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Philosophie des seuils

The notion of Threshold: an investigation into conceptual accompaniment in Aristotle and Hegel

Jason Carter

Résumés

Ce qui est particulier à l’idée de seuil c’est que c’est sa propre définition de générer un contour conceptuel qui va vers un autre champ d’exploration, même si ces images de contour semblent devoir être exclues de sa définition. Hegel a décrit ce phénomène du mouvement de la définition d’un concept vers les termes qui l’accompagnent comme la détermination d’un concept à partir de sa « négation » et a considéré cette méthode comme fondamentale à toute approche philosophique. Ce texte explore le fait que la notion hégélienne de la négation remonte aux idées d’Aristote au sujet de la bonne formulation d’une définition ainsi que ses idées sur les opposés et la connexion entre eux à travers un troisième terme. Il s’applique à démontrer à travers un examen des catégories d’Aristote et de la notion de différence, du mouvement et de la signification de la copule, que la pensée d’Aristote est structurée par des concepts-seuils non-binaires qui sont essentiels à son système philosophique. Enfin, il sera montré comment Hegel a tenté de prendre ces concepts-seuils chez Aristote pour leur donner une fonction méthodologique pour la pensée elle-même ainsi que pour la détermination philosophique des catégories de l’être.

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Texte intégral

  • 1  “And the same is true in the other cases, e.g. if the threshold is characterized by its position, (...)

ὁμοίως δὲ οὐδὲ τῶν ἄλλων οὐθεν, οἷον εἰ ὁ οὐδὸς θέσει, οὐκ ἐκ τοῦ οὐδοῦ ἡ θέσις ἀλλὰ μᾶλλον οὗτος ἐξ ἐκείνης.” -Ἀριστοτέλης1

1The notion of threshold, even when placed under a simple definitional formula - such as Aristotle’s “wood or stone in such and such a position” (Metaphysics VIII.1043a) - inevitably results in two contradictory images. The first of these images is of the term defined, i.e. the threshold as piece of material located at a fixed place at the point of entrance to a house or room. Yet consequent upon this first image (or perhaps lying next to it, as an unnoticed part thereof), a second image of the surrounding boundaries outside of the threshold inevitably follows, an image such as the land bordering the house. But if ‘land’ is included in the definition of the threshold, one then has an accompanying image of a certain type of land, its color, or something else. It seems that while the ‘threshold’ must be given a definitional limit or negation by which to circumscribe it off from other images or terms, at the same time it seems that one cannot properly make sense of the ‘threshold’ if one is not in some conceptual respect already noticing what is beyond the threshold’s limit - both in reference to what the threshold is the threshold of (e.g. of a house), or where its position is in respect of this ‘of’ (e.g. in respect of an entrance, or a door sill in between rooms, or the land outside the house, etc.). Yet this phenomenon is not unique to the definition of ‘threshold’; below we will argue that this phenomenon is indicative of a more general function that occurs in all cases of concept formation, even when it is not thematized as such or noticed by the definer. But granting this, what is unique to the idea of a threshold, especially if one wants to explore its meaning as implying a conceptual model for philosophical investigation, is that it is a threshold’s proper definition to generate or signal an inevitable accompanying (or in Aristotelian terms, ‘accidental’) image or term which serves as a further field of exploration, such that this (seemingly) extrinsic term is actually part of what a threshold is ‘in itself’.   

Movement and conceptual accompaniment

2The problem of conceptual accompaniment is further and intimately related to another phenomenon, which is how a thinker phenomenologically experiences the crossing from term-defined (e.g. the term ‘threshold’) to the accompanying term(s) (e.g. the ‘inner/outer rooms’ connected by the threshold).  David Hume pinpointed this phenomenon with marvelous psychological acuity, arguing that “no sooner one idea occurs to our thoughts than it introduces its correlative, and carries our attention towards it, by a gentle and insensible movement” (Hume, 93). Hume labeled this phenomenon the mind’s ‘principle of association,’ a principle that he found powerful enough to explanatorily rival the notion of real causality. The principle that every cause has an effect, he famously argued, is nothing but an inferred idea formed by habit from the experience of these inevitable correlative psychological associations of any given idea with a further similar or contiguous idea. Hume comes closest to capturing our most basic experience of conceptual movement or the concept production aspect of a threshold, i.e. that once a limit (or idea) is imposed, immediately that limit produces something outside the limit. However, although Hume is working with the same basic phenomenon that we are here, he uses it a basis for his academic skepticism, while here we are investigating it as a potential basis for philosophical knowledge

3While Hume correctly noted this phenomenon at the level of concept formation in personal experience, the conceptual phenomenon of accompaniment (or Humean association) can also be shown to occur at the level of a philosophical system. When Kant circumscribed theoretical knowledge as limited to the realm of possible empirical experience in the Critique of Pure Reason, he was forced to generate an accompaniment within the critical system itself under the idea of the noumena, the realm of things as they are in-themselves. If such a postulation of a system of actual objects outside the realm of our knowledge is a necessary accompanying correlate to Kant’s thesis that we only know appearances of things, then we at least have preliminary grounds upon which to argue that when any concept or system with determinate boundaries is postulated, these concepts or systems either produce or are connected with further concepts or systems that lay outside them, and which are, perhaps, actually a proper part of the original concept or system.

4Even so, Hume and Kant serve primarily a negative function in the problem of conceptual accompaniment, since insofar as both of these thinkers noticed the phenomenon they either ignored it or tried to eliminate it. Hume, for instance, left his notion of association undeveloped (he thought it was a problem that should be left to the new science of psychology), while on the other hand, Kant systematically attempted to keep his noumenal accompaniment ontologically separate from his epistemological system. For this reason, these thinkers will not serve as sources of investigation here. Rather, we will need to turn to philosophers who both notice the phenomenon of accompaniment and also in some sense positively appropriate it. To do this, we need to ‘scale-down’ the scope of the general phenomenon of ‘conceptual accompaniment’, and look at the phenomenon through the idea of a threshold concept.

5A threshold concept we will define as: A concept which, by virtue of its proper definition, has the dual function of both connecting and distinguishing different terms (i.e. as an actual threshold does), while at the same time functioning as a source of conceptual movement (literal or metaphorical) between the terms it connects.

