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.
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.
These are great, please keep them up.
Perhaps the phrases from computer science programming language theory may help too: “Operational semantics” are kind of like your “theory” and “Denotational semantics” are kind of like your “model.”
“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.”
Can we have both? If I could see Krugman-bashing and a decent pop-math treatment of Löwenheim-Skolem, on the same blog in the same day, I’m pretty sure I could die happy.
I think you forgot to close a “bold” HTML tag near the end :)
MORE!!!!
I love this stuff!
The last sentence presents a bit of a false dichotomy.
Some Messiahs are not Krugman.
I’m enjoying this, but Krugman-bashing is also great.
I always appreciate a good Krugmann bashing, but this is more interesting right now.
!”My theory can’t possibly prove that Iran is about to become a nuclear power, because it doesn’t have the vocabulary for that.”
iran is a moderately wealthy nation-state. moderately wealthy nation-states have access to nuclear arsenals.
Five nuclear weapons states from the NPT
United States
Russia
United Kingdom
France
China
Non-NPT nuclear powers
India
Pakistan
North Korea
Undeclared nuclear powers
Israel
Bennett: I think you forgot to close a “bold” HTML tag near the end :)
Thanks. This is fixed now, though probably too late for the RSS feeds.
Dr. L: Please keep this up! It’s a nice post-grad education for me. So is Krugman debunking but this is at least as interesting …
Holy crap, this is a great explanation. I vote for continuing the series!
I am interested in more of these. I am also a substantial number of people, but I can’t prove that.
Aye keep it up please. I enjoy me some good maths for breakfast…
Keep going with this. I can bash Krugman on my own, though it’s not always as entertaining as when you do.
This is good, but don’t let up on Krugman.
This is simply great! Please let us have more such!
Linear assumptions
Make life easy
But sometimes
Things disappear
And the next step
Is nowhere near
Another vote for more of these. It brings back memories of my computational theory and number theory classes years ago.
I’m curious about the purpose of non-standard models; that is, why use them? I get that you make non-standard models to test the breadth of applicability but my take from what you wrote is that the purpose of the standard model is to attempt to describe reality as we experience it as closely as possible. Are there other values to non-standard models?
I like these posts. You’re good at writing clear explanations of math that are fun to read.
This is a bit over my head though I do find it quite interesting.
Honestly, I prefer your policy analysis/moral comparison stuff. Krugman bashing is a plus for sure.
I’m guessing that given you get more responses on the stuff I like than to the unravelling complex math thingamiwhatsis, that is what people in gerenal like more.
But maybe I’m just skewing the evidence to my preferences.
Alan Wexelblat:
I’m curious about the purpose of non-standard models
I’m not sure I understand the question. Theories unavoidably have non-standard models, whether or not they serve our purposes.
If I were to write a post explaining how internal combustion engines work, I might mention that they emit carbon monoxide. You might then ask what is the purpose of this carbon monoxide. I’d be equally at a loss how to answer.
Steven, these posts are fascinating, please keep ’em coming.
One thought: you state that “some true statements about the standard model [of arithmetic]…will turn out to be false when interpreted in one or more of these non-standard models”.
Your careful wording in that sentence (i.e., “some”, “one or more”) raises a couple of questions (to me, at least), such as:
1) Are there ANY true statements in the standard model that are also true when interpreted in EVERY non-standard model?
2) Is there ANY non-standard model such that ALL true statements in the standard model are also true when interpreted in that non-standard model?
If the answer is ‘No’ to either of these, then it seems like the point you are making in your post might be even stronger.
Been away for a while, and just had a brief look so far – need to re-read again, but looks very interesting. One point of detail – are you saying that all even numbers being the sum of 2 primes is actually unprovable, or just that this is the sort of thing that might be?
“Incidentally, the Peano Axioms have a jillion non-standard models in which addition is not programmable on any computer.” Can you elaborate in this?
Harold:
are you saying that all even numbers being the sum of 2 primes is actually unprovable, or just that this is the sort of thing that might be?
The latter.
