Today’s question is: Do numbers exist? The answer is: Of course, and I don’t believe there’s much in the way of serious doubt about this. You were familiar with numbers when you were five years old, and you’ve been discovering their properties ever since. Extreme skepticism on this point is almost unheard of among mathematicians or philosophers, though it seems to be fairly common among denizens of the Internet who have gotten it into their head that extreme skepticism makes them look sophisticated.

Let me be clear that I am not (yet) asking in what sense the natural numbers exist — whether they have existed since the beginning of time, or whether they exist outside of time, or whether they exist only in our minds. Those are questions that reasonable people disagree about (and that other reasonable people find more or less meaningless.) We can — and will — come back to those questions in future posts. For now, the only question: Do the natural numbers exist? And the answer is yes. Or better yet — if you believe the answer is no, then there’s obviously no point in thinking about them, so why are you reading this post?

“Existence” here is used in the ordinary everyday sense of the word, according to which rocks and trees exist, you and I exist, your hopes and dreams exist, and the idea of a unicorn exists. Unicorns themselves do not exist and therefore it makes no sense to study their properties. (Though you can have fun inventing some properties for them.) By contrast, it makes perfect sense for geologists to study the properties of rocks, for botanists to study the properties of trees, for folklorists to study the properties of the idea of a unicorn, and for mathematicians to study the properties of the natural numbers.

An extreme skeptic might deny the existence of rocks. The only possible answers are: a) It’s hard to believe you’re serious, since you’ve been encountering rocks — just like you’ve been encountering numbers — your entire life. b) If you really are serious, I suppose your best strategy is to stop thinking about rocks, and leave them to those of us who find geology interesting. And c) Do not fool yourself into believing that your position is anywhere close to any mainstream school of thought.

Another extreme skeptic might deny the existence of numbers. I’ll leave it to my readers to replace rocks with numbers in the above retorts.

What else might one say to an extreme skeptic? Answer: One might attempt to acquaint him with Godel’s Completeness Theorem. (This is not the same as the far more famous Godel’s Incompleteness Theorem.) Here is (part of) what the Completeness Theorem says: First, without making any assumptions about existence, write down a list of axioms for the natural numbers. For example, write down the Peano Axioms. Then the Completeness Theorem tells you that as long as those axioms are consistent, there must be some mathematical structure that obeys those axioms. (Note that “be” is a synonym for “exist”.) The smallest of those structures (known as “models”) is our good old friend the natural numbers.

In other words, Godel’s Theorem tells you that if the Peano axioms are consistent, then the natural numbers must exist. (Don’t confuse the map with the territory! “Consistency” applies to the axioms; “existence” applies to the natural numbers themselves.)

On the other hand, we can also argue in the opposite direction: If the natural numbers exist, then the Peano axioms, being true statements about existing objects, must be consistent. An accurate map of an existing territory cannot contradict itself.

So — We know that the natural numbers exist because we know the Peano axioms are consistent. And we know that the Peano axioms are consistent because we know that the natural numbers exist. Does that sound circular? It’s not. Here’s the point: We have extremely good reasons for believing in the existence of the natural numbers (beginning with intuition, lifelong familiarity, and the fact that we seem to be able to discover their properties). We have (partly) separate extremely good reasons for believing in the consistency of the Peano axioms (beginning with intuition and the fact that they’ve never yet led us to a contradiction). The fact that our two beliefs reinforce each other — that if either is true, then so must be the other — should build up our confidence that the whole picture hangs together.

Now let’s get back to our extreme skeptic. He denies the existence of the natural numbers. We respond that Godel’s Completeness Theorem proves the existence of the natural numbers, as a consequence of the consistency of the Peano axioms. He now has only two recourses (other than to concede defeat). One is to deny the consistency of the Peano axioms, and the other is to deny the accuracy of Godel’s Completeness Theorem. Let’s see how those strategies are likely to work out for him.

