FOM: Frege hath said it (logical priority)

[William Tait]

On Thursday, March 7, 2002, at 04:25 PM, Dean Buckner wrote:

> Argument: a discussion in which reasons are put forward in support of a
> proposition.  One such reason can be "Aristotle hath said it" (and who
> argued that?)
>
> Here is my argument again.
>
> 1.  Children from an early age (before 2) grasp the concept of "other"
> (here's one foot, point to the other)
> 2.  For there to be one thing and for there to be another thing, is for
> there to be two things.
> 3.  To grasp that there is one thing & another thing, is to grasp there 
> are
> two things (even though can't grasp how to use the word "two")
> 4.  (so) Children under the age of two can grasp something like the 
> concept
> of number
> 5.  But can they grasp Hume's principle?  this is to understand not 
> only the
> concept of "same number", but to grasp a particular condition attached 
> to
> there being the same number
>
> I can see plenty of things to challenge in this, so please challenge it.

Let me stipulate 1: I know nothing about it and moreover think that it 
has nothing to do with what does concern me. 2. Yes. 3. I worry about 
this: if one doesn't grasp how to to use the word ``two'', does one 
grasp that there are two things? Let it pass. 4. is a complete non 
sequitur: to understand the notion of number is not to understand what 
it means that there are two things. It is to understand the finite 
iteration of `one more thing'. 5. The principle mis-named ``Hume's 
principle'' is not in any case an adequate foundation for the number 
concept, as Cantor had already pointed out in his review of Frege's 
_Grundlagen der Arithmetik_. But there is an adequate foundation based 
upon the axioms that express Dedekind's definition of a simply infinite 
system. No: children do not in general grasp this, even long after they 
have grasped far more than the notion of `other'. Indeed, it was a long 
historical process in western culture to the recognition of the right 
foundation of arithmetic.

>  No more Aristotle, no more Frege.

I suggest you read more carefully what is written in response to you.

>  They would not approve.

I think Frege would very much approve of having his ideas attributed to 
him. I have less a sense of Aristotle.
> (What Aristototle actually said was that there is a difference between 
> what
> is logically prior, and what is prior "for man".

Actually, Aristotle wrote in classical Greek. I quoted from Barnes' 
translation of the Post. Analy.; but I like your translation, too.

>  He went on to say is that
> what is prior "for man" is what is particular, i.e. sense knowledge.  
> "The
> most
> universal causes are furthest from sense and particular causes are 
> nearest
> to sense" (72a3).

Indeed he did: but I hope that you do not put too much stock in 
Aristotle's epistemology. What seems to me to be true of his writings 
here---and this is why I felt it appropriate to refer to him---is that 
the principles we take as the foundation of a science are not what we 
first take in.

> He did not mean priority in age!

Good grief! What can you possibly be referring too?

> The proposition I will argue (I won't speak for Heck) is that we can 
> learn
> from children.  If they can grasp the concept of number (or something 
> like
> it) without grasping A, B, C &c, then A B C is not essential to the 
> concept
> of number.)

My goodness , yes, did I learn from my children! But what? Many things 
about cognitive development, but nothing about the foundations of 
science.

`Something like it?' There are peoples who grasped plurality in the 
sense of understanding one, two and many. But they did not understand 
our concept of number. But in any case, to learn how to use number words 
in everyday discourse is not to understand the number concept---how to 
make valid proofs of propositions about  numbers.

> There seems an extraordinary resistance in this group to discussing
> connections between meaning and number.

I don't understand precisely what you mean---or, rather, in the only 
sense in which I can understand it, it is patently false. The whole 
discussion seems to me to be about the meaning of number-theoretic 
propositions. On what grounds are they true or false?

But, in any case,  maybe it is better to worry more about the content of 
one's own contributions to the list than about the response of the 
others. It is certainly a more respectful way to proceed.

>  Maybe that's because my arguments
> seem incoherent?

No, only invalid.

Yours respectfully (and with, I hope, not too much malice),

Bill Tait