[FOM] Mathematical explanation

[Neil Tennant]

On Sat, 29 Oct 2005, Richard Heck wrote:

> The theorem Frege proves (469, but translated into a standard
> formulation of Frege arithmetic) is:
> ~(Ex)(Fx & Gx) & ~(Ex)(Hx & Jx) & Nx:Fx = Nx:Hx & Nx:Gx = Nx:Jx -->
> Nx:(Fx v Hx) = Nx:(Gx v Ix)
> Frege translates the proposition as: "The sum of two propositions is
> determined by them".

Richard,

This is an interesting observation, but I think there is an error in the
English rendering of the thought you attribute to Frege. (I was not
thinking of theorem 469, but rather part of the prose in the Zerlegung
preceding the proof of 469; more on that below.) In the opening sentence
of his Zerlegung, Frege writes

	,,Die Summe von zwei Anzahlen ist durch diese bestimmt",
	in diesem Ausdrucke ist der Gedanke des Satzes unserer
	Haupt"uberschrift am leichtesten zu erkennen ...

The referent of "Haupt"uberschrift" here is to the topmost identity
in the displayed piece of his Begriffsschrift at the top of p.44, the only
one with the spiritus lenis on each side of the identity sign. It
corresponds to the part you render in more modern notation as

	Nx:(Fx v Hx) = Nx:(Gx v Ix).

In the German quote I've just given, the referent of "diese" would then be
to all the other four conditions---the two disjointness conditions, and
the two claims of numerical identity. So a better English rendering might
be "The sum of two numbers is determined by the following conditions" (you
had the phrase "two propositions" rather than "two numbers" as your
translation of "zwei Anzahlen"; and you translated "diese" as "them"
(i.e., the two numbers in question) rather than as (what is clear from the
context) the conditions under which one would say that the number of the
one disjunctive concept was identical to the number of the other
disjunctive concept.

Let me cast more light on my earlier comment about sec.33 in GgII. I had
in mind not the Satz at the top of page 44, which is proved as 469 by page
58, but rather the second sentence, in prose, of the Zerlegung on p.44.
There Frege writes

	Wir nennen hier n"amlich Ne'(------e^z) Summe von
				    (  |      )
				    (  |___e^v)
				    (    |    )
	Nz und von Nv, wenn kein Gegenstand zugleich under den z- und
	unter den v-Begriff f"allt.

The need for the special condition "wenn kein Gegenstand ..." is what
deprives Frege of any claim to have provided a workable definition (by
his own lights) of addition, even on just the natural numbers.

As you rightly point out, the Fregean needs an existential guarantee that
if two natural numbers t and u are in hand, then their sum (t+u) will
definitely exist also. And if the definition of sums has to be via the
disjunctions of disjoint concepts, then we need a way of finding, in
general, disjoint F* and G* respectively equinumerous with any given F and
G, even when F and G might overlap. My own suggested solution to this, as
a neo-logicist, is to help oneself to the operation of orderly pairing as
a new *logical* primitive, so that we can define F* as the concept
under which fall just the ordered pairs of the form (0,f) where f is an F,
and G* as the concept under which fall just the ordered pairs of the form
(1,g) where g is a G. [If you don't wish to use 1 here, use any other
(logical) object that is distinct from 0.] These definitions guarantee
that F* is equinumerous with F, G* is equinumerous with G, and F* is
disjoint from G*.

We can then give the +-Introduction rule as follows:

	u=#xFx   v=#xGx  t=#x(F*x v G*x)
	________________________________
		   t=u+v

These ideas were presented earlier this month to the Midwest
Workshop in the Philosophy of Mathematics VI, at Notre Dame.
The fully written-up paper should be available within the next few weeks. 
Iff accepted, it will appear in a Springer volume titled "Logicism,
Intuitionism, Formalism: What has become of them?" edited by Erik
Palmgren, Krister Segerberg, Sten Lindstr"om and Viggo Stoltenberg-Hansen.

Neil Tennant