Yesterday I answered one of Coupon Clipper’s questions about Godel’s Theorem. Today I’ll tackle the other: Does Godel’s Theorem matter on a day-to-day basis to practicing mathematicians?
And the answer is: Of course not. Mathematicians care about what’s true, not about what’s provable from some arbitrary set of axioms. (Of course this is an overgeneralization; some mathematicians have built distinguished careers on worrying about what’s provable from various sets of axioms. But they are a small minority.) Godel’s Theorem says that not all true things are provable. But for the most part, we’re happy just to know they’re true.
The flashiest example I can give you—and one I’ve used on this blog before—is Fermat’s Last Theorem, which says that no equation of the form xn + yn = zn has any solutions, as long as n is at least 3 and x, y and z are non-zero. Proving this was the was most famous unsolved problem in mathematics for 350 years until it was solved (to much public fanfare) by Frey, Serre, Ribet and Wiles in the 1980’s and 1990’s.
We know from that work that Fermat’s Last Theorem is true. However, we still don’t know whether Fermat’s Last Theorem follows from the standard axioms for arithmetic. And—this is the point—very few mathematicians care very much, at least by comparison to how much they care about the theorem itself. (Here is one of my favorite papers on the subject. Tellingly, the author is a philosopher.)
Continue reading ‘Godel, Fermat, Hercules’
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The occasional commenter who goes by the name Coupon Clipper has emailed me some interesting questions about Godel’s Theorem. I think I’ll answer them here.
The first question is about first-order versus second-order logic, so let me explain the distinction. When we reason formally about arithmetic, we need to clearly specify the ground rules. This means, among other things, specifying the language and grammar we’re allowed to use. A very simple language might allow us to say things like “2 + 3 = 5″ or “8 is an even number”. With a language like that, you could talk about a lot of sixth grade arithmetic, but you wouldn’t be able to say anything very interesting beyond that. To talk about the questions mathematicians care about, you need a language that contains words like “every”, as in Every number can be factored into primes or Every number can be written as a sum of four squares or Every choice of positiive numbers x, y, and z will yield a non-solution to the equation x3+y3=z3 . That language is called first-order logic.
With first order logic we can specify a set of axioms, and then follow a prescribed set of rules to deduce consequences. For example, if you’ve already established that every number is a sum of four squares, then you’re allowed to conclude that 1,245,783 is a sum of four squares. (The general rule is that if you’ve proved that every number has some particular property, then you’re allowed to conclude that any particular number has that property.)
Continue reading ‘First Things and Second Things’
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I’ve just finished reading The Girl Who Played with Fire, the second book in the series that begins with The Girl with the Dragon Tattoo. I’m not giving away any significant plot point when I tell you that there’s a character who works on Fermat’s Last Theorem as a hobby, or that the book was clearly written (or perhaps translated) by somebody with no clue how mathematics works or what Fermat’s Last Theorem is about. I particularly liked the reference to Andrew Wiles using the “world’s most complicated computer program” to solve the problem. It’s my understanding that Andrew barely even uses email. And certainly if you understood anything about the nature of the problem and/or the solution, you’d recognize the absurdity of trying to tackle it with a complicated computer program.
Be that as it may, I finished the novel with a few hours left to spare, so of course I was inspired to work on Fermat’s Last Theorem, or at least on the simplest cases. The problem, if you’ll recall, is to show that there are no integer solutions to any of the equations x3+y3=z3 , x4+y4=z4 and so on, except for the so-called trivial solutions in which one or more variables take the value zero.
Continue reading ‘The Girl Who Played With Numbers’
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If there is a God, this is the closest you’ll ever come to hearing Him sing. Let me explain.
Continue reading ‘The Music of the Primes’
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My colleague Ralph Raimi is witty, acerbic and wise about many things, but particularly about mathematics education. A little time spent browsing around his web page will reap ample rewards in the form of both entertainment and edification. Today I’d like to share a little passage he sent me by email:
I have never tried to count the times I have read a newspaper article explaining that students are bored with math that has no visible practical application, and follows with an example of a teacher, or club, that rectifies the situation in some novel and engaging way.
In the present case a class has built a sculpture that resembles a graph of a modulated wave motion. Of all the practical, real-world
applications of mathematics! It is as practical as a snowman.
Why doesn’t anyone ask for real-world applications of table tennis? What a bore any game must be, that has no real-world application! Why do kids stand for it? Ping-pong again? Ugh.
But I can think of something: Let’s all make a model of a ping-pong ball in the school yard, seventy feet high, blocking all the entrances and thus preventing all us students from entering the (ugh) school. Then we can take our fishing poles and torn straw hats out from under our beds and, with the hats on our heads and fishing poles over our shoulders, all traipse together down the dusty road to Norman Rockwell’s house.
