Infinity, Finitude, and the Good

Plato and Aristotle both attribute great significance to the reality, goodness, and importance for reasoning of limits. Aristotle staunchly opposes assertions of really existing infinity, or of an infinite regress of reasons.

(Despite the opinions of some mathematicians, all that modern mathematics truly requires is “potential” infinity — the ability to construct something arbitrarily large or small through arbitrarily many definite acts of extending, dividing, or whatever of some definite thing. This is consistent, for instance, with the modern grounding of the infinities of calculus in analysis based on the concept of limits. On the frontiers of mathematical research, homotopy type theory and univalent foundations allow the most extravagant “classical” mathematics of higher infinities to be expressed in terms of definite constructions.)

Aristotle’s position on this is closely tied to his central concepts of ends and the good (see also Aristotle on Explanation).

“And since that for the sake of which something is is an end, and this sort of thing is what is not for the sake of anything else, but they are for the sake of it, then if there is any such last thing, there will not be an infinity, but if there is no such thing, there will be nothing for the sake of which it is. But those who make there be an infinite are unaware that they abolish the nature of the good. (Yet no one would make an effort to do anything if he were not going to come to a limit.) And there would not be intelligence among beings; for what has intelligence always acts for the sake of something, and this is a limit” (Metaphysics book small Alpha (II), ch. 2, Sachs tr., p. 31).

This is also central to his argument for the existence of a first cause.

The emphasis on a kind of finitude here should not be taken to imply any dogmatic attachment to particular formulations or representations, such as Hegel for instance objected to. In the following chapter, in part echoing the beginning of the Nicomachean Ethics, Aristotle notes, “Some people expect everything to be said with precision, while others are annoyed by precision, either because they can’t keep the connections straight or because of its hairsplitting pettiness. For precision does have something of this sort about it, so that, just as in business agreements, so also in reasoning it seems to people to be ungenerous. For this reason one must have been trained in how one ought to receive each kind of argument, since it is absurd to be searching at the same time for knowledge and for the direction to knowledge; and it is not possible to get either of the two easily” (ch. 3, pp. 32-33).

“Courses of lectures go along with one’s habits; for in the way that we are accustomed, in that way we think it fitting for something to be said, and what departs from this does not seem the same, but through lack of acquaintance seems too obscure and alien. For we are used to what is familiar. And what great strength the customary has, the laws show, in which mythical and childish things are of greater strength than knowing about them, because of custom” (p. 32).

This last remark clearly shows that Aristotle’s emphasis on the shareability and actual sharedness of values does not at all mean he assumes that what is held to be authoritative by one’s own community or society is always right. Unlike Socrates, who after his eloquent defense meekly accepted his city’s ignorant condemnation, Aristotle at one point fled Athens “lest the Athenians sin against philosophy twice”. This lesson is crucial to the understanding of Hegel as well.

Next in this series: Aporias

Otherness

I wanted to elaborate a bit on what I see “otherness” as doing in the part of Hegels’ text that formed the subject of the previous post. Cambridge University Press provided only a skimpy index, which scandalously includes no entry at all for this key term. I don’t specifically recall “otherness” being literally used in the main body of the Phenomenology, though it may well exist somewhere. What I find googling “Hegel otherness” seems entirely devoted to the relation of self-consciousness to other people. Quick review of top results failed to turn up a supporting quote from Hegel using the literal term “otherness” in that way, however. This leaves it unclear to me whether this more social usage of “otherness” is even literally Hegelian, or is rather a term interpolated by commentators.

Relations to other rational beings are essential to Hegelian self-consciousness, to the point where I have quipped that it might better be called other-consciousness. This social and ethical meaning of otherness is not irrelevant to the current context. However, I take Hegel’s use of “otherness” in the Phenomenology Preface to be primarily “logical” in his special sense, rather than social.

