Mathematical Introduction to Hegel

It is sometimes thought that Hegel’s system of science, and its foundational text the Science of Logic, is an interesting yet ultimately unprovable exercise in metaphysics, which lacks the precision and necessity of empirical science and formal logic, and therefore that the order of the categories in the Logic is arbitrary. This assessment is not true. The structure that Hegel is describing in his Logic is a precise and necessary structure, indeed it is the necessary structure, the ground framework of reality and the basic pattern of all thinking and being. There are perhaps many paths that can be taken to enter into Hegel’s thinking and become convinced of its truth, but in this essay which is intended for beginners I have opted to take the route that Plato took twenty-four hundred years ago in his lecture ‘On the Good’ to the Athenian people, and begin with a detour via mathematics.

Everyone knows since grade-school that there are six basic arithmetic operations: addition and subtraction, multiplication and division, and power and root. Now addition and subtraction can be understood as a kind of multiplication, in which only one number is taken as a potentiator. For example, in the calculation 5 + 4, we obtain the result 9 by counting the numbers contained in the latter addend (1, 2, 3, 4), and for each of these numbers, we augment the former addend (the number 5) by 1. So we start with 5, and then for each of the units contained in 4, we augment 5 by 1, thereby retrieving 9. So addition and subtraction can be understood as multiplication operations in which only one factor is operational. The meaning of multiplication and division is thereby revealed to be this: the mutual potentiation of two numbers. When we multiply 5 times 4, for each of the units contained in but not including the number 4 (1, 2, 3), we augment 5, not by 1 but by 5. The power or potency of one number is combined with the power or potency of another, so that they potentiate each other. Each number is made more powerful by the other. Now the meaning of the square (e.g. 3²) is revealed: the number potentiates, not another number but itself. The stronger the number is, the more it potentiates itself, the more it grows. So we have, in addition and subtraction, a one-directional or one-sided potentiation; and then in multiplication and division we encountered a mutual or reciprocal potentiation; and finally in the square (or power generally) we arrived at self-potentiation. The square can also be analyzed back into its factors through the square root (because it has a kind of necessity which is absent in addition, multiplication, etc.).

What does this have to do with metaphysics? Here is where things get interesting. In the 9th century an Irish philosopher named John Scotus Eriugena wrote a book called Periphyseon or ‘The Division of Nature’. His work bears striking resemblance to Neoplatonic thought (especially Proclus), though he probably did not have access to those texts, and his knowledge of Neoplatonism was mediated by Christian mystics such as Dionysius the Areopagite. Anyway at the beginning of Periphyseon Eriugena says that nature as a whole can be divided or carved up into four divisions: things can be either, 1. created and not creating, 2. created and creating, 3. uncreated and creating, or 4. uncreated and uncreating. The fourth of these divisions is superfluous and can be eliminated. Now if creation is taken as a kind of potentiation, then we have the following triadic series: 1. one thing potentiates another thing, but this other thing is passive and not itself potentiating, 2. one thing potentiates another thing, which is itself potent, and so they are reciprocally or mutually potentiating, and 3. what is potent, but potent with respect to itself, and thus is self-creating, self-potentiating. Examples of the first would be atoms, rocks, mountains, water, space and time, nebulae, etc. Things that are basically just passive material but have been produced as a result of some other process. Examples of the second category would be things that are roughly living, in the broad sense of just active and potent: the sun, weather, plants, animals, nuclear reactions, chemical reactions. These are examples of mutual potentiation. And in the third category we have those natural things which are self-potentiating, which are active and alive, but not with respect to another but to themselves: human beings, states, social groups, etc.

I have put these interesting observations about mathematics and the division of nature into juxtaposition in order to draw out the analogy between them. The structure of nature as a whole is a reflection, that is an image or analogy, of the structure of mathematics, which is quantitative logic. Logic is the structure of the natural world, and the natural world has logic as its skeleton. Thus it is also observed that Eriugena’s division reflects the academic division of the natural sciences into physics, chemistry, and biology. Physics studies the motion of bodies in space and time: outer space, atoms, and other beings that are in external and mechanical combinations with each other. Chemistry studies the inner affinity of substances for each other, their mutual potentiation. And biology studies those substances which have attained a degree of independence and self-subsistence. And at a higher level, Eriugena’s division corresponds to the Neoplatonic triad of being, living, and thinking. Outer space consists of that realm of things which simply are, but are not alive, and thus are external to each other, caused but not actively causative; Earth is that realm in which there is a reciprocal interaction of cause and caused, and living things in general are reproducing themselves through another; and third is the human world: the civilizations that human beings have created for themselves are explicitly self-created worlds, worlds which are self-contained and self-reproducing. Though they are within nature, they have achieved the status of something unconditioned, i.e. uncreated.

