[FOM] Re: Falsify Platonism?
Andrej Bauer
andrej.bauer at andrej.com
Thu Apr 29 18:02:31 EDT 2010
I would like to remind the participants of this discussion of an
alternative definition of natural numbers which is algebraic and
categoric in the sense that it defines the natural numbers up to
isomorphism as an initial algebra. This definition is expressed in
first-order logic.
When discussing an object of interest, in our case the natural
numbers, we may consider two kinds of definitions or
characterizations. An intrinsic one only refers to the structure of
the object itself. Such are the standard Peano axioms PA. An extrinsic
definition may explain what role the object of interest plays with
respect to the rest of "mathematical discourse" or "mathematical
universe" (which need not be set theory or a topos, in many useful
situations it can be a much smaller universe).
We can define the natural numbers as the initial algebra with one
constant 0 and one unary operation S. Specifically, an object N with
morphisms 0 : 1 -> N and S : N -> N in a given category C is *the*
object of natural numbers when, given any object X and morphisms z : 1
-> X and f : X -> X there is a unique morphism g : N -> X such that g
o [0,S] = [z,f] o (id+g), i.e., the following diagram commutes (please
switch to fixed-font to view it)
[0,S]
1+N ------> N
| |
id+g| |g
V [z,f] V
1+X ------> X
This is a first-order definition in terms of the underlying category C
which involves quantification over all objects and morphisms of C. It
determines the natural numbers up to isomorphism. It works well as
soon as C is cartesian-closed and has coproducts (and we can do
slightly better if we are willing to complicate the definition).
When C has enough structure to express an induction principle, the
induction principle can be deduced from the above characterization --
essentially in the form "every sub-algebra of N is N". There will be
two ways of interpreting "every sub-algebra":
(1) externally, as a statement of the form "every monomorphism M --> N
such that 0 and S restrict to M is an identity."
(2) internally, if C supports an interpretation of second-order logic,
as an internal second-order statement "for every predicate P on N, if
P is closed under 0 and S then P holds of all numbers."
In fact, under mild conditions such an induction principle (either
internal or external) can replace the initiality condition.
From this point of view one wonders then why the usual axioms of PA
express induction as a schema, which amounts to an external version of
induction (1) in which we quantify only over those monomorphisms that
are denotations of first-order formulas (what about all the others?).
This is a bit like studying real analysis in which we restrict
attention only to those sequences of reals which are expressible as
terms in some chosen calculus.
A possible criticism for defining the natural numbers as an initial
algebra would be based on the insistence that the natural numbers
ought to be viewed as a structure in itself. After all, modern
axiomatics strives to study objects of interest only in terms of
themselves and their properties:
(1) Hilbert's plane geometry involves only objects directly present in
the plane and their properties
(2) The axioms of a group only speak about a group
(3) Set theory is only about sets
Therefore, it seems preferable to axiomatize the natural numbers only
in terms of themselves. Preferable for whom? Certainly for the
logician who wants to turn the natural numbers into a laboratory mouse
that can easily be subjected to Tarski-style model theory of
first-order logic.
However, the mathematician in the street knows that a mathematical
object comes to life only in relation to and interaction with other
objects. Thus we have subjects called "analytic number theory" and
"algebraic topology". The mathematician in the street will not think
twice before applying cohomology to a number-theoretic problem, and
will even be proud of himself for finding such a cool connection.
It can therefore be argued that the initial algebra definition of the
natural numbers is better than the usual PA axioms because:
(1) it is still first-order
(2) fixes the natural numbers up to isomorphism
(3) places the natural numbers within a wider mathematical universe,
which is the natural approach for the working mathematician
(4) allows for great flexibility since models of natural numbers can
now be studied in any cartesian-closed category with coproducts
(rather than just in the context of model theory of first-order logic)
I propose to the supposed Platonist (are there any?) that he should
accept as the official definition of the natural numbers the one I
reviewed above, where for the ambient category C we take the Platonic
universe, whatever it may be. This will immediately dispense with
strange phenomena conjured up by logicians, such as non-isomorphic
models of natural numbers and strange talk about "natural numbers vs.
the concept of natural numbers".
Doubtlessly, there will ensue a debate between logicians and
philosophers about the primacy of first-order axioms of PA over the
initial-algebra definition of the natural numbers, but that is a
discussion that the Platonic working mathematician in the street may
easily ignore.
With kind regards,
Andrej
P.S. I am not a Platonist (which should be obvious, since I am trying
to sneak in the idea of having many mathematical universes).
More information about the FOM
mailing list