Should he doubt the consistency of the axioms? The Peano Axioms lay out the rules of arithmetic that you’ve used your whole life; they say things like “Every number has exactly one immediate successor” and “x + (y+1) = (x+y) + 1”. People (and to some extent animals) have been applying these axioms, explicitly or implicitly, since long before the dawn of history and no contradiction has ever arisen; moreover, for what it’s worth, the consistency of these simple axioms is instantly clear to most people’s intuitions. If we were to jettison our belief that these axioms are consistent, then we’d pretty much have to give up all quantitative reasoning.

Well, then, should our skeptic doubt Godel’s Completeness Theorem? The theorem is proved using elementary notions about sets — the idea that it’s possible to talk about sets of things and about membership in a set, that it’s possible to form the union of two sets, and so on. This has nothing to do with the more esoteric subject of “axiomatic set theory”; instead, it uses only the most fundamental notions associated with forming collections of things. (These notions, in fact, are prerequisite for axiomatic set theory and therefore cannot depend on it.) Once again, if you were to abandon this sort of reasoning, you’d pretty much have to abandon reasoning altogether.

For anyone who accepts the simplest sorts of combinatorial reasoning, there is no longer an out. The natural numbers are real. Again, this says nothing about where they came from — be it Plato’s heaven, the minds of humans or the mind of God. We’ll get back to that in the next installment of this occasional series.

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Congratulations to the 2010 Fields Medalists, announced yesterday in Hyderabad. Elon Lindenstrauss, Ngo Bau Chau, Stanislav Smirnov, and Cedric Villani have been awarded math’s highest honor. (Up to four medalists are chosen every four years.)

My sense going in was that Ngo was widely considered a shoo-in, for his proof of the Fundamental Lemma of Langlands Theory. Do you want to know what the Fundamental Lemma says? Here is an 18-page statement (not proof!) of the lemma. The others were all strong favorites. Nevertheless:

• Ngo’s Wikipedia page was created in May, 2007 — three years after he had won the prestigious (though not quite as prestigious as the Fields Medal) Clay Research Award.
• Smirnov’s Wikipedia page was created in May, 2009 — eight years after he had won the Clay Research Award.
• Villani’s Wikipedia page was created in May, 2010 — just a few months ago, and two years after he’d won the prestigious European Math Society award. (Moreover, until my friend Tim fixed it this morning, the page failed to come up if you searched for “Villani” on Wikipedia.)
• Lindenstrauss’s Wikipedia page was less than one week ago — four days before he won the Fields Medal, and six years after he won the European Math Society award. .

Now I realize that of all the wonderful free services we get from the web, the one we have least cause to complain about is Wikipedia. Still, I’m struck that as recently as a week ago, a mathematician with the caliber and influence of Elon Linderstrauss did not have a Wikipedia page, and a few months ago the same was true of Cedric Villani.

Are the superstars of other areas equally poorly represented?  Print

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Suppose I show you a map of Nebraska, with as-the-crow-flies distances marked between the major cities. Omaha to Lincoln, 100 miles. Lincoln to Grand Island, 100 miles. Omaha to Grand Island, 400 miles.

You are entitled to say “Hey, wait a minute! This map is inconsistent. The numbers don’t add up. If it’s 400 miles straight from Omaha to Grand Island, then there can’t be a 200 mile route that goes through Lincoln!”

So a map can be inconsistent. (It can also be consistent but wrong.) Nebraska itself, however, can no more be inconsistent than the color red can be made of terrycloth. (Red things can be made of terrycloth, but the color red certainly can’t.)

With that in mind, suppose I give you a theory of the natural numbers — that is, a list of axioms about them. You might examine my axioms and say “Hey! These axioms are inconsistent. I can use them to prove that 0 equals 1 and I can also use them to prove that 0 does not equal 1!” And, depending on the theory I gave you, you might be right. So a theory can be inconsistent. But the intended model of that theory — the natural numbers themselves — can no more be inconsistent than Nebraska can. Inconsistency in this context applies to theories, like the Peano axioms for arithmetic, not to structures, like the natural numbers themselves.

