/sci/ - Science & Math

>>16001426
any number is correct, just pick a random one and fit a polynomial to the sequence.

You are talking to your friends about your favorite subject: maths and why you like it. One reason you like maths, is that it is logical and final. e.g.

"If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics." -- Roger Bacon

"The mathematical type of investigation alone will give the inquirer firm and unshaken certainty through demonstrations carried out by unquestionable methods." -- Ptolemy

"Mathematics alone, if one applies himself diligently to it, will give the inquirer after knowledge firm and unshaken certitude by demonstrations carried out with unquestionable methods." -- Ptolemy

Your friend tries to write a mathematical argument, and comes up with this:

Assumption: This argument is valid.
Conclusion: 1 ≠ 1

He wonders if this is a correct (valid) proof. You say that it is not. For if it were a correct proof, it would have a true assumption but a false conclusion, for 1 = 1. On the other hand, he says that it is in fact correct, for if the assumption is true, then it is a valid proof with a true assumption, so the conclusion would also be true. But this precisely what a valid proof is.

But if it is impossible to tell whether even a two line proof can be valid, how can we possibly tell whether a complicated proof is, and how can math be said to be certain?

According to the Oxford dictionary a sandwich is "An article of food for a light meal or snack, composed of two thin slices of bread, usu. buttered, with a savoury (orig. spec. meat, esp. beef or ham) or other filling."

And a hot dog is "A hot sausage enclosed as a sandwich in a bread roll."

Conclusion: a hot dog both is and is not a sandwich. In classical logic, this implies the Riemann hypothesis.

Many Ph.D. students are taking 6+ years when previously it was relatively common to be done in 3. Meanwhile these projects seem to be the equivalent of (for a physicist) spending several years studying what shape of foil best reflects microwaves in some particular manner.

Meanwhile your chances of getting a faculty job are far less than 10% and that's for postdocs not students, and that's assuming you're from a top university with a good publication history in the top of the top journal(s). Everyone else ends up 44 years old, with a family to feed, and looking for job with a "second rate has-been" label on his forehead.

Even if you intend to get a job out of undergrad, you are getting worse grades or spending a lot more effort getting the same grades.

The Piraha people do not have numbers, and the notion of the actual infinity of all numbers is a product of human imagination; the story is simply made up. The tale of ω even has the structure of the traditional fairy tale: "Once upon a time there was a number called 0. It had a successor, which in turn had a successor, and all the successors had successors happily ever after."

Some mathematicians, the fundamentalists, believe in the literal inerrancy of the tale, while others, the formalists, do not. When mathematicians are doing mathematics, as opposed to talking about mathematics, it makes no difference: the theorems and proofs of the ones are indistinguishable from those of the others. Let us examine the fundamentalist belief in the existence of the completed infinity ω in the light of monotheistic faith. It is part of monotheistic faith, as I understand it, that everything in creation is contingent; I AM WHO I AM is not constrained by necessity. Are we to believe that ω is contingent, that the truths of arithmetic might have been different had it pleased God to make them so? Or are we to believe that ω is uncreated—existing in its infinite magnitude by necessity, as it was in the beginning, is now, and ever shall be? But these are unreal questions, like “can aleph-2 angels dance on the head of a pin?”

It is possible to doubt the consistency of Peano Arithmetic, because of the untamed power of induction -- the Inductive Hypothesis can be quite complicated, making it possible that an inconsistency may creap in.

I took a real analysis class based on the book Principles of Mathematical Analysis by Walter Rudin, and it just seemed to be mostly just reproving the obvious. I feel that the mathematics of the future will involve a lot more pictures, and the non-rigorous mathematics practiced by theoretical physicists e.g. quantum field theory, 1+2+3+... = -1/12, &c. In another field, the "safety" of R.S.A. or elliptic curve crypto-systems is only conjectural, yet these systems are still useful. In fact Chaitlin has shewn that most mathematical facts are either undecidable, or unprovable in real time. And analysis does not seem to introduce new concepts such as sheaves in order to allow humans to access a greater class of theorems.

>>9563047