In this post I explain the intimate connection between Hegelian metaphysics and prime number theory that is, depending on your philosophical attitude, either very surprising or to be expected.

N.B. Please don’t read this unless you’ve previously read Hegelian contradiction and the prime numbers (Part 1)

## Prime numbers

Prime numbers are whole numbers that can only be divided by themselves or 1.

The primes are special in two ways: we can’t make them by multiplying other whole numbers together. And we can make all the whole numbers by multiplying a together a unique combination of primes.

So we can think of primes as the elementary atoms of multiplication.

You might think this is of purely mathematical interest. But really the structure of the primes manifests everywhere in reality.

Say I give you 45 pebbles and ask you to arrange them in a rectangle. No problem, and you quickly assemble a 9 by 5 rectangle.

But now I hand you 2 more pebbles, and ask you to build a bigger rectangle.

No matter how long you try, or how hard, you’ll *never* make a rectangle from 47 pebbles. It’s impossible, for the simple reason that 47 is prime and so can’t be broken down into a multiple of two numbers.

## The disorder of the primes

Now imagine the infinity of the whole numbers stretched out horizontally on a number line.

We see an infinite number of primes “growing like weeds” among the ordinary numbers. But the spaces between primes aren’t uniform. Sometimes the gaps are small, and sometimes really big. They appear at irregular intervals.

In fact the gaps tend to get astronomically bigger as we look at higher and higher parts of the number line. Let’s draw our first prime staircase:

But there are always short gaps. Right now (early 2019) mathematicians know there are infinitely many primes that differ only by 246. So as we ascend the prime staircase, to unimaginable heights, the steps do get longer, but there are always short steps.

This is a very irregular and disordered staircase!

## The Greek filtering algorithm

The reason for the disorderly gaps is, in one sense, perfectly clear and holds no mystery whatsoever. Early Greek mathematicians specified a very simple algorithm, called the sieve of Eratosthenes (“Era-toss-the-knees”) for generating the gaps.

We can program this on a computer. So the simple rules that generate the disorder are entirely transparent.

However, when we take a step back, and look at the overall shape of the staircase, we also see *extreme regularity and order*. And this is when things start to get a lot less simple.

## The order of the primes

To make the order really clear, let’s construct a different prime staircase. This time, as we travel along the number line, we’ll create a step whenever we hit either (i) a prime number or (ii) any number that’s the power of a prime.

So we create steps at 2, 2 squared, 2 cubed, 2 to the power 4, and so on. And we create steps at 3, 3 squared, 3 cubed, and so on.

But let’s also change the heights of the steps. Each step has a height which is the logarithm of the prime factor.

So the step heights at 2, 2 squared, 2 cubed, and so on, are all of size log(2), which is about 0.7. But the step heights at 3, 3 squared, 3 cubed, and so on, are all size log(3), which is about 1.1.

This new prime staircase is known as the Chebyshev function. What does it look like?

The Chebyshev prime staircase seems to approximates a perfect straight line.

At the micro level, the primes and their powers are irregularly spaced. There’s disorder. But when we zoom out, to the macro level, there appears to be almost perfect order.

We can’t check all the way to infinity with computers. So if this straight-line law holds we need to prove it mathematically. But proving this law required nothing less than a revolution in the methodology of number theory.

## Riemann’s revolution

Number theory studies properties of discrete magnitudes and is as old as civilisation itself. Up to the 17th Century mathematicians employed elementary methods in their proofs that employed the basic operations of arithmetic.

But the discovery of the calculus by Newton and Leibniz started to change that. In the 19th Century mathematicians realised that methods that apply to continuous magnitudes, such as differentiation and integration, also applied to number theory, and in fact were more powerful. The modern field of *analytic number theory* was born.

The mathematician, Bernhard Riemann, wrote a paper in 1859, which used the techniques of analytic number theory, to study the primes in an entirely new way. He constructed a special kind of viewing device, called the Zeta function, which revealed hidden properties of the primes.

Here’s one way of writing the Zeta function:

The meaning of this equation will hopefully become clearer as we go along.

