FOM: "when humanity disappears"...numbers and G-d

[Neil Tennant]

Reuben Hirsch writes

"When humanity disappears, there will no longer be any integers as
abstract objects."

What, pray, is so important about *humanity* as opposed to any other
rational creatures that there may be elsewhere in the universe? And
has anyone ever proposed that integers are concrete, as opposed to
abstract, objects? So let's consider the needed generalization, and
express it more economically:

"When all intelligent mathematicizing life in the universe disappears,
there will no longer be any integers."

Thought experiment: suppose this claim is true. Suppose further that
the planet Earth reaches a time (relative to its intertial frame) by
which, as it happens, all intelligent mathematicizing life in the
universe has died out. Then, according to Hirsch *there would no
longer be* any integers.

Now suppose that life evolves all over again on some planet. In due
course intelligent social-living creatures are produced, who develop a
language for the communication of their thoughts. They count, they add
and multiply, and they generalize about integers.

Question for Hirsch: are these the same integers as ours? Or have
these creatures 'invented' their 'own' integers? If they are indeed
the same integers as ours, how did these integers (all infintely many
of them) manage to go out of existence and then came back into
existence? For they are, after all, abstract objects, are they not?

So the suggestion must be that the integers are abstract objects, but
not *timeless* abstract objects. What sort of abstract objects are
these?

Well, perhaps the integers are like colors---or, better, color qualia.
Colour qualia might be thought not to be able to exist without there
being sighted creatures alive and able to have them. So if all sighted
creatures died out, perhaps the color qualia would die out with them.
If, subsequently, by a serendipidous miracle of evolutionary
re-capitulation, creatures with exactly the same biological
constitution were to evolve all over again, we would be very tempted
to say that *their* color qualia (the red ones, say) were the same
kind of qualia as their precursors enjoyed in the visual presence of
red things. (This is a thought experiment, so we do not have to worry
about probabilities here.)

But why would we be tempted to say that there are the same color
qualia before and after the interregnum? An important part of the
answer would be that the conscious experience of red is in some way a
response on the observer's part to something in the physical
constitution of red things---let us say, their reflectance profile.
So, to the extent that the abstract object (i.e. the type of conscious
sensation we call "seeing red") once existed, then ceased to exist,
then existed once more, it is (partly) because there was a physical
invariant---the reflectance profile---to which it could be tied.
Things have reflectance profiles during the interregnum, even if they
provoke no color experience during that period.

Here, though, is the crucial disanalogy with integers. We cannot make
sense of their existing, then ceasing to exist, then existing again,
by reference to any physically enduring objects during the imagined
interregnum. So Hirsch would be in a difficult position to explain the
alleged "on-off" character of the integers' existence.

But perhaps, Hirsch might object, we can identify the "later"
(evolutionarily re-discovered) series of integers with the "earlier"
series of integers because (at least some of) those integers could be
"read into" collections of physical objects, collections which existed
during the "off" period of integer-existence---in a way analogous to
that in which physical objects capable of provoking conscious
experiences of redness might exist during the "off" period for color
qualia when no sighted creatures were alive to have them.

If this is so, then Hirsch faces the objection that the numerosity of
those things during the interregnum (for the integers, for want of any
mathematicizing intelligences to grasp them) is being conceded to be
independent of mathematicizing intelligence. Surely, then, the number
attached to the collection of things in question exists throughout
that interregnum as well?  There is no difference between

	There are nine planets

and

	The numbers of planets is 9.

If it remains true, during the interregnum, that there are nine
planets, then it remains true also that the number of planets is 9.
This claim is immune to the objection that there is no intelligent
being alive during the interregnum to grasp the latter thought. For,
ex hypothesi, there is no intelligent being alive during the
interregnum to grasp the former thought either.  So, the Platonist
will say, the number 9 will exist throughout the interregnum.
For the number 9 exists by virtue of the in-principle-expressibility
of "numerosity thoughts" (such as "There are nine planets") as
"number-identifying thoughts" (such as "The number of planets is 9").

This line of thought might not cut much ice with Hirsch, who is
inclined even to deny the existence of *thoughts* during the
interregnum when there are no living intelligences. But in that case,
he would face the original puzzle in an even more difficult,
generalized form.  For how, now, would one account for the claim that
the newly evolved intelligences after the interregnum are "doing
integer mathematics"?  How would one identify the *thoughts* they were
having when doing their post-interregnum mathematics?  If thoughts
themselves (in the Fregean sense) ceased to exist during the
interregnum, would there be any *facts* obtaining during the
interregnum? (For are not facts simply true Fregean thoughts?) What,
for example, would become of the fact that there are indeed nine
planets, during an interregnum in which there was no intelligent life
to grasp the thought? Would the *fact* be snuffed out along with the
Fregean thought?

It seems to me that Hirsch's innocent-sounding anti-Platonist
naturalism, whether it goes under the label of 'social ocnstructivism'
or not, is in danger of sliding all the way to out-and-out idealism. I
contend that, to the extent that he might succeed in arresting such a
slide, it will be at the expense of a concession, somewhere along the
line, that would re-instate the integers as existing during the
imagined interregnum.

The Platonist takes the timeless character of abstract existence very
seriously. 

As for Hirsch's claim 

"I am compelled to compare [Platonism] with the existence of G-d"

---he would be hard put to link God-talk to talk about ordinary things
in a way that is satisfactorily analogous to the linkage of
number-talk to talk of ordinary things afforded by the Schema

	there are n Fs if and only if the number of Fs = n*

(where 'n' is adjectival, and existentially non-committal about
numbers, while the numeral 'n*' is substantival, and existentially
committal about numbers). The schema just given states an essential
conceptual control on our talk about numbers as objects. There is no
similar schema placing any kind of conceptual control on God (or G-d).

Thus Hirsch's comparison is most unpromising. There is more serious *a
priori* certification for Platonism about numbers than there is for
either monotheism or polytheism. The Fregean argument for the
(necessary)[and timeless!] existence of natural numbers is way better
than any argument (from Anselm, Aquinas, Augustine, Descartes,
Leibniz,...) for the existence of God. 

Neil Tennant