FOM: Reply to Hersch and Tragesser on aliens and abstract objects
rhersh at math.unm.edu
Fri Jan 2 01:53:00 EST 1998
On Thu, 1 Jan 1998, Neil Tennant wrote:
> In an earlier posting on the topic of whether numbers depended on human
> beings for their existence, I asked
> > What, pray, is so important about *humanity* as opposed to any other
> > rational creatures that there may be elsewhere in the universe?
> Reuben Hersch replied:
> 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.
> First, I do not understand how a question can be a dictum; and I note
> that Hersch did not answer my question. I take it that I am the
> referent of Hersch's phrase "this human being". But I fail to
> understand how I might be "deluded" in this matter. I am moved to ask:
> What is it to be "neutral as between earthly humanity, and any other
> [rational] creatures that there may ... be elsewhere in the universe"?
> The existence of such creatures is a contingent matter. This is a
> contingency that philosophers (and mathematicians) have to take into
> account when contemplating the ultimate nature of an intellectual
> enterprise such as mathematics. My contention is that there ought to
> be a sufficiently abstract account of rationality (even if only of the
> instrumental kind, i.e. means-end analysis) for one to be able to
> contemplate extending the epithet "rational" to other, non-human,
I find it astonishing for any human being to ask, seriously,
What is so special about human (or earthly) intelligence?
If someone is not astonished, that would indicate the
distance between our notions of astonishment.
I find I don't understand "ought to be". It could mean,
I wish there was, or it would be nice if there was. With
that I have no quarrel. If it means, is sure to be or
very likely to be, I can only understand that as following
from a Platonic preconception of rationality as eternal,
universal, inhuman or superhuman. Granted that, then the
rest follows. I don't see that consideration of extraterrestrial
or extragalactic intelligences is necessary or helpful to
understand the nature of mathematics as an earthly, human
We may be the only rational creatures on *this* planet (on
> any satisfying account of rationality
It may be word-quibbling, but to me any species of
animal is rational, to the extent that its behavior
serves to maintain the species.
); but we ought to know what we
> would be looking for on the part of aliens elsewhere in order to be
> prepared to regard them as rational.
> For my own part, I think that I would be looking for creatures with a
> sufficiently developed perceptual apparatus, and sufficiently complex
> behaviour, to support the attribution of a belief/desire psychology;
> and a repertoire of kinds of behaviour that could count as
> communicative. I think that even science fiction would be incapable of
> exploring fully the extent of what is physically possible for life
> forms. There are probably unimaginably various possibilities as to
> what might function as a sensory transducer; or as the internal
> cognitive apparatus where sensory information is processed and
> representations are formed; or as an organ or medium of communication.
> A philosopher such as Kant inspires one to think in such ultimately
> general terms about intelligence, rationality and communication among
> embodied beings. It seems to me that the *only* defensible attitude
> here is one of 'neutrality', in the sense of having an open mind about
> such possibilities, and not being willing to rule them out of court or
> scoff at them. Multiculturalism may well one day extend to
And may well not.
> Someone like Hersch might hold that it is hugely unlikely that there
> are any radically different, alien life-forms that are rational, able
> to communicate among themselves, and able to reason mathematically.
> But the conceptual point is that their existence is entirely possible.
> Therefore, any well-developed philosophical view about, say, the
> existence of numbers, should be able to withstand the test of
> thought-experiments that involve the assumption that such beings
Very well for those who think it's worth the trouble. But
in my opinion the human, earthly mathematics that we
actually know is special--for one it's a reality, not
just a hypothesis--for another, it's ours, so it's
specially important to us!!!!
So any philosophy of mathematics ought to be tested
above all by its correspondence or lack of correspondence with
earthly, human mathematics! (Which some call irrelevant.)
> As it happens, a common assumption on the part of those involved in
> SETI (Search for ExtraTerrestrial Intelligence) is that alien beings
> trying to communicate with other planetary civilizations would
> identify themselves as intelligent by exploiting prime-numbered
> features in their 'messages'. This already presupposes that advanced
> extraterrestrial civilizations will have developed a mathematics to
> support their technology. Another assumption that has even been
> expressed in the serious academic literature on SETI is that their
> mathematics would have to be in broad agreement with ours, even down
> to such details as axiomatized set theory!