But the first thing to note is that we feed the Zeta function with a complex number. It then performs some computations, and hands us back a new complex number.

Complex numbers, you’ll recall, have two parts: an ordinary part and a so-called imaginary part, which is some multiple of the square root of -1 (e.g., 3 + 4 i is a complex number where i=√-1).

## The zeros of the Zeta function

We can think of both the inputs and the outputs of the Zeta function as points in the *complex plane*. So Zeta takes any point on a plane surface and moves it somewhere else on the plane.

Riemann discovered that Zeta maps some special input values to the *origin* of the complex plane. For example, ζ(0.5 + 14.1347 i) evaluates to 0. So we call this input value a “non-trivial” zero of the Zeta function.

Here’s a plot of the first 3 non-trivial zeros that Riemann computed:

Riemann knew there are an infinite number of zeros. But he could only calculate a handful with pen and paper. We can easily explore more with modern computers:

## The zeros and the primes

Now this is very pretty, but so what?

Here we get to the crucial point. Riemann discovered a remarkable fact: *the location of the zeros of the Zeta function encodes the distribution of the prime numbers.*

Riemann, in a sequence of remarkable mathematical arguments, derived a formula for Chebyshev’s prime staircase in terms of the zeros of the Zeta function:

We can ignore the log(2π) constant term, since it quickly gets swamped as we ascend the number line. The first term, x, is a big reveal, since that’s exactly what we’d expect to see if the straight-line law was true.

But we don’t simply have ψ(x) = x. We have an extra term, which is an infinite sum of all the Zeta zeros. The explicit formula tell us that the *Zeta zeros control the magnitude of the fluctuations of the primes (and their powers) about a straight-line law*.

Note that we can’t directly relate an individual Zeta zero to a prime or its power. It doesn’t work that way. Instead, *all* the zeros collaborate in “generating” the primes and their powers.

So the Zeta zeros tell us how far the Chebyshev staircase deviates from a straight line across the whole infinity of integers. And, so, the more we know about where the zeros live, then the more we know about how the primes “grow like weeds” amongst the whole numbers.

But here’s the problem. As of today (February 2019), mathematicians *simply don’t know where all the zeros are*. It’s really hard to find out where they all live.

## The Prime Number Theorem

Riemann did know, however, roughly where they live. The non-trivial zeros must lie somewhere in what’s called the critical strip, where every zero has the form x + i y, where 0 <= x <= 1.

But it wasn’t until 1896, over 30 years after Riemann’s original paper, that mathematicians managed to prove that no zeros exist on the line x=0 or x=1 (the edges of the critical strip). Remarkably, this knowledge alone is sufficient to prove that the distribution of primes is indeed governed by a straight-line law.

The proof is now known as the Prime Number Theorem, and is the crowning achievement of analytic number theory:

So, at the micro level, the primes are disordered. But at the macro level, they approximate a simple, straight-line law. The Prime Number Theorem means this law necessarily holds all the way to infinity.

And Riemann’s Zeta function was the key to unlocking this hidden order of the primes.

## The mystery of the Zeta function

At this point we should begin to feel puzzled by this mathematical story, and start to ask some questions.

Riemann’s new way of studying the primes is mathematically unambiguous. But what this new way of looking *is*, and why it should prove so *effective*, is much more mysterious.

Even mathematicians aren’t exactly sure *why* the Zeta function encodes information about the distribution of the primes, only that it *does*.

But why does it? Why is the Zeta function uniquely successful in encoding knowledge about the primes? Why can continuous magnitudes, and imaginary numbers, tell us new things about ordinary whole numbers?

So to try to answer these questions, let’s now get back to Hegel.

# Part 2: A Hegelian interpretation of Riemann’s Zeta function

Hegel’s Science of Logic, you might recall from the previous post, claims to reveal the *necessary structure of anything that exists* (whether in physical reality or in the mind). Hegel calls this necessary structure “determinate being” or “becoming”.

