FOM: Transfinite Logic

[Richard Heck]

Dean Buckner wrote:

>On the theorem that the set of all natural numbers {0,1,...} is equinumerous
>with the set of all even numbers (0,2,...}, Richard Heck writes "There is no
>questioning this result. It is a totally straightforward, and extremely
>simple, theorem."
>
>Agreed, if it is meant that (in plain English) every number has a double,
>and every double is the double of one, and no more than one number, which
>follows entirely from the definitions Heck has set out.
>
>I question his definitions.  If every number has a double, and every double
>is the double of just one number, does it follow that there are "as many"
>doubles (even numbers) as singles (integers), given the plain English
>meaning of "as many"?  Is it correct to define "as many" or
>"equinumerous" using the idea of 1-1 correspondence?  My argument is about
>use of English, not mathematics!
>
George Boolos used to tell the following story. At the end of his thesis 
defense, Putnam asked him, "And tell us, Mr. Boolos, what does the 
analytical hierarchy have to do with the real world?" Not missing a 
beat, George replied, "It's part of it".

As Prof. Davis remarked, we are talking about /numbers/, not about 
/English/. We are of course using English to talk about numbers: The 
language used by mathematicians is part of English (insofar as that is a 
clear notion, which it is not). And, as with all the concepts of 
science, our mathematical concepts have evolved from "folk" concepts 
rooted, in many cases, in our innate conceptual endowment. But these 
concepts are shaped and transformed by scientific investigation, which 
need not simply rest with them. It is no embarrassement that the 
concepts needed for serious scientific investigation are hard for the 
uninitiated to understand. That is true not just of our modern concept 
of cardinality but of more "empirical" concepts, as well.

The argument we are having is not unlike that Galileo had with those 
unwilling to acknowledge the distinction between average and 
instantaneous velocity. His interlocutors too might have said, "We don't 
care about how scientists talk! We only care about the use of Italian!" 
But plainly, that would have been a very strange argument, to which 
Galileo might have replied: I thought we were talking about physics, not 
about language.

I'll return to this comparison. But before I do, let me make one other 
remark. I think we should not be too quick to dismiss the sorts of 
questions Buckner is raising. The study of our "folk" concept of number 
is not without interest: It has the same kind of interest that the study 
of our "folk" physical and psychological and biological concepts have, 
which is considerable. And, indeed, there has been some very interesting 
recent work on the development of concepts of number in children. But 
one should not confuse psychology with mathematics.

Nor is the study of our "folk" concept of number is without interest to 
the philosophy of mathematics. Some of the disputes here, or so it seems 
to me, turn upon questions about the nature of our /concept/ of number. 
(I have in mind, for example, the question of the status of "Hume's 
Principle".) It is an empirical, indeed psychological, question what our 
/concept/ of number is like.

>The argument usually then turns to the challenge of defining "as many as".
>This brings us right to the very beginning, when I argued (in some earlier
>postings) that can think of number as satisfying n-place predicates such as
>"x is a different thing from y", which is satisfied by Abbott and Costello,
>Oliver and Hardy, but not Clemens and Twain, and not the Three Stooges.
>
>Any two collections are equinumerous (have as many objects as each other)
>when there is such a predicate they both satisfy.  Defined in this way, no
>proper subset can be equinumerous with its parent.  The parent, by
>definition, contains objects "different" from any of those in the subset,
>hence the parent cannot satisfy the same n-place predicate as the subset.
>
>It's not open now to argue that "it's different for infinite sets" as many
>of my offline correspondents have tried to do.  Having defined "proper
>subset" and "equinumerous" in a way that no proper subset can be
>equinumerous with its parent, it's not open to say this any more.  The
>correct approach (which Heck alone has sensibly adopted) is to challenge
>this definition of "as many as".
>
But, of course, I don't challenge this definition of "as many as": I 
simply say, as is obvious, that it fails to apply to the case of 
infinite sets (unless you use infinitary languages, but then you get the 
usual results). It does not tell us that "no proper subset can be 
equinumerous with its parent". It tells us only that no proper subset of 
a set /for which it delivers a number/ can have the same number as the 
whole set.

The question then arises: Is there any reasonable way of defining 
equinumerosity for infinite sets? Better: Is there any way of defining 
equinumerosity /simpliciter/ (that is, without regard for whether the 
set is infinite or finite) that returns this familiar definition as a 
special case? The answer, of course, is familiarly "yes": The now 
standard definition, due to Cantor, does just that. (That, indeed, is a 
kind of adequacy condition.)