6The first half of the definition may be asserted to be a description of the accompaniment problem alone, i.e. as a proper and hence desirable function of the concept to indicate and share in something outside of its limit, while the latter half emphasizes ways one might postulate or conceive of how a thinker’s movement occurs between the concepts or terms that a threshold concept connects. Both characteristics are necessary to understand the phenomenon at hand.

The threshold in relation to Hegel and Aristotle

7A first place to look for examples of threshold concepts is most obviously the works of Hegel, since it is he who both classified the accompaniment phenomenon itself as a proper and necessary form of ‘negation’, and who described in narrative fashion the experience of and the necessary reasons for the crossing over of opposing categories into one another through a narrative account of the Absolute Idea’s coming to understand its own identity through its own self-differentiation. For this reason, Hegel will here serve as one extreme for our understanding of the threshold. However, what we want to discuss here how Hegel’s thinking of negation is both related to and informed by earlier ancient precedents, particularly Aristotle’s thought on definitions, and especially those on contradiction and contrariety. We should remember that it was Aristotle who was the first to codify the various rules of negation (ἀπόφασις) in both its linguistic (in De Interpretatione) and ontological functions (in Metaphysics, especially Book IV). For this reason, Aristotle will serve as the other extreme of our investigation.

  • 2  See for instance, (Priest, 2004) and the idea of dialethism.
  • 3 “δῆλον ὅτι ταὐτὸν τἀναντία ποιεῖν ἢ πάσχειν κατὰ ταὐτόν γε καὶ πρὸς ταὐτὸν οὐκ ἐθελήσει ἅμα.”
  • 4 “τὸ γὰρ αὐτὸ ἅμα ὑπάρχειν τε καὶ μὴ ὑπάρξειν ἀδύνατον τῷ αὐτῷ καὶ κατὰ τὸ αὐτό καὶ ὅσα ἄλλα προσδιο (...)

8The assertion of Hegel’s reliance on Aristotle may at first seem surprising, since Hegel’s dialectic and his notion of infinite Aufhebung, has been recently argued by Graham Priest to be a challenge to the traditional notion that a contradiction cannot be true, an idea reflected in the philosophical and logical principle most commonly referred to as the law of non-contradiction2. The law of non-contradiction was first formulated in Plato’s Republic by the formula, “the same thing will not make or suffer opposites in respect of the same part of itself, in relation to the same thing, at the same time (IV.436b),”3 and was upheld in a slightly variant form by Aristotle in similar statements, such as “the same attribute cannot at the same time belong and not belong to the same subject in the same respect” (Metaphysics IV.1005b).4 Part of the difficultly in understanding Hegel is that he does not always seem to affirm this law, particularly in his notion of dialectic. The classical definition of a ‘dialectical proposition’ was given by Aristotle as making a ‘choice between two contradictories’ (Prior Analytics, I.24a). Hegel’s method, however, often seems to be the affirmation of two contradictories.

9Much of this confusion results from neglecting to remember that Hegel is writing before the invention of modern forms of symbolic logic, the first of which was introduced in the mid 19th century by George Boole. Hegel’s logic is still syllogistic, that is to say, Aristotelian. This latter framework provides for a content-oriented (as opposed to formal) understanding of negation and contrariety, insofar as these terms will be understood as functions and descriptions that apply to both the natural world as well as to the semantic world. We might remember, for instance, Aristotle’s dictum that in nature “everything that comes to be or passes away comes from, or passes into its contrary or an intermediate state [i.e. between the contraries]” (Physics, I. 188b), and “For all things are either contraries or composed of contraries, and unity and plurality are the starting-points of all contraries” (Metaphysics IV.1005a). Contrary things, for Aristotle, are further variegated into four kinds of opposition (‘opposition’, for the purposes of this paper, should be read as synonymous with Hegelian ‘negation’): things can be opposed “as relatives, as contraries, as privation and possession or as affirmation/negation” (Categories, 11b). Aristotle tells us that of these species of opposition only the fourth kind is necessary, giving the reason that if one affirms a proposition, it is either the case that the proposition must be true and its negation (or denial) false, or vice versa (which itself is based upon Aristotle’s principle of non-contradiction cited above). The other species of opposition fall under no necessity since 1) relatives are not necessarily always true or false since things do not necessarily have any relation if the relatum does not exist (e.g. a thing does not have to be double, if there is no thing it is the double of), 2) Contraries are not necessary because things can partake of intermediates between contraries, like grey is between white and black, and 3) Privation and possession are not always necessary since these categories concern properly natural possessions of objects, such that while man may necessarily be said either to have the power of sight or lack it, a rock is not properly spoken of as ‘having’ sight or not (Categories, 11b-13b).

10Though it may seem tedious to chart out these Aristotelian types of opposition, they provide an exemplary point of access for understanding Hegel. These semantically determinate forms of opposition are essentially exactly what Hegel means by ‘negation’. Aristotle’s division of types of opposition reveal two things of interest for us: 1) They point out the various ways in which Aristotle’s philosophy allowed negations (or oppositions) to be structurally and intelligibly organized by their meaningcontent, and not just their bare logical form (e.g. the natural contrary of ‘hot’ is not just ‘not-hot’ but ‘cold’), and 2) Since movement or change occurs between contraries (as stated above), this implies that we may attempt to analyze Hegel’s dialectical movement between his categories on an Aristotelian basis.  

11What we will now have to show is that there is a nuanced way in which this double affirmation of seemingly contradictory terms in Hegel is not as unreasonable as it seems at first glance, and is in fact essential to the notion of a threshold concept which we are attempting to explore. Hegel’s dialectic does not gain its moving force by means of an unintelligible collision of absolute contradictories, nor even by something like Nicholas of Cusa’s mystical doctrine of God as containing a coincidenta oppositorum. Rather, Hegel’s dialectic should be seen as an outgrowth and furthering of Aristotle’s interpretation of individual things being naturally and metaphysically structured in reference to contraries which are connected by something fitting the definition of a threshold concept.

