Doron Zeilberger

SAT is the boolean satisfiability problem, a simple NP-complete problem often used to prove the other problems are NP-complete. An instance of SAT is a boolean formula, and the answer to it is yes if and only if the formula can be true for some assignment of variables. 3SAT is the same problem, only with the formula in a conjunctive normal form and each clause having exactly three variables.

Also, it seems the poster is really just trying to get attention. Complexity theorists do know all that and have spent a lot of time trying to see which postulated consequences of P?=NP are necessary and which are not.

(If I were to argue it is an ill-posed question, I'd go more in the direction of CO-NP (which is the set of problems for which "NO" answers have short certificates), and why this really strong emphasys on "yes" certificates for NP problems)

Doron Zeilberger's take on that ( http://www.math.rutgers.edu/~zeilberg/Opinion98.html ) is a lot more interesting.

And, also, the "easy" equals "polynomial time" is (a) a good heuristic, if you assume computational power increases exponentially (hence, a problem will be solvable in the foreseeable future if all it needs is polynomially more processing power for larger instances); (b) it is an easy to study class, since it is composable; (c) not the only thing complexity theorists argue about (ie, P=NP means practical bounds for some problems and collapses a lot of classes that are not trivially seen to be equal, like AM and MA)

Correction, tons of physicists use real values (or complex real values) in their calculations. That doesn't imply that the corresponding values in the world are irrational, transcendental, infinitely-precise, etc.

There are a few ways we can avoid the reals in physics.

Abstractly, there are results such as Skolem's "paradox" ( https://en.wikipedia.org/wiki/Skolem%27s_paradox ) which tell us that theories involving uncountable sets (such as the reals) can be satisfied by countable models.

More concretely, physicists never manipulate real numbers directly; only symbols which may represent real numbers. Such symbolic manipulation could, in principle, be performed by a turing machine, and hence any value obtained from such theories (e.g. experimental predictions) must be in the set of computable numbers. Hence the reals are just an 'implementation detail' of any theory, and their purported presence cannot have any influence on the theory's predictions.

As I said previously, I have nothing against the reals as a useful mathematical tool; I just disagree with the name, and claims that they are physically real. Others go further though, arguing that discrete mathematics is more general and useful than continuous, e.g. http://www.math.rutgers.edu/~zeilberg/OPINIONS.html

Thankyou for writing this. As a humble seer trying to find his way in this world of bean-counters, I really appreciate it.

Doron Zeilberger has written about this aswell:

"""
The usual names are "theory-builders" vs. "problem-solvers", but these two protoypes of mathematicians also appear under various other appellations. Some other names are strategists vs. tacticians, "monkeys that climb up the tree" vs. "monkeys that climb down", globalists vs. localists, and folowing א ח ד ה ע ם, I once called them prophets vs. priests.
""" [1]

> something to help fund the Seers, à la startup incubator

The problem with this is that the Seers tend to sound like crazy people (and alot of them probably are just nuts). Especially to a craftperson, they really struggle to follow the seer. John Baez is a good example of someone who can balance the Seer with the craft, but i've noticed he will team up with someone who is more seer than he is to get those really good ideas.

[1] http://www.math.rutgers.edu/~zeilberg/Opinion95.html

Grothendieck "died" in the sense meant by Paul Erdos (stopped doing new mathematical research) at a surprisingly young age, while Israel Gelfand kept going into his nineties, not "dying" (ceasing to be a mathematician) until he "left" (died in the usual sense of the term).

http://www.math.rutgers.edu/~zeilberg/Opinion62.html

Gelfand was a mathematician who also cared deeply about mathematics pedagogy, and his textbooks are delightful.

http://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773

http://www.amazon.com/Functions-Graphs-Dover-Books-Mathematics/dp/0486425649/

http://www.amazon.com/Method-Coordinates-I-M-Gelfand/dp/0817635335/

http://www.amazon.com/Trigonometry-I-M-Gelfand/dp/0817639144/

Gelfand poses delightful problems that give students a workout in arithmetic (and CAN'T be done with a calculator) and that build conceptual understanding and interest in higher mathematics.

For anyone who might be misled, here are some links that people can use to judge for themselves how "crank" someone is:

Professor Norman Wildberger's homepage:

http://web.maths.unsw.edu.au/~norman/

...a paper by Wildberger tackling some of the issues in this thread:

Set Theory: Should You Believe?

http://web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf

Some interesting works by Gregory Chaitin

Meta Math! The Quest for Omega

https://arxiv.org/abs/math/0404335

Exploring Randomness

https://www.cs.auckland.ac.nz/~chaitin/ait/

How Real are Real Numbers?

https://arxiv.org/abs/math/0411418

People who are interested in ultrafinitism would also want to check out Doron Zeilberger

http://www.math.rutgers.edu/~zeilberg/

and

Edward Nelson

https://web.math.princeton.edu/~nelson/

It reminded me of what Dr. Doron Zeilberger has been repeating ad-nauseam ( eg: http://www.math.rutgers.edu/~zeilberg/Opinion36.html )
Ultimately automated theorem provers are going to rule. Dr.Z actually takes the very radical stand of calling all paper-pencil math proofs of Paul Erdos trivial, whereas "Theorems that only computers can prove...are not quite as trivial". The fact that we have to reach out to a computer to assist in solving min-sudoko is unfortunate but at the same time, inevitable. Dr. Z mentions Four Color Theorem, Kepler's Conjecture, and Conway's Lost Cosmological Theorem as other theorems only computers can solve.