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Yesterday I told you about one of the deepest problems in arithmetic. Today I’ll explain how you can help solve it.
We’re on the hunt for ABC triples. A brief recap: We start with an equation of the form A+B = C, where A, B and C have no factors in common. We find all the primes that divide A, B or C, multiply them together and call the result D. The goal is to find examples where C is bigger than D.
If I start with 2+243=245, the primes are 2 (which divides 2), 3 (which divides 243), 5 (which divides 245) and 7 (which also divides 245), so D = 2 x 3 x 5 x 7 = 220, and C (that is, 245) is bigger than D. Success! We’ve found an ABC triple.
We want more. A full understanding of ABC triples would allow us to solve some of the hardest open problems in arithmetic. More importantly, the reason we’d be able to solve those problems is that we’d understand arithmetic itself a whole lot better.
The first step is to find a whole lot of examples to help researchers guess at the underlying patterns.
That’s where you come in.
Continue reading ‘ABC at (Your) Home’
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Some of the hardest problems in arithmetic are those that relate multiplication to addition. For example: Is every even number the sum of two primes? This is most assuredly a hard problem—mathematicians have been tackling it for centuries and so far nobody’s solved it. And it relates multiplication to addition. As soon as you talk about primes, you’re (implicitly) talking about multiplication, and of course when you talk about sums, you’re talking about addition.
Or: How many ways can you write the number 2 as the difference of two primes? You can write 2 = 5-3, or 2 = 7-5, or 2 = 13-11. That’s three so far. How many more are there? The betting is that the answer is “infinitely many”, but nobody knows for sure. This problem has stumped some of the best and the brightest not just for centuries but for millennia. And again it relates multiplication to addition. (Well, it relates multiplication to subtraction, but of course subtraction is just addition in reverse.)
The ABC problem has only been around for a few decades, but it’s in many ways the most interesting and important of the bunch. Tomorrow I’ll explain how you can help solve this problem. Today I’ll explain what the problem is.
Continue reading ‘The ABC’s of Arithmetic’
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During our brief intermission last week, commenters chose to revisit the question of whether arithmetic is invented or discovered—a topic we’d discussed here and here. This reminded me that I’ve been meaning to highlight an elementary error that comes up a lot in this kind of discussion.
It is frequently asserted that the facts of arithmetic are either “tautologous” or “true by definition” or “logical consequences of the axioms”. Those are three different assertions, and all of them are false. (This is not a controversial statement.)
The arguments made to support these assertions are not subtly flawed; they are overtly ludicrous. Almost always, they consist of “proof by example”, as in “1+1=2 is true by definition; therefore all the facts of arithmetic are true by definition”. Of course one expects to stumble across this sort of “reasoning” on the Internet, but it’s always jarring to see it coming from people who profess an interest in mathematical logic. (I will refrain from naming the worst offenders.)
So let’s consider a few facts of arithmetic:
Continue reading ‘Just the Facts’
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Blogging, as you might have heard, is changing the face of the media. It may also be changing the face of mathematical research. For the first time ever, a substantial mathematical problem has been solved via an accumulation of blog comments, all building on each other. Could this be the future of mathematical research?
Before I explain the problem, let’s talk a little about tic-tac-toe. As you probably figured out long ago, intelligent players of ordinary tic-tac-toe (on a 3 by 3 board) will invariably battle to a draw. But, as you probably also figured out, not every game ends in a draw, because not every player is intelligent.
On the other hand, if we blacken out the three squares on the main diagonal and don’t allow anyone to play there (so the game ends when the remaining six squares are filled, then every game is sure to end in a draw. There’s simply no way to get three in a row when you’re not allowed to play on the diagonal:
Continue reading ‘Blogging, Tic Tac Toe and the Future of Math’
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In high school, we used to play four-dimensional tic-tac-toe. The board looks like this:
Here each four-by-four subsquare is an ordinary tic-tac-toe board (except that it’s four-by-four instead of the traditional three-by-three). You should think of the four subsquares in the first column (or any other column) as stacked above each other in the third dimension. The red x’s form a vertical line in that direction, so if you manage to place four x’s in those positions, you’re a winner.
You should also think of the four subsquares in the first row as stacked above each other in yet another dimension. The red o’s form a diagonal line passing from the bottom left to the top right (using “bottom” and “top” to refer to directions in this fourth dimension). And the black x’s form another kind of diagonal line, passing from one corner to another through all four dimensions. So there are a lot of ways to win this game.