Previously, in Notes on a mathematical interpretation of the opening of Hegel’s Science of Logic, I developed a mathematical model of becoming as a 2-D system of coupled differential equations. I called this model, “Hegel’s contradiction”, since it’s a dynamic unity of the opposition of *being* and *nothing*:

I plan to take Hegel at face value, and assume he’s right: *everything* is indeed ultimately composed of Hegelian contradictions.

In consequence, the integers – paragons of perfect, immutable objects that are impervious to time and exhibit no apparent changes whatsoever – must be, contrary to appearances, fundamentally dynamic objects with internal contradictions that cause them to change and move. The integers must also be Hegelian contradictions.

On the fact of it, this seems to be an insane proposition. But this is what the logic of Hegel’s *Logic* implies.

So let’s start this experimental line of thought by defining what a “Hegelian integer” might look like.

## Hegel numbers

Last time we had one kind of contradiction. Now we need to start distinguishing different kinds.

A single Hegelian contradiction has two properties — (i) the rate, or “speed”, at which being reacts to nothing (and vice-versa) and (ii) the overall “activity level”, or quantity of substance that flows within it.

Different contradictions, therefore, will have different reaction speeds and activity levels.

Define the Hegel number H[ω] as having a reaction speed ω:

In consequence, being and nothing, in the Hegel number H[2], will react twice as fast to each other compared to Hegel number H[1].The 2-D system of coupled differential equations, that define the contradiction, have the same form as before:

As I mentioned last time, although being and nothing oscillate over time they nonetheless obey a conservation law. For simplicity, let’s call this conserved value the*size or scale*of the contradiction, because it relates directly to the quantity of substance flowing within it.

But how “big” should a Hegel number be? I want to postpone this decision and simply declare that its size is determined, in some yet to be specified way, by ω (and we specify it by setting the initial magnitude of being at time t=0).

And that’s it. We’ve now defined “Hegel numbers”. Ordinary numbers and Hegel numbers have a simple 1 to 1 correspondence:

The number ω corresponds to the Hegel number H[ω]

Let’s take a look at some examples.

### Examples: the Hegel numbers H[2] and H[5]

Every Hegel number is a fluctuation of being and nothing over time. Here are two examples where I’ve arbitrarily set f(λ,ω)=1/ω — so “faster” Hegel numbers are “smaller”.

So I hope you’ve got some idea of how different Hegel numbers behave.The ordinary integers can be added, subtracted, multiplied and divided. What kinds of operations can we perform on Hegel numbers?

## Sublating Hegel numbers

Well, there are many possible operations we could perform. But here I’ll focus on just one, which I’ll call the *sublation operator*.

Here’s two Hegel numbers, H[ω1] and H[ω2], ready for sublation:

We want to metaphorically add these two Hegel numbers. That means creating a new causal structure of being and nothing.

There are many possible ways of combining these contradictions. But we want a way that’s consistent with Hegel’s method. We need to reproduce the kind of moves that Hegel made when he originally sublated pure being and pure nothing into becoming. So we need to ensure that:

- Being always “passes over into” nothing.
- Nothing always “passes over into” being.
- Being always affirms nothing (i.e., has a distinct “direction” different from nothing).
- Nothing always negates being (i.e., has the opposite “direction” to being)
- The higher, sublated unity preserves its components as “moments”.
- But the higher unity also “puts an end to” its components and manifests new properties not reducible to them. (The whole is greater than the sum of its parts.)

Principle (1) implies we connect the being H[ω1] to the nothing of H[ω2].

Principle (2) implies we connect the nothing of H[ω1] to the being of H[ω2].

Principle (3) implies the connection x1 to y2 is negative (since y2 negates x1) with a reaction rate that’s some function of the reaction rates of the component contradictions.

Principle (4) implies the connection from y1 to x2 is positive (since x2 affirms y1) with a symmetric reaction rate.

But what should the reaction rate of these new connections actually be?

It turns out that, in order to satisfy principles (5) and (6), our choice is severely constrained. For reasons that will become clearer shortly, the new reaction rate is (ω1-ω2).

