I recently emailed my book, The Hiddenness of God, to hundreds of academic mathematicians, to see whether or not mathematicians would be interested in the proof buried
beneath the foundations of their subject, and I have had some replies already. The following conversation has been edited, but it is fairly typical, in case you were wondering (as I was) what mathematicians would think of my proof.
Mathematician: Russell's paradox (and
Epimenides' before him) demonstrates simply that the concept of "truth
value" that many logicians had assumed to be well-defined on all
statements, and which works well most of the time, must in fact have a few
limitations. When we talk about truth values too loosely, plain English hides
the fact that we're discussing a function from the class of propositions to the
set {T,F} that may not in fact be wholly defined. It's no more mysterious than
the discovery that division by 0 can't be defined except by giving up several
arithmetic properties that are otherwise unproblematic. Russell simply shows a
similar restriction for truth values of self-referential statements. This is
well-understood.
And Cantor's theorem isn't even a paradox: it just shows that if we define an ordering
by "size" on infinite sets, then the rationals and the reals are in
different size classes - and why shouldn't they be? Our ability to
"comprehend" either is ill-defined (this is where plain English lets
us down): we do not know everything even about large finite numbers (which
digit appears most often in 9^(9^(9^(9^(9^9))))?) and we know a very great
deal about the real numbers, more numerous than the natural numbers though they
are.
While Russell's paradox did do that, the heap paradox
and the liar paradox had done it thousands of years earlier. And while Cantor's
theorem is indeed not a paradox, it exists within axiomatic set theory.
Cantor's paradox arises for the numbers that Cantor was working with, which
were essentially the same as the numbers that we learn about at school. There is an obvious and unambiguous meaning to the word "two": two is the number of things in any collection that has as many things
in it as the sum 1 + 1 has units in it.
Mathematician: I think the heap
paradox is most easily interpreted as showing the axiom that one grain less
than a heap is still a heap to be inconsistent. Heapiness is problematic in
other ways as well. If we base our
definition on general opinion, we more or less have to test it by asking an
observer "is this a heap?" and the answer may depend on the observer.
If we don't appeal to opinion, there's no reason not to define a heap as a
thousand grains or more of sand, or sand grains piled at least five deep.
And
while what you said is true for "two" there are more real numbers (in
the usual sense) than there are definitions in finite strings of characters...
and this happens precisely at the spot we're interested in.
Plain English is good enough for the definition of
"two," though; and similarly, for an arbitrary counting number (even
though most counting numbers are too big for us to imagine anything about them
other than that they are counting numbers). And Cantor's paradox arises for
arbitrary subcollections of subcollections of [...] subcollections of counting
numbers. The real numbers are complicated (and Richard's paradox is
interesting) but irrelevant to Cantor's paradox. As for the answer to "is this a
heap?" I think that it can depend on the observer, and that that is one of the reasons why some
piles of sand are only heaps as much as they are not heaps. Insofar as they are
heaps, removing a single grain of sand would make a negligible difference to
that. And for such a pile, "that pile is a heap" would be true only
as much as it was not true. And similarly, the liar paradox shows that there
are self-referential statements that are true only as
much as they are not true. So, Russell's paradox is more like the liar paradox
(and the heap paradox) than Cantor's paradox.
Mathematician: I think the heap
paradox is somewhat different in that it can be dealt with by saying
that "well, it seems that we need to sharpen our definition of a heap. A
heap will be any collection of sand numbering more than ten grains, stable, and
at least a quarter as tall as it is high." That's roughly what
Cantor did with infinities... a fairly small patch on existing math. The first
was a paradox, and not the second, only because people had more
preconceptions about heaps. Cantor's result is more a proof by contradiction,
eliminating a wrong turning in an exploration of new territory. If Eubulides of
Miletus had been researching novel ways to store sand (insight - we don't need
a bucket!) he might have used the sorites paradox similarly. The liar paradox
can't really be explained away by inventing a better liar: it needs the concept
of truth that underlies all philosophy to be redefined. Similarly, Russell's
paradox involved a complete revamping of basic set theory.
I don't think that the heap paradox can be dealt with by
saying that we need to sharpen the definition of "heap" because
similar paradoxes occur with almost all of our words (as Russell observed) and
because our words simply have the meanings that they have: if we redefine what
"truth" means, then we are no longer talking about the truth of our
words. I suppose that Cantor's paradox is the proof by contradiction that you
think it is if there is no God, but is the proof by contradiction that I think
it is (a proof that there is a God) if we should not redefine what
"truth" means in order to avoid an inconvenient proof.
Mathematician: It's true that if
we take "Cantor's paradox" as a standalone result, rather than as the
obvious (in retrospect) conclusion of his construction of sets of demonstrably
different cardinality, it looks more like Russell's paradox. That's not the
angle I'm used to seeing it from, but I think I see your point. Nonetheless, in
Cantor's case we don't have to redefine "truth", we merely have to
redefine "set" so that some things we would have naively called sets
are "classes" with a smaller set of permitted construction rules. As
for the relevance to God: I am not a believer, but quite happy to argue
hypotheticals. I agree with Aquinas that any god that exists must be bound by
the laws of logic. These are the same laws of logic that bind us: and I see no
reason why using a definition of "set" that Cantor showed to be
inconsistent could be a divine attribute, let alone why we should want it to be
so. Aquinas says in effect that, regarding logic, what's good enough for Cantor
(if Cantor is right) is good enough for God. You
don't get around Cantor by supposing "theological unions" of sets
that somehow differ from those of set theory (or, if you do, you must explain
their properties fully and equiconsistently with ZFC or some other well-defined
system).
