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God created the integers. All else is the work of man. —Leopold Kronecker |
The problem with knowledge is that you have to start somewhere. Once you know something, you can start deducing other things. But how do you know the first thing?
Descartes’s famous solution was “I think, therefore I am”. I have direct knowledge of my own thoughts, and this in turn tells me that I exist. Now we can go on.
Mathematics, like all other knowledge, needs a starting point. Most of our mathematical knowledge is deduced from prior mathematical knowledge. I know that every positive integer is the sum of four squares because I know how to deduce this fact from other things I know. But where do I start?
There seems to be a widespread misconception (widespread, that is, among non-mathematicians — mathematicians know better) that all we know is what we can derive from axioms. This is wrong for several reasons. Two of these I’ve blogged about repeatedly in the past (e.g. here, here, and here and here), but the third is even more fundamental:
- To say all we know about mathematics is what we can derive from the axioms is like saying that everything we know about Nebraska is what we can derive from maps of Nebraska. It confuses a partial description of reality with reality itself.
- Godel’s Incompleteness Theorem (among other theorems) assures us that no system of axioms can serve as a complete description of the natural numbers. Any set of axioms that can describe the natural numbers also describes many other mathematical structures — just as a sufficiently undetailed map of Nebraska might serve equally well as a map of part of Montana. If the map fits both territories equally well, then it can’t be said to specify either territory.
- Any axiomatic system presupposes that you know something about numbers and therefore cannot be the basis for your knowledge about numbers. In an axiomatic system, a proof is a list of statements, each of which is either an axiom or a logical consequence of earlier statements on the list. You can’t make sense of that notion until you know what a list is, and you can’t know what a list is unless you know what it means to come first, second, third and fourth in a series.
Since I’ve blogged before about points 1 and 2, here I’ll emphasize point 3: To employ an axiomatic description of the natural numbers, you must already know something about natural numbers. That knowledge must come first.
In fact, any sort of mathematical reasoning at all seems to rely on at least some prior familiarity with both numbers and sets. We write down axioms to formalize some of that knowledge, but numbers and sets are the starting points. The basic properties of numbers and sets are the things we have to “just know”, in the same sense that Decartes “just knew” that he was conscious.
Some people find it unsettling that all knowledge must have a starting point. But nobody’s ever found a way around that problem, and in math, the starting points are numbers, sets, and their basic properties.
But there is a key difference between our knowledge of numbers and our knowledge of sets. The difference is important and profound.
When it comes to numbers, we write down (say) the Peano axioms to formalize some of what we already know. Then we discover (thanks to Godel and others) that the Peano Axioms have multiple models — that is, there are alternative mathematical territories that they describe equally well. Of the many structures that obey the Peano Axioms, we are particularly interested in the standard model — that is, the ordinary natural numbers that you’ve known about since age three.
When it comes to sets, we write down (say) the Zermelo-Frankel axioms to formalize some of what we already know. Then we discover that the Zermelo-Frankel axioms have multiple models — that is, alternative mathematical territories that obey the axioms for a “universe of sets”. Of the many structures that obey the Zermelo-Frankel axioms, we’d like to pick one out and call it the standard model — the ordinary universe of sets that we’ve all been studying since fourth grade.
But when you meditate on that notion, you discover that “the universe of sets” is a much fuzzier notion than “the natural numbers” — sufficiently fuzzy that it’s not clear there is a standard model of the Zermelo-Frankel axioms. Roughly this means the following: When we talk about the natural numbers, we are quite confident that we’re all talking about the same thing. When we talk about the universe of sets, it’s not so clear.
This in turn makes the real numbers into a fuzzier notion than you might expect — because you can’t describe the real numbers without talking about sets, and sets are fuzzy. The rational numbers are fine — a rational number consists of a numerator and a denominator (together with some simple rules for deciding when two rational numbers are equal, and for adding them etc.), and numerators and denominators are natural numbers, which we understand. But to describe an arbitarary real number, you’ve got to talk about sets. The square root of two, for example, is the real number that separates the set of rational numbers whose squares are less than two from the set of rational numbers whose squares exceed two. Other real numbers have more complicated descriptions in terms of sets. So if there is no preferred “universe of sets” — and it’s not at all obvious that there is — then there is no preferred system of real numbers.