Continue reading ‘Tic Tac Toe in Four Dimensions’
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It’s a bit of an odd feeling to be reading a novel and stumble upon yourself as a character. Well, at least a well-disguised version of yourself. The novel is Victor Snaith’s The Yukiad, and the character is a large Scotsman named Pans who tugs at his earrings when he becomes agitated. I am neither Scottish, nor earringed, nor particularly large, but I suspect that Pans, viewed through the haze of poetic license, is I.
When we meet Pans, he is hovering over a glass contraption—a perpetual motion machine, really—consisting of a circular tube containing several colored beads, which travel around the tube, some clockwise, some counterclockwise, all at the same speed, bouncing off each other in perfectly elastic collisions whenever they collide. Pans is currently tugging at his earrings so hard as to cause some concern for the integrity of his earlobes, as he ponders the following question:
But wull tha’ aver gut bark to weer tha’s started, at a’, at a’?
Well, okay, maybe I’m not Pans. Maybe I’m the character Sherloch Humes, a “trim but rather wrinkled gentleman in worsteds”, who calculates for Pans’s benefit that “the configuration of beads is guaranteed to have exactly replicated itself by the year two thousand and nineteen”. I believe that I am the inspiration for one of these characters and that the mathematician Leonid Vaserstein (who is neither Scottish nor wrinkled) is the inspiration for the other, and here is why:
Continue reading ‘The Yukiad, Perpetual Motion and Me’
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Wall Street quants are always trying to dream up new financial products that nobody’s figured out how to regulate. Sooner or later, I suppose, one of them will come up with a bank account that pays imaginary interest. You deposit a dollar and a year later you get an interest payment of i. That’s not “i for interest”; it’s the square root of minus one. I have no idea what that means for economics, but thinking about it is a good way to understand Euler’s (or, the historical record being unclear, perhaps Johann Bernoulli’s) breathtakingly beautiful formula
Continue reading ‘Financial Imagineering’
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Professor Joseph Weiler, who is facing criminal charges in France for posting a mildly negative book review on a web site he edits, has asked supporters to search out and email him copies of even more negative reviews (presumably of academic writing), to submit to the court as evidence that this sort of thing happens all the time.
The review I’ll be emailing is a classic of the genre. It was written by Andre Weil, one of the most influential mathematicians of the twentieth century, and possibly the most erudite person who ever lived. Here’s how I described Weil shortly after his death:
Continue reading ‘The Hunting of the Snark’
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Today is the 165th birthday of Georg Ferdinand Ludwig Philipp Cantor, the mathematician who indirectly inspired me to major in math. In my first few semesters of college, I was at best an indifferent student, finding little inspiration in the humanities majors I was bouncing around among, playing a prodigious amount of pinball, and attaining (according to rumor) history’s first-ever grade of C in Peter Regenstrief’s Poltical Science 101. Then one day, my friend Bob Hyman happened to mention that some infinities are larger than others, and set my life on track. This—the vision of Georg Cantor—was something I had to know more about. Before long I was immersed in math.
What does it mean for some infinities to be larger than others? Well, for starters, some infinite sets can be listed, while others are too big to list. The natural numbers, for example, are already packaged as a list:
The integers, by contrast (that is, the natural numbers plus their negatives) aren’t automatically listed because a list, by definition, has a starting point, whereas the integers stretch infinitely far in both directions. But we can fix that by rearranging them:
So the integers can also be listed.
Continue reading ‘Split Infinities’
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In 1958, the 30-year-old Alexandre Grothendieck stunned the International Congress of Mathematicians with his audacious proposal to remake the foundations of algebraic geometry, vastly expanding the scope of the field, subsuming all of commutative algebra and algebraic number theory, and paving the way for the solution of the elusive Weil conjectures, then considered decades or centuries out of reach. No mathematical vision had ever been more radical or more ambitious. Someday I will blog about that vision. Today’s post is about genius, eccentricity and intellectual property.
Continue reading ‘Bringing in the Sheaves’
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A musician wakes from a terrible nightmare. In his dream he finds himself in a society where music education has been made mandatory…Since musicians are known to set down their ideas in the form of sheet music, these curious black dots and lines must constitute the “language of music”. It is imperative that students become fluent in this language if they are to attain any degree of musical competence; indeed it would be ludicrous to expect a child to sing a song or play an instrument without having a thorough grounding in music notation and theory. Playing and listening to music…are considered very advanced topics and generally put off till college, and more often graduate school.
As for the primary and secondary schools, their mission is to train students to use this language—to jiggle symbols around according to a fixed set of rules: “Music class is where we take out our staff paper, our teacher puts some notes on the board, and we copy them or transpose them into a different key…One time we had a chromatic scale problem and I did it right, but the teacher gave me no credit because I had the stems pointing the wrong way.”