The differentia problem at the ‘bottom’

12If, following Aristotle’s precedent set out in our first quote above, we assert the threshold to be at least initially characterized by its position (θέσις) or place in the architectonic structure of a philosophy, we may also ask for the ‘place’ of an exemplary threshold concept in Aristotle’s thought. In this case, one obvious candidate for a threshold concept would be a structural term in Aristotle’s philosophy which explicitly functions both to distinguish and to connect terms at the same time, as a threshold does. This idea we find to coincide with Aristotle’s description of differentiae. For Aristotle, the differentiae are the predicates which mark out an essential difference in a genus by which to identify a lower species. Ideally, one should be able to ascend from any given species of being by abstracting from its differentia and in order to arrive at the next higher level of genus, or one should be able to descend to a lower level of species if a new differentia is found within a given genus. Aristotle’s entire ontological system of beings is a structured hierarchy of interconnected differences marking out a species under a genus.

  • 5  The following paragraph is heavily indebted to Edward Halper’s essay, “Hegel and the Problem of th (...)

13However, there are at least two major difficulties with Aristotle’s notion of the differentia which are of interest to our investigation into a notion of threshold, both of which have been decisively pinpointed by Edward Halper.5 The first difficulty is whether or not the final differentia - which provides the last determination and division of a substance into its lowest determinate species and is supposed to give one the formula of a thing’s essence, the τὸ τί ἦν εἶναι (the ‘what it was to be,’ or the ‘thing by which a thing is what it is’) - is itself a part of the genus of the substance that it defines or is itself part of another category? We must remember that for Aristotle, all the various ways of being (or categories of being) are dependent on substance for their existence, substance being the most unified type of being in which lesser unities of being inhere.

14There are at least two explicit possibilities concerning which category the differentia falls under, or what type of being the differentia has.

15Interpretation (1): The last differentiais a part of substantial genus it differentiates, i.e. a division in the substance itself qua substance.

16If we take the definition of man to be ‘animal’ with the differentia ‘rational’, under this first interpretation, the predicate ‘rational’ falls under the category of substance, i.e. rational being is a proper part of, or is a substance itself. However, if we assume this to be the case, then it seems to remain unintelligible how the genus ‘animal’ can be conceptually marked out as anything different from its differentia, i.e. there seems to be another differentia required to mark out and distinguish the lastdifferentia (rational) from the genus which it is part of (animal). Otherwise, how could one distinguish the genus as genus, from the differentia as differentia, since they are both of the same kind of being (viewed under the category of substance)? It would be like saying the species ‘water’ is defined by the genus ‘liquid’ which has the differentia ‘liquid which is different’. We need to definitionally specify by what means or content this difference is constituted. Aristotle’s own answer to this, which he treats explicitly in Metaphysics VII, is that the genus is the matter, while the final differentia is the form, and it is this latter which is most of all called a thing’s substance. However, this conflicts with another interpretation of the differentia, also given by Aristotle.

17Interpretation (2): The last differentia “always expresses a certain quality” (Topics, VI.144a).

18Here we find the opposite problem. In this latter case, the thing which characterizes the unitary ‘what’ of a substance is precisely something that is not of the category of substance, but rather another category entirely - a quality. This interpretation comes about because it seems that to find a difference in the substance we need to find a mark different from the substance.

19Though both of these interpretations are problematic, we can see that the interpretation given in (2) of the differentia allows it to meet the requirements of being a threshold concept, since a definition of substance under interpretation (2) allows a substance to have accompanying extrinsic term - a quality - which causes a movement of vacillation between the two sides of the definition in a thinker. Interpretation (1) seems (so far) not to be a threshold concept, since we have a static definition which has either no principle to explain how a difference can be marked out if genus is marked out by a part of itself (i.e. a substance marking out another substance which it itself is), or, if the genus is matter and the differentia is merely a unitary form, then genus seems unnecessary or not essentially connected to the differentia. However, (1) does seem essential to a definition of substance, since it would seem absurd to define something only by what it is not. Essentially, the problem is: why is it that the differentia, which is supposed to provide the definition and the being of a thing, actually ends up being in one way both the thing in question, and in other way, something else entirely?

The differentia problem at the ‘top’

20Under the above aspects, the nature of the differentia can be classified as a threshold problem which resides at the lowest and most concrete level of Aristotle’s philosophy. Yet, this problem is duplicated at the highest level of Aristotle’s general categories of being as well, for there we find that the differentiae conspicuously drop out of his system. Once we are at the highest generic level of Aristotle’s system, it seems logical to ask for the differentiae or characterizing features of each abstract category of being (e.g. substance, quality, relation, etc.). However, here we are disappointed, for the only thing ‘above’ any of the categories is being in general, and this, Aristotle tells us, is not a genus. Thus the categories cannot properly be called species of a genus (indeed, each category is itself a highest genus). However, the problem is even worse: not only does Aristotle not affirm that being is a genus that could have a differentia, but he could not affirm that it is a genus of all the categories, for what differentia could being in general have in order to bring about the various categories as abstract high-level species, since presumably there is nothing other than being?  If one could conceive of ‘being’ as a genus, and find its differentia outside of the category of ‘being’, which seems impossible, this would be the abstract version of interpretation (2) above. The other option, by which one might attempt to posit that the differentia of a category could be occupied by another category, faces the same problem as occurred in interpretation (1) above, i.e. how could one distinguish the category-as-differentia from the category-as-category in order to arrive at the definition of a species-category? Thus the problem of accurately defining or distinguishing the basic categorial forms of being from one another ‘at the top’ exactly mirrors the problem that occurs ‘at the bottom’ at the level of the most determinate species. The threshold concept of the differentia has seemingly resolved itself into two contrary and equally impossible interpretations.

21Hegel, in opposition to this aporia, thinks that there is a solution to these categorial problems (at least for those “at the top”), and his philosophy offers a determinate method by which we can demonstrate the differences among the highest categories of being. We will attempt to show that Hegel’s method, which he calls the dialectic, is in fact a novel attempt to solve the differentia problem of the categories by means of both differentia interpretation (1) and (2). Although Hegel’s solution seems to be original, it can be argued that it is not historically novel. In fact, Hegel’s solution can be understood as an antecedently and logically available modification of only four resources that are already explicitly available in Aristotle: Aristotle’s notions of 1) The copula, 2) Aristotle’s speculations on the essence of divinity, 3) definition by relation, and 4) actuality.

The meaning of the copula – sameness or difference ?

