What Is Knowledge? An immanent critique of education

The vast majority of possibilities are empty possibilities (leeren Möglichkeiten), which is to say, they are only possible (never actual), and for that very reason, equally impossible. But this impossibility is not appreciated by the pupil who memorises, for he is incapable of discriminating between empty possibilities and what is actually possible. Insofar as no immediate contradiction is present, all possibilities are to be taken equally seriously. We can easily see how all sorts of absurdities follow from this:

“Now since every content can be put into this form [of abstract identity] (and that means merely that it is detached from its relations), even the most absurd and incongruous things can be considered possible. It is possible that this evening the Moon will fall to the Earth, since the Moon is a body separate from the Earth and, therefore, can fall down just as much as a stone thrown into the air can. It is possible that the Turkish Sultan becomes Pope since he is a human being and, as such, can convert, become a Catholic priest, and so on.”6placeholder

That the Moon may collide with the Earth or that the Turkish Sultan may become Pope both seem logically possible on the surface, but only insofar as they are taken abstractly and individually outside of any conceptual connection with anything else. In analysing the matter further, however, one soon discovers the implicit contradiction in the two propositions: the former cannot be consistent with the laws of motion, whereas the latter would violate Sharia law with deadly consequences. Of course, none of this is present to the mind that only knows what has been given to it without any further reflection.

The naïve pupil may naturally know many actualities as a result of his education, but what qualifies him as someone truly educated is rather the kind of possibilities he entertains given his knowledge. “The more uneducated someone is, the less familiar he is with the specific relations between the objects to which he directs his attention and the more inclined he is to entertain all sorts of empty possibilities.”7placeholder In this sense, we could say that the pupil who memorises possesses only knowledge of the actual but no actual knowledge. Any child who is literate can memorise the various forms of energy in physics—kinetic, potential, nuclear, chemical, etc.—but this in itself affords no real knowledge. Soon the pupil will ask why there isn’t any mention of “flying energy,” or “talking energy,” or “thinking energy,” which, incidentally, are all equally valid forms of energy insofar as no principle of division has been specified. This is thus the result of construing knowledge as a ready-made object that can be transmitted and accumulated: everything falls into contingency. The pupil may know that what he learns is true, and yet, he cannot explain why things couldn’t equally be otherwise. To demonstrate necessity, then, requires us to reflect on the grounds and premisses that constitute our knowledge. However, this cannot be accomplished within the stage of memorisation, wherein the outer form of knowledge (individual and immediate) is still inadequate to its own inner content (universal and mediated), instead, it belongs properly to the task of understanding.

Understanding

We now turn to Hegel’s discussion of representation:

“Representation has such sensory material [Stoff] for its content, but posited in the determination of being mine, i.e. the determination that such content is in me, and of universality, the relation-to-self, simplicity. – Yet representation also contains, in addition to that sensory material, material that has originated from self-conscious thought, such as the representations of what is right, ethical, religious, or even of thought itself . . . . [T]he distinctiveness of the representation is generally to be located in the fact that in it such content at the same time remains isolated. To be sure, the right, the just, and similar determinations do not occupy the sensory [positions of] being-outside-of-one-another proper to space. They may indeed appear to be somehow successive in time, but their content as such is not represented as being encumbered by time, as transient and alterable in it.”8placeholder

Already it is evident how this second stage corresponds to what we generally call “understanding.” Representation sublates the sensory by positing what is sensory in the form of being mine. This audibly echoes Kant’s famous proposition that “The I think must be able to accompany all my representations.”9placeholder It is only at this stage that thought—albeit purely subjective thought (we will return to this)—enters the picture; in memorisation, there was neither thought nor thinking, just passive reception. If previously the sensory object confronted me only as an opaque abstract otherness, now it is finally something I can call my own. For, it is impossible to think, or more precisely, to represent, without the naturally accompanying recognition that this representation is a product of mine. The same logic applies to understanding: understanding is ipso facto my understanding.10placeholder It is impossible for me to possess somebody else’s understanding without also positing it as being mine. This is why understanding cannot be imparted by means of direct transmission: the teacher’s understanding of a subject is necessarily the teacher’s own understanding which can only be communicated indirectly, through the mediation of explanations, analogies, examples, etc., in an attempt to bring about an equal understanding in the pupil’s mind. The best one can hope for in this case is for the two understandings ultimately to correspond, yet this is different from saying that they are the same understanding.

According to Hegel, the form of representation can encompass two kinds of content: sensory material and thought material. The two can be loosely understood as the distinction between a posteriori and a priori content respectively, or as that which is derived externally (through the senses) and internally (through self-conscious thought). Despite their differences, however, both contents share the same fundamental form as their common mark of finitude. As Hegel explains, insofar as they remain representations, they are isolated. Let us unpack what he means by this from two perspectives: analysis and composition.

Mind maps, the tool of choice for analysis, are all the craze in pedagogy today. Texts are now widely regarded as a primitive form of expression inasmuch as they contain too much undifferentiated and undigested material. Language is, by its very nature, verbose, and this verbosity has to be minimised as much as possible in order to maximise communicative efficiency. For it seems today that language is nothing more than this—an instrument for communication. Accordingly, most are trained from a young age to read with the sole intention of extracting from within a text a set of essential points and arguments made by the author, as if they were treasures deliberately buried in the sand. They are taught, rather dogmatically, that “true reading,” reading for comprehension, consists chiefly in the ability to extract and reconstruct the author’s argument in a “simpler,” more “straightforward” or “natural” logical form. This methodology obviously rests on numerous presuppositions regarding the relationship between thought and language, none of which are self-evident. For a long time, just the opposite view was maintained: language was not seen as a “code” to be deciphered but as the form or essence of things. Its nature was to reveal, not to cloak, in the sense that language is today compared to clothing, as a means to “dress up” one’s naked ideas. No doubt, this is, at least in part, a consequence of the hegemony of the sciences and their ideals of formalisation. There seems to be a strong tendency today to refashion non-STEM subjects in the image of the sciences, employing graphs, diagrams, and taxonomies, which in the end, amount to nothing more than a bad aesthetic.

