Why its Hard to Find Primes
Part 2 of “Should √2 Be a Prime Number?”
In my last article, I discussed the difficulty of finding prime numbers with our modern ℝeal number system. In this article, I am going to ignore √2 and talk about the deeper question of what it means for ℝeal numbers to be uncountably infinite and how it is related to the problem of finding primes. I also want to talk about how these problems are connected to the most fundamental and biggest outstanding questions in modern mathematics, The Continuum Hypothesis. This is the first item on Hilbert’s list of 23 problems he published in 1900.
Let’s ask ourselves the following:
- Is it a problem that the ℝeal numbers are uncountably infinite? How is countability related to cardinality and is this metric important?
- What is the composition of a ℝeal number vs a Natural number and are our assumptions about them correct?
The Countability of the ℝeal Numbers — Cantor’s Diagonal Argument
In 1891 Georg Cantor demonstrated that the ℝeal numbers are uncountably infinite using a simple method of proof called diagonalization. If you are not familiar with Cantor’s Diagonal argument I discuss it briefly here but there are already great articles to help.
Cantor’s brilliantly simple method of proof has stood the test of time as well as caused headaches for himself and fellow mathematicians, i.e. Russel’s Paradox. However, Cantor’s proof may be much more interesting than we have yet realized.
Cantor’s method of proof requires him to order and stack numbers in a table-like format. In the video linked above, Dr. Grime shows that for fractions or Rational Numbers, in order to list them they have to be arranged in a table where diagonal lines can be used to list them all. A graphic depicting this is below.
In terms of constructability, you may be able to mechanically list the positive Rational numbers, fractions, this way but picking any stopping point on this single weaving line would be problematic. You’d end up with missing numbers or have more than you need depending on your stopping point. Perhaps these are the infamous “holes” in the Rational numbers?
In mathematics, we generally understand dimensions to be linearly independent of each other. A point on one axis should be free to move without influencing points on a different axis. However, Cantor shows here that two axes are required to simply list a single Rational number. Using his scheme you can see that this listing method could easily take the form of a cone or Archemadian spiral and yet still not list all the Rational numbers in order.
What I find fascinating about this simple proof is that Cantor seemed to inadvertently prove that the Rational numbers themselves are at least 2D numbers. The example above shows that the positive Rational numbers themselves can’t be listed in any order without reference to a second axes. With a little mental geometry, this 2D field of Rational numbers could be the very same Real numbers we see. If we cut out a block of numbers on the cartesian plane it would never be complete since their order is not consistent with any axis. You’d need a third axes or more to start to find a way to list them in a specific order. This would make primes trickier to find…
If Rational Numbers are are at least 2D numbers then the ℝeal numbers that include them would also be at least 2D. Our complex numbers are supposed to be 2D and they are constructed from the ℝeal numbers, so, what happens?
In my recent experience, I’ve found that Number Theory is one of the hardest branches of mathematics to find information on. It’s too bad because it is incredibly interesting and learning it has been endlessly useful already.
So far what I have presented isn’t terribly complicated but let’s see how much further we can take this argument by getting a better look at the composition of a ℝeal number.
Anatomy of a Real Number
You might think that the x-axis is the ℝeal number line but the ℝeal numbers seem to be instead projected onto the x-axis. This surprised me at first but Set Theory says very little about how sets are applied to complex geometric objects like a coordinate system. So, let's apply a coordinate system to a ℝeal number like the one below.
One interesting trick I learned from Number Theory is that numbers have a verbose form and a terse form. Factors and prime factors are “pieces” of a whole and carry information nested within them. For any Real number, there is a base 1 (arithmetic base) portion and a base 10 (geometric base) portion.
So, what relationship does a ℝeal number have with say, our cartesian coordinate system’s axis? If we (1) project the vertical components to a straight line along the y-axis, horizontal here, and then (2) project that horizontal line to a single point on the x-axis (vertical) then we’d get a point on x. Finally, we just need to (3) rotate the grid we have here 90 degrees clockwise so our point coincides with the x-axis on the Cartesian plane. That would give us the relation our number has with the x-axis as we know it. Ultimately, that comes out to 2 projections and a single 90-degree CW rotation.
