On Hegel’s ‘Science of Logic’ : A Realm of Shadows — part twelve.
‘About the numbers’
by Johanna Kinkel, (1810–1858), née Maria Johanna Mockel
One deedledum, two deedledum,
Three four five six deedledum,
Sev’n deedledum, eight deedledum,
Nine deedledum, ten deedledum,
Eleven deedledum, twelve deedle deedle,
Deedle deedledum, twelve deedledum,
Eleven deedledum, ten deedledum,
Nine deedledum, eight deedledum,
Sev’n six five four deedledum,
Three deedledum, two deedledum,
One deedle deedledum.
‘Von den Nummern’
Eins dideldum, zwei dideldum,
Drei vier fünf sechs dideldum,
Sieb’n dideldum, acht dideldum,
Neun dideldum, zehn dideldum,
Elf dideldum, zwölf didel didel
Didel dideldum, zwölf dideldum,
Elf dideldum, zehn dideldum,
Neun dideldum, acht dideldum,
Sieb’n sechs fünf vier dideldum
Drei dideldum, zwei dideldum,
Eins didel dideldum.
So, limitation of Quantity is where we have reached in Georg Wilhelm Friedrich Hegel’s, (1770–1831), ‘Science of Logic’. Discrete Magnitude is One and is also a plurality of Ones that repel each other and yet each of these Ones is quite the same as any other hence the Ones continue from one into the other. When we focus on the oneness of Discrete Magnitude, we behold an excluding one, a limit in the unity.
‘Discrete magnitude has, first, the one for its principle and, second, is a plurality of ones; third, it is essentially continuous, it is the one as at the same time sublated, as unity, the self-continuing as such in the discreteness of the ones. Consequently, it is posited as one magnitude, and the ‘one’ is its determinateness — a ‘one’ which, in this posited and determinate existence, excludes, is a limit to the unity. Discrete magnitude as such is not supposed to be immediately limited, but, when distinguished from continuous magnitude, it is an existence and a something, the determinateness of which, and in it also the first negation and limit, is the ‘one’.’
- ‘The Science of Logic’
Limit has long been sublated and so discrete Magnitude is immediately not limited but as distinguished from continuous magnitude, a is a determinate being b, c, a something, with the one c] as its determinateness and also as its first negation and limit. Not only is Discrete Magnitude plainly a determinateness, considered as b, c, but even in its isolated form c it is still a determinateness, because discrete Magnitude fully remembers its ideal moment of being the Many Ones. Furthermore even as c is posited as the Many Ones still it is One and, as such, is Limit and first negation to its own being-in-itself b.
Quantum
If we take c, in Discrete Magnitude, (see previous article), as enclosing, encompassing limit, it is self-related and is the negation in Discrete Magnitude [b, c. c is the negative point itself.
‘This limit, besides referring to the unity and being the moment of negation in it, is also, as one, self-referred; thus it is enclosing, encompassing limit. The limit here is not at first distinct from the something of its existence, but, as one, is essentially this negative point itself. But the being which is here limited is essentially as continuity, and in virtue of this continuity it transcends the limit, transcends this one, and is indifferent to it. Real, discrete quantity is thus one quantity, or quantum — quantity as an existence and a something’.
- ‘The Science of Logic’
But Discrete Magnitude is also Continuity by virtue of which it passes beyond the limit, beyond this one c, to which it is indifferent. This speculative moment leads to Quantum of which Hegel explains real discrete Quantity is thus a quantity, of quantum, quantity as a determinate Being and a Something. Quantum is to pure Quantity what determinate Being was to pure Being, as Hegel specifically emphasizes elsewhere:
‘Quantity, posited essentially with the excluding determinacy that it contains, is quantum or limited quantity’.
