Hegelian contradiction and the prime numbers (Part 1)


Audio of the 35 minutes talk: click here (Oxford, November 2018, CCS)

Accompanying handout: hegel-handout

There was a good discussion, which sadly I can’t upload without permission of all the attendees. But I’ve included the audio of my 10 minute response: click here

Below is the transcript, which is a condensed version of the post: Notes on a mathematical interpretation of the opening of Hegel’s Science of Logic


Marx called Hegel a “mighty thinker” and was thoroughly influenced by him. Many of the concepts of Marx’s Capital can be traced to Hegel’s Science of Logic, published in 1812.

But Hegel’s Logic is extremely abstract and obscure. Philosophy is often difficult, but Hegel is especially difficult: he twists you up, down, left and right and the effect is dizzying.

This difficulty of understanding Hegel partly explains the never-ending tension within Marxism between those who think Hegel is absolutely necessary to the tradition, and those who think he’s a liability.

So in this talk I want raise the following questions: What is Hegel’s logic? And is it a logic worth having?

Any logic worth having should tell us something new about the world.

Marxists will point to Marx’s historical materialism as evidence of the efficacy of Hegel’s logic. But Hegel intended his logic be universal, and apply to all phenomena. So I want a put Marx to one side, and instead test whether Hegel’s logic has anything new to say about fundamental reality, in particular some aspects of physics and number theory.

Today, we’ll look at Chapter 1 of Hegel’s Science of Logic. You can find the main argument of Hegel’s Chapter 1 on one side of the handout.

In the second part — which I’ll give next year — we’ll see how Hegel’s logic applies to analytic number theory, in particular the Riemann zeta function and the nature of the primes.

So we’ve got quite a lot to get through. So let’s begin with a few remarks on what Hegel is trying to achieve.

What is Hegel’s project?

Hegel aims to discover the fundamental structure of everything from pure reflection alone. If you’re a hard-nosed materialist you might already be worrying.

But if you are a materialist then you believe we’re constructed from the same “stuff” as everything else in the universe, and therefore it follows that our cognition must share some fundamental properties with everything. So pure reflection might give us immediate access to some kind of knowledge of those fundamental properties.

Hegel claims to begin his Science of Logic with zero assumptions. He wants to take philosophical doubt even further than Descartes. So Hegel’s beginning even rejects the “I” in the “I think, therefore I am”.

Those familiar with modern, formal logic might also start to worry. We can’t deduce anything without axioms and inference rules. So starting with neither doesn’t seem to make sense.

But Hegel isn’t really in the business of deduction. His approach is more like an empirical investigation, where the raw data happens to be accessed via pure reflection. Hegel aims to merely observe what is there – once we drop all our knowledge, all our presuppositions, all our theories, and even the sense of our own existence.

So Hegel’s starting point is, in a way, mystical and psychedelic. He does mention the ego-less state of someone who meditates while looking at the tip of their nose, chanting “Om!”

Hegel hopes that, if we start from pure reflection, we will gain knowledge of some bedrock metaphysical properties, which are absolutely necessary.

So let’s begin where Hegel does. We don’t have to close our eyes and start chanting. But we do need to abstract from all possible thought contents, and consider what remains.

In the beginning was pure being …

And Hegel says that what remains is pure being, or existence itself.

Hegel says the beginning must be abstract since we “may not suppose anything”. The beginning cannot have any properties, or content, or distinctions. So it must, according to Hegel, be “purely and simply an immediacy, or rather merely immediacy itself”.

Hegel says pure being is the “unity into which knowing has collapsed into at the extreme point of its union with the object”. So, at the beginning, there isn’t a knowing subject contemplating its own existence. Instead, there is only pure being that, in some obscure sense, knows that it exists.

I think it’s worth emphasising that when Hegel talks about pure being he isn’t talking about an abstract concept. He’s actually talking about a real phenomenon, an actually existing thing, which he claims we all have immediate access to, if we’re prepared to perform the mental exercise.

Pictures help. So let’s draw Hegel’s self-referential starting point:


The arrow indicates the self-referential nature of Hegel’s beginning. We could say that the outward port is ‘pure being’ and the inward port is ‘pure knowing’, where being contemplates itself.

This doesn’t seem to be a very promising start, since it’s not obvious anything follows at all from this psychedelic insight.

… and pure nothing

But then Hegel points out that pure being is so pure, it lacks any content. It’s basically so abstract it’s is empty, and therefore nothing at all.

