question: Are you guaranteed that this graph stays positive the whole time?

Answer: Yes, duh. But as with a lot of this foundational stuff, the challenge is to say why.

So let’s make the following inductive argument, by first collecting all the inputs 0 through 10 of my function that yield positive outputs into a set. So far, just from the graph, you know that this set contains all the real numbers from 0 to about 2. We’d love to show that actually all the real numbers from 0 through 10 are part of this set.

Now, let’s tee up the conditions for this set being inductive.

  1. 0 is in this set, because duh I showed you that 0 is in this set.
  2. Pick any number that’s already in this set — it looks like 2.2 is in it — then there has to be some larger number in the set i.e. also yields a positive output. That’s because this function is continuous, and so approaching 2.2 from the right, eventually the outputs need to get really, really, REALLY close to the true output when 2.2 is the input to f. That necessarily means that it’s going to have a positive output, since the output of 2.2 is positive. So there needs to be numbers larger than 2.2 that are in this set if the function is continuous.
  3. And suppose that you knew some range from 0 up to but not necessarily including 8 where was positive, i.e. still hanging out above that zero line. Well, even though we “but not necessarily including”-ed 8, 8 has got to be positive anyway. Because go backwards at all from 8 and you should hit positive outputs, but because this is continuous eventually all those positive values need to be (eventually) super-duper-duper close to f(8). So 8 must have a positive output.

And by induction, tada, there’s nothing stopping you from extending this to the entire domain of 0 to 10, it’s positive the whole way through. And since this is just a restatement of the Intermediate Value Theorem, you’ve proven that too, by induction.

I still have questions, but this is very cool.

Is this math?

Look at those lines, that perspective, the symmetry…there is a clear sense of geometry in this picture, and if geometry is math (it is) then there’s a case to be made that this picture is mathematical. Hence the artist is a mathematician. So the picture is math.

Except that’s not quite right, is it? It’s not fair to this piece of art to see it as essentially mathematical. That’s not what the human person who created it was going for. Math doesn’t get to make a claim on any use of parallel lines. Back off, math! Art gets to be art.

I’ve been thinking a lot lately about what counts as math, and what it means to expand what counts as mathematical. Certainly, if you think math is just what’s covered at school, you’re wrong. But then we make these expansions — kids are doing math when they’re playing, they’re doing math when they’re drawing, when they’re braiding hair, when they’re building…

Maybe the issue is in thinking of mathematics as a field or a single subject, as opposed to a mode of thinking that is used by artists, philosophers, lawyers, Wall Street, cashiers, kids, hair-braiders, engravers, everyone. Mathematics isn’t a distinctive activity that you do, it’s a certain way of doing other things.

But then what is that certain way of doing things? Probably not a clear set of criteria, but a kind of family relation: if you’re using numbers, if you’re paying attention to repetition, using the properties of shapes to create something new — you’re not necessarily doing mathematics, but you’re doing whatever it is you are doing in a mathematical way.

But math doesn’t get to claim art.