(See yesterday’s post for more on theories and models.)

Philosopher Alfred Korzybski admonishes us to remember that the map is not the territory. The theory is the map. The model is the territory. The hallmark of Internet crankery in this area is the refusal to distinguish them.

Whenever someone drones on at length about “the consistency (or inconsistency, or possible consistency, or possible inconsistency) of the natural numbers”, you’ll know he’s blathering. The concept simply doesn’t apply. Nebraska can’t be inconsistent. Only a description of it can be inconsistent.

(This brief snarky detour is brought to you by the small but determined band of commenters who consistently and vocally ignore this distinction in order to spout nonsense both here and on other blogs. I am not talking about anyone who’s commented here lately.)

It can be easy — and therefore entirely excusable — to get confused about this issue because in informal discussions of this subject — as in every other informal discussion of every subject in the English language — a single word can have multiple meanings. That sometimes happens with the word “arithmetic”. The phrase “Peano arithmetic” is the name of a theory — a list of axioms. On the other hand, some of us (me, for example) sometimes use the word “arithmetic” (a bit sloppily) to refer to a structure, namely the natural numbers themselves, which form a model of Peano arithmetic. Fortunately, the meaning is usually clear from context. If someone talks about “the consistency of arithmetic” you know that he’s talking about the theory (unless of course you have reason to suspect that he’s badly confused).

Now then. Let’s start with a theory. There are (at least) two sorts of questions you could ask about this theory. First: Is this theory consistent? In other words, is the theory free of self-contradiction? Second: Does this theory have a model? In other words, is there actually some structure that this theory describes?

If you’re given a map, the first question is like asking whether all the distances add up. The second question asks whether this is a map of someplace that actually exists or just a figment of the mapmaker’s imagination.

Inconsistent theories, obviously, have no models. A map that makes no sense cannot be a map of anything.

What about consistent theories? A consistent theory might a priori have either no models, or just one model, or many models.

The first possiblity is ruled out by Godel’s Completeness Theorem, not to be confused with the far more famous Godel’s Incompleteness Theorem. According to Godel’s Completeness Theorem, every consistent theory has at least one model. This is like saying that if you draw a map, and if nothing about the map is self-contradictory, then somewhere there is a territory that corresponds to the map. You should find this at least mildly surprising, but there it is.

If your theory is a theory of the natural numbers — in other words, if the natural numbers constitute a model for your theory — then the Lowenheim/Skolem Theorem says that your theory has a jillion other models as well. In other words, your map applies equally well to a jillion different territories. And there is no way, just by looking at the map, to tell those territories apart.

In other words, no theory — no list of axioms — can be a complete description of the natural numbers. It will always be a partial description, which applies equally well to a jillion other mathematical structures that look a little bit like the natural numbers but mostly a whole lot different.

Today’s moral: The map is not the territory. The map — the set of axioms — is either consistent or it’s not. If it’s inconsistent, there’s no corresponding territory. If it’s consistent, there are many corresponding territories and the map can’t tell you which one you’re in. That’s a fundamental limiitation on the power of the axiomatic method to describe a mathematical structure such as the natural numbers. It means there’s more to the natural numbers than any set of axioms can possibly know about.

Still to come: Are the Peano axioms consistent? Do the natural numbers really exist? And just how much about the natural numbers is any axiomatic system doomed not to know?

And finally: Thanks to those of you who encouraged me to continue this series. Let me know if you want still more.

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The first thing to keep in mind is that mathematics is not economics, and therefore the vocabulary is not the same. In economics, a “model” is some sort of an approximation to reality. In mathematics, the word model refers to the reality itself, whereas a theory is a sort of approximation to that reality.

A theory is a list of axioms. (I am slightly oversimplifying, but not in any way that will be important here.) Let’s take an example. I have a theory with two axioms. The first axiom is “Socrates is a man” and the second is “All men are mortal”. From these axioms I can deduce some theorems, like “Socrates is mortal”.

