FOM: Sazonov on thought experiments etc.

[Neil Tennant]

Sazonov claims that '[it] is fruitful to [ask] the question
"What does it mean?" about *mysterious* concepts such as
non-relativistic absolute notion of simultaneousness of events
or "the unique standard model of PA".'

I fail to understand why he thinks the concepts *mysterious*. They
are, after all, easily deployed even by those who might deny that they
apply to anything. Indeed, it would be hard to grasp what was so
momentous about Einstein's achievement were it not for the fact that
the concept being *displaced* by his new theory was taken by everyone
to be so *unmysterious* as to be guaranteed, a priori, to hold of
physical space. What was wrong about their view was not that the
concept of absolute simultaneity was mysterious, but rather the
component of their view according to which the concept could find
legitimate application in the physical world. (Had the speed of light
turned out to be infinite, presumably it would have found legitimate
application.)

In the case of absolute simultaneity, then, the concept was
non-mysterious but lacked legitimate physical application. That was an
empirical discovery.

In the case of the standard model of the natural numbers (I'd prefer
to say just: 'the natural numbers', and refrain from reifying the
model itself) the concept "x is a natural number" is not at all
mysterious. Moreover, what is intrinsically true of that concept (i.e.
true statements of the language of Peano arithmetic) remains
completely a priori, *unlike* the case of physical geometry. I
advanced the suggestion that even Einstein would have been well aware
of that (philosophical) fact, and Sazonov adduces no evidence to
gainsay my claim. 

[Digression: Before anyone rushes to point out that Go"del's
first incompleteness theorem showed that any axiomatised subtheory of
(classical) Th(N) is a proper subtheory of Th(N), so that perhaps some
sentences in Th(N) might not be a priori accessible, let me say that
that is no problem for the intuitionist, for whom truth consists in
provability in any acceptable system obtained by reflection on the
concepts involved. For the intuitionist, all arithmetic truths are
unproblematically a priori. Indeed, even for the classicist, all
*known* arithmetic truths are a priori. If ever one wishes to
challenge a classical arithmetician by asking 'How do you know that
S?', their answer should always consist in providing an a
priori---albeit perhaps strictly classical---demonstration of S.]

To my question "What fundamental aspects of a conceptual scheme with
identity are to be jettisoned or mangled?", Sazonov replies "What is
*this* "conceptual scheme with identity"?". Note that my question
didn't advert to any conceptual scheme in particular. It was a general
challenge. Perhaps I should spell it out even more clearly: 

	If someone were to be working with *a* conceptual scheme
	with identity, but was to be disabused (by Sazonov et al.) of their
	desire to use Peano arithmetic in conjunction with that conceptual
	scheme, what aspect of that scheme would have to be jettisoned of
	mangled?

Perhaps Sazonov's question reveals a lack of familiarity with the
notion of a conceptual scheme in general, so a word or two of
explanation might be in order.  Here is a simple conceptual scheme:

	x is red
	x is blue
	x is green
	x is hot
	x is cold
	x is a dog
	x is a human being
	x is an apple
	x is a cave
	x is a campfire
	x is a stick of wood

In this primitive scheme, one would have the identity relations 

	x is the same dog as y
	x is the same human being as y
	x is the same apple as y
	x is the same stick of wood as y

The concepts of cave and of campfire might not divide their references
so clearly into re-identifiable particulars. The color concepts do not
divide their references at all. NUMBERS are used only in connection
with concepts that divide their reference (also called 'sortal
concepts' by analytic philosophers in the Fregean tradition). Let F be
such a concept. Then we require that the number of F's be identical to
n* if and only if there are exactly n F's.  Example: the number of
dogs is ss0 iff there are exactly two dogs (i.e. iff ExEy(~x is the
same dog as y & x is a dog & y is a dog & (z)(z is a dog --> (z is the
same dog as x or z is the same dog as y))). NOTE: once such an
extension to number talk has been undertaken, one has a new sortal
concept:

	x is a number

and one can then inquire after the number of numbers with certain
properties.
	
I asked Sazonov to undertake a thought experiment: "ask yourself
what theory of arithmetic would be excogitable by a disembodied
Cartesian soul in a universe with no physical objects in it at all."
To this he replied "Unfortunately, I am not inclined to make such kind
of experiments. (Probably this is the main point where we disagree.)"

In other words, he is saying "I dare not go there; I fear I shall be
led by the nose to the horrible realization that my theory is really
rather fatuous after all." 

Compare these analogous cases, where T is trying to get a stubborn S
to reach a conclusion by making a vital thought experiment:

T: Ask yourself whether you would like to be treated the way Jews were
treated in Nazi Germany.
S: Unfortunately, I am not inclined to make such thought experiments.
[Conclusion: S must be either evil or morally blind.]

T: Ask yourself whether there might not be a world in which some
people behaved exactly as they do in this world, but in which they
have no inner experiences---no sensations, no emotions etc.
S: Unfortunately, I am not inclined to make such thought experiments.
[Conclusion: S is blind to the problem of consciousness and other
minds.]

T: Ask yourself whether there might not be a world in which all
emeralds are green if examined before the year 2000 and blue otherwise.
S: Unfortunately, I am not inclined to make such thought experiments.
[Conclusion: S is blind to Goodman's new riddle of induction.]

I shall refrain from developing such a list ad nauseam.

Sazonov asks 'Also, what does it mean the "disembodied Cartesian soul"
and a "universe with no physical objects"?  This is again something
mysterious. If there are no objects then there is nothing to count.'

Wrong!! Just because there are no *physical* objects does not entail
that "there is nothing to count"! There are *the numbers themselves*
(abstract objects) to count! That's what was happening when each
natural number n was revealed as the number of preceding natural
numbers. That, one wants to stress, was the whole point of the thought
experiment. Little wonder that Sazonov is "not inclined" to engage in
such a thought experiment.

This underscores my earlier allegation of bad faith in making
hand-waving historical gestures to the like of Einstein, in order to
lend undeserved credence and dignity to one's own purported
intellectually revolutionary breakthroughs. Einstein would have been
the last person to refuse to embark on a thought experiment. That's
how he made his breakthrough. Perhaps Sazonov should take note.

Neil Tennant