Related to the instrumentalisation of language is the dubious presupposition that “points” are the fundamental units of thought. Indeed this is what warrants the usage of mind maps since they are designed to decompose complex information into a set of isolated elements and their relations. However, the problem is that thoughts represented in this fashion are ineluctably fixed in external opposition. The elements visually organised remain outside of one another in both a literal and figurative sense: the points literally exist in a spatial relation with each other in the diagram, but they are also figuratively “standing side-by-side in [their] indeterminate space, connected only by the bare also.”11placeholder As valuable as mind maps can be, it is important to recognise the reification of thought that they enact. The kinetic process of thinking is here reduced to stationary points on a diagram, which, to be sure, bear relations to each other (a hallmark of thinking), albeit only external relations insofar as their isolation precedes their relation. In short, mind maps fail to depict thought in action, which is why they are suitable exclusively as tools for understanding.

The same approach is reflected in composition. The predominant method of writing taught to pupils once again consists in carving up the essay into inert and atomised components: point, explanation, evidence, link, etc. This basic formula is then repeated for three or more body paragraphs which are in turn sandwiched in between a standard introduction and conclusion—there we have what school calls an “essay.” In a way, writing is importantly distinct from mind-mapping insofar as the latter represents thought visually, whereas the former employs language as its means of representation. With language, the principal form of representation is no longer space but rather time. Naturally, words occupy space on a page, but insofar as one grasps what they express, language appears as a linear succession in time, as the temporal progression of sense-making. Meaning compounds and sediments as time elapses: the next sentence can only be understood on the basis of the last, and so on. And yet, the succession of sentences found in the average school essay is purely a mechanical succession—the “content as such is not represented as being encumbered by time, as transient and alterable in it”—there is no coordination between the succession of sentences in time and the progressive unfolding of thoughts in words. This is hardly surprising given that pupils are not taught to think as they write and to write as they think, but instead to first “come up” with points spontaneously, to brainstorm haphazardly, to pluck ideas from thin air, and only subsequently to glue them all together with “transition words.” The widespread fetishisation of transition words in writing courses should, above all, be interpreted as a symptom—as the return of the repressed movement of thought. What are transition words? Despite appearing as one among other components of writing, it is clear that their function subverts this identity. The function of transition words is precisely to embody movement as such, as if it were itself an isolated component that could be appended to a sequence. In this sense, they are symptomatic of the method as a whole: in order to suture the array of purely static components, movement itself must finally be included in the guise of its opposite—as inert transition words that simultaneously connect sentences and expose their lack of connection in the first place.

Indeed, the isolated form of representations may appear similar to the determinations of the sensory at first, yet the two are importantly distinct. The passage from the sensory to representation can be seen as a contradictory one: representation is at once greater and lesser than the sensory. It is lesser insofar as it is less clear and determinate than the latter. The object of sensation always appears to us in greater resolution and detail than what our representation manages to capture. The image we have of it is uncompressed since all of its specific individual qualities stand immediately before us. However, not all of this rich material survives the conversion into representation; indeed, much data is lost along the way. The object is reduced to its essential determinations, whereas the rest is omitted, i.e., it is contracted into the form of simplicity, enduring only as simplified, abbreviated, and abstracted. Yet in spite of this, or more precisely, because of this, representation transcends the mere sensory, for less turns out to be, in fact, more. The power of the understanding resides precisely in its ability to compress the multiplicity of data into a manageable set of stable regular patterns. This is achieved through the dual means of abstraction and differentiation whereby the wheat is separated from the chaff, the superfluities and redundancies are eliminated, and the constant is extracted pure and unmixed from the flux of variables. The pupil stops memorising and starts understanding precisely when he is capable of judging for himself what is important and what is negligible, what is universal and what is particular, what is essential and what is alterable. Not everything has to be retained exactly as it is. In this sense, understanding is the negation of memorisation: all understanding begins with the act of forgetting.

It is along these lines that “Representation here meets with the understanding [Verstand],” which Hegel further qualifies, “differs from the former only in that it posits relationships of the universal and the particular or of cause and effect, etc.”12placeholder The understanding is responsible for the production of these finite isolated representations, but still, it is not reducible to them insofar as it “establishes relations of necessity among the isolated determinations of representation.”13placeholder This is an improvement upon the contingency we saw in the previous stage. In memorisation, knowledge of the actual, insofar as it is grasped immediately, is de-centred by its own self-posited possibilities such that it is itself only something possible. These self-posited possibilities are not real possibilities but mere logical possibilities that vanish as soon as we direct our attention to the embeddedness of facts within the concrete world of appearance. Every concrete existence, writes Hegel, exists in “a world of reciprocal dependency and an infinite connection of grounds and grounded entities.”14placeholder And the moment “we delve into the determinations, the circumstances, the conditions of a fact in order to discover its possibility, we do not stop at this formal possibility but consider its real possibility [reale Möglichkeit].”15placeholder The consideration of real possibilities, as opposed to empty ones, is the first sign of genuine understanding and knowledge. It is evidence that one has gone beyond facts in their immediacy to the grounds that mediate and determine them. If one continues in this direction, then one is only a single step away from also grasping the necessity (Notwendigkeit) of certain facts, why they are indeed the way they are and not otherwise. When we supplement a certain motion, e.g., the Moon colliding with the Earth, with the laws of motion and the relevant data that determine it, we discover that it is only the current orbit of the moon that is really possible, and so couldn’t be otherwise, which also amounts to saying that the current orbit of the moon is necessary given all the conditions. This then led Hegel to conclude: “Real possibility and necessity are, therefore, only apparently distinguished; theirs is an identity that does not first come to be but is already presupposed at their base.”16placeholder