22Aristotle lays it down as a rule that every proposition or ‘statement-making’ sentence must contain a subject and a verb, since “falsity and truth have to do with combination and separation” (De Interpretatione, I.16a). The subject, in its most proper use, ontologically represents the unitary substance or individual in which another predicate (be it a quality, quantity, or activity, etc.), inheres. This subject plus verb or copula is the form of the simple judgment, which is Aristotle’s favored type of philosophical expression. Hegel tells us that his speculative logic, conversely, cannot use simple judgments to express speculative truths, because in simple judgments, the copula signifies only some form of identity or sameness between subject and predicate. Hegel thinks that in fact what we are looking for in any given definition is not an identity, but a difference, for only a difference can give us more information about a subject term.

23Hegel argues that if we want to avoid asserting a simple absolute tautological sameness in our propositions (such as A=A), when we attach a predicate to a subject through a copula, we must posit more than identity alone; we also must posit an essential difference (Hegel, SL, 90). Simple affirmative judgments, such as ‘Man is rational,’ according to Hegel, use the copula only in the mode of asserting identical aspects of sameness between subject and predicate, or in Aristotelian language, the copula as asserting sameness between genus and differentia. We see here that Hegel is following differentia interpretation (1) in his interpretation of Aristotle’s logical predication, i.e. either that the differentia is a proper part of the genus of substance, or the predicate is somehow the same as the subject. However, along with this interpretation, Hegel will assert that there is as well an ontological non-identical aspect that subjects and predicates can share through the copula which only speculative philosophy can theoretically elucidate, and it always does so in a way that simple judgments (i.e. Aristotelian judgments), cannot.  Such speculative truth, Hegel tells us, is arrived at by bringing together seemingly contradictory judgments and resolving their antinomies in a speculative judgment that takes into account (or sublates) the identical and non-identical aspects of the contradiction into a third term, which has the result of explaining both of (seemingly contradictory) terms in a different respect or sense, or in more general terms, in terms of a relation to a higher concept. This method, Hegel thinks, brings with it the result of providing an intelligible way to resolve (seeming) contradictions. Hegel thus invests his copula with a new ontological signification; it is supposed to signify ontological identity and difference at once through the idea of a relation, which itself turns out to fit the definition of a threshold concept.

Definition and divine thought

  • 6 “αὑτὸν ἄρα νοεῖ, εἴπερ ἐστὶ τὸ κράτιστον, καὶ ἔστιν ἡ νόησις νοήσεως νόησις.”
  • 7 “οὐχ ἑτέρου οὖν ὄντοσ τοῦ νοουμένου καὶ τοῦ νοῦ, ὅσα μη ὕλην ἔχει, τὸ αὐτὸ ἔσται, καὶ ἡ νόησις τῷ ν (...)

24If we are to ask for an Aristotelian precedent for understanding the copula as a relation of identity and difference at once, we might look to the description Aristotle gives to the nature of divine thought. In wrestling with this problem (probably in reference to the unmoved mover, but at least in reference to the thought of the gods), Aristotle argues that since divine thought must think upon the best things in order to be justly labeled as divine thought, “Therefore it must be itself that thought thinks (since it is the most excellent of things), and its thinking is a thinking on thinking” (Metaphysics XII.1074a).6 The question of course naturally arises whether or not thought-as-subject and thought-as-object are different things under this account? Aristotle answers that, “thought and the object of thought are not different in the case of things that have not matter, they will be the same, i.e. the thinking will be one with the object of its thought” (Metaphysics, XII.1075a).7 Yet Hegel, it seems, does not think we can interpret this pronouncement as asserting that thinking-as-activity and thinking-as-object can be merely identical, otherwise why would we even predicate one of the other, or say one is directed toward the other? We must, it seems, understanding Aristotle’s definition of the divine as a definition by relation which both identifies and distinguishes at once, through some sort of formal activity.

25Understanding the unique definitional and ontological status of Aristotle’s ‘thought thinking itself’ provides a way into understanding the mechanism that Hegel uses when he begins his Science of Logic. Hegel begins this work by giving a series of seemingly impossible categorial transitions; from the empty category of ‘being’, we move to ‘nothing’, and from ‘nothing’ back to ‘being’, and then both of these categories in their vacillation are identified with ‘becoming’, which is then identified with ‘determinate being’ (which Hegel argues is synonymous with ‘quality’). Under the interpretation of the copula as signifying only identity, these transitions seem unintelligible. If we instead interpret the copula as Hegel does, as signifying a relation of sameness and difference within a concept, or as two sides of some higher term’s relation to itself (of which we have yet to see the whole meaning of), these transitions become more tenable.

  • 8  This Hegelian mode of explanation one might tentatively call an ontological mereology (the study o (...)

26We find that this is exactly what Hegel does – he argues that instead of conceiving of two terms as absolutely different (or negatively opposed), or as absolutely the same, we should rather view these opposites by means of a threshold concept which works as a relation between them, and a relation which in fact constitutes the definition of the terms it relates. For Hegel, the import of this is that each category is necessarily only defined relative to another, just as thought’s definition is relative to how divine thought is both identical and opposed to itself as thinking subject, and as object of thought. This however is only the most basic structural criterion for understanding Hegel’s dialectic. His categorial system becomes more and more complicated when these relations to other categories are viewed as parts falling inside or outside of the category to be defined, a challenge which need not concern us here.8

Synopsis of the above: definition by relation

27What is surprising about all of this, even apart from the notion of divine thought, is that we are still completely within Aristotle’s ways of defining entities, since Aristotle has set down the precedent that, “it is perhaps not possible to define some things in an other way, e.g. the double without the half, and all terms that are essentially relative; for in all such cases to be them is at the same time as to be somehow related to something, so that it is impossible to know the one without the other, and accordingly in the account of the one the other too must be embraced” (Topics, VI.142a, italics mine).

28Thus while Hegel makes heavy use of Aristotle’s understanding of the copula as specifying mere identity between a thing and its predicate, he has also dramatically reinterpreted it by inserting into its meaning a non-identical relation that it also represents. This move by Hegel, as we have attempted to show, is given help from Aristotle’s speculations about thought thinking itself, i.e. as a self-relating activity of a thing to itself, which might intelligibly be understood as positing identity and difference at once. With this new signification of the copula, now understood as meaning the self-relation of an identity and a difference, Hegel now has at his disposal a method such that given any abstract term, there is a way for it to be posited as being related to itself and as making its own differentia. This relation can further be interpreted under various Aristotelian modes of relation (having, not-having, doing-to, possessing, etc.). Our final task is to show how Aristotle’s notion of motion, along with his notion of actuality as a complete (but continuous) motion, is connected to this Hegelian method of finding a differentia, and how the idea of motion works together with the idea of the differentia to allow Hegel’s system to operate by means of a single central threshold concept - that of the Absolute Idea, or self-determining Begriff (or Notion/Concept), which fulfills both requirements of our definition of a threshold concept.