So, once we make these new connections, we get the new causal structure: a *sublated unity of two Hegel numbers*:

The dynamic behaviour of H[ω1] ⊕ H[ω2] has a relatively simple form.

H[ω1], in the unity, behaves just like an isolated H[ω1]. And H[ω2], in the unity, is a superposition of the dynamics of each isolated H[ω2] and H[ω1].

Define the *resultant behaviour* as the fluctuations of being and nothing of the final Hegel number in the sublation. Here’s a plot of the resultant dynamics of sublating the Hegel numbers 2 and 5 (H[2] ⊕ H[5]):

Hegel states, in his *Logic*, that sublation both preserves or maintains its components *and* “puts an end to them”. Clearly the sublation operator introduces new properties we’ve not seen before. But in what sense does it preserve its components?

The preservation is obvious when we decompose the resultant trajectory into the components H[2] and H[5]:

So the sublation operator both preserves its component contradictions, and yet also produces a qualitatively new “ceaseless unrest” (repeated fluctuations of being and nothing) that nonetheless is a “quiescent result” (a bounded, repeated trajectory in phase-space).A word of warning about the animated phase-space visualisations. Don’t confuse the map with the territory. The sublation H[2] ⊕ H[5] is *not* moving in space, and its components are *not* rotating. This is just a useful picture to help us understand the dynamics of being and nothing in this causal structure:

The above sublation doesn’t exist or move in space. Rather, at any time, it has 2 activity levels of being (x1 and x2) and 2 activity levels of nothing (y1 and y2). They mutually interact, and the activity levels fluctuate. (So you may like to visualise “lights” at the nodes that wax and wane).

## Higher order sublations

But why stop here? We can sublate the sublation. In other words, repeatedly apply the sublation operator, ⊕, as many times as we want — and to any combination of Hegel numbers.

Each time, we apply the same principles, and “attach” a new Hegel number to the sublation (e.g. attach H[ω3], to H[ω1] ⊕ H[ω2]).

Here are the next four, higher-order sublations. As you can see, the causal structure rapidly gets complex:

Each n-th order sublation defines a 2n dimensional coupled system of differential equations.

The resultant fluctuations of being and nothing get increasingly complex. Here’s a plot of H[1] ⊕ H[2] ⊕ H[3] ⊕ H[4] ⊕ H[5] ⊕ H[6]:

So every time we apply the sublation operator we create a higher dimensional dynamic system.## The totality of Hegelian integers

But why stop here? We can sublate an infinity of contradictions.

So let’s now do the following: let’s *sublate the Hegelian whole numbers to create an infinite-dimensional dynamic system*.

Traditionally, we think of the integers as the infinite set: ℕ = {1, 2, 3, …}. Each whole number is a static quantity that relates to other members via arithmetic operations (e.g., 2 = 1 + 1).

But we can equally think of the integers as the infinite system: H[ℕ] = H[1] ⊕ H[1] ⊕ H[3] ⊕…

The infinite sublation of the Hegelian integers is a dynamic totality. All its members relate to each other via causal relations. The whole sublation moves. We can formally solve this infinite dimensional dynamic system to derive the following equations of motion:

This infinite dynamic system is a Hegelian view of the integers. Are there any advantages of thinking of the integers as H[ℕ] rather than ℕ?

Well, we already know that the answer is a resounding yes, because what I’ll now show is that Riemann’s revolutionary new way of seeing, embodied in his Zeta function, is precisely this Hegelian viewpoint.

Of course, neither Riemann, nor any modern mathematician, adopts a Hegelian interpretation of their mathematics. Nonetheless, *the Zeta function is a method for exploring the dynamics of the sublated totality of the Hegelian integers, H[ℕ]*. The Zeta function is full of being, nothing and becoming; and therefore full of contradictions and movement.

## Back to Riemann: from H[ℕ] to the Zeta function

So let’s take a few moments to demonstrate this connection.

H[ℕ] is a dynamic system but Riemann’s Zeta function, ζ(s), is a static, timeless map that takes points on the complex plane to other points. What have these things got to do with each other?

