FOM: What should feasibilists be asking?

[Neil Tennant]

Anatoly Vorobey writes

> Your attitude towards the Peano-Dedekind arithmetic seems to be
> rather demanding. On one hand, you claim that a disembodied soul in
> an empty universe (nevermind the question of plausibility of this
> gedankenexperiment) should formulate exactly the Peano-Dedekind
> arithmetic. Mere presence of some physical objects, you say,
> cannot therefore show that arithmetic incorrect. On the other hand, 
> you are prepared, above, to equate correctness of this arithmetic (whatever
> this means to you) with its being "the appropriate generalization
> of our intuitive concepts of counting physical objects" - but *that*
> concept definitely *is* dependent on "mere presence of physical
> objects". 

No, I was not "equating" the correctness of PA with its being "the
appropriate generalization of our intuitive concepts of counting
physical objects." The latter phrase is yours. You introduced it
because you wished to imply that perhaps such "appropriate
generalization" could be in conflict with PA. My thesis is that, once
PA is properly understood---that is, once its *conceptual genesis* is
appreciated---one realizes that no physical contingencies could
possibly affect its truth. All that could be affected is PA's range of
useful applications (i.e. the range of possible "physical counts" that
one might be interested in making), not the logical grounds for PA's truth.

> To use a somewhat extreme example, consider a universe with
> just two thousand basic objects. A mathematician somehow "living"
> in such a universe would probably find counting objects
> quite useful. It may be that such a mathematician will, in course
> of his mathematical activity, arrive at the axioms of the
> Peano-Dedekind arithmetic (after all, he's better off than
> a disembodied soul in an empty universe). 

Yes, it may be so. Indeed, that is what has probably happened with us,
living in a universe where your bound of 2,000 just happens to be
somewhat larger. But note that both we and your imagined mathematician
in the slightly less populated physical universe could *both* derive
the whole of PA simply by reflecting on the operations of our/their
conceptual schemes. That is, both could do what I'm saying the
disembodied Cartesian soul in a universe with no physical objects
could do. Moreover, for all three kinds of thinker, that is what one
might call the "ultimately justifying" way of coming to see the truth
of PA. It is a priori. It uses only logic. It has nothing to do with
physical objects, and everything to do with concepts. It guarantees,
however, the universal *applicability* of the natural numbers in
counting, regardless of the concepts being used to effect the count
(provided only that their extensions are finite), and regardless of
how many objects fall under them.

> It well may be that he would find
> them very interesting, and will proceed to study their
> consequences in detail. I don't think that he would have
> to consider the Peano-Dedekind arithmetic somehow "incorrect" - no
> more than *we* would have to consider it incorrect upon
> discovering that our universe is finite, or consider the Euclidian
> geometry incorrect upon discovering general relativity. 

Here I agree with you about arithmetic, but disagree about
geometry. My point all along is that the physical facts can dictate
which (physical) geometry is correct (modulo certain
operationalizations of the geometrical primitives, which are made with
the pragmatic interests of simplicity etc. uppermost in our
minds). The situation with arithmetic, however, is completely
different.  It would be impossible for anyone, in any kind of physical
universe, ever to have grounds to consider PA incorrect. So, while
I agree with you about arithmetic, you can see that I would go even
further than you do in the quote above.

> However,
> it might be unwise for him to consider the Peano-Dedekind arithmetic the
> appropriate generalization of his intuitive concepts of counting
> physical objects (of which he only has two thousand). He might
> prefer to use for *that* purpose a different axiomatic theory, one which
> would recognize the obvious finiteness of his universe. 

Here's where we differ sharply. Since I don't see the justification of PA
as consisting in any sense in some sort of "generalization" from our
experience of counting (perforce finitely many) finite collections
(all of which are therefore bounded above in size by some N), I would
remonstrate with any such "mathematician" who thought that he had
somehow got hold of a "better", "more applicable" theory than PA, and
one, moreover, which he claimed to be *in conflict with* PA. Rather, I
would recommend that he simply use whatever initial segment of the
series of natural numbers he thinks he needs, and heed the general truth
of PA while doing so. Why should he wish for anything more?