In the Preface, Hegel calls Anderssein (otherness; literally, “being-other”) the “element” and the “ether” in which knowing occurs. Hegel is using “knowing” in a very broad sense here, encompassing everything from the mere acquaintance of ordinary consciousness with objects, to the pinnacles of philosophy. He begins to develop otherness by way of implicit contrast with that other element of “familiarity” and “representation” that he mentions as an obstacle to the higher development of knowing.

He explicitly calls otherness the element of “science” (rational understanding) in knowing, while implying that familiarity and representation characterize a contrasting element of immediacy that he sees as an obstacle to “science”. For Hegel, “science” is first and foremost the “logic” that was to form the first part of the “system” the Phenomenology was to introduce, so it could equally be said that otherness here is the unfamiliar standpoint of Hegelian logic, for which the whole long detour of the Phenomenology is intended to gently and patiently prepare us.

Once again, I take a deflationary approach to his rhetoric about “science” and “system”. In general with Hegel, rather than starting with ordinary assumptions about what his terms mean, it is best to interpret them in light of what he does. Here otherness provides a first thematic anticipation of the general point of view Hegel wants to recommend, and in particular of what is at stake in Hegelian “logic”, “science”, and “system”.

As a first approximation then, we have otherness expressed as the “element of knowing” that the Science of Logic will later develop, initially expressed by way of a contrast with a point of view centered on immediacy, familiarity, and representation.

There seems to be a kind of analogy between this contrast and what I read as the Phenomenology‘s other big contrast between the standpoints of consciousness and self-consciousness. I think Hegel’s view is that neither of these latter is ever found entirely independent of the other in real life, but at the same time that the alienation inherent to the relation of ordinary consciousness to objects is eventually to be overcome by dwelling primarily in what he calls self-consciousness and spirit. The higher phases of self-consciousness and spirit will be characterized by an openness to otherness.

The contrast between the feeling of otherness and those of familiarity and immediacy gives us a first starting point that we can grasp even within the standpoint of the most naive ordinary consciousness. The second contrast between the standpoint of otherness and the standpoint of representation brings this into sharper focus.

In the Preface, Hegel only hints at his very strong reservations about the place of representation in early modern mainstream views of knowledge such as those of Descartes and Locke. But in the Consciousness chapter of the Phenomenology, the alienated relation of consciousness to objects broadly captures aspects of the views of Descartes and Locke, who were the two great representationalist promoters of “consciousness” in philosophy (literally in Locke, and its ancestor French conscience in Descartes; see Consciousness in Locke and Hegel).

We cannot communicate without representation, any more than we can exercise higher functions without consciousness. But Hegel’s implicit critique of representation in the Preface and his more developed critique of consciousness in the Consciousness chapter together constitute a vital thread of his argument. His repeated warnings against taking “fixed thoughts” at face value and against taking propositions in isolation are closely tied to this.

Otherness challenges both fixed representations of thoughts and an overly fixed notion of self. Self from the perspective of otherness is a contextual, relational and adverbial term, not an independently contentful noun with a reference fixed once and for all (see The Ambiguity of “Self”).

What I think he is suggesting is a strong conclusion that in explaining meaning, we ought as much as possible to subordinate the point of view associated with representation, consciousness, objects, immediacy, and familiarity, rather than treating all of these as foundational touchstones.

What we ought to subordinate them to is developed throughout the rest of the Phenomenology, but especially involves the actualization of self-consciousness, and of forms of spirit that are not merely what he calls substantial, but are self-conscious, and thus for Hegel depend essentially on relations of mutual recognition.

A fully developed self-consciousness will be “at home” in otherness.

Here in the Preface, I think he is suggesting an argument complementary to that of the Phenomenology‘s main thread. In the Preface, the accent seems to be on knowing as such, whereas I take the overall thrust of the main thread to be primarily ethical in intent. Here too, at least in a general sense the Preface is closer to the concerns of what Hegel calls “logical” inquiry. The critique of the classic early modern concept of representation falls in this area.