The first step is to map Hegel’s *being* to the *real number line*, and map Hegel’s *nothing* to the *imaginary number line*. So we represent the activity level of being and nothing, at a specific moment in time time, by a complex number:

H[ℕ] is a dynamic system that evolves with time. So, in mathematical terms, t is a free parameter. And, if you recall, Hegelian numbers have another free parameter that sets their scale, which controls the quantity of substance that flows in the contradiction. Call this scale parameter, λ.

Next, in the second step, *we represent the time and scale of a sublation as another complex number*, s = λ+i t, where the *scale* is the *real* part of s and *time* is the *imaginary* part.

Next we form a complex-valued function, f(s), that

- takes as input the complex number, s = λ+i t, which represents a scale and time, and
- outputs a complex number, x(λ,t) + i y(λ,t), which represents the resultant behaviour of the sublation with that scale and at that time:

In other words, we plan to use the complex-valued function f(s) to “embed” the dynamics of H[ℕ] in the complex plane *at all possible scales for all possible times*.

Earlier, we viewed a complex-valued function as mapping points in the complex plane to new points. This is a very “syntactic” or mathematical point-of-view.

The Hegelian interpretation gives us a different way of thinking about some complex-valued functions. They tell us the state of being and nothing for a given scale at a specific time. This is certainly a more poetic point-of-view.

There are some additional mathematical technicalities to properly embed H[ℕ] in the complex plane, which I’ll briefly mention. To ensure that f(s) is truly a function of a single complex variable we need to ensure that the Cauchy-Riemann equations are satisfied. And also we must avoid bad infinities, and ensure the dynamics of the sublation are convergent in the domain. In practice, this means that the relationship between the reaction rate of a contradiction and its scale is determined by a particular function with a very specific form (so the arbitrary choice of f(λ,ω) becomes non-arbitrary).

We’ve nearly completed our journey from the sublated Hegelian integers to the world of the static Zeta function. There is one last step we need to take, however.

## An infinite sublation of the logarithm of the Hegelian integers

The last step is to take the logarithm of the Hegelian integers. So instead of working with

H[ℕ]= H[1] ⊕ H[2] ⊕ H[3] ⊕ …

we’ll work with

H[Log ℕ]= H[log 1] ⊕ H[log 2] ⊕ H[log 3] ⊕ …

Why?

The simple answer is that this transformation allows us to make direct contact with the Zeta function. The more complex answer is that Riemann investigates the multiplicative structure of the integers, and logarithms make that easier.

Now we have all the conceptual pieces in place it is easy to show that the infinite sublation of the logarithm of the Hegelian integers is essentially Riemann’s Zeta function.

Let’s restate that final conclusion (with a slight abuse of notation): So,*Riemann’s Zeta function, and the sublated totality of the (logarithm of the) Hegelian integers, are the same object*.

## From time and scale to being and nothing

In summary, we have a “metaphysical” interpretation of the Zeta function:

The Riemann Zeta function embeds the dynamics of the infinite sublation of the (logarithm of the) Hegelian integers for all possible scales at all possible times.

Complex-valued inputs η(λ + i t):

The real value,λ, is the scale of the sublation.

The imaginary value,t, is the time.

Complex-valued outputs x + i y:

The real value,x, is the resultant quantity of being (at this scale and time).

The imaginary value,y, is the resultant quantity of nothing (at this scale and time).

Recall that, when we first visualised the Zeta function, we traversed a blue vertical input line to get a red output spiral. We can now see that traversing a vertical input line simply set the overall “size” of the sublation and then *moved time forwards*. The red output spiral was the *resultant fluctuations of being and nothing* in phase-space.

Here’s an example of doing just that. This is an animation of H[log(ℕ)] generating the first zero of the Zeta function:

## The zeros of Zeta: moments when infinite becoming attains the state of pure being

So, in this Hegelian interpretation, what are the Zeta zeros?

### Hegel’s metaphysics

Hegel’s metaphysical bedrock is pure being and pure nothing. Pure being, as we saw last time, explodes to infinity, and pure nothing implodes to nothing. These pure states can’t exist since they’re unstable. Hence, we have becoming, their sublated unity, which exhibits both order and disorder.

