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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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:

• Every number is either odd or even. This is a tautology. It does not follow that every fact of arithmetic is a tautology.
• 1+1=2. This (depending on what you take as your starting point) is true by definition. It does not follow that every fact of arithmetic is true by definition.
• Every number is a product of prime numbers. This is neither a tautology nor a definition. It does, however, follow from the standard Peano axioms. It does not follow that every fact of arithmetic follows from the axioms.
• There is no solution to the equation (x+1)ⁿ⁺³+(y+1)ⁿ⁺³=(z+1)ⁿ⁺³. (This is the famous Fermat’s Last Theorem.) This is not a tautology, it is not a defintion, and I have no idea whether it follows from the axioms. Neither, as far as I know, does anyone else. All we know for sure is that it’s true.
• In Chapter 10 of The Big Questions, I give an example of a fact of arithmetic that is not a tautology, not a definition, and surely does not follow from the axioms. Here is a sketch of a proof that there must be some such facts.

It seems that people are often led astray by thinking that the “facts of arithmetic” consist solely of statements like “seven squared plus one equals five squared times two”. The more logically interesting statements are those that speak not about specific numbers, but about infinite collections of numbers. These are the statements that begin with phrases like “Every number…” or “There is a number such that…” or “There are only two numbers such that….”. When mathematicians speak of the “facts of arithmetic”, they means facts like these:

• Every number is the sum of four squares.
• Every prime number that leaves a remainder of 1 when divided by 4 is a sum of two squares.
• No prime number that leaves a remainder of 3 when divided by 4 is a sum of two squares.
• For every number n, there are infinitely many squares of the form 1+ny²
• Between every number and its double, there is at least one prime.
• 8 and 9 are the only successive numbers that are both powers of primes.
• For any two prime numbers p and q, the equations p=x²+yq and q=x²+yp are either both solvable or both unsolvable, unless p and q both leave remainders of 3 when divided by 4, in which case exactly one of them is solvable.
• If you want to black out enough squares on a tic-tac-toe board to make winning impossible, then, as you pass to boards of higher and higher dimensions, the fraction of squares you must black out gets arbitrarily close to 100%. This is the density Hales Jewett theorem, and I have intentionally glossed over some technicalities in the statement.

Now to the point: First, if you want to tell a story about whether arithmetic is invented or discovered—in other words, if you want to tell a story about where arithmetic comes from—then your story has to account for where the facts of arithmetic come from. If your story says that the facts of arithmetic consist entirely of tautologies, definitions and logical consequences of standard axioms, then your story is wrong. Try again.

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Edited to add: In comments, Mike H points me (and you) to this even better quiz, which seems to have been the model for the one I linked to. Enjoy your day.

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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.

Here’s one way to think about simplicity versus complexity: Simple things have short descriptions; complex things have only long descriptions. A string of a million zeros is very simple because I can describe it in six words: “A string of a million zeros”. A string of a million random numbers is complex, because it takes a long time to describe all of the content.

Now what about the system of natural numbers? To first appearances, there’s a very simple description: Start with 0, then add 1, then add 1 again, keep doing this forever, and those are the natural numbers. Unfortunately, “keep doing this forever” is a little vague, and the complexity comes in when you try to make that precise.

So if you were setting out to give a complete description of the natural numbers, where would you start? Probably here:

• We have a number called zero, and then every number has a successor.

But that description fits a lot of things besides the natural numbers; it also fits, for example, the integers (the integers, unlike the natural numbers, include negatives). Here’s an attempt to fix that:

• We have a number called zero, and then every number has a successor, and zero is not the successor of any number.

Better, but still no good; this fails to rule out a system where 1 follows 0, 2 follows 1, 3 follows 2, and 1 follows 3, like this:

To fix that, we have to add a clause specifying that no two numbers (such as 0 and 3) have the same successor. But even now, we’ve only just begun.

We still haven’t ruled out the possibility of infinite gaps between numbers. For all we can tell from our description so far, the number system might look like this:

with infinitely many numbers in between 3 and that very large number N. How can we rule that out?

This one is not so easy. We’d like to say that all gaps between numbers are finite. But how do we define “finite”? Usually we say a number is finite if it’s part of the set of natural numbers. Or to put this another way: We’d like to say that no matter where you start (say at N), you can’t count backward forever; eventually you’ve got to hit a stopping point. But what does it mean to count backwards forever? It means counting back more than a natural number of steps. There’s that circularity again.

What we really really need, it turns out, is to add a clause like this to our description:

• Every non-empty subset of the natural numbers has a smallest element.

This will solve our problem, because it implies, for example, that the set of numbers you can reach by counting backwards from N has a smallest element—eliminating the possibility of that infinite gap.