That’s the theory. My intended model for this theory is the real world, where “man” means man, “Socrates” means that ancient Greek guy named Socrates, and “mortal” means “bound to die”.

But this theory also has models I never intended. Another model is the universe of Disney cartoons, where we interpret “man” to mean “mouse”, we interpret “Socrates” to mean “Mickey” and we interpret “mortal” to mean “large-eared”. Under that interpretation, my axioms are still true — all mice are large-eared, and Mickey is a mouse — so my theorem “Socrates is mortal” (which now means “Mickey is large-eared”) is also true.

A model is any “reality” — the actual real world, the world of Disney, etc. — together with a (possibly non-standard) interpretation of the key vocabulary words “Socrates”, “man” and “mortal” — where all my axioms are true. Because all my axioms are true, so are all my theorems.

Now the model I really care about is the real world, and I’d like it very much if my theory could prove every true statement about that model. Well, of course that’s too much to hope for. My theory can’t possibly prove that Iran is about to become a nuclear power, because it doesn’t have the vocabulary for that. But what I’d really like is a theory that can prove every true real-world statement about men, mortals and Socrates.

Alas, my theory fails. It cannot prove, for example, that “Some men are not Socrates”. And part of the reason it can’t prove such a thing is that, while this statement is true in the intended model, it’s not true in every model. It’s not true, for example, in the model consisting of Christian doctrine, where we interpret “man” to mean “Messiah” and “Socrates” to mean “Jesus” and “mortal” to mean “divine”. My axioms are true — All Messiahs are divine and Jesus is a Messiah — but “Some men are not Socrates”, reinterpreted to mean “Some Messiahs are not Jesus”, is false. That tells me that I cannot possibly hope for my theory to prove that some men are not Socrates.

So if I want a theory that fully describes the real world, I need a richer theory — one with more vocabulary and more axioms. But as long as that theory has non-standard models, it will never be able to prove everything I want it to prove about the real world.

Now let’s talk about arithmetic. A theory of arithmetic is a list of axioms — like “Every number has exactly one immediate successor”. The standard model of that theory is the good old natural numbers, the ones you’ve known about since you were five years old (and which people knew about long long before anyone ever invented the idea of writing down axioms). In the standard model, “number” means “number” — like 0 or 1 or 2 or 3 —, “successor” means “successor” in the usual sense — so that 4, for example, is the successor of 3, and so on.

But it turns out that any theory you can write down also has non-standard models, in which “number”, “successor”, etc. all have completely different meanings — though those meanings are assigned in a way that preserves the truth of the axioms.

Some true statements about the standard model — that is, some true statements about the actual honest-to-God natural numbers — will turn out to be false when interpreted in one or more of these non-standard models. As long as they are false in some model, they cannot possibly be provable from the axioms — just as “Some men are not Socrates” is false in the New Testament model, and therefore cannot possibly be provable from my Socrates/man/mortal axioms.

So if I want a theory that can prove every true statement about the natural numbers — or even just every true statement that my theory has the vocabulary to express — I should try to build a theory with no non-standard models. Alas, the famous Lowenheim-Skolem theorem tells me that all theories have non-standard models. Therefore every theory fails to prove all true statements about the natural numbers. And the even more famous Godel Incompleteness Theorem tells me that any theory must fail in a particularly annoying way — by being unable to prove some concrete mathematical statement like “Every even number greater than 2 is the sum of two primes”, as opposed to merely being unable to prove some esoteric metamathematical statement like “Addition is programmable on a computer”.

Incidentally, the Peano Axioms have a jillion non-standard models in which addition is not programmable on any computer. Obviously these non-standard models have very little in common with the standard model, namely the good old natural numbers. That means there’s a lot of stuff about the good old natural numbers that our theory can’t tell us. And the content of Lowenheim/Skolem/Godel is that the same is true for any theory.

