Jerry LRC,
As Kirsti said, the subject line about categories and modes was
a long thread about Peirce's 1903 classification of the sciences.
I plan to post a copy that text, my diagram about it, and related
quotations by Peirce on my web site.
But I changed the subject line for the topic of pure & applied math.
everything that is imaginable can be described by some theory
of pure mathematics.
How can one describe a “feeling” in pure mathematical terms?
You can't. That would require applied mathematics.
How can one describe a large bio-molecule, such as Nicotinamide
Adenine Dinucleotide (NAD) in pure mathematical terms?"
For any theory of applied math, there is a simple procedure for
finding a corresponding theory of pure math. And it's based
on the point you mentioned:
Simply quote W.O Quine: “To be is to be a variable.”
1. Start with whatever applied theory you have. Let's assume
that it's stated in some mixture of mathematical formulas,
chemical symbols, chemical formulas, and English statements.
2. Leave every name or symbol in pure math unchanged. Replace
every name or symbol in the application with some distinct, but
non-obvious name -- for example, relation names R1, R2, R3...;
function names F1, F2, F3...; and entity names E1, E2, E3....
For variables, use non-obvious names: x1, x2, x3...
3. Then translate every statement or formula in any notation
to predicate calculus (Peirce-Peano algebra). This would be
systematic for the formulas in math & chemistry, but it may
take some thought and rewriting to force raw English into
predicate calculus. But if the English is precise (or can
be restated precisely), the translation can be done.
4. But your theory probably depends on many other theories
of chemistry and physics. Repeat the above steps with all
of those theories -- and be sure to maintain a record of
the way each name was translated -- consistent translation
across all the theories is essential.
5. After you finish that, throw away the crib sheet that says how
the original names were mapped to the R, F, E, and x symbols.
You now have a theory about which Bertrand Russell would say
"We don't know what we're talking about or whether what we're
saying is true."
That's pure math.
Of course, nobody would ever attempt to translate a complex
theory with many complex dependencies by the above procedure.
Scientists and engineers normally adopt and adapt pure math theories
one at a time, as they are needed. They often create applied theories
from scratch, without looking for a prefabricated theory in pure math.
But all such theories can be translated to pure math by the above
method or some variation of it.
John
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