We discussed earlier that knowledge is always universal, and how in spite of this or rather because of it, it initially appears as something singular and contingent, as one particular universal among other universals. Intelligent educators are well aware of this reversal, which is why they refrain from testing pupils directly what was taught in class. This is to prevent them from learning by heart all the ideal answers to questions likely to be asked in the examinations—a practice as meaningless as it is popular. Instead, it is much more preferable for teachers to test their pupils’ knowledge of the universal via the particular case. The pupil is here required to exercise not his memory but instead his judgment and skills of application. The two abilities come hand in hand: to judge means to recognise and subsume a particular case under a universal, which is in turn followed by applying the universal idea to illuminate its particular example. This approach for testing may seem odd at first due to the asymmetry of the universal and the particular: the universal is always more encompassing than the particular, in fact, it is practically inexhaustible insofar as it can be instantiated in an infinite amount of particular cases (hence the endless supply of exam questions). An infinite divide thus seems to separate the two sides: no particular case can instantiate the universal in its totality, and no universal can be exhausted through a particular. And yet, one learns from pedagogy that it is only through the particular case that one can accurately gauge the pupil’s understanding of the universal; one cannot aim at universality directly, for that is surely the way to miss it. It is only by creating this differential between the universal and the particular that the universal exhibits its universality, without which it would revert to something particular and arbitrary. This is why, in keeping with the dialectic of universality, Hegel writes: “The universal determines itself, and so is itself the particular; the determinateness is its difference; it is only differentiated from itself. Its species are therefore only (a) the universal itself and (b) the particular.”17placeholder

Knowledge at this juncture thus bifurcates into knowledge qua universal and knowledge qua particular. The French at this point may have a better chance at conceptualising this bifurcation since they have more than one word in their language for referring to knowledge, the two most notable terms being “le savoir” and “la connaissance,” though the distinction is hard to maintain in English. Colloquially speaking, le savoir refers to a form of knowledge that is formally articulated and acquired through study (e.g., knowledge of the Pythagorean theorem), whereas la connaissance refers to knowledge that a particular individual has internalised (e.g., knowledge of how to speak a language). For our purposes, however, we will have to redefine the two slightly differently.

Perhaps it will be helpful to compare the difference between the two knowledges with another pair of French terms. Consider Saussure’s famous distinction between langue and parole. Both terms here refer to language but with profoundly different senses: langue is used to refer to language as an abstract system of signs (the proper object of structural linguistics), whereas parole refers to language as actually spoken by individuals in concrete situations. It would appear at first that the two aspects complement each other nicely, but the reality is not so straightforward. As Deleuze correctly observed18placeholder, linguistics, in order to study language scientifically, must necessarily posit language as a static system in equilibrium, that is, from the perspective of universality. All the variations and difference, then, are attributed to individual acts of speech, whose status is thereby reduced to mere epiphenomena of language. Once the system of language has been laid down, along with its grammatical rules and signifying relations, then the remaining role for speech can only be secondary, amounting to nothing more than the mere choice of words and ordering of sentences according to subjective whim. This view of language is certainly a very powerful one, and indeed once a revolutionary one, but linguistics is not the be-all and end-all of language. In fact, the entire situation is inverted if we simply look through the other end of the telescope, that is, from the viewpoint of literature. Deleuze was right to point out that linguistics and literature are, in reality, opposed to each other despite sharing the same subject matter. The student of literature does not study language as an abstract structure but, first and foremost, as a text—a text that has been written by a real writer. The tables here are completely turned: speech is no longer something subordinated to language (not entirely, at least); on the contrary, language is something that is shaped in and through speech. It owes its spirit, so to speak, to the texts that comprise its literary canon.

As we can see, the division of language into langue and parole does not result in the two aspects coexisting in harmony but instead gives rise to an antinomy. We may ask, in the simplest terms, from the standpoint of linguistics: “What is Proust without French?” But we may equally ask from the standpoint of literature: “What is French without Proust?” As we shall see, the division of knowledge into le savoir and la connaissance similarly gives rise to an antagonism that fails to be reconciled.

Le savoir is perhaps best encapsulated in English by the expression: body of knowledge. Indeed, this commonplace expression is rather peculiar upon closer examination, for it borders on an oxymoron: combining thought and extension into thought as extended. How should we make sense of this? To say that le savoir is a body of knowledge means that it is substantial knowledge, collective knowledge, knowledge that is inscribed and recorded in the locus of the Other19placeholder, endowed with an existence of its own, external to our minds. It is composed of thought insofar as it is the externalisation/alienation (Entäußerung) of representation from itself. It is extended inasmuch as it embodies an accumulated archive, or even better, a corpus—meaning “body” in Latin—to which many have contributed throughout the course of history. In its most natural presentation, le savoir takes the form of textbooks and encyclopaedias: knowledge is here exposited as a self-identical system of ready-made elements that confronts the pupil as radically Other, as a substance to be learnt, understood, and mastered. It is knowledge expounded from the standpoint of disinterested objectivity, addressed to a universal audience without any specific interests in particular. The mode of presentation is thus thoroughly decontextualised and deindividualised, emphasising only polished results that have been purified from the circumstances from which they arose. Accordingly, le savoir is never curious, never fascinated, never confused, never stumped, never caught in the midst of thinking, for everything has already been thought over and smoothed over. As merely a body of knowledge, it has no inherent purposiveness, no thought or movement, no tension or problem; it is, rightly put, headless.

Whatever has been expunged from the point of view of universality—purpose, thought, problems, context, etc.—now returns as relegated to the viewpoint of particularity, which is, in turn, relegated to the domain of subjectivity. La connaissance is knowledge of the particular case as determined by the universal. It follows that its status is merely accidental with respect to le savoir: it has no real implications for knowledge itself, only for the subject immersed in concrete particular situations. The particular case has long been solved by le savoir, albeit only implicitly; the task remains for the subject to uncover this implicit solution through understanding the universal, judging the particular, and applying the universal to the particular—three steps that we can group under the heading of “subjective thinking,” i.e., psychological thought. Thus la connaissance is placed in clear opposition and subordination to le savoir, the relation between the two being one of substance and accident. The application of knowledge “for us,” for the purpose of our concrete problems, nevertheless remains external to knowledge, as it were, “in itself,” in its universal formulation. The latter persists in its abstract self-identity beyond the concrete diversity of its instantiations, whose status is thereby reduced to merely subjective and contingent epiphenomena.