>Except I question that he has successfully done so.  My theory, if you like,
>is that natural language has built into it a system of numbering, and a
>concept of number that is at odds with "transfinite" ideas about number.
>This is why - given that natural language underlies the way we all think (or
>at least learn to think as children) - people sometimes find these ideas
>"difficult" (as Heck points out).
>
Now, I think, we get to the point. As mentioned above, I of course do 
not disagree that we have various "folk" ideas about number. And I agree 
further that elements of these "folk" ideas are part of what make it 
hard for some people to assimilate Cantorian ideas about the 
transfinite. But the question is what to make of that fact. I say we 
should make of it pretty much what we make of the fact that children, at 
a now well studied point of their development, have a hard time 
assimilating the distinction between instantaneous and average velocity. 
And we should attempt to explain it in a similar fashion. Our "folk" 
ideas about number merge /two/ notions of number, as our "folk" ideas 
about velocity merge two notions of velocity: There are, in particular, 
both ordinal and cardinal elements in our "folk" concept of number.

Recent work (being done by Susan Carey, among others) gives us reason to 
think that there may  be one system involved in the cognition of very 
small numbers (say, up to five or so) and another involved in the 
cognition of larger numbers. The latter employs something like 
/counting/. It seems to be to this system that the ordinal elements of 
our concept of number trace. The former system employs something that is 
more like pattern recognition: Think of Mill's remark that a "three" is 
a group of objects that can be arranged in a certain fashion. The 
elements of our concept of number that emerge from this system are 
cardinal in their character: The standard definition of cardinality in 
set-theory is, after all, one of a set whose members /can be ordered/ in 
a certain way. (Frege's pithy response to Mill, that we're lucky not 
everything is nailed down, doesn't apply here, since the modality can be 
eliminated in favor of quantification over functions, a reply 
unavailable to Mill.) The "pattern-recognition" system has as much claim 
to be part of our "ordinary" notion of number as the counting-based 
system does: In fact, it probably has more of a claim, as it appears to 
be more primitive and active from an earlier age.

There is also, as I've discussed elsewhere ("Counting, Cardinality, and 
Equinumerosity"), a role for a notion of there being "just enough to go 
around". That's a notion very similar to one-one correspondence. It too 
has every right to be considered part of our ordinary notion of number.

Our ordinary concept of number, as I said, has all these facets. And we 
ordinarily suppose, quite without thinking about it (and probably by 
design), that the various systems involved here will always deliver the 
same sorts of results, just as children ordinarily suppose, pretty much 
innately, that the car that is blurrier (has the higher instantaneous 
velocity) will also get there first (has the higher average velocity). 
For much of what they encounter with velocity, and all of the cases of 
number that we meet in day-to-day life, these presumptions are 
satisfied. But not always. Galileo got himself into a lot of trouble by 
producing cases that frustrate our naive expectations in the physical 
case. Cantor and Dedekind got in less trouble, fortunately, but they 
were just doing the same thing, so far as I can tell, with the case of 
number. Piaget, for his part, recognized that there is something worth 
studying here: Although he had, of course, no interest in denying the 
reality of Galileo's physical distinction, he realized that the very 
fact that grown people /did/ and small children /do/ have a hard time 
assimilating it reveals something about our conceptual development 
(though not, as it happens, quite what he thought it did). I am 
suggesting much the same thing: The fact that people find it hard to 
assimilate the concept of cardinality that we find in transfinite set 
theory tells us something interesting about our ordinary concept of 
number. That's of interest to psychology and, I think, even to the 
philosophy of mathematics.

>This may also be why it is difficult to explain transfinite ideas, as
>embedded in mathematical symbolism, since to "explain" anything is to
>translate such symbols into plain everyday language.  If (as I've argued) we
>can't do this, then there's a problem.  Particularly, as we saw, for writers
>of elementary textbooks.
>
So I've offered an explanation of this fact, one quite different from 
that suggested here. I see no reason to think that "explanation" must 
always mean /translation/ into ordinary English. Explaining the concepts 
of string theory does not involve such "translation". And that is 
certainly not how we acquire the distinction between instantaneous and 
average velocity, nor how Galileo conveyed it to his contemporaries.

Richard Heck