(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
============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org