However, there is an obvious reason why this clear-cut division is untenable: it pretends as if a body of knowledge somehow acquires its existence independently of us, as if we didn’t establish it in the first place, as if we are merely subject to and didn’t partake in the construction of knowledge itself. But this is exactly what happens in practice: textbook knowledge often appears to pupils as knowledge that has fallen from the sky, insofar as its origins have been thoroughly effaced and thereby mystified. Nowhere is there any admission of uncertainty or ambiguity; everything is known categorically—an illusion that ultimately leads young minds to draw the worrying conclusion that knowledge is always categorical, or worse, that knowing consists in certainty. The truth is, however, that it is impossible to separate le savoir from la connaissance, or more generally, to hold apart knowledge in its positivity from its moment of negativity or its becoming-other.20placeholder To do so would be to deprive knowledge of its creativity and genesis. Knowledge, in this case, can only be given from without by some higher authority, whereby the task of the pupil is restricted to one of passive understanding—passive insofar as it is merely receptive.

Accordingly, the subjectivity that the present stage of knowledge produces is none other than the expert—a figure that must be carefully distinguished from the intellectual. The expert represents the ideal of absolute mastery, the subject who has been fully absorbed into substance. The expert is someone who has fully appropriated and mastered le savoir and whose interests now turn to its applications in concrete circumstances and for particular purposes. In this sense, the expert does not produce any new knowledge; his only role is to utilise established knowledge to solve present-day problems. This isn’t to say that experts are incapable of conducting research however, only that such research is often directed towards the investigation of second-rate problems which does not augment le savoir but only expands la connaissance. Such research may end up being significant for specific purposes and particular interests, but its implications would be relatively insignificant for knowledge itself.21placeholder The production of genuinely new knowledge is generally reserved for the work of the intellectual, who, in many ways, proceeds contrary to the expert. Where the expert stands firmly, the intellectual senses the ground tottering. Where the expert knows for certain, the intellectual ventures to think, and think again. No new knowledge can be produced under the subjugation of la connaissance by le savoir, or by opposing thought to knowledge, particular to universal, accident to substance, negative to positive, difference to identity. The consequence of this series of oppositions is the complete estrangement of subjectivity from its essential role in the genesis of the objective, to the extent that it can no longer recognise itself in this otherness. To affirm knowledge is to deny one’s own subjectivity—this is the alienating conclusion we are forced to draw if we remain within the stage of understanding. Its error resides in the refusal to confront the contradiction that underlies the objective and which betrays the insubstantiality of knowledge—that it owes its genesis to the subjective and is furthermore something in itself subjectivised.

Thinking-Through

After representation finally comes thought. Although there is much talk about the importance of “critical thinking” in education today, not nearly enough has been said about the radical implications of thought, as well as the complications of embracing objective thought in education (perhaps because they’ve yet to think it through). The difficulty lies in the antagonism between knowledge and thought. Pupils are not supposed to think things through; they are supposed to know by reflex, to think on their feet. They are expected to respond with the right answer without delay when called upon in class. The “smartest” pupil is hence the one who has to think the least, the one who always knows the right answer without fail. Stopping to think and to ponder instantly betrays signs of dimwittedness or obtuseness. For, if one knew, then there would be no need to think. And conversely: if one thinks, then it means that one does not yet know. There is therefore no coincidence between knowing and thinking, between knowledge and thought; their relation is one of disjunction.

This disjunction has its roots in the rigid dichotomy set up by the understanding (Verstand). Here knowledge is placed firmly on the side of substance, of positive self-identical being, to the exclusion of thought qua subject which stands for the moment of negativity, of doubt, of scepticism, of becoming. The two poles are thus held apart, substance and subject, each repelling the other and thereby falling into abstractness. This brings us to Hegel’s enigmatic formula in The Phenomenology of Spirit. There he writes: “everything turns on grasping and expressing the True, not only as Substance, but equally as Subject.”22placeholder But what does this mean exactly? How do we think the two extremities together? In reality, there is no le savoir without la connaissance (no knowledge that falls from the sky) and no la connaissance without le savoir (no knowledge without objective externalisation); the two are strictly interdependent. But still, this doesn’t go far enough yet. A reciprocal relationship is not yet a speculative one insofar as it unites opposite poles without transforming them. It isn’t enough to construe negativity as the principle of the positive’s genesis and evolution insofar as this negativity in the end concerns only the subject whose task it is to update obsolete knowledge. The point isn’t simply that knowledge has no independent existence out there waiting to be discovered, that it is constructed and shaped through our subjective activity, grounded in our understanding, for this would amount to a conflation of the subjective process and its objective result. Indeed, the result depends on the process without which it would be a caput mortuum just as much as the process depends on the result without which it would be an aimless drive, but there is no reason why after the process is completed, the result shouldn’t achieve an independent existence separate from the process. Thus, the reconciliation of substance and subject cannot consist in the mere acknowledgement of the interdependence of the two sides; rather, one must affirm the inherent subjectivity of the objective itself, how substance has its own moment of becoming subject, how knowledge by itself is engaged in a process of understanding itself. Therein resides the proper distinction between understanding and thinking-through: the latter concerns the objective expository process of knowledge, the self-movement of the true.

To accomplish the passage from substance to subject therefore requires negativity to already be at work in substance itself, derailing it from within. But where would we locate such a principle of negativity within knowledge itself? Precisely in the existence of what can be rightfully given the term “problems,” by which we mean something quite specific. Problems—in the strict sense of the word—must never be conflated with mere exercises or puzzles. Exercises refer to practices designed to train the application of a specific set of knowledge, skills, or techniques that one has already learnt. Textbooks are full of these, which can be found concluding every single chapter. These exercises are, as a rule, trivial insofar as they amount to purely ad hoc problems fabricated with the sole intention of forcing pupils to recall knowledge and/or practise techniques which they’ve been taught in class, so that, by the end, everyone “will have at least learnt something.” Puzzles, on the other hand, are a little less trivial since they usually require additional (a priori) reasoning skills. It is because of this that they are found most abundantly in mathematics and the sciences. For better or for worse, being “good” at mathematics or science in school is largely characterised by nothing more than the ability to solve a range of “math-themed” or “science-themed” puzzles. The difficulty of puzzles, in general, can vary depending on the occasion: school-level puzzles are usually quite simple insofar as they primarily involve methods that will have been covered in class, whereas competition-level puzzles can be significantly more challenging in that they are designed to be unsolvable by standard techniques, thereby compelling participants to devise new and creative solutions on the spot. But the logic in both cases is identical: puzzles are always engineered to obscure the path to the solution as much as possible, ensuring that one must jump through a great number of hoops before arriving at the answer. The setup is thus completely artificial and contrived. The difficulty resides solely in straightening out the distortion fabricated by the puzzle, which eventually leads to a simple answer that is meaningless in itself. This is why, as challenging and stimulating as they may be, puzzles cannot be considered problems in the strict sense; they merely have the property of being tricky, which is not to be confused with being profound.