I’m planning to continue this series of mini-lessons at the rate of, oh, about two a week or so. But only if a substantial number of you are interested, of course. Let me know whether you’d like to see more. If not, I can always go back to bashing Krugman.

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Start with this P=NP-inspired question:

I don’t need you to answer that question. I just want you to answer an easier question:

If you said yes (as would be the case, for example, if you happen to be sane), then you have recognized that statements about arithmetic can be either true or false independent of our ability to prove them from some set of standard axioms. After all, nobody knows whether the standard axioms of arithmetic (or even the standard axioms for set theory, which are much stronger) suffice to settle Question 1. Nevertheless, pretty much everyone recognizes that Question 1 must have an answer.

Let’s be clear that this is indeed a question about arithmetic, not about (say) electrical engineering. A computer program is a finite string of symbols, so it can easily be encoded as a string of numbers. The power to factor quickly is a property of that string, and that property can be expressed in the language of arithemetic. So Question 1 is an arithmetic question in disguise. (You might worry that phrases like “quickly” or “substantially faster” are suspiciously vague, but don’t worry about that — these terms have standard and perfectly precise definitions.)

What’s true is not the same as what’s provable. This is an elementary observation, and I think you’d have to look long and hard to find a mathematician (or even a philosopher) who considers it controversial. So you might wonder why I’m bothering to harp on it. Answer: I’ve learned from the response to previous blog posts that there’s a small population of extremely vocal commenters who are both deeply confused and curiously passionate about this issue, and who have somehow talked themselves into believing that in mathematics, nothing is true unless it can be proved.

My usual response to these commenters is to pose a question along these lines:

Nobody knows the answer, but pretty much everyone agrees that there is an answer. And moreover, the answer is what it is regardless of whether our favorite axioms can prove it. In fact it’s entirely possible that our favorite axioms can’t settle the question either way.

That’s pretty clear to almost everyone. But occasionally, commenters will take the bizarre stance that Question 3 might have no answer — because, they say, numbers aren’t real anyway, so it makes no sense to ask questions about them (or something like that).

Maybe instead of pointing those commenters to Question 3, I should try pointing them to Question 1. Will they really want to argue that there might be no answer to the question of whether it’s possible to write a program that factors numbers quickly? That’s a pretty concrete question. We can all agree that as of today, nobody knows the answer, and a pessimist might believe that nobody ever will know. But to say that there might be no answer puts you on pretty strange metaphysical turf. I’m not even sure what it would mean for there to be no answer.

If there is an answer, then that answer is a truth that holds independent of what we can prove. If our axioms can’t determine that answer, then so much the worse for our axioms — but there’s still an answer.

This matters because it’s the first baby step toward understanding whether numbers were created or discovered, and why they could conceivably form the fabric of the Universe. Over the next couple weeks, I’ll make occasional blog posts about these issues. Of course you can learn a lot more by reading The Big Questions, but here I’ll stick to addressing simple confusions that have shown up in earlier blog comments.

I’ll try to stick to one simple observation per blog post. Today’s moral is that well formed statements about arithmetic are either true or false, regardless of whether they have proofs or disproofs. I expect that the mathematicians in the crowd will consider this too obvious even to mention, but my blogging experience tells me that it’s not obvious to everyone. I hope this post fixes that.

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A little over a week ago, HP Research Scientist Vinay Delalikar claimed he could settle the central problem of theoretical computer science. That’s not the momentous part. The momentous part is what happened next.

Deolalikar claimed to prove that P does not equal NP. This means, very roughly, that in mathematics, easy solutions can be difficult to find. “Difficult to find” means, roughly, that there’s no method substantially faster than brute force trial-and-error.

Plenty of problems — like “What are the factors of 17158904089?” — have easy solutions that seem difficult to find, but maybe that’s an illusion. Maybe there’s are easy solution methods we just haven’t thought of yet. If Deolalikar is right and P does not equal NP, then the illusion is reality: Some of those problems really are difficult. Math is hard, Barbie.