Hegel, in his Logic, continues, and claims that becoming must individuate into separate things, which relate to each other in sublated unities of higher and higher complexity.

This universal process finally culminates in a state of absolute knowledge, which overcomes the original contradiction between being and nothing, and where God finally comes to fully know itself. So in Hegel’s philosophy there is some kind of limit, or end-point of final reconciliation.

Perhaps surprisingly, the mathematics of the Zeta function has a similar structure.

### The zeros as pure being or pure nothing

Mathematically, as we sublate Hegelian integers, they become increasingly causally entwined, and we create higher and higher complexity. The Zeta function encodes the infinite limit of this process.

The Zeta function exhibits order and disorder. In fact, the fluctuations of being and nothing are chaotic in the strictly mathematical sense. The disorder of the infinite sublation is more disorderly than any single component.

But order emerges from this chaos. It appears that the Zeta function generates trajectories that forever fluctuate about a special, zero state.

The zero state is very special indeed.

In the Hegelian interpretation, a Zeta zero is a moment when both being and nothing are identically zero. Or, if we apply the reciprocal map from previously, a moment when they are identically infinite. So either the final lights in the infinite sublation blaze bright, or they’ve blinked out of existence.

This means that: *The zeros of the Zeta function are moments in time when becoming, which is an infinity of contradictions, attains a state of pure being or pure nothing*.

An individual contradiction can never do this. So the order manifested by the infinite sublation is more orderly than any single component. But these pure states of perfect order are achieved by infinite chaos. So, once again, they are unstable and therefore transitory, and now merely moments of an infinitely complex process of becoming.

Riemann, in his remarkable paper, demonstrated that the zeros encode the distribution of the prime numbers. The primes are irreducible atoms of the number system, they are the mathematical bedrock.

Hegel’s logic implies that these zeros are moments when becoming reduces to pure being or pure nothing. So the zeros represent the irreducible atoms of Hegel’s Science of Logic. They are a metaphysical bedrock.

This means that: *the arithmetical irreducibility of the primes is a manifestation of the metaphysical irreducibility of pure being and pure nothing*.

So now let’s return to the questions I originally posed. Can we make sense of Riemann’s revolution? And why is the Zeta function so uniquely successful in revealing hidden properties of whole numbers?

# Conclusion: The metaphysics of Riemann’s revolution

I think it’s pretty clear, at this stage, that we have more questions than answers. But we can make some general remarks.

Riemann moved number theory into the complex plane. This revealed entirely new phenomena, which have yet to be fully understood.

The success of Riemann’s project is strong evidence that the whole numbers – which we think of as static, unchanging quantities – are really some kind of shadow or projection of the Hegelian integers. The Zeta function reveals more because it *represents whole numbers as what they actually are*, that is dynamic contradictions of being and nothing.

But, in addition, the Zeta function represents the whole numbers as a sublated unity, where the entities internally relate via the exchange of a conserved substance. And this whole moves and changes with time. This is quite unlike the vision offered by set theory.

In the 1970s physicists noticed that the distribution of the Zeta zeros follow the same statistical law as the distribution of energy levels of systems of subatomic particles (see Hilbert–Pólya conjecture). For many, this connection was surprising and even shocking – for there seems to be no reason why fundamental physics and number theory should be intimately connected.

But Hegel would expect to see such connections, for the simple reason that he believed thought and being are identical, and conform to the same underlying laws, laws which he attempted to elucidate in his Science of Logic.

Of course, Hegel’s Logic did not invent analytic number theory or fundamental theories of physics. Rather, Hegel’s logic implies that harmonic phenomena are a necessary consequence of the fundamental ontological contradiction between being and nothing. The appearance of harmonics in the most fundamental structures of physical reality, and the most fundamental structures of Platonic thought, is a remarkable, and thoroughly comprehensive clue that Hegel’s logic is not only a logic worth having, but a logic worth developing.

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