Of course we all know how to specify a real number — it’s a (possibly infinite) string of digits, punctuated by a decimal point. But which infinite strings are permitted? Well, all of them course. But what are “all of them”? What infinite strings are there? Ultimately, infinite strings are defined in terms of sets, which brings us full circle back to the fundamental ambiguity.
The real numbers, then, are a little fuzzy. The natural numbers are sharp. And their sharpness derives, ultimately, from our direct intuition of them, from which all mathematical knowledge flows.
(Nothing really to add here)
Fantastic post Steve. The sort of thing – the down and dirty – that I wish had its own chapter in ‘The Big Questions’.
A slightly related question: Where to prior probabilities come from? I didn’t find an answer to this in a couple minutes of Googling. Suppose I see what appears to be a bald eagle fly past the window. Either I saw a bald eagle or I did not. In order to estimate the probability that I saw a bald eagle, I would need to know: 1) The prior probability that there would be a bald eagle flying past my window at that moment, 2) The prior probability that I might have just been hallucinating, etc. But how could I ever know the answer to these? For 1), I could look up the migration patterns of bald eagles, etc., but again I would be faced with estimating the prior probabilities that my sources were valid, etc.
Bottom line: in a Bayesian analysis, you have to estimate a prior probability. If you don’t know it, you can estimate it based on other prior probabilities. But at some point, you have no choice but to just posit, with no evidence whatsoever, a prior probability associated with at least one condition. Where is the justification for this?
All you are saying here is that writing knowledge in terms of axioms is only useful if the axioms are meaningful. That is why meaningful axioms are used.
One of the most interesting thing about the ZF axioms is C — the axiom of choice. This is ‘obviously true’ but it leads to ‘obviously false’ conclusions! C is independent of the other axioms, and different mathematicians had different views on it. As I recall Godel thought it was obviously true and Cohen thought it was obviously false.
It is even hard to decide if C is an axiom that logically creates sets or logically forbids certain kinds of sets. Very fuzzy indeed!
I appreciate Landsburg’s distinction between our understanding of rational numbers and our understanding of sets. I haven’t read Principia Mathmatica, but I’ve heard that it attempts to create a foundation for mathematics based in set theory. And that critics (and eventually the authors?) concluded that set theory is not the firmest of foundations.
That said, to reprise our prior discussions: I have understood mathematics as a skill – much like reading — not as a field of knowledge. I still have not identified any practical purpose to characterizing mathematics otherwise.
Assuming the truth of this statement, what does that say about mathematics? More specifically, how does the field of mathematics change if we say “Natural numbers are TRUE” rather than “Assume natural numbers”? As far as I can tell, all the same conclusions flow from the second statement as from the first, except that the conclusions that flow from the second statement require the qualifier “assuming natural numbers obtain” or something similar. Is that so different that the (implicit) qualifying statement “assuming my intuitions are accurate”?
Awesome post! I like these math/logic posts.
@nobody
Well there is a theorem that states all consistent theories have a model. So if the theory is consistent then it is *true of that model*. So natural numbers are true of *something*. The question is, are they true of the world around us? Saying they are true is just saying the world is a model of them.
Do you regard the statement “The world is a model [example?] of the natural numbers” an empirical conclusion? A logical deduction? A statement of faith?
Thought experiment: What would we say if we discovered some inconsistency between our experience and the natural numbers? Would we conclude that our experience is untrue? Or that natural numbers are untrue? Or would we transcend our inclination to privilege one account over the other? That is, would we simply conclude that our world provides some useful experience for drawing conclusions about natural numbers, and vice versa, but that ultimately natural numbers and the world differ?
If we would adopt the latter conclusion, then I’d suggest we should adopt it even in the instance when we know of no inconsistencies between the world and the natural numbers. That is, I prefer to say that the world provides some useful experience for drawing conclusions about natural numbers and (more importantly to me) vice versa. I don’t find any additional value added by introducing the concept of “truth” into this discussion.
nobody.really:
Do you regard the statement “The world is a model [example?] of the natural numbers” an empirical conclusion? A logical deduction? A statement of faith?
I’ll go with “a nonsensical string of words”.
Very interesting post. One thing I found out recently is that parrots have a basic understanding of numbers. I think this is pretty useful to counter the “math is invented by humans” idea.
