Too many interesting comments to follow up. But to Lee, Friam probably
doesn't forward attachments. I didn't get the article with your earlier
message either. There is an entry in the Stanford Encyclopedia of
Philosophy on Evolutionary
Epistemology<http://plato.stanford.edu/entries/epistemology-evolutionary/>.
It seems from first glance that it makes sense. We -- and all animals --
evolve epistemological capabilities that improve survival. At that level it
seems almost tautological.
Secondly, your answer to my
"question about finding a characterization of "the domains to which
mathematical thinking is successfully applied", namely, that they are (or at
least necessarily include) the domains for which "effective [i.e.,
mathematical] thinking" promotes species survival in "the external world in
which we have evolved".
seems to be contradicted by your own good ideas. Does knowledge about the
domains to which they apply promote species survival? (They certainly
promote individual survival as a successful mathematician, but that's
another matter.)
Does knowledge generated by any so-called "pure science" promote species
survival? Only by chance, it seems. Besides why should improved species
survival be related to the possibility of interesting theorems? The
importance to us of a domain is certainly a function of its role in species
survival. But why does that suggest that the domain is likely to give rise
to sophisticated mathematics? I don't see the connection.
-- Russ
On Sun, Apr 25, 2010 at 2:07 PM, Eric Smith <desm...@santafe.edu> wrote:
> (expressions of ignorance to follow:)
>
> I wonder in all this whether there is anything interesting to be said
> by looking at the relation of syntax to semantics in mathematics,
> perhaps not in the sense of "applying" language, but rather in the
> sense of recognizing that mathematics shares syntactic elements with
> other constructs, which are more primitive than either, and have to do
> with applying formal descriptions to models of onesself.
>
> To be less random and cryptic (with luck):
>
> 1. We perform repetitive operations all the time, so our actions
> "embody" the inductive aspect of the natural numbers in some vague
> sense. But the natural numbers as a formal construct come into
> existence when we represent addition-by-one as a syntactic
> operation. (Here showing my ignorance of what Conway, Knuth, and
> other number theorists do to show how "real" all these formalisms
> are.) The point was, one is never supposed to ask "what comes
> after Z" in the alphabet, while the transition to realizing that
> one must ask "what comes after 26", sometime between three and four
> years of age, is the human transition to "understanding"
> arithmetic, which chimps and monkeys never make, even though they
> share some of the quantity-sense that is part of the semantic
> dimension of arithmetic.
>
> 2. So now we have the natural numbers as syntactic as well as semantic
> constructs. Why isn't that all, or why isn't every consequence of
> it immediately available to us?
>
> 2a. [Back to behavior] We break collections into groups all the time,
> and we compare groups for equivalence. Again, operationally, our
> actions embody ("en-corp-orate") multiplication and division. When
> the natural numbers have been created, they present an opportunity
> for us to do that to them, too. I think of that opportunity as a
> semantically created thing. Once numbers exist, we can do to them
> the same things we do to other objects, because they exist in a
> representation that allows us to think of them as objects.
>
> 2b. But grouping and comparing groups of numbers may not yet be
> multiplication and division. Those become parts of arithmetic when
> they are assigned a syntactic representation, so that operations
> are well-defined "without reference" to their semantic antecedents,
> if I understand the goal of Russell and Whitehead, with all of its
> reversals etc. The theorems derivable from rules of multiplication
> and division go from semantic possibilities that could be tested by
> action, to formal constructs within language, when multiplication
> and division are made parts of the syntactic construct.
>
> 3. From there we encounter a topic that has shown up on this list
> several times in discussions of emergence: the primes. What
> brought them "into existence", and why are their identities and
> properties not immediately available? An algorithm generated
> inductively from a small number of rules, and guaranteed to stop in
> finite time, makes prime/non-prime a well-defined distinction.
> Presumably that distinction doesn't exist in the purely semantic
> world of action, because it refers to the properties of the
> algorithm that apply to each particular number; action can only
> test cases. So primeness seems to rely on the ability to refer
> syntactically to operations, even though the opportunity to
> distinguish what can or can't be done to (any particular) set with
> a certain size is semantically created.
>
> So, having taken too much space to say something either obvious or
> ignorant, should we understand the growth of mathematics as partly an
> appeal to semantics, to guide us to unforeseen syntax? Then theorems
> use the new syntax to compactly represent connections that may not
> have been representable with anything short of simulation, in the
> earlier syntactic layers.
>
> Anyway,
>
> Eric
>
>
>
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