> It is
> bizarre that *you* would consider such a choice on his part as
> claiming the Peano-Dedekind arithmetic to be "incorrect", since
> it is you who claims this arithmetic cannot be disproved by mere
> presence of physical objects. 

Here I think you have the dialectic of this exchange wrong. I've
already said that I would have no quarrel at all with someone who
simply wished to *append to* PA some theory consistent with PA that
was more directly concerned with the finite limitations of the actual
physical universe. 

> > Moreover, the burden of explication of the notion of
> > incorrectness is properly on the finitists or feasible
> > arithmeticians. 
> 
> I am neither a finitist nor a feasible arithmetician. I am simply
> concerned about what seems to me as your desire to use the big
> scary word "incorrect" as a scarecrow against finitists/feasible
> arithmeticians. They typically talk about a finite universe and
> limitations on the concept of counting as a consequence, not
> about "incorrectness" of Peano-Dedekind arithmetic (quoting Sazonov:
> "Is the chess game correct or not?"); you are the one
> who says they claim such incorrectness, so the burden is on
> you to explicate this notion. 

Unlike Sazonov, I do not see assertions in arithmetic as analogous to
making moves in a chess game. Nor is deciding to play chess like
deciding to adopt PA. PA is not an intellectual option, the way
deciding to take up chess is a recreational option. PA is a necessary
constraint on any kind of thought that involves the use of concepts
that divide their reference.

> > If Vorobey's way of putting matters does not amount to the momentous
> > claim that Peano-Dedekind arithmetic is incorrect, then we are owed an
> > explanation as to why this is so. 
> 
> I hope that the above may serve as such an explanation.

If I've understood you correctly, you are then not claiming that the
finitist or feasibilist is in any way an "arithmetical *reformist*"?
Rather, he is just someone interested in tacking onto an already a
priori branch of knowledge (PA) some further pieces of theorizing that
address the question of limitations on possible physical processes of
counting and of computation. 

Or, perhaps more abstractly, he may want to take on some extra
theorizing a la Friedman, in which one first takes some fixed bound N
(determined, independently, from conjectural knowledge about the
distribution of matter and energy in spacetime) and then, within
mathematics (that is, in purely a priori fashion), proves theorems of
the form 

	"There is no proof in such-and-such formal system, 
	of length less than N, of the statement S". 

Such pure mathematical (or perhaps one should say: metamathematical)
results then provide some reasonable sense for the feasibilist's
assertion that the truth of S cannot be feasibly recognized (using
only the resources of the system in question). The problem is,
however, that Friedman usually has some other way of accessing the
truth of S---using impredicative methods, or large cardinal
hypotheses, or whatever. So what would the feasibilist have to say
about that? Perhaps there might be some Friedmanesque result of the
following kind, showing that one could really be "feasibly in the
dark" about the truth of S:

	"There is no proof *or refutation*
	in such-and-such formal systems [and here
	one lists all currectly acceptable systems used by
	mathematicians], of length less than N, of the
	statement S."

Are there any such results? *Could there be* any such results? If so,
could one prove, further, of such S, that

	"...nevertheless, in such-and-such acceptable system,
	there is (in principle, but not feasibly discoverable)
	a proof *or a refutation* of S (we cannot feasibly know
	which), whose length of course exceeds N, which means
	that we shall never in fact we able to discover it."

But even a strong result of this kind cannot rule out the possibility
that some *future* acceptable system might contain, not merely a proof
(or refutation) that is accessible-in-principle, but one that is also
*feasibly* accessible.

Do the feasibilists offer any characterization of statements for which
such optimism is demonstrably absurd?

I think these are the more interesting foundational questions that a
feasibilist or finitist should be raising.

Neil Tennant