One thing I’ve always wondered about is the axiomatic systems for set theory other than ZFC. Do they produce any radically different results? Also, are the Peano Axioms the only axiomatic system for the natural numbers? They seem pretty solid, but were there other contenders or alternatives?
@Roger Schlafly, I don’t think that’s at all what he was saying.
On Descartes…
To be more specific Descartes was trying to determine if anything was real and if even he himself existed. He decided, actually, that he could doubt all these things. It was in this despair of doubt that he found his existence — the one thing that he did not doubt was that he could doubt. I think; therefore, I am.
Parrots and other animals perhaps can understand numbers. Computers too understand numbers in an operational sort of way. Who said human intelligence is needed to understand numbers? An alien intelligence out there, which Steve is arrogant enough to claim cannot exist, also understands numbers, I expect. Numbers are a concept of an information processing system, and to the extent that information processing is part of nature, numbers are part of nature.
i think any animal that has a concept of ‘self’ must at least understand three or four numbers. one – ‘me’, two – ‘you’, three – ‘us’ and zero – ‘that dead critter over there’
that being said i think numbers are fantastical constructs that do not exist anywhere in nature and therefore…do not exist.
@Steve Landsburg:
Is it a meaningless string of words to say that the real world is a model of Euclidean geometry? I think not, I think it perfectly meaningful, and false. Every consistent theory has a model. Saying the theory is “true” rather than “true of” some specified model means the model fits the world. There are models where the axiom of choice holds, and there are models where it does not. So in one model it is true and in the other it is false.
If someone said that intgers modulo 13 was amodel of how the world works I think you would point to the existence of the baker’s dozen and conclude the claim was obviously wrong. But that is not the same as concluding it is meaningless; quite the opposite in fact.
@Steve Landsburg:
I confess I am still at a loss to understanad why “Proposition P is true in Model M” which a model theorist might write M |= P is not the same as “M is a model for P”, or how one statement can be meaningful and not the other.
Ken B:
Is it a meaningless string of words to say that the real world is a model of Euclidean geometry?
A model is a mathematical structure. The real world, according to most people, is not a mathematical structure. Therefore the real world is not a *candidate* to be a model of anything.
“M is a model for P”,
Models are models for theories, not for propositions, but I do not object to “M is a model for P” as informal shorthand for “Propostion P is true in Model M”.
The rabbits in Watership Down could count to four; numbers beyond that were deemed “a thousand.” For what it’s worth….
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“The square root of two, for example, is the real number that separates the set of rational numbers whose squares are less than two from the set of rational numbers whose squares exceed two. Other real numbers have more complicated descriptions in terms of sets.”
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I’m not sure I know how to ask a meaningful question about this, but . . .
What about time?
Suppose, for instance, that I could figure out a way to avoid experiencing every instance of time that corresponds to a rational number of hours during the next hour. The instance of time equal to 1 hour is gone. The instance of time equal to 1/2 hour is gone. The instance of time equal to 345/1234 hour is gone. Etc. If I could do this, how much time would I experience during the next hour?
Based on my training in mathematics, I would integrate time from 0 to 1 and remove the measure of the set of rationals in that interval, so I’d be left with the full hour. Does this have any meaning in the context of this post?
What I’m trying to get at is the issue of whether or not real numbers exist in the same way that rationals do, and time is one thing that I can’t imagine modeling without the existence of real numbers.
I don’t think about this topic much, so maybe there is too much wrong with this question to warrant a response.
@Steve:
Well it’s a minor point but isn’t it true that a first order theory has a model iff every finite subset has a model? So technically M models the set with one member, proposition P That is M|= {P} rather than M |= P. In any case I recall being given assignments to prove M |= P for some given proposition P many long years ago, so the mild abuse cannot be that uncommon.
Thomas Bayes:
I agree that the irrational numbers between 0 and 1 have measure 1. It does not follow that we have pinned down what those irrational numbers *are*.
In particular: ZFC (the usual set of axioms for set theory) has a countable model — that is, a countable candidate for the “universe of sets”. So ZFC cannot rule out the possibility that *every* set is countable (including, of course the set of real numbers). Within that model, one can still prove that the reals in the interval [0,1] have measure 1, and hence so do the irrationals in the interval [0,1]. (And you can prove every other theorem you know about the real numbers while you’re at it, because the axioms all hold, hence so do the theorems.) But when you envision the real numbers, you probably don’t envision a countable set.