The question now runs: what constitutes a real problem then? It has nothing to do with breadth or depth, difficulty or scope. There is no reason why a problem cannot be simple or circumscribed. Instead, the most fundamental determination of a problem is to be sought in the relationship it entails between the subject and knowledge. Problems stricto sensu are, first and foremost, problems that concern the order of knowledge and not the subject. This is in sharp contrast with exercises and puzzles which only ever concern the subject who attempts to solve them. The purpose of an exercise is always to test the pupil’s grasp of the material, just as the purpose of a puzzle is always to challenge the pupil’s ingenuity and acuity—which is accordingly why both of them are narcissistic in their structure. What is at stake in these pseudo-problems is always the competence or ability of the subject: it’s all about whether or not I can solve it. As Hegel put it well: “For instead of getting involved in the real issue, this kind of activity is always away beyond it; instead of tarrying with it, and losing itself in it, this kind of knowing is forever grasping at something new; it remains essentially preoccupied with itself instead of being preoccupied with the real issue and surrendering to it.”23placeholder True problems require one to surrender to the matter itself, not to assert one’s genius over it. This is not to say that exercises and puzzles don’t have their place. After all, teachers assign their pupils exercises not because they pose intriguing problems but because they need to assess them. Likewise, we do not deny that puzzle-solving can be greatly rewarding and addicting for those who have an affinity for it; yet it is important to recognise at the same time that this enjoyment has virtually nothing to do with knowledge and everything to do with one’s private experience. For these reasons, exercises and puzzles belong to the earlier stages of memorisation and understanding, and have yet to ascend to the standpoint of thinking-through, insofar as they separate substance from subject.

A real problem is never a subjective problem but always an objective one. A problem is that which problematises its object, or better still, subjectivises its object. Here it is no longer a question of an empirical subject who is called upon to solve the problem but of knowledge itself which rises to the occasion by becoming subject. “What has just been said can also be expressed by saying that Reason is purposive activity.”24placeholder That is: purpose grasped from the standpoint of reason (Vernunft) does not impose itself upon knowledge from without, as a consequent application of knowledge by the subject, but on the contrary, inheres within knowledge itself as its principle of negativity. Knowledge does not exist naturally as a “headless” corpus but as a substance that thinks through problems and thus reveals itself to be subject. To borrow an example from mathematics—which isn’t very difficult given the history of mathematics is full of great problems—a real problem would be something like the one famously posed to Leonhard Euler, namely whether it is possible to cross each of the seven bridges of Königsberg once and only once without doubling back. In what sense is this a real problem? How does it differ from an exercise or a puzzle? Precisely in the manner that it problematises the field of mathematics itself such that either new lasting insight is produced or a new branch of mathematics has to be invented. And as we all know, this was exactly what happened. It was Euler’s celebrated analysis of the Königsberg bridge problem that led to the invention of graph theory—the study of networks of interconnected objects called graphs. But there is nothing special about the particular example we picked; in a way, every branch in mathematics begins with and subsists on the engagement with objective problems. One is therefore fully justified in saying that it is only in and through such problems that mathematics qua subject comes to know itself in and for itself.

And yet, upon translation into le savoir, into textbook knowledge, we find that these problems are not foregrounded but instead erased. They must be erased in some sense if mathematics is to appear as a substantial body of knowledge. Hence, instead of taking Königsberg as its point of departure, the textbook presentation of graph theory begins with the definition of a vertex, the definition of an edge, and the first theorem which states that the sum of the degrees of the vertices is always twice the number of edges. This is followed by a set of examples and exercises contrived to “apply” this knowledge, in a desperate attempt to fabricate a sense of utility and significance that is absent. While the pupil in this case may have no issues in understanding the necessity of this knowledge, this necessity only amounts to what Hegel terms “relative necessity” (relative Notwendigkeit), for “it has a presupposition from which it begins; it takes its start from the contingent.”25placeholder “This first becomes apparent because real necessity, although something necessary according to form, is still something limited according to content, and derives its contingency through the latter.”26placeholder Put differently: although the pupil understands why he has to multiply by two here and divide there, he hasn’t the faintest clue why anyone would invent such an elaborate game or study such arbitrary relationships. He will confess to us that, even having understood everything, he does not truly understand what he understands. The necessity that he grasps pertains only to the formal characteristics of the body of knowledge, i.e., its internal consistency and validity, it does not extend to the content of the knowledge, i.e., to the facts or the terms in which the knowledge is stated, which nevertheless appears to him as contingent. The body of knowledge itself lacks a raison d’être. Yet this reason isn’t to be found somewhere outside of knowledge, in an other. We cannot simply ascribe it to the fact that “all men by nature desire to know,” or some other external motivation. What is necessary must be grounded in itself: the purpose of knowledge must be found to inhere within knowledge itself, such that, in relating to its purpose, knowledge relates only to itself. The name for this purpose is therefore a problem.