Foundational uses of representation are based on strong presuppositions about the identity of represented things (the “fixed thoughts” to which Hegel is objecting). Representationalist theories of meaning focus on the ways in which representations are supposed to unambiguously refer to objects, which basically reduces meaning to a kind of implicit pointing at things that are presumed to be unambiguously identifiable. But this is a huge presumption that Hegel wants to question.

Alternatively, the meaning of representations can be explained in terms of form, value, internal structure, and inter-relations, all of which I think for Hegel are potentially articulable complete in themselves “in the element of otherness”, without any pointing or presumption required. Otherness thus appears to stand for coherence over reference and difference over identity in the explanation of meaning. Again, that is not to suggest that reference is absent, just that it ought not to dominate or primarily drive our explanations.

Finally, Hegel would remind us that even pure difference or pure coherence also needs to be considered from the point of view of its becoming and not just one static view. Otherness as an orientation toward difference and coherence in their becoming gives us a first approximation of the concerns Hegel means to bring to the fore when he speaks of dialectic. (See also Pure Negativity?; Teleology After Kant.)

At Home in Otherness

This is part 3 of my direct walk-through of the Preface to Hegel’s Phenomenology. It seems that the phrase “being at home in otherness” originated in my own notes on H.S. Harris’ commentary, and literally occurs neither in Hegel nor in Harris. Nonetheless, I still want to suggest that the underlying idea is central to the perspective Hegel wants to recommend. Hegel speaks at length about what might be called thinking in the element of otherness, and provocatively ties it to the overcoming of alienation, thereby seeking to transform our pre-existing notions of what that might mean.

More conventionally, the overcoming of alienation has been represented as the recovery of a lost possession or lost innocence that we originally had, like a figurative return to the garden of Eden. The German Romantics of Hegel’s time had popularized this sort of comfortable and reassuring notion. Hegel wants to give it an altogether different and much more challenging meaning.

He points out the inherent weakness of all isolated theses and unelaborated statements of principle.

“[A]ny further so-called fundamental proposition or first principle of philosophy, if it is true, is for this reason alone also false just because it is a fundamental proposition or principle. — It is consequently very easily refuted. Its refutation consist in demonstrating its defects; however, it is defective because it is only the universal, or, only a principle, or, it is only the beginning. If the refutation is thorough, then it is derived from and developed out of that fundamental proposition or principle itself — the refutation is not pulled off by bringing in counter-assertions and impressions external to the principle. Such a refutation would thus genuinely be the development of the fundamental proposition itself” (Pinkard trans., p. 15).

No matter how good the principle, a shallow statement of it will be “false”.

“Conversely, the genuinely positive working out of the beginning is at the same time just as much a negative posture toward its beginning; namely, a negative posture toward its one-sided form, which is to be at first only immediately” (p. 16).

Everything that Hegel would recognize as genuine development and improvement begins with thoughtful criticism of what went before.

“[Spirit] must be, to itself, an object, but it must likewise immediately be a mediated object, which is to say, it must be a sublated object reflected into itself” (ibid).

“To sublate” translates German aufheben, a famous Hegelian term that means simultaneously to absorb and to transform (literally, “to on-lift”).

“Pure self-knowing in absolute otherness, this ether as such, is the very ground and soil of science, or knowing in its universality. The beginning of philosophy presupposes or demands that consciousness is situated in this element. However, this element itself has its culmination and its transparency only through the movement of its coming-to-be. It is pure spirituality, or, the universal in the mode of simple immediacy. Because it is the immediacy of spirit, because it is the substance of spirit, it is transfigured essentiality, reflection that is itself simple, or, is immediacy; it is being that is a reflective turn into itself” (pp. 16-17).

In a very characteristic gesture, he begins to point out that in human life, even mediation and immediacy don’t just stand alongside each other as statically independent opposites. Rather, we end up with all sorts of mixed forms of “mediated immediacy” and “immediatized mediation”. This interweaving is especially typical of what he calls “spirit”.

