[FOM] Which is clearer, "integer" or "symbol"?

[Richard Heck]

On 01/03/2011 04:33 AM, Vaughan Pratt wrote:
>
> On 1/2/2011 4:57 PM, MELVYN NATHANSON wrote:
>> Everyone, even infants, understands one, two, three,..., but what's a symbol?
> But do they understand one, two, three before they understand their
> first dozen words?  According to
> http://firstwords.fsu.edu/parentsReportMeasures.html
> children have typically acquired 30 words by 21-22 months.
>
> Yet according to
> https://wesfiles.wesleyan.edu/home/ashusterman/web/pdf/Shusterman_Gibson_Finder_BUCLD2010_largefont.pdf
> children do not understand the exact meaning of "two" until they are
> nearly three years old.  Before then they only know it as meaning "more
> than one."
>
> It seems to me that these data make a compelling case for defining
> numbers in terms of words than vice versa.
>
Why? Temporal priority doesn't show very much.

Moreover, the paper to which you refer is primarily about children's 
acquisition of number WORDS, and does not really speak to their 
acquisition of number CONCEPTS. There is quite a lot of research that 
shows that even very young children have concepts of small cardinal 
numbers as, for that matter, do many other animals, e.g., chimps and 
parrots, neither of whom have language in the way we do.

>    From a pedagogical standpoint it might be better to start with the
> letter O than I, so that one can write O, OO, and OOO, pronounced
> respectively Oh, Ooh, and Oooh.  The first two can go hand in hand with
> learning to read words like MOM and BOOK.
>
It's a lark, but are you aware that there is very good evidence that one 
of our number-estimating mechanisms involves something like a bucket 
being filled with sand at a constant rate? The amount of sand in the 
buckets correlates well enough with the number of things being 
"counted", if they are "counted" (e.g., scanned) at a constant rate.

It looks as if there are, in fact, three distinct such systems, which 
come on line at different ages: (i) a kind of "pattern matching" system, 
which we share with many lower animals, and which is strikingly like 
what Frege parodied as Mill's "gingerbread arithmetic"; (ii) the 
accumulator system just mentioned, which we also share with many lower 
animals; (iii) something akin to a counting-based system, that has 
something (it's not clear what) to do with one-one correspondence, and 
which we do not share with many other of our fellow animals, if any. 
This latter system is the one most naturally tied to words, and it is 
also, as you will note, the only one that seems to lend itself to recursion.

The interesting hypothesis, it seems to me, isn't so much about natural 
numbers as it is about finitude or, better, the notion of a recursive 
process. Clearly, our understanding of natural language puts us in touch 
with some such notion. (To borrow from George Boolos: "This is boring", 
"This is very boring", "This is very, very boring", etc, etc.) So one 
might suggest that our understanding of recursion arises somehow from 
our reflective awareness of our innate linguistic capacities. An 
empirical question, of course. But it does not tell us how to define 
anything.

Richard

-- 
-----------------------
Richard G Heck Jr
Romeo Elton Professor of Natural Theology
Brown University