Two projections and 1 rotation is 3. Does this suggest that Real numbers are 3D numbers?
The ℝeal decimal numbers we all use are base 10. This is why taking the base 10 logarithm is important in scientific applications. Taking a logarithm removes base 10 exponentiation and converts it to base 1. Without removing the base 10 exponentiation you’d mistakenly believe that your dataset is non-linear without knowing that it is in fact the ℝeal numbers themselves that are not linear.
I suspect that Real numbers might each be 2D unit circles or equally right triangles scaled to the size of each number. This is how I think we can insert irrational numbers via Dedekind cuts. Infinitely thin 2D planes have no problem sitting close to each other along the number line. But, how do we know how long the number line is? I don’t know yet myself but it is on my list.
Also interesting, from basic algebra we have the definition of y*x=1. This tells us that x and y are functions of each other. The x and y values are both needed to define a whole. I imagine an axis split in two down its centerline, one portion is the x-axis and the other portion is the y-axis. This is precisely why primes are easier to find on the line of y=x. These are where the whole numbers are. The numbers on the x-axis matching primes are purely coincidental. Numerically they have the same face value but they mean something very different. The numbers on the x and y axes individually are only partial coordinates which is specific to the Real numbers only. This would also mean that finding primes along the x-axis is purely a coincidence and that the historical meaning of primes was not inherited by our Real numbers.
So far this is very fascinating but also very concerning. How is it that we know so little about how our numbers work? Unfortunately, we still don’t know how the greeks constructed our Natural numbers and by association, we don’t fully understand our Real numbers. It would be nice to check our “Theory of Real Numbers” but, that’s missing also.
Cardinality, Information and Modeling the Continuum — The Holy Grail of Mathematics
Even without a theory of Real numbers we have still been using them ubiquitously so let’s check the foundations we’ve established in Set Theory and see how these issues might tie into the Continuum Hypothesis.
The Continuum is the field of ℝeal numbers. This field is what we do most of our modern mathematics on and looks like one of the coordinate systems we use. Cantor was the first to formulate what is known as the Continuum Hypothesis, CH. It states that there is no set of ℝeal numbers and function that can place ℝeal numbers and Natural numbers in 1:1 correspondence.
We have gone as far as to say that the continuum hypothesis can’t be answered within Set Theory. Despite my best efforts, I have found no resource that provides an opinion on whether this is a problem we should be concerned about or the implications of a comprehensive solution.
The Cardinality of Real and Natural Numbers
What we want to do is talk about cardinality and how the Real and Natural numbers differ. Cardinality is a measure of the number of elements a set has. Cardinality can also measure infinite sets who may have infinities nested inside them and order them relative to other infinite sets. This is how we can have an infinite set that is listable like Natural numbers but also know that an infinite set like the Real number is larger.
Within Zermelo-Fraenkel set theory (ZFC), our most popular axiomatization of Set Theory, the continuum hypothesis is equivalent to the following:
The Hebrew character shaped like an “N” is called Aleph and with the sub of zero, “N_0", it is called aleph-null. Aleph-one, “N1”, is sub one and so on.
In Cantor’s hierarchy of aleph numbers, Set theorists don’t know where ℝeal numbers fit in. What we assume is that Natural numbers have a cardinality of aleph-null (N_0) and we assume that aleph-null, with base 2, is the cardinality of the ℝeal numbers.
I almost immediately take issue with this. Base 2 makes sense because we use binary-computing to describe everything. But, what about our base 10 like we advertise our ℝeal decimal numbers to be? Where does that fit in here?
How about the “2” in that base, is that a ℝeal number? If it is then its true form without base 10 exponentiation would be 2/10. We have our base 10 back but now we have a fractional base?