We thought we had pure existence, but this kind of existence is nothing, a complete void, nullity. As Hegel says, pure being is just ’empty thinking’.

So we can equally draw the beginning as pure nothing.


This is the same picture, we’ve just labelled it differently.

Yet there is a difference: being has a connotation, which is existence. But pure nothing has a different connotation. It’s about absence.

So it seems like we’ve got two things. When we consider pure being, the most abstract concept of all, we find that, as soon as have it, we’re left with nothing at all.

The beginning is a paradox

But pure nothing of course exists, because we’ve just observed it.

So although it’s pure nothing it has the property of existing. So the concept of pure nothing seems to turn immediately back into pure being, just existence without any content. We seem to swap back-and-forth between two viewpoints of the same very abstract beginning.

At this point in his argument, Hegel suddenly says a number of remarkable things, which I’ll now quote:

Pure Being and pure nothing are, therefore, the same. What is the truth is neither being nor nothing, but that being — does not pass over but has passed over — into nothing, and nothing into being. But it is equally true that they are not undistinguished from each other, that, on the contrary, they are not the same, that they are absolutely distinct, and yet that they are unseparated and inseparable and that each immediately vanishes in its opposite. Their truth is therefore, this movement of the immediate vanishing of the one into the other: becoming, a movement in which both are distinguished, but by a difference which has equally immediately resolved itself.

Hegel, in this quotation, pulls a rabbit out of a hat — which is a new concept he calls becoming — while, at the same time, trashes the conventions of formal logic, by saying that being and nothing are both ‘the same’ and ‘they are not the same’, and, just to add to the dizzying effect, also introduces the idea of dynamic change, where the concepts supposedly move and vanish into their opposites.

Many philosophers, at this point, accuse Hegel of not grasping the basic rules of formal logic.

Yes versus no

So let’s add the distinction between pure being and pure nothing to our picture. I’ll use plus and minus to indicate that pure being connotes existence whereas pure nothing connotes non-existence.


Obviously, we have to use a ready-made language to describe anything. Hegel uses German and Latin. But he warns that languages come with presuppositions and semantic baggage, which may not want to apply to the phenomena, and therefore can mislead.

Obviously, Hegel’s Logic has been translated into many languages. So we don’t have to use German and Latin. In fact, I’ll translate Hegel’s statements into a simple mathematical model, which I think makes the structure of his argument clearer.

Hegel talks of change, of vanishing and movement. Change implies a sequence of states. So we need a way to denote individual states in a sequence. I’ll use the symbol ‘t’ for this purpose.

This choice suggests time. But at this stage, try to think of it merely as way of referring to states in an ordered sequence.

So, if pure being ‘vanishes into its opposite’ then it must change in some way. We can write this as:


where dx/dt denotes how being changes from one state to the next.

But how does it change? We know that it must change in a way that reflects its difference from nothing. And being, unlike nothing, affirms existence. So let’s say being changes in a positive direction:


But what does pure being change with respect to? What is the cause of its change?

The answer here is clear: we have thought contemplating itself. In other words, being changes positively with respect to itself:


Now let’s perform the same exercise, but for pure nothing. Again, its change must reflect its negativity. And, again, pure nothing changes with respect to itself. So we can write:


So we now have two, quite simple mathematical statements that describe two different ways that thought can contemplate itself:


Coming-to-be and ceasing-to-be

Hegel claims that being and nothing are different and yet somehow ‘vanish’ into each other and therefore are ‘the same’. Let’s tackle this issue by solving these equations, and seeing how they change.

I’ll skip the math. The result is:


which is exponential growth. And so can plot the states of pure being indexed by t.


And so we find that being explodes towards infinity. It never gets to infinity in finite time. It exhibits what Hegel calls a coming-to-be.

So it’s definitely not vanishing, it’s getting bigger and bigger!

But Hegel doesn’t really say that being vanishes. He says it vanishes into its opposite. But that doesn’t seem to be happening either.

Let’s now solve the equation for pure nothing:


and this is exponential decay. So it implodes towards zero. Again, it never gets to zero in finite time.


Pure nothing is vanishing: it’s getting ‘smaller’. It exhibits what Hegel calls a ceasing-to-be.

So we’ve got being and nothing moving, in their distinct ways. But we haven’t seen them vanishing into each other just yet.

