mathematics and universal truth
Every person who studies mathematics (and here I’m not talking about the fresh-faced achievers from the fraternity/sorority system outfitted in your choice of Abercrombie/Hollister/Old Navy you slept next to in College Algebra; rather, I am talking about math majors and mathematicians, outfitted in the same clothes they’ve worn for the last three days) must confront the following question: Does mathematics express some intrinsic universal truth, or is it simply a semantics game played with man-made symbols?
I saw a wire story today that I read as evidence for the former. Duke University conducted a test with twenty infants aged seven months old to measure whether the infants were able to match the number of voices they heard with the number of faces they expected to see. The Duke researchers found that they could:
The study of 20 infants by researchers at Duke University was similar to a previous experiment done to demonstrate that monkeys show numerical perception across senses.
In the new study, babies listened either to two women simultaneously saying the word “look” or three women saying the same word.
At the same time, the infants could choose between video images of two or three women saying the word.
As they had found with the monkeys, the researchers said the babies spent significantly more time looking at the video image that matched the number of women talking.
“As a result of our experiments, we conclude that the babies are showing an internal representation of ‘two-ness’ or ‘three-ness’ that is separate from sensory modalities and thus reflects an abstract internal process,” researcher Elizabeth Brannon wrote.
“These results support the idea that there is a shared system between preverbal infants and nonverbal animals for representing numbers,” she said.
“What we do know is that somehow, very quickly, they’ve (the babies) acquired this ability to perceive number and divorce it from the sensory information,” Brannon said.
This story might not have even caught my eye or rated more than a passing glance had I not just spent the last several evenings reading about the life and work of one of history’s greatest mathematicians — Princeton’s Kurt Gödel.
Kurt Gödel proved, in his famous Incompleteness Theorems, which say essentially that
- Any formal logical system powerful enough to express arithmetic in terms of the system’s axioms is incomplete, i.e., contains a true statement that is unprovable from the axioms; and
- No formal logical system powerful enough to express arithmetic in terms of the system’s axioms is both complete and consistent, i.e., contains no contradictions. The concept of consistency is important because in an inconsistent formal system, any statement at all — even false statements — can be “proven.” An inconsistent system essentially says nothing.
The phrase “a system powerful enough to express arithmetic” can be generally understood to mean “a system in which construction of the set of natural numbers (= {0, 1, 2, …}) is allowed.” These theorems, of course, necessarily include any system based on such a system; which, as it turns out, is nearly all of mathematics. It doesn’t matter how complex your system is or how many axioms it has — the system is either inconsistent (and hence useless) or incomplete. Also, mathematics contains at least one statement that is both true and unprovable. Does it represent universal truth? Consider this: if you add this true and unprovable statement as an axiom to your system, and if it is logically consistent with the other axioms, then your new system will be consistent; hence it would be incomplete and thus contain another true and unprovable statement! What does this represent?
In short, every logical system which is useful for something must contain at least one statement that is both true and unprovable. So, how do we know that the statement is true, if it cannot be arrived at from first principles? There are competing schools of thought on that, and while that discussion is interesting — and relevant to this post — I will not discuss them here.
What does all this have to do with babies — and monkeys! — matching a given number of voices to the same number of faces? Infants are far too young to be socialized and can only communicate with adults on a rudimentary level; how did they acquire concepts of “two-ness” and “three-ness”? How did monkeys — who cannot verbally communicate with adult humans at all — acquire these concepts? Perhaps the German mathematician Leopold Kronecker was right when he said that “God made the natural numbers; all else is the work of Man.”
I prefer to believe that there are universal truths which we can — and eventually will — discover. It is clear from what I have seen that Gödel himself believed this as well, which would have made him a distinct minority in the circles in which he travelled. Consider Einstein’s relativity theory (which says, among other things, that an absolute measurement of time does not exist. Incidentally, Einstein was a close friend of Gödel) and Heisenberg’s uncertainty principle (which says, among other things, that one cannot accurately measure both the location and speed of a moving particle). All three of these theories insert some sort of uncertainty into the universe, something that transcends human ability to quantify and measure. Instinct and intuition enters in.
That’s all that infants and monkeys have — instinct. Something they know is true; the properties of “two-ness” and “three-ness”. Scientists can show that they have these concepts.
My question is this — How did they get them?
I will discuss what I think would be the answers from competing groups in a later post.
UPDATE: The scientist quoted by Reuters would totally disagree with me. See if you can figure out how I knew that.
UPDATE 2: This post has been revised, due to some rhetorical inconsistency on my part made known to me by Josh. I realize this is a bit out in left field, but don’t make him do all the work!
02.14.2006 @ 13:55
The fact that babies (or monkeys or apes) have a concept of “2″ or “3″ doesn’t necessarily imply that math, or even those concepts, are truly universal. It could be that these are concepts which the primates evolved and which we share at some deep developmental level. I think I once saw something about crows and counting, and that would be a better test. If species whose intelligence can be seen as independently evolved show similar mathematical abstractions, that suggests something about the nature of the concepts (corvids are super-smart, but not all birds are, nor are various mammals and reptiles which branch off in between primates and corvids).
Of course, there’s a huge gulf between counting and the rules of logic which govern math. It’s always amazed me what you can derive given 0, 1, logic, and addition (maybe addition comes naturally with 0 and 1, I can’t remember now).
Even if numbers exist ab initio, it need not imply that math as we conceive it must all be universally true. Even if it is a set of true statements about universal truths, that doesn’t mean that it’s the only way to discuss those truths. Natural scientists get into this debate in deciding whether natural laws exist, and if so, whether they are unique, or whether one could express the concept of thermodynamic laws in only two laws, or segment them into three parts differently than we now study them. Fermat’s last theorem probably represents some deeper truth about numbers, and we find particular arrangements interesting, so we express that truth in terms of a^n+b^n=c^n, but that need not be unique, and the expression of the truth may not be universal, even if the truth is.
It’s an interesting topic.
02.15.2006 @ 10:43
You wrote:
That is, of course, true in a sense. On some level mathematics is a creation of pure reason.
On another, the question is what does a given mathematical system (i.e., the axioms it rests on) express? Does it express some fundamental property of the universe (a “universal truth”), or is it a Man-made word game? I believe it is the former.
I don’t think that all of mathematics expresses universal truth. (I now realize why you might think I did; I’ll revise it later tonight.) Goedel’s theorems seem to say that at least some part of it does; in fact, that’s what he himself believed.
The truth of any formal system rests on the axioms on which it’s built, and Goedel’s result seems to say to me that at least some part (not necessarily the totality) of mathematics rests upon its axioms, at least one of which ought to be one of Goedel’s true but unprovable statements. What I’m saying here is that there is some part of mathematics which expresses a “universal truth”, i.e., a fundamental property of the universe.