By “science”, once again, he means mediated rational understanding. “Absolute otherness” is the antithesis of the identity-oriented simplicity and rigidity of the point of view of ordinary consciousness. What we mainly encounter in life are mixtures of these two, with a tilt toward the ordinary. I’m inclined to think there could be no human experience at all without some admixture of otherness. A stronger otherness disturbs our complacency and takes us out of our comfort zone, but Hegel wants to gently suggest that this can be a good thing.

“However much the standpoint of consciousness, which is to say, the standpoint of knowing objective things to be opposed to itself and knowing itself to be opposed to them, counts as the other to science — the other, in which consciousness is at one with itself, counts instead as the loss of spirit — still, in comparison, the element of science possesses for consciousness an other-worldly remoteness in which consciousness is no longer in possession of itself. Each of these two parts seems to the other to be an inversion of the truth” (p. 17).

Here he acknowledges that what he is recommending must seem incredibly strange from the perspective of ordinary consciousness.

He continues, “For the natural consciousness to entrust itself immediately to science would be to make an attempt, induced by it knows not what, to walk upside down all of a sudden. The compulsion to accept this unaccustomed attitude and to transport itself in that way would be, so it would seem, a violence imposed on it with neither any advance preparation nor with any necessity. — Science may be in its own self what it will, but in its relationship to immediate self-consciousness, it presents itself as an inversion of the latter…. Lacking actuality, science is the in-itself, the purpose, which at the start is still something inner, at first not as spirit but only as spiritual substance. It has to express itself and become for itself, and this means nothing else than that it has to posit self-consciousness as being at one with itself” (ibid).

Hegel’s own favored attitudes, like rationality or “science”, are not exempt from the general requirement of development. To simply try to foist “science” or our favored view of rationality or the value of otherness on the public as ready-made conclusions differs little from attempts to socially impose any arbitrary prejudice. It is a means not at all suited to the ends of philosophy.

In speaking of “immediate self-consciousness”, he applies another paradoxical mixed form. The very essence of self-consciousness for Hegel is mediation, or the opposite of immediacy. But even the most highly mediated form can also be named, pointed at, presented, represented, or recalled in a more immediate way. Every level of development has its own characteristic reflection in relative immediacy.

He continues, “This coming-to-be of science itself, or, of knowing, is what is presented in this phenomenology of spirit” (ibid).

“Knowing, as it is at first, or, as immediate spirit, is devoid of spirit, is sensuous consciousness. In order to become genuine knowing, or, in order to beget the element of science which is its pure concept, immediate spirit must laboriously travel down a long path…. In any case, it is something very different from the inspiration which begins immediately, like a shot from a pistol, with absolute knowledge, and which has already finished with all other standpoints simply by declaring that it will take no notice of them” (pp. 17-18).

Immediate spirit is devoid of spirit in the deeper sense that travels down a long path. But still it contains a beginning.

“The aim is spirit’s insight into what knowing is. Impatience demands the impossible, which is to say, to achieve the end without the means. On the one hand, the length of the path has to be endured, for each moment is necessary — but on the other hand, one must linger at every stage along the way, for each stage is itself an entire individual shape” (p. 19).

Rational understanding has to grow organically — to be actively taken up and worked over by its participants — to realize its value. Once again, it is never enough to just present summary conclusions and expect the world to agree, no matter how right they are. A long, patient working out is essential to achieving the goal he has in mind.

“In this movement… what still remains is the representation of and the familiarity with the forms” (ibid).

“The element thus still has the same character of uncomprehended immediacy, or, of unmoved indifference as existence itself, or, it has only passed over into representational thought. — As a result, it is at the same time familiar to us, or, it is the sort of thing that spirit has finished with, in which spirit has no more activity, and, as a result, in which spirit has no further interest” (ibid).

Familiarity is an issue because it leads us to take things for granted and become inattentive. Hegel contrasts all forms of static representation of knowledge with the kind of active coming-to-be of knowing he is aiming at.

He continues, “However much the activity, which is finished with existence, is itself the immediate, or however much it is the existing mediation and thereby the movement only of the particular spirit which is not comprehending itself, still in contrast knowing is directed against the representational thought which has come about through this immediacy, is directed against this familiarity, and it is thus the doing of the universal self and the interest of thinking” (ibid).