So the collectivity of all the theorems you can prove is not enough to pin down your vision of the what the real numbers are. It’s not clear that anything else can pin it down either, or that your conception is even clear enough for “pinning it down” to have any meaning.
Hmm. This last response to Thomas Bayes has a wording issue.
“So ZFC cannot rule out the possibility that *every* set is countable .” The word countable has switched meanings in midsentnce here. I can prove in ZFC that the power set of the integers is “uncountable”, that is not ‘finite’ and not susceptible to being mapped 1-1 with the integers. In a countable model of ZFC (everyday meaning of countable)– which MUST exist — that must still be true, there must still be such an “uncountable” set. But that might not mean quite what we think of as “uncountable” the way we normally think of the real numbers. As Steve notes, it might not have more than what we think of a a countable number of members, but to inhabitants of the model it would be “uncountable”.
Ken B: Yes, exactly. The reals would be internally uncountable but externally countable — which means that inhabitants of the model would see them very differently than those outside the model. But since it’s not at all clear what model we inhabit (or whether there’s even any meaning to that notion), it follows that our vision of the reals is at least somewhat indistinct.
Is the reason we can’t learn everything there is to learn about the natural numbers just by studying the Peano axioms the same reason we can’t learn everything there is to learn about R^3 by studying finite dimensional vector spaces?
I’m afraid I don’t understand the theory/model relationship very well. Is the example I gave of the correct form?
mcp:
Is the reason we can’t learn everything there is to learn about the natural numbers just by studying the Peano axioms the same reason we can’t learn everything there is to learn about R^3 by studying finite dimensional vector spaces?
This is in spirit exactly right. The theory of finite dimensional vector spaces has many models, one of which is R^3. The theory of Peano arithmetic (that is, the theory described by the Peano axioms) has many models, one of which is the natural numbers. But R^3 has special properties not shared by other finite dimensional vector spaces, and the natural numbers have special properties not shared by other models of Peano arithmetic.
Great example.
@nobody.really
great book. topics like this often give me tharn. (if my memory serves and that usage isnt butchery)
btw..how do you add the quotes? i recall seeing someone mess it up on one post and i meant to remember it..but i mustve forgotten.
[“]
@mcp:
This theorem, the Lowenheim-Skolem theorem, is like Godel’s theorem, really about the inherent limits on formal methods. As it turns out however necessary and useful formal theory is it can never be quite adequate to deal with what is really true. Godel’s theorem show that provable and true are never quite the same. This theorem shows that meaning and truth are never quite as clear as you might like either. (Insert caveats about all of Peano’s axioms, and countable axioms here.)
@Steve:
Yes our notions of the reals have a lot of problems. Not least the continuum hypothesis or the axiom of choice and measure. The older I get the more sympathy I have with the Brouwer crowd. But I still think pure existence proofs like the one where you can prove there exist 2 irationals a, b such that a^b is rational are valid. [For th math inclined: Form root2 to the root2 to the root2. If root2 to the root2 is rational then done, else call it a and root2 b. Done. Brouwerites reject this basically because you cannot say WHICH case holds — the proof is non-constructive. But it convinces me.]
I’ve always had trouble with the concepts of countability and uncountability. I know there can’t be a one-to-one mapping from say the rationals to the irrationals, but I don’t know how to reconcile that with the density of each set in the other.
It seems to me that in order for there to be “more” irrational numbers than rational numbers, there have to be at least 2 irrational numbers x and y such that there is no rational number a so that x > a > b. But this violates that the rationals are dense in the irrationals and vice versa.
I think there’s a flaw in the way in the way I think about cardinality, but I don’t know what it is.
@JLA:
Yes there is a flaw, but the subject is a difficult one. Think of the rationals as a mesh, like you might have in a screen door. You make the wire ever thinner but keep doubling the number of lines. If you ignore minor issues like atoms then you can continue this forever, with the same amount of metal being used. At each stage you get more air than mesh.
The key issue is taking limits. We might THINK we know about the reals but at a certain level we can talk about them ONLY by constructing limits.
To go back to your example, here is a little trick to make it seem more plausible. Pick a natiural number d as a denominator. Any d but it is fixed. Then you can always find 2 irrationals with no rational whose lowest form has d as denominator between them. But the converse is not true: you can always find irrationals between any 2 x/d numbers — or any irrationals too.