The Riemann Hypothesis, Part 3 – Independent?

The analogy between Heisenberg’s Uncertainty Principle and the Reimann Hypothesis leads me to a conjecture about RH itself. This is pure speculation, based on my—by now evident—tendency to find patterns across diverse areas. Those patterns, of course, may not mean anything.

There are three commonly considered outcomes to the search for a proof of the Riemann Hypothesis. It could be proven to be true; it could be proven to be false; or no proof might ever be found about its truth value at all. (The 3rd possibility is not counted very highly, as mathematicians are hopeful beasts.)

I wonder about a 4th possibility that is sometimes discussed. Maybe it will be shown to be neither true nor false, but independent. Like the sentence “This statement is false” discussed in “Reality, Part 4 – Impossible Things” in Pencils 2019, maybe it doesn’t have a truth value, or its truth value changes depending on how you think about it.

This is different from no proof being found. It means there will be a proof found that the RH doesn’t have a well-defined truth value. As with Gödel’s Incompleteness Theorem or Turing’s Halting Problem, the end of the road isn’t an end—not even a dead end—it’s the edge of cliff over a bottomless pit.

With Zeno’s paradox we can make decisions about its truth based on our observations of our world in order to operate in our world. We can’t do that with the Uncertainty Principle, because the world of subatomic particles isn’t at all like “our world”. Still, we can use it according to what works in our world even though it makes no sense in our world.

RH is different because it’s pure math and, as noted before, math has a very problematic connection to our world. We use it when it suits us and drop it like a hot iron when it produces nonsense—most of the time. Sometimes the nonsense is hard to ignore or so clearly NOT nonsense that we are left feeling very uncomfortable. Occasionally, this leads us to new insights about our world. Other times, we just stay uncomfortable.

The rational conclusion is that we don’t yet understand things well enough, and eventually, we—the human race—will understand it. That’s getting harder to accept. Not because we’re running into physical limits on our observational abilities, but because some things just aren’t understandable. Understanding itself has limitations.

I don’t know whether the RH falls into this category, but it won’t be the first such upheaval in number theory. At the end of the 19th century, Georg Cantor threw such a huge logical monkey wrench into the notions of counting and infinity that he was reviled by another mathematician as a “corrupter of youth”. Look him up.

I think RH might end up this way partly because of the slipperiness of prime numbers, which is its focus. There are many seemingly simple questions about primes that either have defied an answer or required really dense mathematics to prove. RH itself came about as part of an attempt just to find the number of primes in the natural number line up to any chosen number and locate their approximate positions.

It’s even hard to answer the question as to whether primes appear randomly or not depending on how you define random*, and the best information we have about them is statistical.

Stop and consider this. Prime numbers are ordinary positive integers. They aren’t fractions, or functions, or irrational numbers (which some of the Greeks considered to be evil!). They aren’t complex numbers, or transfinite numbers, or even negative numbers. The idea that something about ordinary positive integers can only be determined statistically is a little strange. That they might be illuminated by something as bizarre as the Uncertainty Principle is beyond strange. Once again, it makes me wonder if we aren’t just looking in a mirror. (“Reality, Part 3 – Navel Contemplation” in Pencils 2019)

The truth value of the Riemann Hypothesis (true, false, or independent) may be decided in the next few years. There are some really smart hard-working young number theorists out there inspired by some recent successes. (e.g. Terrance Tao. Look him up, too!). On the other hand, at the beginning of the 20th century, mathematicians predicted it would be decided before the end of that century. It wasn’t, so who knows.

I’m just a cheer leader for that effort. My reason for diving into this, which one of my readers (my only one?!) likened to diving into a swimming pool with no water, is this: If the RH is independent, it means it has bumped up against the limits of rationality itself in a big way.

Like explorers in a cave chamber in total darkness, we are mapping the boundaries of our rational universe by feel rather than by sight. Those boundaries are there, whether or not the Riemann Hypothesis touches one of them. However, if it does—if it is shown to be independent—we will know a lot more about our world than we did before, just from discovering one more thing we can’t know.

Hugh Moffatt
Watertown, Massachusetts
March 25, 2020

*Prime numbers stretch our notion of what random means. There is no known way to predict exactly where the next one will appear as you go up the number line. That indicates randomness, but there are ways to predict approximately where it will appear. RH is the most accurate, if it’s true. It works for the numbers we know, but that’s not all of them, and...it’s still approximate. The Prime Number Theorem (the nth prime is approximately equal to n*ln(n)) is less accurate but still useful, and it’s been proven. So prime numbers are random in particular and approximately determined generally. Does that make them random or not? The answer depends on how you define randomness and involves statistical considerations. Sounds like quantum mechanics, doesn’t it? And discussions of the definition of randomness tend to make that concept foggier rather than making the primes clearer. Heading the wrong direction!