In more Aristotelian language, once an understanding is acquired, it becomes passively available for easy use. The mode of this availability and easiness is a kind of habit. Habits have a great utility for action and responding to the world, but in exercising a habit we are not learning anything new. The active becoming of knowing, on the other hand, demands continuous learning.

“What is familiar and well-known as such is not really known for the very reason that it is familiar and well-known. In the case of cognition, the most common form of self-deception and deception of others is when one presupposes something as well known and then makes one’s peace with it. In that kind of back-and-forth chatter about pros and cons, such knowing, without knowing how it happens to it, never really gets anywhere. Subject and object, God, nature, understanding, sensibility, etc., are, as is well known, all unquestioningly laid as foundation stones which constitute fixed points from which to start and to which to return…. Thus, for a person to grasp and to examine matters consists only in seeing whether he finds everything said by everybody else to match up with his own idea of the matter, or with whether it seems that way to him and whether or not it is something with which he is familiar” (p. 20).

“To break up a representation into its original elements is to return to its moments, which at least do not have the form of a representation which one has merely stumbled across, but which instead constitute the immediate possession of the self. To be sure, this analysis would only arrive at thoughts which are themselves familiar and fixed…. However, what is separated, the non-actual itself, is itself an essential moment, for the concrete is self-moving only because it divides itself and turns itself into the non-actual” (ibid).

Actualization as a process is not just the tranquil extension of what is already actual. The emergence of new actuality essentially depends on what is currently non-actual.

He continues, “The activity of separating is the force and labor of understanding, the most astonishing and the greatest of all the powers, or rather, which is the absolute power” (ibid).

Hegel is better known as a sharp critic of the limits of the understanding that divides and sees only fixed things. But here, against the Romantics he defends analytical understanding’s creatively disruptive role in unsettling our complacency.

He continues, “The circle, which, enclosed within itself, is at rest and which, as substance, sustains its moments, is the immediate and is, for that reason, an unsurprising relationship. However, the accidental, separated from its surroundings, attains an isolated freedom and its own proper existence only in its being bound to other actualities and only as existing in their context; as such, it is the tremendous power of the negative; it is the energy of thinking, of the pure I” (ibid).

Just as new actualization depends on what is non-actual, the complacency of substantial existence is only spurred to new learning by what first appears as accident.

“Spirit only wins its truth by finding its feet in its absolute disruption” (p. 21).

To “find its feet in absolute disruption” is to be at home in otherness.

He continues, “Spirit is not this power which, as the positive, avoids looking at the negative, as is the case when we say of something that it is nothing, or that it is false, and then, being done with it, go off on our own way on to something else. No, spirit is this power only by looking the negative in the face and lingering with it” (ibid).

“Negation” for Hegel is not the simple thing that it is in Boolean logic. Boolean negation is purely formal, and yields the exact opposite of its input. For Hegel, every manifestation of otherness is a sort of “negation”. Personally, I prefer the language of otherness. Thus I would say, “looking otherness in the face and lingering with it”. This involves looking beyond fixed thoughts and everything that has the form of givenness.

“[I]n modern times, the individual finds the abstract ready-made…. Nowadays the task before us consists not so much in purifying the individual of the sensuously immediate and in making him into a thinking substance… It consists in actualizing and spiritually animating the universal through the sublation of fixed and determinate thoughts. However, it is much more difficult to set fixed thoughts into fluid motion than it is to bring sensuous existence into such fluidity” (ibid).

Ready-made abstractions are the bane of deeper understanding. It is far easier to announce that we ought to overcome them than to actually succeed in doing so.

“Thoughts become fluid by pure thinking, this inner immediacy, recognizing itself as a moment, or, by pure self-certainty abstracting itself from itself — it does not consist in only omitting itself, or, setting itself off to one side. Rather, it consists in giving up the fixity of its self-positing as well as the fixity of the purely concrete…. Through this movement, pure thoughts become concepts, and are for the first time what they are in truth: self-moving movements” (pp. 21-22).

