FOM: extra-terrestrial math etc.

[Neil Tennant]

Re: Gardner on Hersh, Pratt on Gardner on Hersh ...

Let us conceded the fluidity, topicality, fashionableness, or
whatever, of various portions of the mathematics thus far invented or
discovered. Let us agree also that it is likely that there are areas
of new mathematics yet to be discovered and which we at present cannot
anticipate. Let us imagine that there are extraterrestrial
intelligences, and that it is likely that extraterrestrial
mathematicians may be 'doing kinds of mathematics' of which we at
present can have no inkling.  Thus we concede even the possibility
that they (the green or grue critters on Quasar X9) are not 
interested in pi or e or Pythagoras's theorem or whatever. 

ALL THIS is compatible BOTH with a realist/platonist view AND with
certain idealistic brands of intuitionism or constructivism. The
realism/platonism that I have in mind is one according to which there
is an abstract realm of mathematical entities and structures giving
rise to eternal, necessary truths about them. At least some of these
truth will be intellectually accessible. Whether in addition some of
the (necessary) truths might be intellectually *inaccessible* is a
further, optional extra for such realism or platonism. The idealistic
brands of intuitionism or constructivism that I have in mind are ones
according to which mathematical entities come into existence only in
response to intellectual probings, and all truths about them arise
only from intellectual processes that are historically contingent
(i.e. processes that might not have taken place).

If Plato's heaven is there, waiting to be discovered, then
mathematicians in different historical periods on earth and
mathematicians in different civilizations of other planets will in all
likelihood discover different parts of it.  On the other hand, if
Plato's heaven gets its denizens 'elected', so to speak, by ongoing
mathematical activity on the part of intellects all over the cosmos,
those denizens should all be sitting in their proper places for later
inspection (like a reformed House of Lords in Britain). There will be
no squabbling over an elected position in the series of natural
numbers. One is not going to have two little guys in their abstract
ermine robes, one legitimately called "7+5", the other legitimately
called "7+6" (according to the standard meanings of the expressions
involved), fighting over the twelfth seat in that portion of Plato's
heaven set aside for the natural numbers. It is no argument against
this to point to the historical accident that the symbol "7" refers to
the seventh non-zero natural number. Historicity of symbolism is
absolutely beside the point here. What matters is what thought is
expressed, regardless of the symbolism employed to express it.

What CANNOT be squared with mathematical experience, and with the
norms governing mathematical investigation generally, is the 'social
relativistic' thought that what we have PROVED about certain
mathematical entities (be these 'discovered' or 'created' by our
mathematical activity) might be coherently DISPROVED by some fellow
human being (differently situated socio-historically), or by little
green critters on another planet (or vice versa).  

Let us take even the wildest Brouwerian, who earnestly believes that
the series of natural numbers came into existence only when they were
first 'thought of' by some human mathematician who gave a means of
generating or constructing any natural number you might wish to
consider. Indeed, let us even take the wildest version of this
Brouwerian position, according to which only those natural numbers
exist as have actually been constructed via the inscription or
utterance of an appropriate numeral. I want to maintain that EVEN THIS
wacko philosopher of mathematics would quite rightly assert the
NECESSITY of the claim that 7 plus 5 makes 12.

Vaughan Pratt may be right that the little green critters could base
their mathematics on (what we would optimally have to INTERPRET as)
category theory rather than set theory; but so what?  A correct
interpretation of their mathematicising as such would have to be based
on a prior identification of areas of necessary AGREEMENT on major
portions of "obvious" mathematics as not just true, but necessarily
true---such as the arithmetic of natural numbers.  When interpreting
those aliens, one would be looking for their analogues of such
necessary truths as "7 + 5 =12"; or "3 squared plus 4 squared equals 5
squared". One would ALSO be looking for their analogues of the
CONCEPTUAL CONTROLS linking mathematical discourse with discourse
about ordinary things. (My favourite such conceptual control is the
schema "There are n Fs if and only if the number of Fs is [numeral]
n*".) If you couldn't find anything like this on which to agree with
the aliens, then you couldn't even begin to make sense of what they
were doing, or even begin to render what they were doing as allegedly
interpretable mathematics.  Even more so, there would simply be no way
to interpret them so definitively as to render them as 'correctly
claiming' (!) that, say, 2 plus 3 does NOT equal 5.

It astonishes me that some mathematicians can be so willing to
misrepresent so grotesquely the unique nature of mathematics as a
deductive, a priori enterprise.  Mathematics is centrally concerned
(at least in areas such as the theory of the natural numbers and the
theory of the continuum) with the pursuit of necessary truths about
certain abstract structures, about which we can attain certainty via
proof based on self-evident axioms. The very philosophical interest of
sophisticated brands of *nominalism* is how they could possibly deal
with this over-ridingly important 'phenomenological' feature of
mathematical theorizing. If we are NOT after truths about an intended
abstract structure (such as the natural numbers, or the real
continuum), why is it that it strikes us so forcibly that that is
indeed the best way to conceive of what we are doing?

Even after Goedel, the cliched Platonistic account of the nature of
mathematics (at least in the two areas I have given as examples) goes
begging for any intellectualy respectable alternative. There just
isn't one. Even the structuralist or the set-theoretical reductionist
still agrees on both necessity and univocity of intended structure.

[Please do not reply that the lesson of Goedel's incompletness theorem
for arithmetic is that there could be 'alternative arithmetics'.  For
in response to such a suggestion one can make two fatal points.
First, ALL these arithmetics would have to contain the same
quantifier-free statements such as '7+5=12'.  Secondly, we all agree
that the independent Goedel sentence for a given (intuitively sound)
arithmetical theory is TRUE in the intended model.] 

Note also that the ontological Platonism I have indicated is one that is quite
congenial even to an intuitionist. An intuitionist can maintain, on
meaning-theoretic grounds, that the correct logic for mathematics is
intuitionistic logic, while never having to give up the conception of
mathematical theorizing as directed at giving an account of what is
knowably true about a certain abstract structure that exists
independently of our mathematical activity. That is, the semantic and
ontic strands of traditional intuitionism are quite separable.
 
That the theories of the naturals and of the reals nail down necessary
truths about certain abstract structures might be a philosophical pill
bitterly swallowed by those who would like to be able to cause a stir
with outlandish relativistic or social-constructivistic claims about
the nature of mathematics. But, to adapt a Catholic cardinal's caustic
comment about the stir over the papal encyclical on birth control,
"such claims would be no more than a dot or a comma in the two
thousand year history" of mathematics.



Neil Tennant