Motion and actuality

29Aside from the notion of a differentia, Aristotle’s notion of motion should also be classified as a threshold concept, because it both connects and differentiates potentiality and actuality without fully being either of them, although in a way it partakes of both. Aristotle tells us that all substances or things contain an actuality which, when that substance is in contact with another thing that has the potential to receive it, injects its actuality into the potentiality of the second thing. While the form is in the process of being transferred, or when a potential thing is coming-to-be actual in the thing, there the phenomenon of motion occurs (Physics, III.202a).

30Yet the question arises, how is this actuality (ἐνέργεια) or formal-activity of the mover to be conceptualized and distinguished from the motion (κίνησις) it produces? Aristotle’s answer is that although actuality is a type of motion, we must say that it is complete motion, in that its activity is at one with its telos, i.e. the final goal for which any particular motion is aiming at (Metaphysics IX.1048b). Motion qua motion is incomplete or unfulfilled activity, since while a thing is in motion it is still having its various possibilities fulfilled (i.e. motion assumes that some possibility in the world hasn’t been made actual yet). Thus, when a builder is engaged in the activity of building something, the telos aimed at is the completed building (an actuality). However, before the building is constructed, there is a process in which this potential telos is coming-to-be fulfilled or actualized by the builder and the material, and this is the process of motion. When something has become a self-containing or fulfilled activity, with its telos completely fulfilled in the activity, we have a transition to an activity working-in-itself (ἐντελέχεια).

31Aristotle tells us we can notice a distinction between a motion and an actuality by comparing a thing’s present activity with its completed activity. This idea is immensely important for understanding Hegel’s final formulation of what we will designate as his central threshold concept, so I quote it here in full:

  • 9 “οὐ γὰρ ἅμα βαδίζει καὶ βεβάδικεν, οὐδ' οἰκοδομεῖ καὶ ᾠκοδόμηκεν, οὐδὲ γίγνεται καὶ γέγονεν ἢ κινεῖ (...)

“For it is not true that at the same time we are walking and have walked, or are building and have built, or are coming to be and have come to be – it is a different thing that is being moved and that has been moved, and that is moving and that has moved; but it is the same thing that at the same time has seen and is seeing, or is thinking and has thought. The latter sort of process, then, I call an actuality, and the former a movement. (Metaphysics IX.1048b, italics mine)” 9

32There is both an ontological and a linguistic justification for this statement. The ontological explanation is that what differentiates an actuality, such as ‘seeing’, from a motion like ‘walking’, is that according to Aristotle the present activity of the former contains its own final purpose or telos at each moment of its present activity, while the latter’s purpose must take a period of time in order to reach its final goal, i.e. a builder must build for a period of time before a building actually exists complete. The linguistic component is more difficult. Aristotle seems to be asserting that for actualities, their present tense activity and their perfect tense activity (which usually implies the past tense) exist at once in the same thing. How is this to be conceptualized? For Aristotle, the oddity is that such a dual predication would seem to commit him to a violation of the law of non-contradiction - in this case a thing’s having and not having a property at the same time in respect to the same thing - since the perfect tense implies an action that has ended, i.e. a not-having, and the present tense, a having. What I want to argue here is that Hegel ingeniously exploits this linguistic present tense/perfect tense definition of actuality by finding an intelligible and non-contradictory explanation for it in the gerund/infinitival noun form of a verb.

Infinitival self-prediction as the solution to categorial differentiation

33The ‘thing that is the same’, in both the present tense predication and the perfect tense predication, for Hegel, is the verb’s infinitive form reified to have a thing-like ontological status. Although it is linguistically unproblematic for German to use the infinitive of a verb as a noun (reflected in English by using a present participle or gerund, formed by adding ‘ing’ to the stem of a verb, e.g. eats, can be turned into the noun eating, walks, walking, runs, running), it is philosophically problematic if one asks how we are to conceive such infinitival nouns apart from a concrete subject (and Aristotle usually argues that we must always posit a concrete material for such an activity to inhere in). However, if Hegel is taking his categories, (i.e. being, nothing, becoming, something, other, etc.) as ontological infinitives-as-nouns which do not need a separate substance in which to inhere (in fact, he thinks they are the substances, albeit abstract), it is possible to make sense of Aristotle’s claim that for certain activities one can predicate both the present tense and the perfect tense of the verb to that activity’s infinitive or gerund, and the statement will be both intelligible and true. How can this be so?

34Let us take an example. It seems that one can assert ‘thinking has thought’ and that ‘thinking is thinking,’ at the same time and without contradiction, insofar as we view the first predication as a logical moment or part of the second predication, the latter of which is viewed as an ongoing activity. More formally, this method can be mapped out as follows:

  1. Given a Hegelian category [A], [A] should be understood as a verbal noun/infinitive noun (even if [A] is not naturally a verb).

  2. The infinitive [A] is then given its definition or determination by being predicated with its own activity as a verb in the present tense.

35Now the question remains, what results from this bizarre self-referential predication of an infinitive-noun to its own activity in present tense? The answer is surprising. In certain cases (most assuredly not all), such a predication quite naturally leads to the concept or image of the perfective form of the verb, or what we viewed above as the temporal slice or moment of the activity. However, if one views the perfective form as not merely a temporal slice, but the last “slice” of the activity, or as the very telos of a verb’s activity (as Hegel seems to do), one has a different image for the result. This latter image is that of the fulfillment or completion of the verb’s activity as activity. Put more simply, what this means is that if one takes certain activities or verbs which imply a negation or passing away, say, “dying”, and then turns this verb into a gerund/infinitive, one can then predicate the present tense form of the verb to the gerund/infinitive and ask, “what concept or state of affairs is implied by the statement “Dying dies?” The answer is that the activity (here reified into a gerund) can be understood either as 1) A continual process of dying, or more paradoxically, 2) A gerund which has been predicated in such a way as to imply its own negation or ceasing of activity by means of that very activity that it is. This latter interpretation is exemplified in the perfect tense form of the verb, ‘has died.’ In this second case, the implication of the statement that ‘Dying dies,’ is that ‘Dying has died.’ Though this might appear nonsensical at first, it can be made semantically intelligible. To affirm the truth of the phrase ‘Dying has died’ is minimally to affirm that there is no longer any such activity as dying.  

