$$ \newcommand{\rmap}[3]{#1:#2\rightarrow #3} \newcommand{\lmap}[3]{#1:#2\leftarrow #3} \newcommand{\map}[3]{\rmap{#1}{#2}{#3}} \newcommand{\reals}[0]{\mathbb{R}} \newcommand{\xreals}[0]{\mathbb{R}\cup\{\infty\}} \newcommand{\ub}[1]{\rm{ub\ #1}} \newcommand{\lb}[1]{\rm{lb\ #1}} \newcommand{\glb}[1]{\rm{glb\ #1}} \newcommand{\lub}[1]{\rm{lub\ #1}} \newcommand{\ftom}[4]{\glb{ \left\{#2#1 |\, {}^*#2 = #3\ \rm{and}\ #2^* = #4\right\}}} \usepackage{mathrsfs} \newcommand{\alg}[1]{\mathscr{#1}} \newcommand{\complexes}[0]{\mathbb{C}} $$

Why Is Mathematics?—An Exploration of All Things Mathematical

Welcome!

I've been a student and researcher of mathematics and its applications for essentially all my life. I was 15 when I began my second semester as a sophomore majoring in Physics, 16 when I taught my first college math class, and have never stopped. In the meantime I've been a professor, at Stanford and then Columbia, a software entrepreneur, an inventor.

I've noticed a few things which I'm finally starting to put into a consumable form. I hope you'll enjoy.

This is the beginning of a work in progress. I'd love to hear your comments and questions.

Cognitive Functions

Making Love and Feeling It

I made some interesting discoveries, in collaboration with Susan Lawlor, about how personality works, and how that impacts relationships. In particular, we were able to identify how personality determines how we perceive love, and what makes us feel loved, and what makes us feel unloved. It's mildly surprising that those involve two distinct mechanisms. If you're familiar with Gary Chapman's concept of love languages, it was inspired by that work, and elaborates on it, showing how some of its subtleties are best understood in the context of Jung's eight cognitive functions. In addition, it provides a roadmap for discerning what specific kinds of things will be received as love and appreciation, without reference to love languages. In Making Love and Feeling It, a work in progress, I describe the discoveries, and show you how you can use the insights to make your relationships stronger, more resilient, and more satisfying to you and them, whether it's a romantic relationship, a friendship, a partnership, or your children or grandchildren.

Computing

Monads

I wrote a blog post sorting out several equivalent definitions of computational monads, and pointing out a curious structural fact related to the twin association structures of a category.

Optimization

I've recently done some work with optimization, and wrote about a way of bringing it transparency. Sometimes, it turns out that the lack of transparency was by design, and it can be a bad career move to be the one who shines the light. I also recently made a tweak to an optimization algorithm with big positive revenue consequences; it's a nice small illustration of a philosophy of Lax' about how to do numerical analysis.

A One-pass Linear Solver

Not too long ago I was designing an analytics language. The team wanted to illustrate it with logistic regression. The standard linear solvers which find $x$ satisfying $Ax=b$ where $A$ is linear, and $x$ and $b$ are vectors, are a bit wasteful for that problem, because they are optimized for having multiple $b$ values for which $x$ is needed. In logistic regression, $A$ changes with each new $b$. I thought that the same recursion that yields the Cholesky decomposition would yield a one-pass algorithm; it does.

Vector fields and finite state machines

Some time ago now, exploring the axioms for a mathematical category, for reasons having to do with Quantum Mechanics, I realized that one particular axiom embraced many other parts of mathematics. One slightly unexpected piece, is the notion of quiver, which generalizes graphs by encompassing multiple paths. As such, on them you can define machines. But then those machines end up looking like differential equations.

A Simple Expression Parser

By no means should you implement a parser. That machinery already exists. But having said that, I was thinking today about implementing one, and wasn't remembering all the details of recursive descent, or precedence parsing. Rather than look them up, I hit on an alternate approach, which seems, at least to me, to be much simpler.

An iterator on a tree - a Python generator case study

I had occasion to write a program for traversing a tree. The trick was that I needed to create a function which could be called repeatedly to retrieve values as needed. I saw that this could be a nice use of generators, but I didn't think it would be too painful to do without them. I was wrong. I'll spare you the pain, and show you how slick the generator-based way is. There's a tree-specific part in the form of a Tree class, and a sample driver program that uses it.

Physics

It's standard in Physics courses to show that Newton's laws of motion and gravity produce Kepler's laws of planetary orbits. It's possible to go the other way. I think that Helaman Ferguson told me that Wayne Barrett had shown him that calculation, but Helaman doesn't recall the conversation. At any rate, it got me to thinking, and I worked out what I think is a plausible explanation for how Newton came up with his laws. Most of it is just a calculation. Nevertheless, it may turn out that Jupiter was an important clue.

Mathematics

Why is Mathematics

If you throw away all but one of the axioms describing a category, and keep only the one describing associativity, you obviously get a more general structure. It will obviously encompass categories, and it's trivial that it also describes semigroups, and sets. I found it surprising that it also includes quivers, so in particular graphs, as well as things like group actions. Moreover, all of these structures can be seen to arise from a structure theory. This illustrates a central theme in mathematics. For example, if you have a group (say permutations), if it has something called a normal subgroup, that gives you a way of breaking the group up into simpler pieces, although the way those pieces interact could stil be quite complicated. This strategy won't work on a group that has no normal subgroups. Such a group is called simple. But now we can view the property of not having normal subgroups as a new axiom—for simple groups. That new axiom is so powerful that it has proved possible to completely classify all the finite possibilities. Similarly, in exploring this axiom, various possibilities for decomposing into simpler structures arise. The negation of those possibilities spawn new axioms, and pretty much all of mathematics rises out of the mist. I had worked this all out quite some time ago, but had been puzzled by a missing piece. I didn't see how topology fit into the picture. I also hadn't yet realized how central rings are within the concept of category. With those pieces having fallen into place, a nice coherent picture emerges.

Symbols

Mathematics uses lots of symbols. If you're not familiar with the Greek alphabet, it's easy enough to become so. Check it out.

I'm Andrew Winkler. I hope you've enjoyed exploring these ideas with me. Thanks for your time. If you've got any questions, send an email to 4af502d7148512d4fee9@cloudmailin.net.

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