Additionally, a base would only describe the growth rate of aleph-null. If it were base 3 or base 6 it would grow much more quickly to the same infinite size as the Natural numbers. The only exception and difference being that we can’t efficiently track, index, and count our numbers on this slope… My question is, does anything still count if it can’t be counted? I am not sure if inventing things too big or complicated to understand is something to brag home about…
Mischaracterizing Cardinality for Information?
Regardless, we use Natural numbers to build sets like the Rational numbers or the ℝeal numbers. If they were somehow mischaracterized at the outset then this would trickle up and impact everything. This may be the case.
If Cardinality is a measure of the number of elements in a set then we may need another metric and measure to keep track of distinct objects. We can have infinitely many elements in a set but if they all describe the same object then infinite divisibility would only make data representation more verbose. If we take a set for a distinct object over the number of elements then this could be a way to roughly measure and gauge information density per object. We’d want to minimize this as much as possible which touches on the P vs NP problem.
For geometric objects, this would be quick and easy. How many points, lines, and planes does the object ideally have? This would give us a maximum and minimum representation efficiency to work with as well.
Because ℝeal numbers use the same algebra to do work with points as it does for physical dimensions there is a dimensional mismatch between algebraic numbers and ℝeal numbers. A Natural number can count something that is physically real and real things exist in 3 dimensions. We generally avoid counting things that can be broken down because ℝeal numbers can’t handle 3 spatial dimensions. ℝeal numbers are base 2 and so can only count up to 2D objects like planes and lines. However, if you take a look at a few algebraic operations you can see that our algebra can vary in 3 dimensions and so is 3 dimensional.
This would mean Natural numbers are intrinsically 3D. This should be the case because they can count physically real things. This makes sense in a historical context too because modern mathematics is much more abstract than we expect the Greeks to have had.
ℝeal numbers used with algebra are limited to 2 spatial dimensions because they start operations on half of a single point. This shifts all the dimensions down by 1. Broadly, this is almost entirely the result of a simple indexing error. Points are traditionally zero-dimensional but Real numbers count them as one so we can use algebra with them.
What does this mean?
Ultimately, we did mischaracterize our Natural numbers, and as a result, created a headache for ourselves. As for why it is hard to find primes?
Primes are hard to find because we are using 2D numbers to find 3D numbers on a 1D line…
I am also suspicious that this is why we see our characteristic 2:3 ratio in our physics and mathematics. We told the universe to be base 3 but we are using 2D ℝeal numbers to try and describe a 3D universe. This would make the need for complex numbers in physics make perfect sense. We have 3D numbers with only 2 dimensions of it being indexed or countable. Fixing this would probably simplify all things computing and perhaps open doors to new physics.
The Grail of Grails…
Last thing before I go. I wanted to take a second to talk about why addressing how we model the continuum is the “holy grail” of mathematics. Any other claims to the holy grail will be put to rest here. This problem is important because there is no generally accepted theory of Natural numbers. After all this time we still aren’t sure how the Greeks generated the Natural numbers. In modern set theory, we borrow the symbols or glyphs that are the Natural numbers and use those glyphs to build a set of Natural numbers. This is why Set theorists have to define and prove operations on each set. The aim is to rebuild numbers in the way we think the Greeks did. This flexibility leads to some interesting mathematics and metamathematics so it isn’t all bad. However, as a result, there is no truly complete or logically rigorous theory of numbers.
Fixing this problem could easily simplify all of our mathematics including the kind we teach at school to the computational physics we use to simulate the universe or the weather. I know this because those are my other projects I am working on.
Again, here is my favorite pure mathematician that explains the problems with the continuum. A bit lengthy but worth the watch!
Personal note: I am an amateur physicist and mathematician and have uncovered a few problems that are far too big for me to handle and solve on my own. If you have time, advice/guidance, know someone who can help, or know how to contact anyone who’d be interested in this, please contact me at JaisonEngineers@gmail.com
