FOM: Reply to Hersch and Tragesser on aliens and abstract objects

Neil Tennant neilt at
Thu Jan 1 19:14:32 EST 1998

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,
creatures. We may be the only rational creatures on *this* planet (on
any satisfying account of rationality); 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

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

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 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.

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. 

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




But this really cannot do. For one thing, it has already been conceded
that the substantival (noun) use ushers in the abstract objects. For
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. 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. 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

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