> I hold no brief for ZFC being galactically invariant as mathematical
> practice! I also think there are a great many overly naive assumptions
> (expressed by such influential figures in SETI as Arthur C. Clarke)
> about the imagined ease with which we might be able to 'decode' alien
> messages if we were to receive them 'in vacuo', without being able
> directly to observe the alien life forms in their own habitats and
> social settings. (See my paper 'Do we need extra terrestrial
> intelligence in order to search for extraterrestrial intelligence?',
> available from my website
> www.philosophy.ohio-state.edu/tennant_pubs.html) But I have no trouble
> at all imagining alien intelligences being apprised of the natural
> numbers (AS ABSTRACT OBJECTS---for that is what they are!), and having
> developed much the same grasp of arithmetical truth as we have.
Having no trouble at all imagining something is a
weak argument for the existence of that something.
> Why? Because I believe that the natural numbers are deeply embedded as
> a necessary possibility (even if only via extension) of any conceptual
> scheme for individuating, classifying, and discriminating physical
> (and abstract) objects. I believe that there is no alternative, for
> intellection anywhere in the universe, to the use of a conceptual
> scheme involving reference, predication, identity and quantification. In
> this way, first-order logic is a very profound conceptual kernal
> indeed. The way the numbers come in is by providing the alternative
> analysis of thoughts about numerosity to which Frege first drew our
> attention. We can say that there are nine planets; or we can
> re-express this as the thought that the number of planets is
> (identical to) 9. Any first-order conceptual scheme can be extended by
> means of a term-forming variable-binding operator # that satisfies the
> there are n Fs iff #xF(x)=n*
> where n* is the numeral for the number in question.
> Any rational intelligences, anywhere in the universe, ought to be able
> to attain this basic conceptual control on number-attribution.
again "ought to"
It is the re-analysis of the thought as "the number of Fs is identical
> to n*" that brings in reference to numbers as abstract objects. I do
> not mind if Tragesser regards this brand of Platonism as "painfully
> over-simplified"; for it is as much Frege's as mine.
> In response to my question
> > has anyone ever proposed that integers are concrete, as opposed to
> > abstract, objects?
> Hersch replied
> 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 ...
> But this really cannot do. For one thing, it has already been conceded
> that the substantival (noun) use ushers in the abstract objects.
Not abstract. Conceptual, intersubjective, real as shared
thoughts or ideas or concepts in a human community
> another thing, there is the fundamental difficulty that there may be
> only finitely many physical objects in the whole universe; while there
> are of course infinitely many natural numbers requiring to be
> identified as distinct objects.
That's no difficulty once you distinguish between the adjective
(or counting) numbers and the noun numbers or mathematical
numbers or pure numbers or abstract numbers if you like (using
the word abstract is not a concession to Plato-realism.)
The abstract numbers are conceived of or understand as
infinitely many. The counting numbers, as you point out,
are finite, not only because there may be only finitely many
objects in the universe (a statement of doubtful meaningfulness)
but especially beause at any epoch in the history of humanity
we will have counted only to some finite bound.
THe counting numbers and the mathematical numbers are so
closely connected histolrically and psychologically that
the disctinction between them is rarely drawn. In fact,
most practical calculations require freely passing back and
forth between them. But philosophical clarification of
the number system requires recognizing this disctinction.
Counting is primitive and basic. Abstract arithmetic
is sophisticated. It's essential in practise to see the connection
between the two. It's important philosophically to see
Arithmetical truths involve universal
> quantifications over those infinitely many distinct numbers, so there
> is no hope of getting by with finitely many physical surrogates for
> the numbers themselves.
I hope this question is answered in the previous explanation.
Could Hersch indicate how this difficulty is
> to be overcome, on his view? (Remember that it is a difficulty even if
> it turns out that the universe contains an infinity of material
> things; for even if that were so, that *need not* have been the case,
> and such a modal fact would be highly relevant here.)
> Neil Tennant
I am enjoying this conversation. I would enjoy
it even more if you would stop sticking a c into
my last name.
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