Summary: The Phenomenology of Spirit (page 10)

Hegel proceeds in two stages.

First, he shows that mathematics, like philosophy, is subject to dialectic. Far from merely stating a bare result, the geometer must present a demonstration, linking the result to the reasons that lead to it. Thus, there is no definite answer ¹ in mathematics in the sense of an immediate solution given without any dialectical process.

In mathematics, the demonstration is an essential moment of the truth—even if we are not always aware of this: Even in mathematical knowledge, the essential character of demonstration is still far from having the meaning and nature of being a moment of the result itself; on the contrary, in that result it appears as something that has passed away and vanished.

With demonstration, dialectic appears—both as the internal movement linking a result to the reasons behind it, and as the movement uniting the subject (the mathematician) and the object (the theorem) in the process of knowing.

Indeed, it is through the demonstration that the geometer is convinced of the result and gives it inner assent:

We would not regard as a geometer someone who knew Euclid's theorems by heart—that is, externally—without knowing their demonstrations, without having inwardly assimilated them.

Here we find again the dialectical relationships between subject and object, between consciousness and its object. Thus, even bare truths of the kind cited as examples are not exempt from the movement of self-consciousness—the dialectical movement described earlier.

The same holds for historical truths—for example, Caesar's birthdate: It is only in knowing them together with their reasons that they are held to have true value, even though […] only the bare result is supposed to be at issue.

In conclusion: the 'bare result', the 'definite answer'—just like immediate knowledge or any other supposed escape from dialectic—is a myth.

In the second stage, Hegel argues that the dialectic at work in mathematics is essentially impoverished. As such, this discipline can claim only a lower form of truth—one of the earliest stages in the pursuit of truth, not its final one, as many would believe.

Mathematics prides itself on the self-evidence of its results. But for Hegel, this very self-evidence is simply a sign of the poverty of its object:

The self-evidence of this defective kind of knowledge of which mathematics is so proud—and which it also displays in order to flaunt itself before philosophy—rests solely on the poverty of its aim and on the defective nature of its material, and thus belongs to a kind that philosophy can only disdain.

The proper object of mathematics is number: measuring, counting, performing operations, reducing figures to magnitudes. But what can we say about this from a dialectical point of view?

The goal it pursues, or its concept, is magnitude—precisely an inessential relation, one without concept. This is why the movement of knowledge takes place at the surface, never touches the thing itself, never touches essence or concept, and for this reason, is not a grasping-through-concepts.

Moreover, the proper object of geometry is the study of spatial figures. One might be tempted to argue that space is something concrete—but a brief reflection is enough to realise that it is, in fact, an abstract, ineffective element. And let us recall: for Hegel, truth resides in the concrete, the actual.

The material with respect to which mathematics displays its delightful treasure of truths is space and the One. Space is the existence in which the concept inscribes its differences as if in a void and lifeless element, where they are equally motionless and without life. What is actual is not, as mathematics assumes, spatiality.

As a result, neither concrete sensible intuition nor philosophy can be satisfied with this ineffectiveness that characterises the objects of mathematics.

This abstraction even undermines the dialectical nature of mathematics that Hegel had acknowledged earlier—a qualification he now introduces:

In this kind of ineffective element, there is, accordingly, only ineffective truth—that is to say, fixed, dead propositions; one may stop at each of them; the next begins anew, without the former having carried itself forward into the latter, and without there emerging from the nature of the thing itself any necessary connection. [For this reason,] knowledge runs along the line of identity. For what is dead, not moving of its own accord, does not arrive at essential differences, does not reach opposition or essential non-identity, nor therefore the passage of one opposite into the other—the qualitative, immanent movement, autonomous movement.

What Hegel seems to be affirming here is that mathematics contains only a lower or impoverished form of dialectic, reflecting the poverty of its object—number—and of the type of truth to which it gives access.

By virtue of their abstract nature, mathematics represents only one of the earliest stages of knowledge—not its ultimate form. In truth, this discipline fails to rise to the level of concept, which is the proper object of philosophy, and which, in the final analysis, determines even mathematical relations themselves:

For it is only magnitude—the inessential difference—that mathematics considers. It abstracts from the fact that it is the concept which divides space into its dimensions, and which determines the connections among and within them.

Whereas: Philosophy […] does not examine inessential determinations; it examines determination inasmuch as it is essential. Its element and its content are not the abstract or the ineffective, but the actual—that which posits itself and lives in and through itself—existence in its concept.

We now better understand why Hegel had stated earlier that the nature of this kind of truth is different from that of philosophical truths. Mathematical truth cannot pertain to the concept—it is philosophy alone that holds this privilege. As such, the two domains do not contradict one another, and mathematics can in no way constitute an objection to philosophy—nor serve as its model.

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