Jeff Rubard
2010-02-06 04:02:38 UTC
Mathematics and Human Interests
Now, for a piece of philosophy of mathematics. It’s hard to see how
you could take a “sideways-on” approach to mathematics: it is true if
anything is, right? Well, maybe we can stretch that out a little
further than what is involved in the usual Third Realm calisthenics. A
piece of mathematics is something that will be true for all time:
mathematical insight means seeing into the ages, what was and ever
will be. Conversely, lack of mathematical acuity — our inability to
solve a problem — is caused by the deceptions of the age, our follies
and delusions.
As regards my own very modest mathematical output, I have adduced
considerations that P cannot equal NP in ”A Cool Million (Some
Thoughts on P and NP)”, ”Further Thoughts on P and NP“, and ”Why P
Cannot Equal NP“. They do not have the form of a formal proof and have
failed to convince the contemporary complexity community (as a whole),
but it is my honest conviction that the very simple logical
difficulties associated with reducing the complexity of DEXPTIME-hard
NP-complete problems are insuperable. The considerations are “dumb”,
but honestly limitative results always are — the Incompleteness
Theorem is a piece of simple trickery that would hold no interest were
it not true.
Really, I think the hope that a brand new algorithm will crack the
problem is a pipe dream; the essential nondeterminacy of the
Satisfiability problem is just untouchable. Why does it appeal, then?
Because computer science is about technical control, and the
temptation of a computational Eden where all cryptographic algorithms
can be cracked and mathematical proofs grow on trees is just too
strong. (The harder-headed claim that problems from oracles — which
are rather moldy, as they can be read about in the original edition of
Hopcroft and Ullman – and other considerations may make the P and NP
question impossible to solve, but it may just be our own damn fault
for not defining our terms clearly. Always hard to tell.)
Now, for a piece of philosophy of mathematics. It’s hard to see how
you could take a “sideways-on” approach to mathematics: it is true if
anything is, right? Well, maybe we can stretch that out a little
further than what is involved in the usual Third Realm calisthenics. A
piece of mathematics is something that will be true for all time:
mathematical insight means seeing into the ages, what was and ever
will be. Conversely, lack of mathematical acuity — our inability to
solve a problem — is caused by the deceptions of the age, our follies
and delusions.
As regards my own very modest mathematical output, I have adduced
considerations that P cannot equal NP in ”A Cool Million (Some
Thoughts on P and NP)”, ”Further Thoughts on P and NP“, and ”Why P
Cannot Equal NP“. They do not have the form of a formal proof and have
failed to convince the contemporary complexity community (as a whole),
but it is my honest conviction that the very simple logical
difficulties associated with reducing the complexity of DEXPTIME-hard
NP-complete problems are insuperable. The considerations are “dumb”,
but honestly limitative results always are — the Incompleteness
Theorem is a piece of simple trickery that would hold no interest were
it not true.
Really, I think the hope that a brand new algorithm will crack the
problem is a pipe dream; the essential nondeterminacy of the
Satisfiability problem is just untouchable. Why does it appeal, then?
Because computer science is about technical control, and the
temptation of a computational Eden where all cryptographic algorithms
can be cracked and mathematical proofs grow on trees is just too
strong. (The harder-headed claim that problems from oracles — which
are rather moldy, as they can be read about in the original edition of
Hopcroft and Ullman – and other considerations may make the P and NP
question impossible to solve, but it may just be our own damn fault
for not defining our terms clearly. Always hard to tell.)