I agree that we should be bound by the laws of logic, and
I take that to mean that we cannot just make those laws up. And I am certainly not trying to get around Cantor by
supposing theological unions (whatever they are). I am questioning his
assumption that mathematical collections must exist timelessly. Cantor chose to
believe in the existence of collections that were inconsistent, rather than
give up that assumption! Mathematicians can of course use any definition of “set”
and “class” that they like, but there is still the paradoxical behaviour of
mathematical collections (in the logical sense) to explain. Cantor’s paradox
showed that his conception of set was inconsistent, but his conception included
the assumption that mathematical collections exist (insofar as such things can
be said to exist) timelessly. Incidentally, although Russell found his paradox while
he was thinking about Cantor’s paradox, I don’t think that Cantor’s paradox is like Russell’s paradox.
Mathematician: My view is that
the word "exist" is not used in mathematics in the sense that
Mount Everest is and Alma Cogan isn’t (as the guy on the Monty
Python record put it). It's an axiomatically-defined predicate in mathematical
theories and metatheories (parallel lines exist in the Euclidean plane, they do
not exist in the projective plane). From this viewpoint, I don't see
time/timelessness as having anything to do with mathematical existence (I
suppose one could take a time-dependent Platonist view where pi really was
three in Old Testament times, but that is not how I see it).
For most mathematicians nowadays, mathematical existence
is indeed existence within an axiomatic structure, and for such structures it
is consistency that matters. And within set theory, there is only Cantor’s
theorem. But for numbers like the counting numbers and the number of all the
counting numbers, and so on, it is logic that matters: such numbers are essentially
properties of logically possible collections (you and I are two people, and we
would have been two possible people had we never existed, and the properties of
that “two” are logically prior to any axiomatic model of them). And if it is logically
possible for there to be a God, then there are all the numbers (in that
sense) that give rise to Cantor’s paradox. That is how I have been able
to show that if it is logically possible for there to be a God then there is a
God, because it is only if there is a God that such numbers could possibly be
getting more numerous (and it is only in the last hundred years that
mathematicians would have denied that such numbers were part of mathematics).
Mathematician: The statement that
"numbers are getting more numerous" is, if not downright false,
highly ambiguous. Our mathematical knowledge may encompass more numbers, but a
given axiom system implies the same numbers yesterday, today, and forever, even
if nobody alive at some time understands that. Furthermore I hold, with (for
instance) Aquinas, that it is a logical necessity that no deity could change;
so, claiming that the creation of new numbers within a fixed axiom system implies
the existence of a god is true only ex falsi quodlibet. Apart from that
major objection, if your argument did prove the existence of some entity X, I
think (again, hypothetically) that it would fall far short of showing that this
X was what was generally called "a god," let alone a specific faith's
God.
The numbers in “numbers are getting more numerous” do
not exist within any axiom system, but as a consequence of there being numbers
of things in the world (such as us two). Axiomatic models of them are timeless,
but they themselves are properties of logically possible collections of things,
so it is a matter of objective fact whether they are timeless or not. And while
we naturally assume that they (and logical possibilities generally) are
timeless, it is conceivable that they (and some other logical
possibilities) are not timeless if there is a God who is not timeless. As for your belief that if there was a God then
that God would have to be above and beyond time and change, I suppose that you have
a good reason for believing that, but as I do not know what that reason is, I
cannot say why it is not a valid reason (and similarly for your reason for
believing that X could not be called a God, unless it is the same reason). I
have thought a lot about the reasons that are in the literature, and none of
them are valid when it comes to the God that Cantor’s paradox shows exists (which
did not surprise me because a lot of the religious believers who take God to be
above and beyond time and change would also say that He is above and beyond our logical abilities).
Mathematician: You would seem to
be saying that there's an argument showing, on the basis of some axiom system,
that some number (call it Stigma) exists... and that at some time in the past
the same argument was not valid, or was valid but did not show that Stigma
existed. A fun science-fiction idea, but in reality if we pick at it, expanding
the argument out to a long but finite list of axiomatic steps and going through
it a step at a time, there's a step that somehow didn't work then and does now.
But that step is supposedly an instance of an axiom, so the axiom set has
changed. Gods whose powers vary in time (depending
on who's stolen whose hammer today) are more at home in comic books than in
philosophical arguments; when I said "god" I meant the sort of god
that modern philosophy usually considers, whose view of the universe is in some
sense ultimate and synonymous with reality. If the power of such a god were
greater today than yesterday, it would have to have been less than it might
have been yesterday. Which, as Spinoza would have said, is absurd.
I too meant the God whose view of the universe is the
universe. And I agree that the power of such a God cannot increase, or decrease. However,
the knowledge of such a creator would increase as a matter of logical necessity whenever any particular thing was created (as I
show in the first “chapter” of my first email). As for your interpretation of
what I was saying in terms of an axiom system, the existence of the most
basic numbers (1, 2, 3 etc.) does not have to be existence within any axiomatic
system, even if there is a God. The existence of such numbers could be the logical
possibility of there being collections of that many things (which is why my
argument is a logical argument based on Cantor’s original paradox, which he discovered before mathematicians and philosophers axiomatized numbers and collections) [...]