The logical and causal aspects of vanishing

At this point, I want to emphasise that Hegel’s vanishing has two aspects, one logical aspect, and another causal.

From a logical point of view, the vanishing of being into its opposite, nothing, and vice versa, is similar to the famous liar paradox.

The paradox arises when we consider the statement: `This sentence is false’.

Obviously, if the sentence is true, then it must be false. But if it is false, then the sentence is in fact true. And so we go, around and around, repeatedly flipping back and forth between thinking the sentence is true or false.

In a similar way, Hegel’s pure being implies pure nothing, but pure nothing implies pure being, and so on, repeatedly. Both viewpoints are equally applicable.

So Hegel’s ‘vanishing into its opposite’ has this logical aspect of implication (between the concepts).

But, the vanishing isn’t only logical. Being and nothing themselves move into their opposites. He talks about them as actual processes, which have their own intrinsic dynamic.

So Hegel’s ‘vanishing into its opposite’ also has this causal aspect.

And, if we look a little closer, we can identify both the logical and causal aspects of Hegel’s vanishing in the equations I’ve written on the whiteboard.

The logical aspect of vanishing

First, notice that the trajectories of pure being and pure nothing, and notice they are mirror images of each other.







In fact, we can define a function that maps between these trajectories. And, in consequence, the trajectories are mathematically isomorphic to each other.

So we can repeatedly flip back-and-forth between being and nothing by applying the isomorphic map. Pure being and pure nothing are opposites, but the coming-to-be of pure being is, from a higher viewpoint, identical to the ceasing-to-be of pure nothing, and vice versa. In this, logical sense, they vanish into their opposites.

So, although being and nothing are different, in another sense they are the identical.

The causal aspect of vanishing

Now let’s consider the causal aspect of vanishing.

Being explodes towards infinity. Any natural or mechanical systems with this internal dynamic would eventually fall apart. They could only exist for a finite time.

Similarly, pure nothing implodes towards zero. And again, any real systems would quickly cease and become inert.

So being and nothing, as purely self-referential systems, are unstable. They cannot permanently exist in their pure states of self-reflection.

In both cases, being and nothing themselves vanish, either by exploding or imploding.

But, although suggestive, this isn’t vanishing into an opposite. But we can make sense of this too.

Let’s consider being in the state epsilon close to zero, where epsilon is a number that we can make as small as we wish.  This state of being is close the asymptote of pure nothing, which is zero.

Now being in the state close to pure nothing rapidly vanishes towards the asymptote of pure being, which is infinity.

In other words, when we try to contemplate pure nothing there’s always an irreducible element of existence, which when noticed, takes over, and flowers into pure being.

And we can look at this the other way. Consider nothing in a state close to infinity (think of the number 1/epsilon, where, again, epsilon is as small as we wish). Nothing in the state close to pure being rapidly vanishes towards the asymptote of pure nothing, which is zero.

In other words, when we contemplate pure being, there’s an irreducible element of non-existence or nothing that arises, which when noticed, takes over and decays into pure nothing.

So, in this causal sense, being and nothing do indeed vanish into their opposites.

So Hegel’s talk of identity and difference, and vanishing into opposites, makes perfect sense if we interpret Hegel as essentially talking about isomorphic positive and negative feedback loops.

At the beginning of Logic we shed all our assumptions, including our own ego, and simply contemplate thought itself. At this point we enter a self-referential feedback loop. This mental state has paradoxical properties. We seem to be contemplating being, but this is identical to something different, which is nothing. Both points-of-view make sense.

Yet both points-of-view are unstable. The pure states they strive towards can never be reached.

So Hegel’s beginning appears to be logically inconsistent and also causally unstable. This beginning doesn’t make sense and cannot exist. And Hegel acknowledges: the beginning is indeed incomprehensible.

So, should we throw up our hands at this point? Is it a dead end?

Well, this is where Hegel gets very interesting.

The sublation of being and nothing

Up to now, we’ve been thinking of being and nothing as separate things. We have two separate models of ‘thought contemplating itself’.

So far, we’ve noticed how they relate to each other. But Hegel says that we need to get out of the way, and notice how being and nothing themselves relate to each other. They don’t need our help to do it.

The beginning is a paradox only because we’ve been thinking about parts, not wholes. We thought we were observing two independent things. But really they are aspects of just one thing.

Hegel says being and nothing constitute an irreducible whole, which he calls becoming.

Let me quote from Hegel at this point.