But do we not thereby relapse into the same difficulty as before? It will be objected at once that objective problems equally function as contingent starting points insofar as nothing grounds their necessary existence; after all, why pose a problem concerning the Königsberg bridges and not something else? There is certainly some truth to this objection, but it doesn’t rule out the possibility of two further replies. First, we must concede that it is not always obvious at the outset whether a problem is objective (real) or subjective (an exercise or puzzle). Indeed, when the Königsberg bridge problem was first posed to Euler by Carl Gottlieb Ehler, he initially dismissed it as a trivial puzzle containing no insights for mathematics.27placeholder It was only by working through the problem that he discovered that it wasn’t so trivial as it was unsolvable by the geometry available at the time and therefore called for innovations in the field. Sometimes problems are only revealed to be objective problems retrospectively, however, retrospection doesn’t always mean from the vantage point of a solution. It would be a mistake to consider only those problems that yielded solutions real or objective problems. Virtually no problems in philosophy possess established solutions, but they count as objective problems nevertheless insofar as they manage to burrow a hole in knowledge itself—a fact that can be demonstrated by thinking those problems through. Secondly, and relatedly, it follows that necessity cannot be isolated immediately but has to demonstrate itself through its own immanent development. Nothing is just necessary tout court but only, as it were, a “becoming-necessary.” As Hegel notes in his 1831 Lectures on Logic:

“Necessity is defined as the unity of actuality and possibility. On the whole, this is correct—necessity is the third [term], but it is not well expressed. Necessity is this very course. What is difficult here is coming to know transition. The transition is a manifold one, but these manifold transitions then constitute a single course.”28placeholder

Necessity is the unity of manifold transitions. So, what are these “manifold transitions” that Hegel is referring to? In his Encyclopaedia Logic, Hegel outlines three essential moments of necessity: the condition (Bedingung), basic matter (Sache), and activity (Tätigkeit).29placeholder It is true that all objective problems succumb to contingency at the outset insofar as the act of posing a problem is always subjective and arbitrary.30placeholder But this isn’t the end of the matter. For, as contingent, the problem contains two aspects within itself: on the one hand, it is an immediate actuality that exists concretely for itself without regard for anything else; but on the other hand, it is also potentially the possibility to be sublated as the real possibility of something else, i.e., to serve as the condition of another subsequent actuality. The name for this subsequent actuality is the “basic matter,” or, in our case, knowledge. Every problem, to the extent that it is a real problem, presupposes the existence of new knowledge as a real possibility within itself, and is in turn posited by knowledge as its respective condition to be sublated. The transition between the two moments occurs through a mediating term which Hegel calls “activity,” or what we will now properly introduce as “thinking-through.” Hegel defines the activity as “the movement only of setting the basic matter forth from the conditions (in which it is on hand in itself ) and by way of sublating [Aufhebung] the concrete existence of the conditions, providing the basic matter with concrete existence.”31placeholder Knowledge initially exists as something only implicit in a problem. It must be made explicit by thinking, where what is meant by “thinking” here is not simply a passive subjective means of uncovering what is already substantially formed yet hidden from view. In other words, we are not dealing with the case of subjective thought that relates le savoir to la connaissance. Instead, thinking now conducts itself objectively as an activity that translates the problem into new universal knowledge and vice versa. The problem qua condition is thereby sublated insofar as it is both negated by the knowledge that resolves it and preserved as the content of this knowledge. This point is crucial: the content of knowledge does not materialise ex nihilo but develops out of the problem itself. Every problem naturally contains within itself a bundle of scattered information and questions.32placeholder Thinking-through accordingly consists in unfolding the problem into its differentiated content and reshaping it into knowledge, where knowledge is nothing less than this process of thinking.

It is in this sense that we can say necessity is a “single course” which comprises “manifold transitions.” The three moments of necessity outlined above do not merely confront each other as separate self-sufficient components that also depend on one another. What the concept of interdependence fails to capture is the fact that the condition, basic matter, and activity are three aspects of one and the same process, each already being the whole in itself. We can therefore summarise the preceding exposition in the following proposition: knowledge is the thinking-through of a problem. Like all philosophical propositions, Hegel reminds us that it must be read twice: first ordinarily, then speculatively. What the ordinary reading cannot avoid is fixing the subject and predicates as mutually opposed. First, there is the aforementioned opposition between knowing and thinking: to know is not to think, and to think is not to know. Secondly, there is equally an opposition between knowledge and problems: problems are extinguished the moment new knowledge is created. Once the problem has been thought through and resolved, the body of knowledge reconstitutes itself flawlessly, leaving no scars behind. Now, the speculative reading doesn’t intend to deny any of this (which is why it always comes second). All it suggests is that we make sense of this contradiction and reconcile ourselves with it.

If education struggles to embrace knowledge as thinking-through, it’s because it cannot endure this contradiction. Once again, the barrier in question is not merely pedagogical but epistemological. If we examine more closely the opposition of knowledge vis-à-vis thinking-through and problems, we discover that it is actually rooted in an even more fundamental opposition between the true and the false. Knowledge is, by definition, on the side of the true. And that leaves thinking-through and problems on the side of the false. This isn’t to say that either of them are in themselves falsehoods however, only that they involve error, falsity, and negativity. Now, this dichotomous view may have sufficed for the previous stages of knowledge: in memorisation, the pupil either reproduces the exact fact correctly or he does not; in understanding, his understanding either corresponds to the textbook’s elaboration or it does not (although even here the situation is complicated by the existence of other determinations, e.g., it is equally valid to speak of distinct or confused understandings). But this dichotomy will not suffice for our present stage, where it is necessary to finally move beyond this stubborn either-or and to grasp truth and falsehood in their speculative identity. To this end, we must give error its due.

“Why bother with the false?”33placeholder asks the truth-lover. Why mustn’t we seize the true straightaway? Why must we confer upon error not only existence but being itself? In response to this, Hegel provides, what is to my estimation, one of the most profound answers in the history of philosophy:

“The false (for here it is only of this that we speak) would be the other, the negative of the substance, which as the content of knowledge is the True. But the substance is itself essentially the negative, partly as a distinction and determination of the content, and partly as a simple distinguishing, i.e. as self and knowledge in general. One can, of course, know something falsely. To know something falsely means that there is a disparity between knowledge and its Substance. But this very dis­parity is the process of distinguishing in general, which is an essential moment [in knowing]. Out of this distinguishing, of course, comes their identity, and this resultant identity is the truth.”34placeholder

Too often, we regard truth as a mere property possessed by a special subset of propositions. According to this view, propositions fall into two immediate categories: those that do happen to correspond to reality which are labelled as “true,” and those that fail to correspond to reality (due to our ignorance) which are labelled as “false.” While it might seem as if Hegel is proceeding along the same lines here by describing the false as a “disparity between knowledge and its substance,” i.e., as non-correspondence, and the true in turn as their “identity,” i.e., as correspondence, a closer reading invalidates this hasty interpretation. Far from concurring with the correspondence theory of truth, Hegel’s point is exactly the opposite: truth is, first and foremost, non-correspondence. The reversal resides in the details where Hegel writes that it is only the “resultant identity” of knowledge and its substance that is, properly speaking, the true, i.e., that this identity does not exist prior to or independent of disparity but, on the contrary, it is only produced in and through this disparity as a result. This is classic Hegelian logic: identity is always the identity of identity and difference.

