FOM: "when humanity disappears"...numbers and G-d
rhersh at math.unm.edu
Mon Dec 29 14:36:49 EST 1997
On Fri, 26 Dec 1997, Neil Tennant wrote:
> 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?
ANOTHER ASTONISHING DICTUM! This human being pretends to
be neutral as between earthly humanity, and any other "rational"
(meaning ??) creatures that there may or may not be elsewhere in the
universe. What astonishing self-delusion, excuse my emotion.
> has anyone ever proposed that integers are concrete, as opposed to
> abstract, objects?
YES KORNER HAS FOR EXAMPLE, AND I HAVE. THE POSITIVE INTEGERS,
UP TO SOME VAGUE UPPER REGION, ARE CONCRETE AS ADJECTIVES AND ABSTRACT
AS NOUNS, AS EXPLAINED IN MY LAST 2 POSTINGS.
"There are 9 planets" and "The number of the planets is 9" do
not mean the same thing, the second statement says, "There is some
thing called the number of the planets, and that number is 9."
Which is not identical or equivalent to "There are 9 planets."
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
> 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
> 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
> 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
Social consciousness and social practise are not the same
as disembodied, bodiless "ideas." So I am not sliding, rolling, falling
or moving in any fashion into idealism.
As far as this ingenious and far-fetched scenario of intelligent
creatures disappearing and reappearing--sorry, it just leaves me cold.
You can make up fantastic stories and demand that I account for them,
but this is not a game that seems interesting or worth the trouble
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