Becoming is the unseparatedness of being and nothing, not the unity that abstracts from being and nothing; as the unity of being and nothing it is rather this determinate unity, or one in which being and nothing equally are. However, inasmuch as being and nothing are each unseparated from its other, each is not. In this unity, therefore, they are, but as vanishing, only as sublated. They sink from their initially represented self-subsistence into moments which are still distinguished but at the same time sublated.

So Hegel asks us to notice a new phenomenon, which he calls ‘sublation’. In this sublation, being and nothing are joined in a unity, where they still vanish into their opposites, but in a new way.

The sublation gets rid of ‘their initially represented self-subsistence’. So rather than pure being and pure nothing relating only to themselves, they now relate to each other.

We can model Hegel’s idea of sublation by literally joining the feedback systems together. The output of being becomes the input of nothing, and the output of nothing becomes the input of being.


In this unity, being no longer affirms its own existence, but now affirms nothing or non-existence. So being now changes positively with respect to nothing.

And, on the other side, nothing no longer negates its own existence, but now negates being (and therefore changes negatively with respect to being).

We’ve now got a new system of coupled differential equations. And I’ll refer to these equations as ‘Hegel’s contradiction’.

Hegel states that becoming contains being and nothing as reciprocally referring to each other, but that these relations are ‘of unequal value’. And, in our interpretation, we can immediately see that, in general, dx/dt will not equal dy/dt.

Hegel also says that being ‘passes over into nothing’ and nothing ‘passes over into being’. And, in this coupled system, we have a ‘substance’ that actually flows from being into nothing, and the ‘substance’ leaves nothing and enters into being.

The sublation preserves the positive and negative aspects of being and nothing. So they are still different from each other.

Hegel says that being and nothing have ‘different directions’ and are ‘so different they interpenetrate and paralyse each other’.

But in what sense do they paralyse each other?


Hegel now introduces the idea of an order and chaos in another paragraph that is a bit mysterious:

The equilibrium in which coming-to-be and ceasing-to-be are poised is in the first place becoming itself. But this becoming equally collects itself in quiescent unity. Being and nothing are in it only as vanishing; becoming itself, however, is only by virtue of their being distinguished. Their vanishing is therefore the vanishing of becoming, or the vanishing of the vanishing itself. Becoming is a ceaseless unrest that collapses into a quiescent result.

Here we have another typically Hegelian claim: becoming is both a ‘ceaseless unrest’ and a ‘quiescent result’.

So the ‘vanishing’ that previously implied that being and nothing would cease to exist, now, in this sublated state, ‘vanishes the vanishing itself’ such that we now have ceaseless unrest that paradoxically collapses into a stable result (that presumably doesn’t vanish by either exploding or imploding).

Is it possible to make any sense of this? We can, by solving Hegel’s contradiction. I’m going to skip lots of steps, and simply state the result.

In this mathematical interpretation of becoming, being changes according to cos(t) and nothing changes according to -sin(t). The exploding and imploding has disappeared, and we’ve got some new behaviour.


Let’s plot this behaviour and take a look.

Becoming as ceaseless unrest

becomingPlotWe see that being and nothing oscillate between finite limits, forever.

They oscillate in exactly the same way, but are permanently out of phase.

When being achieves its maximum then nothing is at its minimum of 0, and vice versa. In fact, we can show that Hegel’s equations satisfy a simple conservation law.

So becoming is a process where coming-to-be affirms ceasing-to-be, and ceasing-to-be negates coming-to-be – forever. And this continual dance of co-operation or conflict never settles down.

The unity of being and nothing is unstable because states never settle into steady values. The opposing concepts pull in different directions and any equilibrium is immediately undermined.

So Hegel’s contradiction does generate ceaseless unrest. But in typically Hegelian fashion, becoming is also a ‘quiescent result’. Can we make sense of this too?

Becoming as quiescent result

Every moment of becoming has two internal states, which is x(t) and y(t), that exist together. Let’s consider every possible pair of values of and plot them.

unityThis plot shows the state-space of the dynamic system defined by the equations. It shows all the possible configurations that becoming can be in.

And we see that it traces a perfect circle in state-space. That circle is a direct result of the fact that the contradiction satisfies a conservation law.

So although becoming is ceaseless unrest, that unrest is always bounded.

Pure being, which merely self-related, explodes, and pure nothing, implodes; in this sense, neither can exist. In contrast, their sublated unity is a stable dynamic system that neither explodes or implodes, and therefore it’s a ‘quiescent’ or stable result that reproduces itself indefinitely.

