The Riemann hypothesis

Why is the RH referred to as a "hypothesis", not "conjecture" or "theorem", as many other problems are?

Looking at pic related, it almost sounds like these people are talking about whether or not god exists, or something of that nature.

I think there is way more to this question than meets the eye, however I'm having a real tough time getting at what that truth might be.

How am I to think along the right lines if I don't know that the right line is even the right line to begin with?

Hypothesis and conjecture mean the same thing. The Riemann hypothesis is called a hypothesis because of historical reasons and because that sounds cooler than the Riemann conjecture.

A theory has been formally proven and is known to be 100% true.

For example, the Pythagorean therm has been formally proven and is known to apply to all right triangles. If the Pythagorean theorem had not been formally proven, all we'd have is a bunch of right triangles that all seem to follow the a^2 + b^2 = c^2. However, we might wonder if triangles that dont follow a^2 + b^2 = c^2 exist. In this case, a^2 + b^2 = c^2 would be known as the Pythagorean conjecture, or hypothesis. However, this is not the case. We have proven the Pythagorean theorem and we do know all right triangles follow a^2 + b^2 = c^2.

Hypothesis, however is more used in the context of scientific experiments, not mathematics, and more has the connotation of something meant to be tested.

What I'm trying to get to is that it is possible that the assumption that the RH may not even be true is possibly a mental barrier to proving it.

I dont think so. Certainly, the assumption that the RH may not be true is not a mental barrier to disproving it.
Assuming that the RH was true or false would be a grave mistake, one that would result in mistakes in other proofs.

Is conjecture to theorem as hypothesis is to theory? Is a hypothesis an empirical conjecture and a theory an empirical theorem?

In mathematics, conjecture = hypothesis. From wikipedia "In mathematics, the Riemann hypothesis is a conjecture..."

I would say I'm right in that that's the common usage, even if it's not a strict categorization. One term certainly seems to be used more for empirical observations than mathematical proofs.

>the assumption that the RH may not be true is not a mental barrier to disproving it.
Again I think that the way "Riemann Hypothesis" sounds cooler than "conjecture" or "theorem" is telling of what might the nature of the problem be.

When we speak of disproving a "hypothesis" we usually think of conducting an experiment the results of which might verify or refute said. For example, one way to "prove" 1 + 1 = 2 "experimentally" would be to hold up one finger, raise another, and see that now I have two, however the proof of the mathematical statement is something else altogether.

Now how would one go about disproving the "Riemann Hypothesis"? Assume that zeroes off the critical line exist, and stand apart with my feet somehow corresponding to the position of a zero pair?

>Now how would one go about disproving the "Riemann Hypothesis"?
Produce a zero of the Riemann-zeta function that is not a negative even integer or a complex number with real part 1/2.

Or disprove one of the many conjectures shown to be equivalent to the riemann hypothesis

Well giving a concrete counterexample surely isn't the way to disprove something... take the Skewes number as an example.

What I'm trying to speak of is mental barriers one might have by thinking of the RH as some hypothesis to be tested as opposed to a mathematical statement to be analyzed. Now what is it one might take to mean "producing" such a number? Write down on a piece of paper 1/2+/-y+/-ix ? Scream at the right frequency?

>When we speak of disproving a "hypothesis" we usually think of conducting an experiment the results of which might verify or refute said.
In mathematics, you propose a hypothesis when you see a pattern in "experimental" data. For example, you might be interested in the function y(x) = x + 1. You notice, by plugging in several numbers, that y(x) is always seems to be larger than x. You hypothesize that y(x) is greater than x for all x. However, you do not have the tools to formally prove it. You now have a conjecture, or hypothesis.

>For example, one way to "prove" 1 + 1 = 2 "experimentally" would be to hold up one finger, raise another, and see that now I have two.
In science, it is impossible to prove a hypothesis. It is only possible to disprove hypothesis.

>In mathematics, you propose a hypothesis when you see a pattern in "experimental" data
But isn't this really science? For example in math I wouldn't need to keep dividing the sequence 1, 1/2, 1/4, 1/8, ..., 1/(2^n)... to know it never actually reaches 0 and that 0 is the limit.

>However, you do not have the tools to formally prove it. You now have a conjecture, or hypothesis.
What proof would I have to give that x+1>x?

>In science, it is impossible to prove a hypothesis. It is only possible to disprove hypothesis.
Then why can't I just say something that is impossible to disprove by current methods and assert it to be true?

>thinking of the RH as some hypothesis to be tested as opposed to a mathematical statement to be analyzed.
No mathematical worth his salt would consider a mathematical statement in terms of empiricism just because it was named "hypothesis."

>Write down on a piece of paper 1/2+/-y+/-ix ?
If that was a number that was a zero of the Riemann zeta function that was not a even negative number or a complex number with real part 1/2, yes absolutely. Your proof could be one line: "ΞΆ(1) = 0 but 1 is not an even negative number or a complex number with real part 1/2, QED" and you'd be celebrated in mathematics for years to come.

Hypothesis are just inspiration for real math. Its sort of like "hey, I wonder if this is true" and some people think that's interesting and try to prove it.

>What proof would I have to give that x+1>x?
There are several. Here's one. "Clearly, 1>0. Add any number x to both sides and you get 1+x > 0 QED."

>Then why can't I just say something that is impossible to disprove by current methods and assert it to be true?
Most people consider hypothesis that cannot be easily proven unscientific. A hypothesis must be predictive to be useful. See Russel's teapot.

>No mathematical worth his salt would consider a mathematical statement in terms of empiricism just because it was named "hypothesis."

Yes, but such an interesting function naturally raises questions as to whether it has anything to do with physics, even if just as a possible approach to solving it (take the Hilbert Polya conjecture for instance).

And in a trivial sense any function can be made to have a relation with physics. Building a rollercoaster with the shape f(x)=y for example.

>you get 1+x > 0
Er, you'd get 1+x > x.
>hypothesis that cannot be easily proven
I mean "disproven." One criticism of String theory is that it is not disprovable with current technology.

>any function can be made to have a relation with physics
Not exactly. Even the most carefully made physical triangle is still a hopelessly complicated collection of jiggling atoms; it changes its size from one minute to the next. Math, I'd say, is entirely divorced from the physical world.

But can anything be truly random? Clearly one thing can't just lead to another without having to obey some principle, right?

Also, one "pure" form of mathematics that connects to the physical world is the way humans think. For instance we naturally have the ability to perceive harmonics.

As to my original question:
>How am I to think along the right lines if I don't know that the right line is even the right line to begin with?

Anyways some people say that RH is impossible to prove/disprove in terms of set theory, perhaps we could interpret this in a different sense.

All the possible thoughts that we can think can be described in terms of set/graph theory with things being in categories and things naturally leading to others, and if there were some innate assumption that all of our thoughts were based on that is incompatible with the answer to a certain question being definitely true or false, then it is possible we may never think of it.

What I'm suggesting is that there is some sort of mental barrier that prevents us from going there. In quantum mechanics, for instance, we are introduced to the notion that some things are inherently unknowable. Perhaps knowing whether RH is true or false is just incompatible with our notion of reality in some way.

What are the implications if it being true or false?

There are many statements as to the distribution of primes that depend on its truth, but I have a feeling it goes way deeper than that.

I see.

I'll do some "research" on this.

Hypothesis and conjecture are mostly synonymous in mathematics, but hypothesis is often used to emphasize that there are many conjectures which depend on the hypothesis. Indeed, many conjectures would become theorems if RH is true (or false).