Sol:
“Incidentally, the Peano Axioms have a jillion non-standard models in which addition is not programmable on any computer.” Can you elaborate in this?
Not easily. In these models, addition (and multiplication) are too complicated to be described to any computer, and therefore probably too complicated to describe in any way that’s comprehensible to a human being.
Take your favorite example of a statement about the natural numbers that’s true but not provable from the Peano Axioms. (In ##tbq, I gave an example of such a statement—look under “Hercules and the Hydra”.) Then there is a non-standard model in which the interpretation of this statement is false, and in which both addition and multiplication are too complicated to describe. We know that model exists, but there’s no good way to say much about it.
Harold – Even numbers being the sum of two primes is Goldbach’s Conjecture: http://en.wikipedia.org/wiki/Goldbach%27s_conjecture. It’s unproven, and it’s not known if it can be proven. It’s not known to be unprovable, either. Some number theorists are still working on it.
Dr. L – If every even integer greater than 2 can be expressed as the sum of two primes, is there a number n such that every even integer greater than n can be expressed as the sum of two primes in two or more ways? And is there a number m such that every even integer greater than m can be expressed as the sum of two primes in three or more ways, etc.?
P.S. Keep it up. I started reading for the economics, but I like the math and physics better.
Al V: is there a number n such that every even integer greater than n can be expressed as the sum of two primes in two or more ways? And is there a number m such that every even integer greater than m can be expressed as the sum of two primes in three or more ways, etc.?
Almost surely yes (based on non-rigorous but highly convincing arguments). I am not sure, however, whether this is known. (I expect that any expert in analytic number theory, which I am not, could answer this instantly.)
Steve, you have a funny view of reality. The way you use it to distinguish between math and economics is especially confusing. In both mathematical logic and economics, a theory is a methodology for determining what is true, and a model is an instantiation of the theory.
I love this stuff, but I think you could go further in combining mathematical logic with Krugman bashing. For example, is there an interesting logical/mathematical formulation of Krugman’s arguments if they were primarily motivated primarily by vitriol and desire to be recognized as an authority, rather than by a systematic search for truth?
So by your terminology the theory of evolution may be used as a model of memes? That’s reasonable I guess.
Phil Maymin, If you’re saying what I think you are, I would agree. Genes (or more specifically, DNA) define a limiting set of biological outcomes. However, there are a number of different models that are compatible with DNA, but what has actually evolved is dependent on our specific initial conditions: a particular temperature range, frequencies of light, chemical composition of our water, soil, and atmosphere.
I remember reading once that when plants first evolved, our atmosphere and oceans had a much higher concentration of CO2. Plants consume CO2 and exhale O2, so the proportions changed, and when animals evolved we evolved to consume oxygen. There could be a model for life that evolves under different conditions, and therefore with different outcomes.
I’m enjoying this series, but like many others I find Krugman bashing fun too.
I love these mini lessons.
My only question is whether Steve or anyone else has a good book to recommend to someone interested in learning about this stuff in a more technical way.
I’m becoming fond of the mathematics. I think you rob a little from the nature of models in your comparison of economic models and mathematical ones.
You say that economic models are an approximation of reality where as in mathematics the model is reality itself. But if you were to restate your description of mathematical models this way, “mathematical models are reality itself to the extent that we know it at the time” wouldn’t you have a better description?
And in a very real sense that makes the difference between mathematic and social models a little less severe. I mean that because everything is not known about reality there is some necessary gaps in the mathematical model. With social models, especially economics but also decision theory and risk theory and others, we have to assume the same limits on what is known about the mathematics in our assumptions about human behavior. Plus, even when human behavior can be modeled accurately the complexity makes the models unwieldy.
So, my thinking on this is that we intentionally approximate reality in social models beyond even what is required because it allows us to get our work finished on time. And mathematicians endure approximate models because they have not yet transcended the boundaries of human ability. But neither are ever working with the “real reality”, not in any real sense.
A nice related discussion about why we need proofs at all:
http://rjlipton.wordpress.com/2010/08/18/proofs-proofs-who-needs-proofs/