The analogy between mathematics and the structure of nature is highly edifying, because it demonstrates that the sciences have a much higher degree of systematic interconnectivity than is usually assumed. In fact we name the study of this systematic interconnectivity of science, philosophical science. And Hegel is that philosopher who carried this science further than anyone else. The Science of Logic describes the content of formal logic, mathematics, systems theory, metaphysics, syllogistic logic, and theology in terms of the basic triadic structure which was seen above in the concept of arithmetic. The whole of the formal sciences (logic in the broadest sense) may be divided into 1. those that bring content in general into external connection and combination, without considering the operation itself as internal to the content, and 2. those operations by which the content brings itself into relation to other content intrinsically, so that the inner affinity of the content is brought out, and 3. that formal operation which is operative on itself, so that the form is the content and the content is the form, and there is a development of this thinking from out of itself, self-developing thinking. Hegel calls these: 1. being, 2. essence, and 3. concept respectively. Being is that phase of reality according to which things simply are, without any internal potency. So any kind of activity or operation on being must come to it from outside, as we described above with addition and subtraction (which was external or one-sided potentiation). Examples of formal logical operations appropriate to mere being would be conjunction and disjunction, unions of sets, and counting. Essence is that category of cognition according to which things are intrinsically potentiating, but potentiating other things. The basic metaphysical categories here are cause and effect. In formal logic, these are conditionals and biconditionals. The connection between atoms or nebulae in space falls in the sphere of being, because they are just externally conjoined. The relation between microorganisms, seeds and trees, etc. falls in the sphere of essence, because they are mutually potentiating. The third category of reality is called concept, which is that potency which is actual as potency because it is self-potentiating. It actualizes into itself, because its essential nature is self-reversion (Proclus’s ‘epistrophe’, and Fichte’s ‘self-positing ego’). Its existence just is this reversion into itself. And this is the human mind. The concept is basically virtual reality: human thought, cognition as such. The virtual which is real as virtual. This is a self-reversion which is active in respect to itself. The formal logical structures which correspond to the concept are modality: possibility and necessity. Because what is necessary is self-causing, related only to itself and not dependent on another (as we saw with square and root above). The concept is a causal relationship bent back into a circle.

At the highest level, all of science divides into a triadic structure: 1. formal science, which deals with pure thinking, i.e. logic, reason, the Logos; 2. natural science, which is the thinking of external objects and created beings, and 3. human science, which is the thinking of thinking things, i.e. of the Logos or reason as returned back to itself within nature, the creature which has the Logos or reason. Now the whole of science can be imaged (reflected) as a tree. 1. Logic (formal science) is like a seed, the universe in itself, as a hard and inflexible singularity. 2. Nature is like the tree which emerges from the seed, for which the seed was only a formal blueprint or ground plan, which develops out of itself into externality, where there is a degree of contingency and fungibility. 3. Spirit (human science) is like the fruit, which is the seed returned back to itself in the externality of the tree (e.g. man is the rational animal).

A final comment may be made before we bring this brief beginner’s introduction to a close. We started by considering the series of ordinary arithmetic operations. Three things need to be noted about this series: firstly that it is a triadic development which goes from simple to complex, or from abstract to concrete. Secondly, that the operations come in pairs, where one is the opposite of the other. And thirdly that the operations can be deduced from each other: multiplication can be deduced as an extension of the principle which was implicit in addition and subtraction, and the power and root can be deduced as an extension of what was implicit in multiplication and division. Putting this all together we may conclude with this remark: the philosophical comprehension of the operation which is truly involved in simple mathematics, and the philosophical deduction of the content of pure arithmetic in and of itself, consists in a dialectical derivation which proceeds through the interconnection of opposites, and proceeds from what is most abstract to what is most concrete, and by progression from external combinability or one-directional potentiation to reciprocal potentiation and achieves completion in self-potentiation. That is all.