So. Deolalikar presented (where “presented” means “posted on the web and pointed several experts to it via email”) a 102 page paper that purports to solve the central problem of theoretical computer science. Then came the firestorm. It all played out on the blogs.

Dozens of experts leapt into action, checking details, filling in logical gaps, teasing out the deep structure of the argument, devising examples to illuminate the ideas, and identifying fundamental obstructions to the proof strategy. New insights and arguments were absorbed, picked apart, reconstructed and re-absorbed, often within minutes after they first appeared. The great minds at work included some of the giants of complexity theory, but also some semi-outsiders like Terence Tao and Tim Gowers, who are not complexity theorists but who are both wicked smart (with Fields Medals to prove it).

The epicenter of activity was Dick Lipton’s blog where, at last count, there had been been 6 posts with a total of roughly 1000 commments. How to keep track of all the interlocking comment threads? Check the continuously updated wiki, which summarizes all the main ideas and provides dozens of relevant links!

I am not remotely an expert in complexity theory, but for the past week I have been largely glued to my screen reading these comments, understanding some of them, and learning a lot of mathematics as I struggle to understand the others. It’s been exhilarating.

Why is this momentous? In some ways, there’s nothing new about any of this. It’s not terribly uncommon for a serious looking paper to address a major outstanding problem, and it’s de rigeur for experts to comb through those papers, searching simultaneously for new paradigms, irreparable flaws, and salvageable insights. We call it peer review.

But what’s new is that this played out in public, and that it took a week instead of the usual months or years, and that the dozens of conversations taking place all over the world were melded into one giant conversation where every idea was available for everyone to hear—and for everyone to shoot down. Many very smart people said very smart things that turned out to be wrong, and in the world before the Internet, they might have gone on believing those things for weeks or months. The Net made it very difficult to believe wrong things for more than an hour.

It is a cliche to say that the Internet has changed everything, and in particular that it’s changed the way we do science. But what I saw this week seemed to me to be a whole new grand leap forward. The icing on the cake is the growing public record of everything that’s been said. Most of that record is a monument to the passion and dedication of a community obsessed with finding truth. In those thousand or so comments, I see almost nothing that smacks of self-aggrandizement, almost no instances where the proponent of an idea fails to back off instantly in the face of a better idea. Sometimes we in the academic community lose sight of how extraordinary are the high standards we routinely demand of ourselves and each other. Sometimes those outside of academics have no concept of how high those standards are. It’s inspiring to be reminded.

This week, there’s been no better inspiration—and no better education, and no better entertainment—than to read Dick Lipton’s blog.

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That central question is called the “P versus NP” problem, and for those who already know what that means, his claim (of course) is that P does not equal NP. For those who don’t already know what that means, “P versus NP” is a problem about the difficulty of solving problems. Here’s a very rough and imprecise summary of the problem, glossing over every technicality.

Deolalikar’s paper is 102 pages long and less than about 48 hours old, so nobody has yet read it carefully. (This is a preliminary draft and Deolalikar promises a more polished version soon.) The consensus among the experts who have at least skimmed the paper seems to be that it is a) not crazy (which already puts it in the top 1% of papers that have addressed this question), b) teeming with creative ideas that are likely to have broad applications, and c) quite likely wrong.

As far as I’m aware, people are betting on point c) not because of anything they’ve seen in the paper, but because of the notorious difficulty of the problem.

And when I say betting, I really mean betting. Scott Aaronson, whose judgment on this kind of thing I’d trust as much as anyone’s, has publicly declared his intention to send Deolalikar a check for $200,000 if this paper turns out to be correct. Says Aaronson: “I’m dead serious—and I can afford it about as well as you’d think I can.” His purpose in making this offer?

I could think of only one mechanism to communicate my hunch about Deolalikar’s paper in a way that everyone would agree is (more than) fair to him, without having to invest the hard work to back my hunch up.