Existence: its necessary metaphysical structure

So let’s summarise what I think Hegel is saying, before coming to a conclusion.

Our most abstract possible concepts seem paradoxical. Pure being and pure nothing are the same, and yet different. They imply each other, but they also contradict each other. And they necessarily imply each other.

Becoming is the name Hegel gives to this unity.

The sublation of being and nothing preserves their ‘vanishing’, but instead completely vanishing the “vanishing is vanished”, and they now ‘interpenetrate each other’ and mutually ‘vanish’ into each other by exchanging their substance in an oscillatory but conservative manner. Becoming is therefore a ceaseless unrest that nonetheless remains stable over time.

So the beginning is a dynamic and contradictory unity.

And this beginning, according to Hegel, reveals properties that must be shared by everything. So this fundamental structure of becoming must be present in anything that exists at all.

What’s very interesting about Hegel’s metaphysical argument is that it implies that negativity, or nothing or non-existence is not the absence of being but a necessary and irreducible kind of being. So Hegel is a substance monist, but his substance has two fundamentally different aspects.

Harmonic oscillation all the way up, and all the way down

So Hegel makes very strong claims about the necessary structure of everything that exists. So is there any evidence for it? Or is Hegel’s Logic merely some nice metaphysical poetry?

I skipped all the steps when solving Hegel’s equations. But there’s an intermediate step in the solution where we represent the contradiction as two second-order differential equations, which is an equivalent way of stating the dynamics of the contradiction:


So in the sublated state, being and nothing in fact still relate to themselves, just in a different way.

Those with a physics background will recognise these equations describe simple harmonic oscillation. So it’s no coincidence that becoming exhibits oscillatory waves.

You’ve heard of simple harmonic oscillators because they are the bread-and-butter of physics courses. And that’s because they’re ubiquitous in nature. They really are everywhere, both in the microcosm, where they appear in quantum mechanics, and in the macrocosm, where they appear in general relativity.

For example, quantum field theory, the currently dominant theory of fundamental particles, is essentially simple harmonic motion taken to increasing levels of abstraction.

So it’s completely uncontroversial to state that simple harmonic motion is a fundamental structure that appears, again and again, at all levels of physical reality.

Obviously, physicists have observed harmonic oscillation, and they’ve developed a formal theory to describe it. They don’t need any help from Hegel to do this.

But physicists tend not to ask, and perhaps couldn’t answer, why wave motion is everywhere in nature.

Hegel’s metaphysics, in contrast, gives a candidate explanation of this empirical phenomenon: According to Hegel, everything that exists is necessarily a unity of being and nothing and therefore – according to this interpretation of his work – must exhibit harmonic motion.

So it’s very remarkable that Hegel’s mystical starting point, which is purely conceptual and abstract – and makes no reference to physical reality or empirical knowledge whatsoever – nonetheless implies a structure of ‘becoming’ that is equivalent to the fundamental structure found everywhere in physical science.

Next time (and conclusion)

So let me wrap up.

Next time, I want to examine whether Hegel’s Logic can tell us something new about a domain that seems particularly static and impervious to change, which is the realm of natural numbers. Surprisingly, Hegel’s contradiction appears in the study of prime numbers, although number theorists don’t think of their own work in this way.

But for now, that’s it.

Part 2 of this post: Hegelian contradiction and the prime numbers (part 2).



  1. […] specifically points to the two- (three-?) part essay “Hegelian contradiction and the prime numbers”, which I can’t vouch for yet. But the intro post and my light skimmings look promising. With a […]


  2. Thanks for writing this, very interesting. On the mathematical analogy, I think it could be meaningful to represent the structure of everything with a modular arithmetic with modulo 1. In this arithmetic, 1 is being, 0 is nothing. Of course 0+0=0, but also 1+1=1, since being is all that exist. But also 1=0, since being can be seen as nothing, as argued (similarly to how the absolute value of say an electric potential has no real meaning, the value is arbitrary. What matters are differences, but there cannot be differences, if being is all that exist).
    Remarkably, one can start from this to construct much more (“parts inside the being”), for instance 1+1=2 and we can associate 2 to some physical property (say negative electric charge, assuming that electric charge is a fundamental property of a definite theory of quantum gravity, or whatever describe our universe) . Of course 2=0 (and so on), implying that fundamentally nothing really exist!


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