If I tried to submit a paper like this to an actual scientific journal, I would be laughed out of the room, probably for not following protocol, secondly for not being one of them. However this is how it looks to a fledgling student of “science studies” to look at the debates currently going on in physics and mathematics and to be dumbfounded that ideas and theories that are entirely plausible to still receive some skepticism because they aren’t proven “indubitably and undeniably true”. Let’s enter into a discussion about scientific convention through the process of analyzing two current scientific debates, one in physics and one in mathematics- string theory and the Riemann hypothesis.
Now I’ve already done some stuff on string theory debate previously (see my earlier posts), so let’s tackle the Riemann hypothesis. A brief synopsis:
The Riemann hypothesis, called by some the “most important unsolved problem in mathematics” is about a function involving complex numbers with real and imaginary parts, called the Riemann zeta function. Bernard Riemann was one of the most famous mathematicians who basically invented the geometry of higher dimensional spaces (non-Euclidean geometry) that laid the foundation for Einstein’s general relativity. So if what Riemann said about the function is correct, that all the non-trivial zero values of the function lie on a critical line 1/2+i(t) and not outside that line, then the hypothesis is correct, and the function can be used to predict the distribution of prime numbers. It is the distribution of primes that makes this function useful and interesting, not that it is evaluated using real and imaginary numbers, that is a ubiquitous feature of complex analysis. The function itself is interesting in itself, in that it is a convergent series. I’ll show you what I mean:
This is the actual Riemann Zeta function. So if you plug in 2 for s, you get 1/1^2 +1/2^2+1+3^2, etc. I.e you get the perfectly understandable 1+1/4+1/9…i.e the sum of the reciprocals of the squares. Here’s the interesting part (there are actually many interesting parts of the Riemann Zeta function, and you can go down the rabbit hole with how interesting it is). At s=2, the answer converges to something astonishing-
pi^2/6!! Where did pi come from? What does this function have to do with a circle? And that’s not all!
So to recap, the function is just a function, it exists by definition. What Riemann’s hypothesis is is that all the zeros of the function (except the ones on the x-axis) are on a critical vertical line when using complex numbers and imaginary numbers are given by the y dimension.
Ok so what does all this math mean? It means that Riemann came up with an astonishing theory that shines a light on something fundamental in mathematics- prime numbers, which seem to be randomly distributed, but actually according to this theory can be predicted using a formula. The only problem? The hypothesis isn’t proven.
Why? That’s a good question I’m still trying to figure out. The best we’ve come up with is we haven’t been able to disprove it. There is no logical/mathematical proof as of yet of the Riemann hypothesis. The theory rests on the fact that there are no zeros outside a certain line- despite having found BILLIONS of zeros (actually over 10 trillion) using supercomputers- and they are all on the line so far!!
So mathematicians, even when there are “probabilistic proofs” of the theory (given by Denjoy) still have to say “the jury is out” because math is not a science of probability, but requires, in the words of basically every mathematician, “absolute knowledge”. Things like the weak Goldbach conjecture, which were “first proved using the generalized Riemann hypothesis” were also later proved unconditionally true, but this is too indirect a line of evidence for mathematical minds.
My argument to the mathematicians- if we can’t do brute calculations to infinity, no matter how many computers we have, maybe its time to call a spade a spade. The “consensus of survey articles” is that it is probably true. That sounds good enough to me.
Maybe mathematics should take a page from quantum theory and accept imperfect knowledge a la the uncertainty principle. I’ve watched many talks now about how the cutting edge of mathematics is coming from physics. It sounds like mathematics needs to import some of that physics “can-do” mindset and drop the Platonism.
In addition, when it comes to string theory, “indirect” evidence for string theory is also very strong, almost implied by some observed phenomena about particle physics. It doesn’t take a genius to understand this- even a layman can understand this and sift through the morass to find the answers. String theory is probably correct, so is the Riemann hypothesis. I’d bet money on it.
A definitive proof of the Riemann hypothesis gets the mathematical prize of 1 million dollars from a certain institute. This is proof of the value, the actual monetary value, placed on absolute proof in the field of mathematics. In a sense, it is what everything in it is based on- geometric proofs for example, or just the simple fact that you get one answer to a math problem. 2+3=5 dammit, and nothing else! Now, you can get two values for a particular equation, but that equation still has One answer. But what I’m saying doesn’t contradict this grade school logic. All I’m asking is that alternative lines of proof, including mathematically rigorous lines of evidence, from a “probabilistic” perspective, be given credit. It seems always that in these debates, something is left unsaid to the general public. What is left out here for the RH is “its basically been proven already”. What’s been left out for string theory is that “we already have a theory of quantum gravity, it doesn’t even require string theory, etc.” More on that later.
For now, just realize that these “definitive proofs” that we lack of unsolved problems in physics and mathematics have many dimensions to them. It is more indicative to me of a cultural issue, an obsession with Absolute Truth, and not being satisfied with relative truth. Maybe we can go ahead and say that relatively, the Riemann hypothesis should be assumed to be correct. We already know that primes aren’t randomly distributed- they make spirals and diagonal lines when you chart them:
The black dots represent the primes. If this doesn’t represent proof that they aren’t random, then call me Ishmael.
Edit: sorry for the typos before, I wrote this late at night