In Hegel’s usage, a “concept” is not a fixed thought but an active rational disposition. Further, he suggests that real immersion in active thought implicitly involves letting go of a fixed presupposed self separate from the activity of thinking. At the same time thoughts, instead of being identified with inert fixed contents, become “self-moving movements” (see Ideas Are Not Inert).

“[I]t ceases to be the type of philosophizing which seeks to ground the truth in only clever argumentation about pros and cons or in inferences based on fully determinate thoughts and the consequences following from them. Instead, through the movement of the concept, this path will encompass the complete worldliness of consciousness in its necessity” (p. 22).

The “complete worldliness” of consciousness is the overcoming of the habitual duality of consciousness and object in which consciousness “sets itself off to one side” from everything else.

“Consciousness knows and comprehends nothing but what is in experience, for what is in experience is just spiritual substance, namely, as the object of its own self. However, spirit becomes the object, for it is this movement of becoming an other to itself…. And experience is the name of this very movement in which the immediate, the non-experienced, i.e., the abstract (whether the abstract is that of sensuous being or of ‘a simple’ which has only been thought about) alienates itself and then comes around to itself out of this alienation” (pp. 22-23).

“The inequality which takes place in consciousness between the I and the substance which is its object is their difference, the negative itself. It can be viewed as the defect of the two, but it is their very soul or is what moves them” (p. 23).

Here inequality manifests otherness. Notably he refers to it “taking place” rather than simply existing.

Even the core defect of the standpoint of ordinary consciousness — its duality, in which consciousness stands “off to one side” of its objects — in its capacity as a source of unrest already points beyond itself, kicking off the whole long movement that the Phenomenology aims to characterize.

“However much this negative now initially appears as the inequality between the I and the object, still it is just as much the inequality of the substance with itself. What seems to take place outside of the substance, to be an activity directed against it, is its own doing, and substance shows that it is essentially subject” (ibid).

Unqualified “substance” in Hegel’s sense really encompasses everything there is, even though we imagine that we are somewhere off to the side. Thus the apparent duality between us and substance that we think about turns out to be internal to substance itself. What seemed to be “our” separate activity turns out to be equally the activity of substance that is no longer “just” substance. The substance that is thought of loses its fixity and becomes an active thought.

“Why bother with the false at all?…. Ordinary ideas on this subject especially obstruct the entrance to the truth…. To be sure, we can know falsely. For something to be known falsely means that knowing is unequal to its substance. Yet this very inequality is the differentiating per se, the essential moment. It is indeed out of this differentiation that its equality comes to be, and this equality, which has come to be, is truth. However, it is not truth in the sense that would just discard inequality, like discarding the slag from pure metal, nor even is it truth in the way that a finished vessel bears no trace of the instruments that shaped it. Rather, as the negative, inequality is still itself immediately present, just as the self in the true as such is itself present” (pp. 23-24).

Hegel’s usage of “knowing” is much more inclusive than the strict Platonic or Kantian sense that I have been recommending here.

Here we reach another delicate point. What is false, he is saying, is not purely and simply false, because it also creates the unrest that is the impetus for further development. But this is very easily misunderstood, and can lead to complete nonsense.

To avoid this kind of misunderstanding, he continues, “For that reason, it cannot be said that the false constitutes a moment or even a constitutive part of the true. Take the saying that ‘In every falsehood, there is something true’ — in this expression both of them are regarded as oil and water, which cannot mix and are only externally combined. It is precisely for the sake of pointing out the significance of the moment of complete otherness that their expression must no longer be employed in the instances where their otherness has been sublated. Just as the expressions, ‘unity of subject and object’ or of ‘the finite and the infinite’, or of ‘being and thinking’, etc., have a certain type of clumsiness to them in that subject and object, etc., mean what they are outside of their unity, and therefore in their unity, they are not meant in the way that their expression states them, so too the false as the false is no longer a moment of truth” (pp. 24-25).

