Hegel and Non-Euclidean Geometry

Hegel and Non-Euclidian Geometry: Stereographic projection in 3D

For Hegel, space is self-externality; that is to say, it is the embodiment of the idea of externality in what we call reality. However much I try to analyze space, to divide any distance or spatial structure, I will get another spatial distance or form. I will never be able to reach a level that is the last spatial one beyond which I cannot find any spatiality.

Hegel is no longer interested in whether space is objective or subjective because he positions himself on another level, that of identity between the human mind and God’s mind, i. e., the fundamental structures of reality. Since there is a fundamental categorical identity between these two poles, neither subjectivity nor objectivity of space have any relevance any longer.

What would it mean to say that space is objective? It would mean that my knowledge of it derives from perceiving it as it is, independent of my knowledge. In this way, we set our knowledge of space as subsequent to or dependent on the perception of space, which then should be the first element of our knowledge.

But in that case, the principle of identity, on the side of which Hegel posits himself, would no longer be the ultimate benchmark. It is necessarily something derived from the perception of space and from thinking that our perception corresponds to reality.

In other words, the accent would not lie in what we understand when we speak about space (on the concept of space) but in the structure of our perception, that is to say, of our sensibility. The question would then be whether this perceptive structure corresponds to reality or not, a question which, in itself or at this level, is undecidable.

This is why Hegel considers the controversy between the subjective and the objective character of space the ‘most well-worn of all metaphysical questions’ (Hegel 2007, p. 225): something which, because it has no answer, has prompted over and over again the same disputes. Entering this controversy would downgrade the approach from the level of pure understanding to a matter of human sensibility. The same is true for the opposite approach.

Obviously, the principle of identity between mind and reality lies at a more profound level than the question concerning the subjective or objective nature of space, the settlement of which derives from whether one affirms or denies that principle of identity.

Something strange happens with this approach of Hegel. The understanding of space as a pure externality does not entail anything about the structure of space. On the contrary, insofar as it simply states that the spatial points must be external to each other, it leaves open a more concrete determination of space which must derive from that ultimate feature.

Although acknowledged by Hegel, the tridimensionality of space is not (and cannot be) a logical result of its self-externality. It is only a ‘fact’ that we notice during our interaction with reality.

Tridimensionality is indeed based on the ‘determinations of the Notion,’ but this determination is similar to the historical determinations of human history. Also, here, necessity must be thought of as the necessity of history is thought of: not as a series of stages that emerge logically from each other, but as a series whose necessity must be thought of as being based on its end, as a series interpreted backward, starting from the present.

For Hegel, past necessity is not based on previous events but on what the ‘necessary’ event contributed to the present-day constitution of human reality. In this respect, from the event of Jesus’ appearance in history, one cannot derive the necessary conclusion that the Roman Empire would transform into a Christian Empire. Jesus is necessary only if you try to explain why the Roman Empire transformed into a Christian Empire. You can only understand the latter by assuming the presence of Jesus in its past history.

This is why Hegel’s understanding of space is entirely compatible with non-Euclidean geometry and with Einsteinian or even quantum physics and why Hegel and Non-Euclidean Geometry is a worthwhile topic. In non-Euclidean geometry, the fundamental understanding of space as a totality of potential points external to each other is maintained.

However, this space’s formal structure, as described by Euclidean geometry, is abandoned. Non-Euclidean geometries imagine different types of externality than those corresponding to our familiar intuitions; those types are different ways things can spatially relate to each other.

In Euclidean geometry, one determined these relations according to one’s natural experience of the surrounding world. In non-Euclidean geometry, one removes the conditions of this experience but not also the condition of self-externality.

Because the latter is only a pure concept, without any perceptual or sensible schema associated, it can receive an indefinite number of concretizations. Spatial self-externality can thus be tridimensional, four-dimensional, five-dimensional, and so forth, depending on how many dimensions I can mathematically build.

In this case, self-externality is a concept as general as rationality or living being. There is an indefinite number of forms of rationalities and perhaps of living beings.

Self-externality of space means that all its contents must be located beside each other and that a spatial content can never occupy the same space as another content if the latter continues to be there in that space. It can do this only if it removes it from that place. This feature doesn’t say anything about the fact that there are ultimate contents of space or about how spatial contents relate to each other.

Something can be beside something else in very many ways. For example, when Euclidean geometry postulated that one could draw a single parallel to a straight line through a point external to that line, it thought of externality as starting from the intuitive model of a two-dimensional plane.

However, if you think of that plane as being curved, things change, and the parallel line’s uniqueness is no longer true. Although these geometries are non-intuitive, unlike the Euclidean one, the spatial relationships present in them can be built through a mathematical toolbox.

And in the same way, in which numbers and equations can apply to an infinity of concrete cases (2+2=4 independently of the fact that the summed entities are apples, goats, planets, or phones), the mathematics of non-Euclidean geometries can create a variety of mathematical models of space.

The latter are not intuitive but logically and mathematically coherent.


Hegel’s Philosophy of Nature, edited and translated by M. J. Petry, volume I, Routledge, London and New York, 2007.

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