Examples of categorical self-predication in Hegelian dialectic

36As we said above, Hegel’s logic is never merely formal. Each ‘logical’ move in his dialectic is always intrinsically bound up with the meaning of his terms, as is Aristotle’s. Because of this, the method demonstrated above will not work with just any verb or category. If one were to predicate a gerund like ‘walking’, for instance, with ‘walks’, as Aristotle himself notes, such a sentence does not seem to imply the perfective form ‘has walked’– presumably because the meaning of walking does not have the generalized sense of a negation or a ceasing-to-be . Yet even within this limited scope, we can see this above defined method exemplified in multiple places in the Science of Logic. Let us take two examples.

  • 10  “Ihr Verschwinden ist daher das Verschwinden des Werdens oder Verschwinden des Verschwindens selbs (...)

37Hegel defines ‘becoming’ as: “the vanishing of being in nothing and of nothing in being and the vanishing of being and nothing generally” (Houlgate, 199). Given that ‘vanishing’ is definitionally synonymous with ‘becoming’ for Hegel, Hegel can now treat ‘vanishing’ as an infinitive-noun, predicate it with its own present tense form, such that ‘vanishing (per definition!) vanishes’. Then he can reasonably argue that this definition conceptually leads to ‘the vanishing of the vanishing itself,’10 a process whose outcome gives the ‘stable result’ of a new category - that of ‘determinate being’ or quality.  Hegel’s argument is that when vanishing (as infinitive-noun) fulfills its definition in the present tense, it naturally reaches its perfect form - linguistic and ontological - and has vanished; it is thus conceived to have stopped its own activity of mediation between being and nothing. Hegel can then argue that we have the conceptual accompaniment or idea of a new categoryas a result (ibid, 198-199). Being and nothing then, as the terms connected by becoming, stop vanishing.

  • 11  “Aber es kommt darauf an, ob in der Ansicht beim Sein der Endlichkeit beharrt wird, die Vergänglic (...)

38A second and just as striking an example is found at the point of transition from the category of finitude to that of infinitude. Hegel defines the category of finitude by the activity of ‘ceasing to be,’ insofar as all things finite have an end to them because they cease to be at some given limit (be that a temporal limit, or a spatial limit, etc.). Again, Hegel treats this ‘ceasing-to-be’ as an infinitive-noun, and asks if ‘ceasing-to-be’ should properly ‘cease to be’ at its limit (Hegel, SL, 130)?11 The answer for Hegel is of course yes – and if ‘ceasing to be’ (a.k.a. finitude) should itself cease to be, what is our conceptual accompaniment or implied result? The answer of course is an Aristotelian contrary – the infinite (that which does not cease to be) is the result, for the finite now no longer ceases to be at its limit, but continues past its limit to infinity. If one were to ask where the threshold concept is in this example, we should note that the threshold concept is not to be identified with the ‘limit’ of finitude at which it borders infinity. Rather, it is the finite itself which is functioning as a threshold concept, insofar as it is postulated as having being outside of its limit, while still having being at the limit.

  • 12  If we were to venture a broad and overly generalized claim here, we might venture to say that whil (...)

39This model, as should be apparent, can be fairly easily translated back into the language of Aristotle’s differentia.  For Hegel to define a new abstract category (or in Aristotelian terms, a new higher level species), he needs a genus and a differentia. To do this, he can start with a single category viewed as an infinitive or gerund, and predicate the proper activity of the infinitive (i.e. its proper definitional being) to itself in the present tense. Thus when we take vanishing [of being into nothing/nothing into being] as genus + vanishes (differentia), we get the category of determinate being-and-nothing-together (as quality) which is the new species. This, we assert, is a clear example of how a category can provide its own differentia through a self-predicating process.12

40Thus we finally see that Hegel has shown how interpretation (1) and interpretation (2) of Aristotle’s differentia can work together to produce a novel solution to categorial differentiation. Following interpretation (1), Hegel has shown that the differentia of a category indeed falls under the same genus as the category it differentiates, since the differentia just is the category’s own activity, but under interpretation (2), the differentia will also fall under a different category than the category it differentiates, since as product of the category it is something different from the category itself.

Hegelian categories as necessarily enduring thresholds

41One problem remains with this account. While it ostensibly seems that once a Hegelian category has been determined into its perfected tense we can infer that category’s passing-away (as well as the generation of another category), this cannot be the right interpretation. If it were, Hegel’s system would have only given us one category at most (all previous ones being destroyed at the end of the development). Thus, we must assert that it is not the category that passes away, but the thinker who has passed on to a view of the more determinate whole. The categories as activities in fact cannot be destroyed. One again we have a stark precedent in Aristotle. In his discussion of God as the eternal cause of motion (motion and the universe being co-eternal for Aristotle), he notes that “God therefore…fulfilled the perfection of the universe by making coming-to-be uninterrupted; for the greatest possible coherence would thus be secured to existence, because that coming-to-be should itself come-to-be perpetually is the closest approximation to eternal being” (On Generation and Corruption, II.336b-337a). Here Aristotle anticipates Hegel by nearly two thousand years in predicating an activity of its own infinitive. However, since this infinitive-noun has a positive valence, Aristotle can infer the perpetuity of the activity rather than its negation. Aristotle amends this description with one more thoughtful detail, and it is this latter detail which is the more philosophically powerful one for Hegel. Aristotle says that: “It follows that the coming-to-be of anything, if it is absolutely necessary, must be cyclical, i.e. must return upon itself” (ibid, II.338a). According to Aristotle, this necessary activity we can see physically exemplified in the revolution of the heavens, since those things that depend on the circular motion of the heavens “come-to-be of necessity, and of necessity will be” (ibid, II.338a-338b). Hegel, it seems, postulates that this necessary cyclical movement is also the path of thought itself, such that it must necessarily go out of itself as from a point, and after traveling its circumference, return to its origin of departure.