Here he is employing an Aristotelian “said in many ways” distinction to avoid confusion and nonsense. It remains the case that everything for Hegel being more than it “just” is requires a great wakefulness on the part of the reader, to avoid slipping into just the kind of nonsense he is warning about.

Incidentally, he suggests that “otherness” is a better alternative to talk about the unity of subject and object, finite and infinite, being and thinking, etc.

Wrapping up this part of the argument, he continues, “The dogmatism of the way of thinking, in both the knowing of philosophy and the study of it, is nothing but the opinion that truth consists either in a proposition which is a fixed result or else in a proposition which is immediately known…. [E]ven bare truths… do not exist without the movement of self-consciousness…. Even in the case of immediate intuition, acquaintance with them is linked to the reasons behind it” (p. 25).

Searching for a Middle Term

“But nothing, I think, prevents one from in a sense understanding and in a sense being ignorant of what one is learning” (Aristotle, Posterior Analytics; Complete Works revised Oxford edition vol. 1, p. 115). The kind of understanding spoken of here involves awareness “both that the explanation because of which the object is is its explanation, and that it is not possible for this to be otherwise” (ibid). To speak of the “explanation because of which” something is suggests that the concern is with states of affairs being some way, and the “not… otherwise” language further confirms this.

Following this is the famous criterion that demonstrative understanding depends on “things that are true and primitive and immediate and more familiar than and prior to and explanatory of the conclusion…. [T]here will be deduction even without these conditions, but there will not be demonstration, for it will not produce understanding” (ibid). The “more familiar than” part has sometimes been mistranslated as “better known than”, confusing what Aristotle carefully distinguishes as gnosis (personal acquaintance) and episteme (knowledge in a strong sense). I think this phrase is the key to the whole larger clause, giving it a pragmatic rather than foundationalist meaning. (Foundationalist claims only emerged later, with the Stoics and Descartes.) The pedagogical aim of demonstration is to use things that are more familiar to us — which for practical purposes we take to be true and primitive and immediate and prior and explanatory — to showcase reasons for things that are slightly less obvious.

Independent of these criteria for demonstration, the whole point of the syllogistic form is that the conclusion very “obviously” and necessarily follows, by a simple operation of composition on the premises (A => B and B => C, so A=> C). Once we have accepted both premises of a syllogism, the conclusion is already implicit, and that in an especially clear way. We will not reach any novel or unexpected conclusions by syllogism. It is a kind of canonical minimal inferential step, intended not to be profound but to be as simple and clear as possible.

(Contemporary category theory grounds all of mathematics on the notion of composable abstract dependencies, expressing complex dependencies as compositions of simpler ones. Its power depends on the fact that under a few carefully specified conditions expressing the properties of good composition, the composition of higher-order functions with internal conditional logic — and other even more general constructions — works in exactly the same way as composition of simple predications like “A is B“.)

Since a syllogism is designed to be a minimal inferential step, there is never a question of “searching” for the right conclusion. Rather, Aristotle speaks of searching for a “middle term” before an appropriate pair of premises is identified for syllogistic use. A middle term like B in the example above is the key ingredient in a syllogism, appearing both in the syntactically dependent position in one premise, and in the syntactically depended-upon position in the other premise, thus allowing the two to be composed together. This is a very simple example of mediation. Existence of a middle term B is what makes composition of the premises possible, and is therefore what makes pairings of premises appropriate for syllogistic use.

In many contexts, searching for a middle term can be understood as inventing an appropriate intermediate abstraction from available materials. If an existing abstraction is too broad to fit the case, we can add specifications until it does, and then optionally give the result a new name. All Aristotelian terms essentially are implied specifications; the names are just for convenience. Aristotle sometimes uses pure specifications as “nameless terms”.

Named abstractions function as shorthand for the potential inferences that they embody, enabling simple common-sense reasoning in ordinary language. We can become more clear about our thinking by using dialectic to unpack the implications of the abstractions embodied in our use of words. (See also Free Play; Practical Judgment.)