On the one hand, this seems like quite an effective way for Scott to communicate the strength of his hunch, which is obviously something he very much wants to do. On the other hand, I’m a little baffled by Scott’s remark (in the comments to the linked post) that “If P≠NP has indeed been proved, my life will change so dramatically that having to pay $200,000 will be the least of it.” I’m sure that if P≠NP has indeed been proved, it will dramatically change the life of a complexity theorist like Scott Aaronson, but I’m not sure why it will change it in a way that makes $200,000 less valuable.

But that’s of course his call. I just wanted to share this and invite your comments.

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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.)

In Chapter 10 of The Big Questions, you’ll find a description of a game called “Hercules versus the Hydra”. It turns out that Hercules always wins the game, no matter how stupidly he plays. This fact (that Hercules always wins) is known to be true and known to be unprovable from the standard axioms. Fermat’s Last Theorem, by contrast, is known to be true and might or might not be unprovable from the standard axioms. If that question gets settled, most mathematicians will be interested enough to sit up and take notice. But unlike the Last Theorem itself, few are motivated to work on it.

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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.)

Second order logic expands the language we’re allowed to use, by allowing us to apply words like “Every” not just to numbers, but to sets of numbers. So in second order logic, we can say things like “Every set of numbers has a smallest element”. In first-order logic, that sentence would be ungrammatical and hence meaningless.

Now: Godel’s Theorem, as it’s sometimes stated, says that no first order logical system can prove all the truths of arithmetic. Start with any true axioms you want, and there will always be other true things you can’t prove—not just because you’re not smart enough but because there really are no proofs within your system.

Coupon Clipper’s first question is (I am paraphrasing, accurately I hope): So what? Why not just use second order logic instead? He also guesses accurately at the answer, which is that Godel’s Theorem applies just as well to second order logic as it does to first order logic. There will still be some true statements in arithmetic that your system can’t prove.

That answers the question, but it raises another question: Why do mathematicians prefer to avoid second order logic?

Second order logic certainly has its advantages, and here’s the big one: It let’s us nail down what we’re talking about. What we’re talking about, of course, are the natural numbers: 0,1,2,3, and the rest. And nothing more! We don’t want our system to contain numbers that are infinitely big or infinitely small or exotic in other ways. First order logic can’t rule that stuff out. Second order logic can.

In other words, in any system of first order logic, all the theorems you can prove are true statements about the natural numbers, but there will always be other more exotic systems of “numbers” of which your theorems are also true. That means there is no way, within the language, to distinguish between the honest-to-god natural numbers and some of these other systems. There is no grammatical way to say “No, no, I mean the *true* natural numbers, not those impostors!”

With second order logic, that problem goes away. The theorems you can prove are true statements about the natural numbers, and they’re not true statements about anything else. There’s no ambiguity about what you’re describing.

But the offsetting disadvantage is huge: In first order logic, I can tell you what all the rules are. (Remember, for example, the rule that says that if you’ve established that every number has some property, you’re allowed to conclude that any particular number has that property.) In second order logic, I can’t. Neither can anybody. Neither can any computer. It is a theorem that no computer program can generate all the valid rules of inference in second order logic. That’s in some sense a much bigger deal than Godel’s theorem. Godel’s theorem says that (in either first or second order logic) no computer can follow the rules and discover all the true statements of arithmetic. But now I’m telling you that in second order logic, no computer can even figure out what the rules are!

Hence the oft-repeated slogan that “second order logic is not logic”, and hence our reluctance to rely on it.

Coupon Clipper’s second question is “Does any of this matter for the actual practice of mathematics?”. That’s a much easier question with a much shorter answer, but I think I’ll save it for tomorrow.

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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.

