FOM: reply to Shipman on arithmetic and geometry

Neil Tennant neilt at
Wed Oct 14 21:45:34 EDT 1998

Joe Shipman writes:

> if the universe were actually finite, or (even if infinite) it
> imposes some absolute constraints on the calculations we can perform or
> the theorems we can derive, how might one ever talk about such
> constraints formally and rigorously?  A system of "bounded arithmetic"
> or "feasible arithmetic", *without in any way attempting to displace
> Peano-Dedekind arithmetic as "the" theory of integers considered
> abstractly*, might still be valuable for describing or talking about the
> universe we actually live in.

(1) Why would one *want* a so-called formal and rigorous description of
those constraints?

(2) What aspects of an answer to (1) oblige mathematicians, rather
than physicists, to provide the theory?

> one can go further and ask for a system in which a statement like "n(3)
> exists" is actually "false", where "false" is interpreted as "there is
> no feasible integer with the property defining n(3)".

This last quoted claim boils down to "n(3) is not feasible". Why
couldn't a statement like that be accommodated in a consistent
*extension* of PA, wherein one talks about feasible computability
modulo certain specified resources?  Perhaps, as Harvey has
suggested, one could define "k is feasible" as "there is a proof in
such-and-such formal system of the claim Ex(x=k), and the length of
the proof is bounded above by f(R)", where R is a specification of the
contingently available resources, and f represents some (uniform?) way
in which the limitations of those resources translate into "feasible
proof lengths".

> Consider the following sequence of
> theories:
> 1) Presburger Arithmetic
> 2) Exponential Function Arithmetic
> 3) Primitive Recursive Arithmetic
> 4) Peano Arithmetic
> 5) Predicative Arithmetic
> 6) Sentences of Arithmetic provable in ZF
> 7) Th(N): The set of true sentences of Arithmetic
> All but the last are incomplete.  

Agreed---provided the language involved contains + and x. By the way,
these are the *classical* versions, not the intuitionistic ones, I
take it.

> All can be obtained (though the sense
> of the word "obtained" weakens from "decidably" to "enumerably" to
> "definably" as you move down the list) by the epistemological process
> i) observe regularities in the empirical behavior of "objects" and
> "counting"
> ii) create a formal system that represents these regularities
> iii) extend the formal system in a logical (or "logicist") way based on
> a priori intuitions, mathematical elegance, simplificatory rounding out,
> or any other considerations you like.

I'm not so sure that what you say here applies to (7): Th(N). Your
process (i)-(iii) strikes me as perhaps approximating Th(N) from
below, but never reaching it. 

> When you refer to "Peano/Dedekind Arithmetic", do you mean 4) or 7)?

I mean (4): the deductive closure of the Peano axioms. Moreover, I am
also talking about the very weak fragment of just successor
arithmetic! In this case, of course, PA=Th(N), which is why, perhaps,
your question arises.  If we add both addition and multiplication to
the picture, then by PA I mean the deductive closure of the Peano
axioms, where now induction will have instances involving + and/or x.

> If
> you mean 4), then are you claiming that this (incomplete) theory has a
> privileged epistemological status compared with the ones before and
> after it on the list? If so, why? If you mean 7), then Sazonov and
> Mayberry say "but what do you mean by this, what is this standard model
> '{1,2,...},S,+,*' ?".

With the previous clarification, perhaps now this last question does
not arise. It's a complete theory, and everything in it is accessible
(in principle) to a reflective intellect capable of numerical
abstraction and utterly basic logical moves. (The same holds if either
+ or x is added in isolation. Only when both are added does Go"delian
incompleteness arise.)

> I'm trying here to shift the discussion from *alternatives* to Peano
> arithmetic to *weakenings and strengthenings* of it, in order to
> continue comparing arithmetic and geometry.  

Sensible move.

> If you dump the parallel
> postulate you get an *incomplete* theory of geometry which *as far as it
> goes* seems to describe the physical world.  

Yes, but still only upon suitable operationalization of the notion
"straight line" etc. in physical space. But the natural numbers
require no such operationalization. When they are used in counting,
the physical phenomena do not play any role in determining an answer
to the question "Which arithmetical theory should we use to 'describe'
physical reality (via counting processes, feasible computations,
etc.)?". This question simply never arises. It would rest on a sort of
category mistake.

> Similarly, with Peano
> arithmetic (or any of the other theories except the last on my list) you
> get an *incomplete* theory of arithmetic (Presburger arithmetic is
> complete only for sentences which don't have the multiplication symbol)
> which *as far as it goes* seems to describe the physical world.  

Here I repeat that I have only ever been concerned with successor
arithmetic, which is complete.

> there are
> no *natural* examples of multiple incompatible extensions [of, say,
> PA with + and x].  Or are
> there?

As far as I know, there aren't any.

> I would guess this last point is the critical one for you.  Because I
> can propose the theories PA + MCC ("a measurable cardinal is
> consistent") and PA + NMCC ("a measurable cardinal is inconsistent").
> What's the difference between viewing these two theories as alternative
> extensions of PA, and viewing Euclidean and non-Euclidean geometry as
> alternative extensions of "geometry without the parallel postulate"?

There's a great difference. EG and non-EG are 'alternative extensions'
or core geometry only in the Carnapian sense of 'formal geometry'. For
the purpose of physical description, however, it is to be presumed that
Nature will answer as to whether we should use EG or some non-EG to
describe physical space.

> You can say "only one of MCC and NMCC is in Th(N), and I can reply "but
> only one of PP and NPP is in Th(R^n)", where PP is the parallel
> postulate and NPP is its negation, using a standard method of reducing
> sentences in elementary geometry to sentences about real numbers.  

That standard method in effecting identifies the straight lines as the
graphs of linear equations, if one uses Cartesian coordinates. That
forces PP upon you as part of Th(R^n).  The analogous choice between
MCC and NMCC is much harder to make. But still, whatever reasons are
ever regarded as conclusive upon further reflection (regarding the
adoption of MCC v. NMCC), these reasons will, presumably, be entirely
a priori. But perhaps no decision will ever be reached in the future
of mathematics as practised by any living intelligences able to
communicate with us or any of our descendants. In other words, we
might just have to reconcile ourselves to having to refrain from
asserting MCC, and from asserting NMCC. As an intuitionist, I wouldn't
base any result on their disjunction, either---not until one of them
had been established by a priori methods.

> Since
> you admit that both Th(N) and Th(R^n) are a priori...

Careful! Only knowledge (or potential knowledge) can be a
priori. Thus, among truths, only the knowable ones can be a
priori. Now for an intuitionist, all truths are indeed knowable, so he
can say that all arithmetical truths are a priori. But note that what
he calls 'arithmetical truth' is not necessarily (classical) Th(N).

Neil Tennant

More information about the FOM mailing list