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.”

Screenshot 2019-02-13 at 11.38.10 AM.png

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?

Master of the Die

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.

The Real Satisfaction of Mathematics

“In short, mathematics only exists in a living community of mathematicians that spreads understanding and breaths life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others.”

I return often to this quote from Bill Thurston. Here’s what I’m getting out of it today: the mathematical researcher is just looking for something new to share with others, a distinctive piece of the mathematical universe that they can teach the world. The search is for something new, but only because new ideas are something worth sharing.

Mathematical teachers, professionally speaking, are chasing the same goal via the opposite strategy. We don’t bother seek new mathematics, which after all is only one way among many to find something worth sharing. But what we do aim to share, we aim to share more widely than any researcher could.

I wonder whether there’s an alternate history of mathematics out there. Rather than focusing on those who were mathematically innovative, it would be a history of learners and sharers of mathematics par excellence. Would that be the same history? I don’t think it would.

Thinking about Francis Su’s “Mathematics for Human Flourishing”

Francis Su:

What I hope to convince you of today is that the practice of mathematics cultivates virtues that help people flourish.  These virtues serve you well no matter what profession you choose.  And the movement towards virtue happens through basic human desires.

I want to talk about five desires we all have.  The first of these is play.

Yeah, so the first of these fundamental human desires is play, and the whole list looks like this:

  1. Play
  2. Beauty
  3. Truth
  4. Justice
  5. Love

The way Su sees things, if you use math to chase these desires then you end up cultivating virtues. Here are some of those virtues:

  • Hopefulness
  • Perseverance
  • Joy
  • Transcendence
  • Rigorous Thinking
  • Humility
  • Circumspection

(Justice and Love are desires that don’t end in virtues for Su, it seems, as he doesn’t name any virtues in those sections of his talk. Instead, pursuing love is supposed to enable your other pursuits because it results in seeking meaningful human connections: “Love is the greatest human desire.  And to love and be loved is a supreme mark of human flourishing.  For it serves the other desires—play, truth, beauty, and justice—and it is served by them.”)

(So, then what’s the role of Justice? Maybe for Su it’s sort of like a necessary consequence of valuing all these other things. To truly desire all these other things is to desire them for everyone, and that necessarily is the pursuit of justice? I’m just making stuff up here, I’m not sure what Su thinks.)

Anyway, basically none of Su’s talk resonates with me. I don’t mean that I think he’s wrong.* I mean it doesn’t resonate — it doesn’t hit my frequency, make me hum.

* Well, I guess I don’t have any confidence that the study of math itself can impart any of these virtues. I don’t know if he’s claiming that they will, though I think he is.

OK, fine, but then what do I think?

One of my favorite games is “long response to short thing OR short response to long thing.” Let’s play “short response to long thing”:

Michael Pershan’s Version of Math for Human Flourishing:

Five Fundamental Desires:

  1. Understanding
  2. Belonging
  3. Growing
  4. Teaching
  5. I’m not sure but I came up with four

Virtues that one MIGHT develop in math, but I make no commitments about how frequently or reliably even an idealized version of math education could promote these in our students, since virtues are BIG things and math is just ONE thing:

  • Humility
  • uhhh this is also hard
  • I’m totally stuck

Maybe not “virtues,” but ways in which I think math makes my life richer:

  • Math helps me know that, to really understand something or someone, I need to give it my full attention.
  • It gives me an arena in which to grow.
  • It gives me questions to share with others.
  • Is there a name for that thing where you’re walking down the street and you can’t not see parallel lines or tessellations or similar triangles? The math curse? I love that.
  • I love theory. I love that thing where you take a perfectly conventional idea and flesh it out, completely. I recently read that topology came, in part, in response to Cantor’s proof that the line and the plane were equinumerous. But clearly there is a difference between the two — how can we capture that? “How can we capture that?” is one of my favorite questions to ask.
  • I love that math gives me things to help other people think about.