The preceding argument is not as abstract as it sounds. The only abstract claim here is the claim that truth designates an immediate relation between a proposition and a state of affairs. In reality, what is at stake between abstract truth (which excludes falsehood) and concrete truth (which emerges through falsehood) is reflected in the distinction between a bad teacher and a good one. The bad teacher is the undialectical teacher, who treats what is true and what is false, as Hegel puts it, “like oil and water, which cannot be mixed and are only externally combined.”35placeholder Suppose that a pupil were to make a false statement. This statement would immediately be negated by the teacher and replaced with the correct answer. But what sort of negation is being spoken of here? For Hegel reminds us that there are, in fact, two kinds of negation: abstract negation and determinate negation. The failure of contemporary education generally resides in its reluctance to recognise negation as anything other than abstract negation, which is to say that it “only ever sees pure nothingness in its result and abstracts from the fact that this nothingness is specifically the nothingness of that from which it results.”36placeholder The all-too-familiar process by which only right answers are affirmed while wrong ones are rejected and supplanted without further consideration can only be futile in practice insofar as it results in pure abstract nothingness from which one cannot advance further, “but must wait to see whether something new comes along and what it is, in order to throw it too into the same empty abyss.”37placeholder No true knowledge can be gained by negating what is false only to begin again from scratch, by this simple alternation between the true and the false that omits their passage. We can imagine that pupils are equally dissatisfied with this procedure since, in this case, the teacher seems to import the true from nowhere and allow it to stand beside the false. The issue here is not so much that these pupils consequently fail to understand their teacher’s corrections—they may, in fact, understand them very well—but rather that they are not led to these truths themselves and therefore find the true external to their cognition.

A good teacher, by contrast, does not negate his pupils’ errors in the abstract, but instead guides them from these errors towards the true. In this sense, the good teacher is the dialectical teacher, who adopts an immanent approach whenever possible, as opposed to a transcendent approach which pits truth against falsehood from without. Let us suppose once again that a pupil were to make a false statement. How would the good teacher respond? Not by refuting it on the basis of some external truth, but by pushing the pupil to elaborate and develop his position as far as possible whilst questioning him critically (just as Socrates did at his finest). In this way, the pupil’s reasoning can be brought to a point where it defeats itself without the need for introducing alternative ideas or external measures to bear.38placeholder It would simply collapse under the pressure of its own inherent inconsistencies or self-contradictions. The teacher does not transmit or explain any knowledge in this case; in fact, knowledge is here on the side of the pupil, not the teacher. It is the pupil who claims to have knowledge (albeit false knowledge). All the teacher does is play the role of the ignorant Socrates, claiming that all he knows is that he knows nothing—a statement whose intention is, in truth, more strategic than confessional. The teacher, of course, knows full well that the pupil’s knowledge is false, but such knowledge, if asserted outright, would produce little to no effect on the pupil. The pupil’s knowledge may be false at the outset, but this falsity remains only in itself. In other words, the false qua in itself is not false for the pupil but only false for the teacher. The goal of the teacher is accordingly to direct this falsity to become for itself, i.e., to bring the pupil to recognise the false as false. But this resultant identity, namely “the false is false,” is nothing less than the truth we’ve been searching for all along. That the false is false is itself a true statement, and what is more, it is a statement that can be derived from the false immanently. Through its own self-contradiction, the false reveals itself to be false, but once we recognise this as such, we are no longer mired in the false but have stepped outside it into the true. Truth’s identity is therefore not a pure affirmation but much more a case of what Hegel called “negation of negation,” the result of which does not lead to a reaffirmation but a unity in contradiction. This is also why the true qua result is not without disparity, “as if the disparity had been thrown away, like dross from pure metal, not even like the tool which remains separate from the finished vessel”39placeholder; rather, the disparity is “still directly present in the True as such,”40placeholder as opposed to being superseded as a mere “moment of the True, let alone a component part of it.”41placeholder It is still directly present in the true because the true is nothing without the process of becoming itself. And as with all becoming, it entails a simultaneous affirmation of both being and nothing, both truth and falsehood, since to become is to be what something is not yet and to not be what something is now.

Thus, to construe knowledge as thought is really harder than expected, for the positing of this identity requires the prior acceptance of a contradiction difficult to stomach, namely, the fact that the true and the false are consubstantial—“The false . . . as the content of knowledge is the True.” But this is precisely what contemporary education cannot accept. At most, it is only able to acknowledge the inevitability of error, but already here we have reached the limits of its tolerance; to then consider the immanence of error is strictly out of the question. From the point of view of education, knowledge can only be that which knows with certainty—a definition which seems a tautology. But in this definition, we have thereby excluded the possibility of all (objective) thought, for thought entails uncertainty in knowing. It is precisely because we are uncertain that we finally resolve to think, and it is also because we think that we once again fall into uncertainty for having thought too much. Put differently, the activity of thinking-through always entails a traversal of error and a tarrying with the negative. At stake in the identity of knowledge and thought therefore is the very objectivity of this negativity and error. If to know is to think, then to know is also to know how not to know—this is the only consistent way of defending objective thought in education. Indeed there would be nothing to think through if truth could be grasped immediately; we would simply know, although we would only know dogmatically. In fact, this is exactly how Hegel defines dogmatism, as “the opinion that the True consists in a proposition which is a fixed result, or which is immediately known.”42placeholder For Hegel, the dogmatist is not someone who knows falsely but exactly someone who knows nothing of the false while holding fast to the true. The dogmatist does not think because to think is to lose hold of one’s certainty of the truth, it is to pass over into error, to make mistakes and to feel stupid. And yet, without this peril of thought, whatever truth otherwise ascertained can only be empty insofar as it withdraws too easily into error and loses itself, which is why the dogmatist must work so vigorously to resist any and all opposition in order to secure his fragile relationship to truth. Genuine knowing, by contrast, is not afflicted by such paranoia insofar as it is capable of recognising itself even when absolutely estranged in the false.43placeholder It fears no error, finally, as it is itself constituted by error. Who’s afraid of error then? Only the dogmatic educator and his dogmatic pupils, whereby it must be asked, following Hegel, if this fear of error is not tantamount to fear of truth itself.

