Today let’s talk about consistency.
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.
Continue reading ‘Basic Arithmetic, Part III: The Map is Not the Territory’
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Today’s mini-lesson in the foundations of mathematics is about the key distinction between theories and models.
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.
Continue reading ‘Basic Arithmetic, Part II’
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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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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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This quiz amused the hell out of me. I hope it does the same for you.
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 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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A reader has just emailed me a link to a Washington Post story about North Carolina workers losing their jobs to foreign competition. Presumably he believes there’s a larger moral here, because his subject line is “Wrong again, Steve”. Here is a slightly edited version of my emailed response:
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.
Continue reading ‘Trading Up’
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