Summary: The Phenomenology of Spirit (page 9)

First of all, one must not oppose truth and falsity as if they were two absolutely distinct realities, without any relation. We must remember that Hegel proposes a new ontology – a dialectical one – no longer grounded in the principle of identity or contradiction.

On the contrary, he argues that identity includes difference – the negative – and that a thing must become its opposite before returning to itself and thereby finding its identity: Only that identity which reconstitutes itself, or reflection into otherness within itself – and not an original or immediate unity – is the true. ¹ And between two opposing terms, there is always a synthesis which establishes their relation, and finds their identity in a new term that both transcends and includes them.

This dialectical movement also applies to truth and falsity – and to their opposition:

Truth and falsity are among those determinate notions which, in the absence of movement, are taken to be separate essences, each always on the opposite side from the other, with no community between them and each holding fast to its own position.

Against this, we must understand that falsity corresponds to a crucial moment – that of difference. No identity can be constituted without difference; this is why the false is an essential moment of the true:

To say that one knows something falsely means that knowledge is in non-identity with its substance. But this very non-identity is the act of differentiation as such – which is an essential moment. Certainly, out of this differentiation there comes their identity, and this resulting identity is the truth. But it is not the truth in the sense that non-identity has been cast aside, like the dross separated from pure metal, nor like a tool discarded after the vessel has been made: rather, non-identity, as the negative, as the self, is still immediately present in the truth itself.

Hegel sums it up as follows: One must not […] consider the truth as a lifeless positive resting on its own side.

He is, however, careful to add that it is no longer as false that the false is a moment of the truth. Once considered within their unity, truth and falsity no longer signify what they are outside of that unity. We must beware of the traps into which language can lead us: Just as the expression of the unity of subject and object, of the finite and the infinite, of being and thought, etc., has the drawback that subject and object, etc., are taken to mean what they are outside of their unity – and thus are not grasped in the sense of the unity the expression is meant to convey – so too it is not as the false that the false is a moment of the truth.

But given that the true contains within itself the false (properly understood – that is, in their dialectical relation), the mode of presenting truth that Hegel proposes in the Phenomenology of Spirit is appropriate.

What is set out here is the way in which Spirit gradually appears to itself. Now, ‘appearance’ is not to be understood as mere illusion, in the sense of deceptive semblance, but as an essential moment of truth:

Appearance is the coming-to-be or passing-away which itself neither comes to be nor passes away, but is in itself, and constitutes the actuality and the movement of the life of truth.

Once again, the project of a phenomenology – a science of appearance (of Spirit to itself) – is thereby legitimised.

After this general reflection on truth, Hegel turns to consider a specific kind of truth: mathematical truth.

Indeed, mathematical propositions seem to lie outside the dialectical process. What is 2 + 2? What is the square of the hypotenuse in a right-angled triangle equal to? It seems that such questions call for a definite answer – one that rests on a classical conception of identity and the principle of non-contradiction, escaping all dialectic.

‘2 + 2 = 4’, ‘the square of the hypotenuse is equal to the sum of the squares of the other two sides’ – the problem is solved, without requiring any dialectic whatsoever. The same goes for historical truths (e.g., the birth date of Caesar).

It is clear that Hegel must respond on this point: if mathematics – this sovereign discipline – escapes dialectic, does this not call the entire dialectical method into question? Would Hegel’s ontology, revolutionary in its novelty, not collapse in the face of such a confrontation?

Dialectic, and the new ontology it entails, seeks to reconfigure the entire field of knowledge. It must provide a new map of knowledge, define the place and limits of each science, and show in what ways these sciences are traversed by a dialectical process – or, if they are not, explain why.

This is a task of immense scope, full of promise but also fraught with difficulties that could seem insurmountable. Is it not absurd to attempt a critique of mathematics, given its astonishing successes – and to do so without being a professional mathematician?

But of course, Hegel does not aim to deny the truth of mathematics. Rather, he seeks to show that the nature of such a truth is different from that of philosophical truths.

And so the question arises: what kind of truth do mathematical propositions actually express?

¹ Our translation. The references for the quotations are available in the book Hegel: A Close Reading