42This evidence allows us to see that for Hegel, each category is essentially a threshold into another category, either forward or backwards, such that no category has true temporal priority over any other one (only logical priority). In that sense, each of Hegel’s categories has a present existence which excludes other categories only when viewed or thought in respect to former categories, but they also have a perfect past existence (i.e. they have-been and are not presently) when viewed or thought about from the index of later categories. It is then the thinker who perceives a category’s passing away or transition, but in actuality (for Hegel) these categories are necessarily eternally accomplishing and accomplished from the highest categorial perspective.

43Each Hegelian category should then be viewed as part of an eternal form, which Hegel images along with Aristotle as “the circle, the line which has reached itself, which is closed and wholly present, without beginning and end” (Hegel, SL, 149). At the same time, this enclosed activity must be asserted to be both presently occurring, and eternally past. Thus we read in Hegel’s shorter Logic, “The Good, the absolutely Good, is eternally accomplishing itself in the world: and the result is that it need not wait up on us, but is already by implication, as well as in full actuality, accomplished” (Hegel, Logic, 274), or again: “The consummation of the infinite End, therefore, consists merely in removing the illusion which makes it seem yet unaccomplished” (ibid, 274). The illusion we live under, according to Hegel, is that sometimes we see a lower categorial development in its present tense, sometimes its negation or passing away into another form, but usually not their interconnection or their essential relational identity which subsists at a higher level as an eternally active actuality, which he will eventually label the Absolute Idea. Thus Hegel’s answer to the question, “Why is there something rather than nothing,” is the answer which comes out of the depths of Aristotelian circular necessity, which states, “Because it could not have been otherwise, and always has been.”

44It is clear that Hegel’s categorial development of the Concept or Begriff has both the components of connecting and differentiating the categories it produces (by means of the infinitival form viewed as a type of abstract substance), but it also functions as the source of active movement between itself as infinitive and the activities it predicates of itself. This is in fact a threshold concept writ large - so large in fact, that Hegel will identify this self-predicating infinitive as God himself coming to know himself through his own self-differentiating activity. While we will not claim anything as lofty as this, Hegel’s claim should serve as an indication that the scope of threshold concepts are not confined to a certain structural level of philosophical system, and are in fact powerful enough to both generate and unify an enormous amount of philosophical information.  

Transition from threshold concepts to phenomenological soglitude

45We are now in a position to assert that without the use of threshold concepts which define by means of being both the identity and difference of concepts, neither Hegel nor Aristotle would have been able to generate their key philosophical insights. If such threshold concepts were lacking, Aristotle could not have formulated his principle of the scientific definition of entities by means of a differentia, nor could he have explicated his main principle of nature, the principle of movement, which is boththe same as and differentfrom a potentiality and an actuality. Similarly, without the threshold concept of a category’s relation to its own negation, or a category which posits its own fulfillment by predicating itself with its present tense activity, Hegel would have no principle by which to assert the simultaneous identity/differentiation of a category from its opposite, nor the self-determined nature of his logical categories, which is the sole motor of his dialectic. What this essentially means is that threshold concepts, as we have defined them, play a much more structurally important role in philosophical thought than is normally noticed, and operate by means of an intelligible, meaning-based triadic logic (though one is tempted to say ‘trinitarian’, since unity is just as essential as the numerical threefold structure).

46In noting this, we will end with a final unresolved problem. We have been attempting to explicate various textual exemplifications of major threshold concepts in Aristotle and Hegel, yet we have conspicuously left unaccounted what a threshold concept might be apart from its relation to a given philosophical system of terms and ideas. Does it have a logical existence? Is it merely a mental concept? Can we define it in isolation from the type of textual instances which we explored above? Hegel’s answer, which we did not endorse, was that these threshold concepts were nothing more than the shapes of thought itself, explained by means of an absolute idealism in which every person’s thought is literally a part of the thought of the God himself as the Absolute Idea thinking through the individual. A more somber potential answer to this question comes from Edmund Husserl, who may be said to provide a theory of a philosophical threshold which is justified not just by textual or merely formal instances (as we have done here), but is rather explicitly identified in a more existential register as individual consciousness itself.

47In Husserl’s later phenomenology, he argues that consciousness is not only constituted by its intentional act/intentional object structure, as we mentioned above, but also that every conscious act, even when directed at a singular object, inevitably experiences the object within a horizon of ‘co-present surroundings [Umgebung],’ which themselves are interpreted as a further plurality of possible objects of investigation (Husserl, Experience and Judgment, 103). The horizon itself is interpreted as the conscious anticipation of further temporal experiences of objects. This phenomenological postulation of consciousness as horizonally-structured, such that every object occurs in a horizon of further possibilities and actualities that point beyond any perceived object itself, is essentially the claim that consciousness is itselfthe threshold which connects and distinguishes itself from objects, and objects from further objects. Thus, our initial characterization of the phenomenon of conceptual accompaniment that occurs in attempted definition, or which occurs whenever we notice an idea which is immediately followed by a further correlative idea, is by Husserl explicitly identified with the essence of consciousnessitself. To investigate this very relation qua relation of ourselves to objects in ourselves as horizon, Husserl finds to be a particular mode of consciousness which is identified as “relational contemplation” (ibid).

48If this identification of the threshold with consciousness itself (or its essential structure) is correct, and we offer it here only as a suggestion, we can see that our original definition of a threshold concept, which we exemplified through textual and structural examples in Aristotle and Hegel, contains the possibility of being widened into a broader theory of threshold-thinking, a theory of the individual qua threshold, which could be accurately captured in neologism in the term soglitude, derived from the Italian word for ‘threshold’ – ‘soglia’ (pronounced similarly to the English ‘solitude’, insofar as the ‘g’ in ‘soglia’ is left un-vocalized). Semantically, the neologism could be defined in reference to the moment when an individual’s consciousness realizes that its movement towards an accompanying thought, or its experience of phenomenological horizon, is its proper identity. Individual consciousness, the nature of which may be said to be mirrored in our philosophical concepts, always seems to be bordering on an ‘other’ to account for its experience of conceptual movement, even in its most isolated moments. However, if this is the fundamental situation of consciousness, then Aristotle’s initial characterization of the physical threshold must be incorrect as a characterization of the threshold as consciousness. Rather, the formula should be inverted; we must be open to the possibility that it is not the position which produces a threshold, but rather the threshold that produces the position. This idea, however, must be investigated further.