This is relatively easy to prove in the n=4 case (that is, for the equation x⁴+y⁴=z⁴), and in fact I was able to reconstruct two separate proofs, one using elementary algebra and the other using a little geometry. (”Reconstruct” means that there was a time in my life when I knew these proofs well—and even taught them at a graduate level—but that was long long ago.) And I was able to reconstruct Lamé’s flawed proof, which, when supplemented with some more work, can be converted to a correct proof for a large class of exponents (beginning with n=5). The attempt to understand when Lame´’s argument can (or can’t) be patched up inspired a century of progress in algebraic number theory. Alas, that work reveals that there are plenty of exponents for which the proof is irreparable, beginning with n=37. The only known proof, associated in the popular imagination with the great Andrew Wiles, but more properly attributed to Frey, Serre and Ribet, is nothing like Lamé’s (and about one octillion times more difficult).

But what really surprised me was that I didn’t have a clue how to solve the case n=3. And even now, I have no memory of ever having known how to solve the case n=3. I was aware that it took Euler to solve it in the first place, and that I am not as smart as Euler (by a factor of about one octillion), but I was also aware that I know a lot of fancy techniques that Euler didn’t have. So, like the character in the novel, I thought I’d give it a go.

My first idea was to use Fermat’s favorite technique: Pretend you’ve got a solution, and show that from that solution, you can construct a smaller solution. Keep repeating and your solutions get smaller forever, which is quite impossible with integers. (If your first solution involved x=100 and x gets smaller each time, you’re going to get stuck after 100 iterations—x can’t go below zero). This means you never had a solution in the first place. (Fermat called this the “Method of Infinite Descent”.)

So I pretended I had a solution—that is, a set of numbers x, y, z that satisfy x³+y³=z³—and used a little geometry to construct a new solution. I did this using what is, for a geometer, the obvious idea. Namely:

• Set X=x/z and Y=y/z, and observe that X³+Y³=1
• Observe that (0,1) and (X,Y) are both points on the curve defined by the equation x³+y³=1
• Draw the line connecting these two points. Because the curve is defined by a third degree equation, that line will hit the curve three times. We already know it hits at (0,1) and (X,Y). Compute the third point. Because everything else in sight is a rational number, that third point will have rational coordinates.
• Write the coordinates of that point as (a/c,b/c), where a, b and c are integers. (You can always make the two denominators equal by choosing a common denominator). Then because this point sits on the curve, it satisfies the equation (a/c)³+(b/c)³=1. This in turn implies that a³+b³=c³.

So starting with one solution (x,y,z), we get a new solution (a,b,c). If (a,b,c) is in any reasonable sense smaller than (x,y,z), we can keep repeating till we get a contradiction.

When I did this, I got a = x(1+y³), b = -y(1+x³) and c = x³-y³. (You can check by hand that if x,y,z solve the Fermat equation then so do a,b,c.) Sadly, this doesn’t help because the new solution is not smaller than the old solution in any reasonable sense that I can think of. (I’d expected as much, because if something this simple had any chance of working, it wouldn’t have taken Euler to solve the problem.)

So I futzed around with a few other ideas that didn’t work (e.g. instead of drawing the line that connects two points, you could draw the tangent line at the point (X,Y)) and finally looked up Euler’s proof, which I must say, rang absolutely no bells with me, meaning either that I must have been curiously uncurious about this when I was younger or that my memory is failing even more precipitously than I realized. On a side note, I also learned (for the first time, as far as I can recall) that Euler’s first published attempt was incorrect.

Well, at least I got a blog post out of this, and more importantly it was fun. Sometimes it pays to have a short memory. Every now and then (especially when I’m stuck in a boring meeting) I compute the sum of the infinite series 1 + 1/2ⁿ + 1/3ⁿ + 1/4ⁿ + … for various values of n, which is another problem that Euler got to before I did. The main idea stays with me, but the details are new every time.

Edited to add: For those who are playing along at home—I copied incorrectly from my notes. The a, b and c announced above come not from the line that connects (0,1) to (X,Y), but from the tangent line at (X,Y). If you use the line connecting (0,1) to (X,Y), you get a=-x, b= z, c=y, which is even less useful.

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