Conclusion

The question now confronts us anew: so, what is knowledge? Our analysis thus far has enabled us to identify three fundamental determinations44placeholder, the succession of which, in retrospect, can also be grasped as the progression from being-in-itself to being-for-itself to being-in-and-for-itself, or correspondingly, as the passage by which knowledge’s existence comes to be identical with its essence. The essence of knowledge, as we’ve shown, consists of three aspects: concrete universality, absolute necessity, and simple becoming—all of which are finally embodied adequately in the existence of knowledge as the thinking-through of a problem. This does not imply, however, that the prior stages of knowledge stand negated from the vantage point of knowledge as thought. Both the stages of memorisation and understanding nevertheless constitute valid and necessary moments of knowledge insofar as they are naturally posited by the concept of knowledge itself. But they are not the highest form of knowledge, which is why they must ultimately be overcome through sublation (not negation).45placeholder

Of course, we are aware that not all knowledge that can be characterised as such exists in the form of thinking-through. That Napoleon was born on the island of Corsica in 1769 is a brute historical fact that one either remembers or doesn’t; there is hardly anything to understand or think about here. Similarly, for the natural sciences in general, what the pursuit of knowledge signifies is the establishment of a codified body of knowledge, an encyclopaedia of everything knowable in the universe, which, again, leaves little room for thinking-through. It almost seems as if the only discipline that treats knowledge as pure thinking-through is philosophy. Take the example of Plato’s dialogues. Nobody reads The Republic solely for the definition of justice proposed by the end. Indeed, if Plato had intended for this to be the sole purpose of the dialogue, then he should’ve written a treatise instead. The knowledge that one gains from reading the dialogues is, in fact, a knowledge that is inseparable from the very unfolding of the dialogue itself which traverses both true and false, a knowledge that is identical to the activity of thinking-through.46placeholder

Having said this, we must also resist the temptation to associate the different stages of knowledge with the preoccupations of different disciplines. Indeed, physics exists chiefly in the form of le savoir, but being a physicist requires much more than a mere adequate understanding of the knowledge of physics. For, the physicist, even in his capacity solely as a physicist, can never be reduced to physics as a canonical body of knowledge, otherwise the field itself could never innovate. And just to be clear: this doesn’t mean that the physicist, in order to be a good physicist, must necessarily possess knowledge of other fields; it simply means that the knowledge of physics itself cannot be reduced to empirical facts or theories—what eludes the latter, again, is the thinking-through of problems. “The wrong view of science,” says Popper, “betrays itself in the craving to be right; for it is not his possession of knowledge, of irrefutable truth, that makes the man of science, but his persistent and recklessly critical quest for truth.”47placeholder In reality, any sufficiently complex discipline will naturally encompass all three determinations of knowledge: they will each possess their own facts, build their own theories, and think through their own problems in their own fashion.48placeholder

A final word on education and pedagogy. We initially embarked on our critique not with the intention of prescribing a course of action for the future of education but in order to comprehend the conditions for such revolutionary change. Our wager is that these conditions are, first and foremost, epistemological in nature, and only secondarily pedagogical. So long as we remain within the dominant discourse of education which speaks of knowledge in terms of “acquisition” and “mastery” and learning in terms of “standardisation” and “examinations,” so long will we remain within the limitations of the current preconceived notions about education, unable to fathom the existence of radically new possibilities. Education in its modern form finds itself either stuck at the stage of memorisation or understanding which presupposes that knowledge is some-thing that can be possessed once and for all, or that somehow one can master knowledge in its entirety. Knowledge construed thusly remains an insuperable authority over against the pupil, and in this way, dissimulates its fundamental insubstantiality. To grasp substance as subject means to accept knowledge first and foremost as an active process and that this process necessitates a traversal of error. An education established upon this principle would imply an education that values the activity of thinking in and for itself—not just as a Wittgensteinian ladder to be kicked away after climbing but as constituting the essential content of knowledge.

And yet, isn’t there a danger that, by foregrounding thought in education, we thereby push pupils to run before they can properly walk? To this, we reply: do not confuse the level of development of knowledge with the level of difficulty of knowledge. The two do not correspond. The level of difficulty derives from the difficulty of the subject matter, not the form of knowledge. It would be undialectical to separate the three stages of knowledge into the three stages of education and thereby set them in mutual opposition. Memorisation and understanding are certainly prerequisites for thinking, but there is a profound difference between undertaking them purely for their own sake and undertaking them for the purpose of thinking. Thinking is not in itself “more difficult” than memorising or understanding; it is the most adequate form of knowledge. Only thinking is valid in and for itself. Memorisation and understanding are only valid relative to thinking. Indeed, as Paul Lockhart argues in A Mathematician’s Lament, we would find it absurd if we taught any other discipline in the same soulless manner we currently teach mathematics, by dividing it into disparate “progressive” stages. Imagine if music education were divided into the same ternary structure: primary education being devoted solely to basic music theory and notation (e.g., memorising one’s circle of fifths); secondary education to lessons in scales, harmony, counterpoint, and so on (e.g., understanding how to transpose melodies into different keys); and finally tertiary education to actual playing, composition, and appreciation of real music since these topics are considered so “advanced.”49placeholder While it can be argued (pretentiously) that the complexity of a Bach can hardly be appreciated until one has memorised and understood all the music theory behind it, it still doesn’t follow in any case that the progression of education should correspond to this logical hierarchy. All we end up accomplishing by this is alienating the students who could’ve been great musicians to the point that they renounce their education altogether and risk setting up those who do manage to proceed to graduate school for the painful realisation that they’ve been mistaken about what knowledge is all along, and that they don’t have any real talent whatsoever.