…
Sadly, our present system of mathematics education is precisely this sort of nightmare.
So begins Paul Lockhart’s scathing critique of how mathematics is taught in this country, A Mathematician’s Lament. The book is an expansion of Lockhart’s essay of the same title. I encourage you to read the essay, buy the book, and share your thoughts in comments.
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Yesterday we started a conversation about whether mathematics is invented or discovered. Today I’ll give you my three best arguments for “discovered”. And to focus the discussion, I’ll talk not about mathematics generally but about the natural numbers (0,1,2, and so forth) in particular.
I believe the natural numbers exist, quite independently of whether anyone’s around to think of them. Here’s why: First, we perceive them directly. Second, we know non-trivial facts about them. Third, they can explain the Universe. In more detail:
Continue reading ‘Real Numbers’
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Is mathematics invented or discovered? In my experience, applied scientists often think of mathematics as a human invention, while actual mathematicians (with a few notable exceptions) feel sure that mathematics was always there to be discovered. (Of course, it’s sometimes hard to tell how much of this is genuine disagreement and how much is a language barrier.)
I’ve just finished reading an excellent book by Mario Livio which is entirely about the invention/discovery question, though he’s chosen the (somewhat unfortunate) title Is God a Mathematician? Much of the book is a lively romp through mathematical history, with a well chosen mix of biography and exposition. Although he parts company with them in the last chapter, Livio gives a more than fair hearing to the many great mathematicians who have insisted that they are discoverers, from Pythagoras through Galileo, G.H. Hardy, Kurt Godel, and the contemporary Fields Medalist Alain Connes (among others). Here, for example is Connes:
Continue reading ‘Jellyfish Math’
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The Intelligent Design folk tell you that complexity requires a designer.
The Richard Dawkins crowd tell you that complexity must evolve from simplicity.
I claim they’re both wrong, because the natural numbers, together with the operations of arithmetic, are fantastically complex, but were neither created nor evolved.
I’ve made this argument multiple times, in The Big Questions, on this blog, and elsewhere. Today, I aim to explain a little more deeply why I say that the natural numbers are fantastically complex.
Continue reading ‘Non-Simple Arithmetic’
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On his blog A Blank Slate, Vishal Patel posts a cute little brain teaser (with a hat tip to the Cosmic Variance blog):
Jack is looking at Anne, but Anne is looking at George. Jack is married, but George is not. Is a married person looking at an unmarried person?
(a) Yes
(b) No
(c) Can not be determined
This reminded me of one of my favorite little “zinger” math proofs. (If you think about the brain teaser long enough, you’ll see the connection.)
Continue reading ‘Rational Irrationality’
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The mathematician John Baez has been dazzling science lovers on the web for over 15 years with his weekly Finds in Mathematical Physics. (He was a blogger long before there were blogs). Baez recently gave a lovely series of talks on his favorite numbers (they are 5, 8 and 24) in which he mentions Euler’s observation that if you sum up all the positive integers (1 + 2 + 3 + 4 + …) you get -1/12. (I promise, this is not a joke.)
Baez’s “proof” uses a little calculus, but I’ve reworked it into a form you can share with your middle schoolers—and better yet, have them share with their teachers.
Continue reading ‘A Little Arithmetic’
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Among the things you’re sure of, which are you surest of? For Richard Dawkins, writing in the Wall Street Journal, it’s the theory of evolution:
We know, as certainly as we know anything in science, that [evolution] is the process that has generated life on our own planet.
Now, I would be thunderstruck if the theory of evolution turned out to be fundamentally wrong, but not nearly so thunderstruck as if arithmetic turned out to be inconsistent. In fact, I can think of quite a few things I’m more sure about than evolution. For example:
1. The consistency of arithmetic. (This amounts to saying that a single arithmetic problem can’t have two different correct answers.)
2. The existence of conscious beings other than myself.
3. The fact that the North won the American Civil War. (That is, historians are not universally mistaken about this. I am not interested in quibbling about what constitutes a “win”; I mean to assert that the North won in the everyday sense of the word, as reported in all the history texts.)
Continue reading ‘What Are You Surest Of?’
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In The Big Questions (pages 18-19) I talk (channeling the physicist Eugene Wigner) about the apparently unreasonable effectiveness of mathematics in revealing truths about the physical world. In Wigner’s words, “It is difficult to avoid the impression that a miracle confronts us here.”
But the physicist Peter Landsberg (no relation!) observes that sometimes the miracle runs in the opposite direction, and offers a curious use of physical reasoning to reveal a purely mathematical truth!
Continue reading ‘The Unreasonable Effectiveness of Physics’
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