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Bibliographie

ARISTOTLE, 1984, The Complete Works of Aristotle (Volume One and Two), Edited by Jonathan Barnes, Princeton, Princeton University Press.

HALPER, Edward, 2004, Form and Reason: Essays in Metaphysics, New York, State University of New York Press.

HEGEL, G.W.F., 2005, Hegel’s Logic, Translated by William Wallace, Oxford, Clarendon Press.

HEGEL, G.W.F., 1989, Hegel’s Science of Logic, Translated by A.V. Miller Amherst, New York, Prometheus Books.

HUME, David, 1999, AnEnquiry Concerning Human Understanding,  Edited by Anthony Flew, Chicago and La Salle, Open Court Publishing Company.

HOULGATE, Stephen, 2006, The Opening of Hegel’s Logic, West Lafayette, Indiana, Purdue University Press.

HUSSERL, Edmund, 2001, Logical Investigations (2 Volumes), Translated by J.N. Findlay, London and New York, Routledge.

HUSSERL, Edmund, 1997, Experience and Judgment, Investigations in a Genealogy of Logic, Translated by James S. Churchhill and Karl Ameriks, Evanston, Northwestern University Press.

KANT, Immanuel, 1996, Critique of Pure Reason, Translated by Werner S. Pluhar, Indianapolis/Cambridge, Hackett Publishing Company, Inc.

PLATO, 1997, Plato: Complete Works, Edited by John M. Cooper, Indianapolis/Cambridge: Hackett Publishing Company.

PRIEST, Graham, 2004, Beyond the Limits of Thought, Oxford, Oxford University Press.

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Notes

1  “And the same is true in the other cases, e.g. if the threshold is characterized by its position, the position is not produced by the threshold, but rather the latter is produced by the former” (Metaphysics VIII.1043b).

2  See for instance, (Priest, 2004) and the idea of dialethism.

3 “δῆλον ὅτι ταὐτὸν τἀναντία ποιεῖν ἢ πάσχειν κατὰ ταὐτόν γε καὶ πρὸς ταὐτὸν οὐκ ἐθελήσει ἅμα.”

4 “τὸ γὰρ αὐτὸ ἅμα ὑπάρχειν τε καὶ μὴ ὑπάρξειν ἀδύνατον τῷ αὐτῷ καὶ κατὰ τὸ αὐτό καὶ ὅσα ἄλλα προσδιοπρισαιμεθ’ ἄν, ἔστω προσδιωπρισμένα πρὸς τὰς λογικὰς δυσχερείασ’.”

5  The following paragraph is heavily indebted to Edward Halper’s essay, “Hegel and the Problem of the Differentia,” in his book, Form and Reason: Essays in Metaphysics. Though Halper’s analysis of Aristotle’s differentia problem and how Hegel offers a working solution is correct, Halper’s assessment of Hegel’s solution misunderstands the nature of Hegel’s dialectic at a crucial point when he argues that Hegel’s solution “destroys” the categories. I will argue below that although Hegel’s narrative explanation of the dialectical development seems to destroy the categories, his philosophical explanation of the Idea as self-determined determinacy accounts for their continuous preservation.

6 “αὑτὸν ἄρα νοεῖ, εἴπερ ἐστὶ τὸ κράτιστον, καὶ ἔστιν ἡ νόησις νοήσεως νόησις.”

7 “οὐχ ἑτέρου οὖν ὄντοσ τοῦ νοουμένου καὶ τοῦ νοῦ, ὅσα μη ὕλην ἔχει, τὸ αὐτὸ ἔσται, καὶ ἡ νόησις τῷ νοουμένω μία.”

8  This Hegelian mode of explanation one might tentatively call an ontological mereology (the study of the wholes and parts of being/beings), in the sense that a given categorial concept (viewed as a whole) may be related to its own definition (i.e. the other categorial parts and wholes which determine it) in various ways, just as a circle may be related to the semi-circle which resides in it. Hegel’s most radical whole/part claim is that a concept’s determinate negation, i.e. what a concept is not in terms of its semantic opposite, is in fact a proper part of that concept’s definition (as a whole). Thus the Hegelian category ‘something’ cannot be defined unless the category which it is opposed to, the category ‘other’, is included in definition of ‘something’ as a non-being-part of the definitional whole.

9 “οὐ γὰρ ἅμα βαδίζει καὶ βεβάδικεν, οὐδ' οἰκοδομεῖ καὶ ᾠκοδόμηκεν, οὐδὲ γίγνεται καὶ γέγονεν ἢ κινεῖται καὶ κεκίνηται, ἀλλ' ἕτερον, καὶ κινεῖ καῖ κεκίνηκεν: ἑώρακε δὲ καὶ ὁρᾷ ἅμα τὸ αὐτό, καί νοεῖ και νενόηκεν. τὴν μὲν οὖν τοιαύτην ἐνέργειαν λέγω, ἐκείνην δὲ κίνησιν.”

10  “Ihr Verschwinden ist daher das Verschwinden des Werdens oder Verschwinden des Verschwindens selbst.”

11  “Aber es kommt darauf an, ob in der Ansicht beim Sein der Endlichkeit beharrt wird, die Vergänglichkeit bestehen bleibt, oder ob die Vergänglichkeit und das Vergehen vergeht.”

12  If we were to venture a broad and overly generalized claim here, we might venture to say that while Heidegger is the philosopher of the presence of the present tense, Hegel may be said to be the philosopher of the perfection of the infinitive through the present tense.

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Référence électronique

Jason Carter, « The notion of Threshold: an investigation into conceptual accompaniment in Aristotle and Hegel », Conserveries mémorielles [En ligne], #7 | 2010, mis en ligne le 14 avril 2010, consulté le 21 octobre 2017. URL : http://cm.revues.org/431

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Auteur

Jason Carter

received his B.A. in Philosophy at the University of Georgia in 1999. He spent four years touring the U.S. with his band in pursuit of musical fame before coming back to his first love, philosophy. He is currently living in Athens, Georgia with his beautiful wife Lauren, and pursuing his Ph.D in Philosophy at the University of Georgia, with a focus on medieval theories of intentionality and early modern theories of intellectual intuition. He currently works as a teaching assistant in the philosophy department.

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