But this assumption, stated in this way, opens a can of worms that almost nobody wants to open. Here’s why: For the first time, we’ve been forced to talk about sets of natural numbers, as opposed to natural numbers themselves—and even to talk about all those sets at once. In technical jargon, we’ve left the world of first-order logic and entered the world of second-order logic. But that’s a very strange world indeed. In ordinary (first-order) logic, we have a small number of rules of inference that allow us to proceed, for example, from “Socrates is a man” and “All men are mortal” to “Socrates is mortal”. But in second-order logic, not only are the rules of inference not finite; they cannot even be printed out (even in an infinite amount of time) by any computer. That’s why the great logician Willard van Ormand Quine insisted that second order logic is not logic, and why mathematicians usually prefer to avoid it.

All is not lost, though. Instead of adding one second-order axiom, we can add infinitely many first-order axioms, viz:

• The set of odd numbers has a smallest element.
• The set of numbers greater than 7 has a smallest element.
• The set of numbers that can be reached by counting backward from 100 has a smallest element.

And so on.

Okay, our description of the natural numbers just got infinitely long, but at least it’s infinitely long in a simple sort of way. We’ve added an infinite number of axioms, but they all fit the same simple pattern—a pattern that you could easily train your computer to recognize.

Unfortunately, though, we still have a long way to go to get to a full description of arithmetic. First, we have to add rules for addition and multiplication. (If we don’t do this, then we won’t be able to talk about interesting subjects like prime numbers.) Now we’ll want to add even more axioms. But now we come up against the content of Godel’s incompleteness theorem: NO description suffices. No matter what axioms you add, your description will always fail to distinguish the natural numbers from any of an infinite number of other structures. (Those other structures are usually called “non-standard models of arithmetic”).

When I say that “NO description suffices”, you might reasonably ask what counts as a description. Here’s what counts: A description is some (possibly infinite) list of axioms that some computer program is capable of recognizing. So if, for example, you try to describe arithmetic by listing every true statement about it, I will cry foul, because no computer program is capable of recognizing every true statement about arithmetic. (This is not just an observation about the state of the art in computer programming. It is a theorem about all possible computer programs.)

That’s the sense in which arithmetic is fantastically complex. Not only do the natural numbers have no finite description; they have no description that is recognizable by any computer. If “simple” means “capable of a short description”, then the natural numbers are about as far from simple as you can get. Not only do they have no short description, they don’t even have an infinite description.

Other mathematical structures are simpler. Euclidean geometry, for example, can be fully described by a first-order theory, and there is a computer program that can distinguish true from false statements in that theory.

Likewise for the (first-order) theory of the real numbers. There are axioms for the real numbers that suffice to prove all true first-order statements about the real numbers, and there is a computer program that can distinguish the true statements from the false. In that sense, the real numbers are far simpler than the natural numbers. (There are still non-standard models of the real numbers, but through a first-order lens, they are indistinguishable from the real thing.)

(You might be tempted to think that because the natural numbers sit inside the real numbers, they must be simpler. But of course any sequence of arbitrary complexity sits inside the very simple sequence 010101010101….., if you can pick and choose what to keep and what to throw away. Complexity can reside quite comfortably inside simplicity.)

Did that help?

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It would be dishonest for me or anyone else to defend free trade by pointing to its advantages while ignoring its disadvantages.

It is equally dishonest to oppose free trade by pointing to its disadvantages while ignoring its advantages.

What you need is a framework that accounts for all the advantages and disadvantages, together with enough of a logical structure to instill confidence that nothing imporant has been overlooked. Thats what economic theory supplies. You can find that theory in the economics textbooks. You can also find (I think) a pretty good summary of it in The Big Questions.

My correspondent wrote back with a pointer to a website with fifty years of what he calls “extrapolatable stats” that he thinks supply the necessary framework. This misses the point entirely. There is no way a hodgepodge of numbers can settle the question of whether something’s been left out. For that you need a theory.

Here, in brief, is the theory: If Joe the American sells blankets to Mary the American for $15 each, and if an opening to trade allows Mary to buy Chinese blankets for $5 each, then three things happen:

• Mary is better off by $10.
• Joe is worse off by at most $10—because Joe can always match the Chinese price if he wants to, taking a $10 hit. On the other hand, he also has the option of getting out of the blanket business, which he’ll choose only if he prefers it to taking that hit.
• Frieda, another American, who might not have been willing to pay $15 for a blanket, picks up a Chinese blanket for $5 and goes to bed warm tonight.

Of these, only the second effect is bad for Americans, and it’s got to be outweighed (or at least matched) by the first effect. The third effect is pure gravy.

That, in essence, is the argument for free trade. There are plenty of obvious objections, and plenty of somewhat less obvious responses—again, all easily found in textbooks. But if you’re going to argue that trade is bad, then this is the argument you’ve got to confront, because this is the argument on which the vast majority of economists rest their case.

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