In his five dialogues on Cause, Principle and Unity, the Renaissance philosopher Giordano Bruno cites the famous idea that a circle of infinite size is a straight line. He uses this aphorism, attributed to Nicolas of Cusa, as a justification of his own theories of the universe consisting of infinite space.
At first hearing, this is quite an impressive idea, but it has a problem. The problem is that it is nonsense, and this nonsense goes to the heart of Bruno’s flawed thinking about physics and metaphysics (which he seems to have no time for).
Why do I say so? The real issue here is the definition of infinity. At many points in his work Bruno uses the term synonymously with limitlessness, absolute unity, and the unconditioned. This is the crux of the problem, for when we are talking of space, we are necessarily discussing limit, multiplicity, and conditionality. Except by a kind of linguistic ambiguity, there can be no truly ‘infinite’ space, because the nature of space as such is conditioned. The inescapable conditions of space are the six cardinal directions, which cannot be reduced or equivocated. They give space its definition, and therefore its finitude. This is not to say that, hypothetically, space could not extend indefinitely, but it would always be defined by those limits of directionality, or, in a word: form.
A circle cannot cease to be a circle except by a change in its form, and this has nothing to do with quantity. It is defined by its quality, which is not conditioned by matter. Bruno’s problem is that he subordinates everything to his conception of space which he regards as absolutely simple. He conflates the idea of infinity and limitlessness with the idea of matter as the substrate of all being. Since matter, naturally, is conditioned by space (we would say), this naturally entails for Bruno an infinite extension of space in a field or substrate which encompasses all things.
While he does have recourse to the Pythagorean idea that all of reality is number – which he favours over a geometrical symbolism – he cannot account for the emergence of the many out of the One. For Pythagoras, the point in geometry symbolised the unmanifested, and was therefore analogous to God, or the spiritual principle which holds together all things. All extension is generated out of the point, which occupies no space in and of itself.
In the traditional symbolism of the cross, we find the four cardinal directions emanating out of a central point, which is sometimes taken to represent the “quintessence” or the fifth element, aether. Bruno takes this symbolism to suggest that space is an “infinite” field, consisting of an indefinite number of points, and it is out of this substrate (which he regards as plain matter, without defining it as aether per se) that everything else is generated, as out of a ‘womb.’ The problem is, in classifying form as mere ‘accident’; as a superstructure which emerges out of simple matter; he cannot account for qualitative differentiation. That is why he cannot distinguish between an ‘infinite circle’ (a nonsensical idea) and an ‘infinite straight line’ (likewise incoherent). Thus the foundation of Bruno’s physics cannot serve for his purposes.
His rejection of metaphysics is what cripples his thought, for at base he is really dealing with a kind of nothingness. Since the point occupies no space, and there is nothing beyond space, the universe may be said to emerge in an absolute void. Yet Bruno refuses to recognise this void which undergirds his hypothetical substrate of raw matter in an absolute, undifferentiated form. He wants to suggest that both act and potency are equally present as an undifferentiated unity in this substrate. Even when he supports the idea of an invisible world, he still speaks of it in terms of materiality – for, as I have said, he subordinates all metaphysics to his conception of space.
It is more reasonable to take the Christian view that God, who infinitely transcends the created order, brings into being a universe out of nothing; in which case the void underlying all matter points towards the dependence of creation on God. It makes perfect sense that the divine mind could create the conditions of our universe – not least of all the conditions which define space. At the same time, we may recognise that God in His own essence is unknowable, and cannot adequately be defined by us as complex or (absolutely) simple. Recognising the limits of our own created intelligence does no damage to metaphysics, though it does force us to admit our creaturely status.
Since I have mentioned Descartes in the title, I ought briefly to trace the connection between his thought and that of Bruno. Descartes, while he is taken as a great rationalist, was motivated in no small degree by an ideological desire to purge science of any savour of magic or alchemy. He was, like Bruno, prejudiced against metaphysics, too, as a relic of the Medieval, scholastic past. While he has no sympathy for the Hermetic side to Bruno’s thought, he utilises the same idea of space as a unified field. In Descartes’ mechanistic physics, therefore, all bodies are determined by extension alone, because he fails to recognise the qualitative distinctions which define space as such. In other words, he doesn’t acknowledge that space is already something which participates in form; a metaphysical principle.
It was René Guénon’s writings that first alerted me to this problem in Descartes’ physics, which Guénon regarded as a triumph of quantitative thinking in the sciences – and thus an abandonment of metaphysics. As Guénon asks, posing the problem of the purely quantitative view of science:
[W]ould any geometrician, however deeply imbued with modern conceptions, dare to maintain for example that a triangle and a square of equal area are one and the same thing? He would only say that they are ‘equivalent’, but he would clearly be leaving out as being understood the words ‘in respect of size’, and he would have to recognize that in another respect, namely that of shape, there is something that differentiates them; and the reason for which equivalence in size does not carry with it similitude of shape is that there is something in shape that precludes its being reduced to quantity.1The Reign of Quantity and the Signs of the Times
This brings us back “full circle” (if you will excuse the pun) to our starting point, with the nonsensical idea of an infinite circle being ‘one and the same thing’ as a straight line. We can see that, far from originating with Descartes, the reduction of all quality to merely quantitative thinking is already evident in Giordano Bruno’s philosophy. No doubt it preceded him, of course, but it may be enlightening to trace its influence back by stages in the developments of Western philosophy and scientific thought. I hope that this brief article might serve to open further such lines of enquiry.
1 René Guénon, The Reign of Quantity and the Signs of the Times, tr. Lord Northbourne (Sophia Perennis: 2001), p.34