The Cubic Formula and Galois Theory

I read a couple of books fairly recently that discuss the cubic equation: Barry Mazur’s Imagining Numbers: (particularly the square root of minus fifteen) and William Dunham’s Journey through Genius. They got me interested in this formula, which had always been an afterthought (to the extent to which it had been a thought at all) in my mathematics education.

The cubic and quartic formulas were perhaps the greatest accomplishments of Renaissance mathematics, and they were a key spur to Evariste Galois’ beautiful study of the symmetry inherent in polynomial equations. Yet I got a PhD in mathematics without knowing much more than the bare fact that such formulas exist. It should certainly be possible to understand these formulas in light of Galois theory, but such explanations are not easy to find.

I wrote this paper to pass on what I learned about the cubic equation, and how it can be derived in the context of Galois theory.

Continue reading “The Cubic Formula and Galois Theory”

Errol Morris Resurfaces

Errol Morris, the noted documentary filmmaker, has recently published a book entitled, The Ashtray (Or the Man Who Denied Reality). It’s about my father, Thomas Kuhn, and the views on the history and philosophy of science that he initially set forth in his 1962 book, “The Structure of Scientific Revolutions.” As you might gather from the title, it is not a sympathetic account; the philosopher Philip Kitcher says, in his excellent review in the Los Angeles Review of Books, that “Morris disarmingly confesses [that] this book is a vendetta.”

Since the book’s publication, Morris has been appearing here and there on the radio, and friends have been asking me about my take on the book. Rather than saying the same thing over and over, I have decided to put my thoughts together in one place. I haven’t read Morris’s book, for reasons that will become clearer to those who read further, but I was a careful reader of his five-part 2011 series on the New York Times Opinionator website, which the book is based on. I find Morris’s reading of my father’s work a vast oversimplification, to the point where the straw man is easy to knock down. Many people have taken that approach over the years, but Morris has a higher profile than most. There are many people more qualified to deal with his criticisms on whatever merits they may have, and I will mostly leave it to them.  What impelled me to write this is my belief that the episode from which the book draws its title, in which my father supposedly threw an ashtray at Morris, never actually happened.

Is it important whether it happened or not? Everyone will have to come to their own conclusion on that. To be very clear up front: I would love to be wrong about this. I would love to hear from Morris’s classmates, family, or anyone else, that he told them that my father threw an ashtray at him at or close to the time that it allegedly happened. Because as misguided as I think his attacks on my father are intellectually, I believe that they come from a place of wanting to defend truthfulness. Defending truthfulness was important in 2011, and it is painfully obvious that it is more important than ever in 2018. I have great respect for Morris’s film work, and I think I share much of his political orientation. But just as the Buddha says hatred does not cease through hatred, falsehood does not cease through falsehood, and I believe that some of our current ills as a society stem from our delight in calling out falsehood among our opponents while ignoring it or even condoning it on “our team” and, more important, in ourselves. Continue reading “Errol Morris Resurfaces”

Links to writing on “Black Panther”

I’m not a big fan of superhero movies, and I generally find any efforts to appear “thought-provoking” are pretty tinny.

But  I would say that Ryan Coogler’s Black Panther is actually thought-provoking. Coogler and his cast and crew have accomplished so many things in this movie that it’s hard to know where to start. But one thing they’ve done is to evoke some basic moral/philosophical dilemmas and conversations in a way that doesn’t seem completely, uh, comic-book. Continue reading “Links to writing on “Black Panther””

Guide to My Bread Posts

A number of people have expressed interest recently in my experience making bread. I’m happy to talk about it, but I also refer them to my posts here. There are enough of those posts now that I wind up rewriting the same little email about how to get started. Rather than writing the same email over and over, here is a slightly more polished version. [Added 30 July 2018: I have often said that baking bread is what has helped me survive the current political scene; here is a fleshed-out version of that thought, suggesting that it can also help you survive the on-line world that we are consumed by.] Continue reading “Guide to My Bread Posts”

Make awesome waffles (or pancakes) with that extra sourdough starter

We love this waffle recipe. A visitor from Germany said these waffles were the best she’d ever had, and I think of Europeans as knowing their waffles. Don’t tell people how healthy they are, just serve them and tell them later (or not). This is adapted (i.e. mostly copied) from this recipe, but it is doubled and it generally serves 4. When I make one-and-a-half of this recipe (triple the original), it makes exactly 10 waffles on my 8″ Belgian waffle maker. I haven’t tried making pancakes with it, but it’s supposed to work. Continue reading “Make awesome waffles (or pancakes) with that extra sourdough starter”

Ack! I’m not ready to use my sourdough starter!

OK, so someone (maybe me!) gave you some sourdough starter and it is sitting in your fridge. You might want to use it someday… like maybe soon… or maybe not… to make bread. In the meantime, you don’t want it to die. (If you do want it to die, that is OK. Just throw it out.) You also don’t want to wade through that long post about natural leavening. Well, you have come to the right place. Continue reading “Ack! I’m not ready to use my sourdough starter!”

Bread-Making Equipment (with occasional remarks on technique, and a couple of edits, 3 years later)

Compared to photography, or opera, or boating, bread-making is a pretty inexpensive hobby. Nevertheless, there is a somewhat dizzying array of implements available to help (and sometimes hinder) you, all available in a vast range of prices.  Many of the implements have fancy French names, and come to think of it, if you figure it on a per-fancy-French-name basis, bread-making must be about the cheapest hobby there is.

This post describes the equipment I have settled on at the moment as most helpful for me, with some comments. Continue reading “Bread-Making Equipment (with occasional remarks on technique, and a couple of edits, 3 years later)”

Making Bread with Natural Leavening (Sourdough Starter) Needn’t Be Complex or Time-Consuming

OK, let’s get a few things out of the way.  I don’t really like to call bread made with natural leavening “sourdough” because a lot of people immediately turn off, saying, “I don’t like sourdough bread.”  Naturally leavened bread doesn’t necessarily taste sour. The phrase “naturally leavened” is a little leaden and pedantic, so I instead favor the French word “levain,” which sounds pretentious and, uh, pedantic. Oh well, “You pays your money and you takes your choice,” as they say.  “Starter” works pretty well when it’s not ambiguous.

Here are the top six reasons why I bake bread using natural leavening, in rough order of importance:

1. It tastes better.

2. It tastes better.

3. It tastes better.

4. It stays fresh longer.

5. It is healthier (lower glycemic index).

It seems amazing and delightful to me that the only store-bought ingredients in my bread are flour and salt. The water comes from the tap and the yeast comes from the starter, and ever-renewable resource.

Continue reading “Making Bread with Natural Leavening (Sourdough Starter) Needn’t Be Complex or Time-Consuming”