‘Addition. Quantum is the way that quantity is there, whereas pure quantity corresponds to being, and degree (which will come next) corresponds to being-far-itself. — As for the details of the advance from pure quantity to quantum, this progress is grounded in the fact that, whereas distinction is initially present in pure quantity only implicitly (as the distinction between continuity and discreteness), in quantum, on the other hand, distinction is posited. It is, indeed, posited in such a way that from now on quantity appears always as distinguished or limited. But as a result quantum also breaks up at the same time into an indeterminate multitude of quanta or determinate magnitudes. Each of these determinate magnitudes, as distinct from the others, forms a unit, just as, on the other hand, considered all by itself, it is a many. And in this way quantum is determined as number’.
- ‘The Encyclopedia Logic’
And chapter five is to Quantity what chapter two was to Quality, a display of dialectical Reason. Quantum is, in effect, determinate Quantity. Does Speculative Reason work in Quantum in the same way it did in the Quality chapters? Recall that, at first, the extremes modulated back and forth. Speculative Reason then named the movement and produced the middle term. Later, the extremes turned on themselves and self-erased, the Finites. Speculative Reason named this self- erasure as the True Infinite. Now it appears that speculative Reason has operated on b, c without considering the role of a.
Hegel ends the chapter by correcting this misapprehension. Reverting back to [3] for a moment, Hegel holds that this one which is a limit includes within itself the many ones of discrete quantity but these Many Ones are sublated. c serves as a limit to Continuity, which Continuity leaps over with ease. Since Continuity a leaps over b and enters into c, c likewise leaps back into a, which is just as much discrete Magnitude as it was continuous Magnitude. The extremes equally leap out of themselves and so speculative Reason somewhat akin to an orchestra conductor still signals the activity it witnesses in the extremes.
And so to Quantum. We now proceed upon a lengthy section on this subject. At the end of chapter four, Hegel derived Quantum. Quantum becomes Number, quantity with a determinateness or limit in general.
‘Quantum, which in the first instance is quantity with a determinateness or limit in general, in its complete determinateness is number. Second, quantum divides first into extensive quantum, in which limit is the limitation of a determinately existent plurality; and then, inasmuch as the existence of this plurality passes over into being-for-itself, into intensive quantum or degree. This last is for-itself but also, as indifferent limit, equally outside itself. It thus has its determinateness in an other. Third, as this posited contradiction of being determined simply in itself yet having its determinateness outside itself and pointing outside itself for it, quantum, as thus posited outside itself within itself, passes over into quantitative infinity’.
- ‘The Science of Logic’
Quantum/Number will melt, thaw, and resolve itself into a pair of terms unfamiliar to the modern ear — Extensive and Intensive Quantum, sometimes called Extensive and Intensive Magnitude. Intensive Quantum is also called Degree. Degree is indeed the ladder to all high design. Quantum’s intensity will yield Quantitative Infinity and the infinitely small or large number, which can never be named. When we reach this unnameable thing, Quantum has recaptured its Quality. Quality is independence from outside determination. Whereas as the middle chapter of Quality saw Being chasing away its own content, the middle chapter of Quantity will do the opposite — it will recapture some measure of its content.
And so to Number. Quantum has Limit, but only in ideal form. The very nature of quantity as sublated being-for-self is ipso facto to be indifferent to its limit. But equally, too, quantity is not unaffected by the limit or by being quantum; for it contains within itself the one, which is its limit.
‘Quantity is quantum, or has a limit, both as continuous and discrete magnitude. The distinction between these two species has here, in the first instance, no significance. As the sublated being-for-itself, quantity is already in and for itself indifferent to its limit. But, equally, the limit or to be a quantum is not thereby indifferent to quantity; for quantity contains within itself as its own moment the absolute determinateness of the one, and this moment, posited in the continuity or unity of quantity, is its limit, but a limit which remains as the one that quantity in general has become’.
- ‘The Science of Logic’
So Quanta are Continuous and indifferent to Limit but they also have Discreteness. Ten is distinct from nine. But ten what? The number ten has no content except that it is not nine or eleven.
Quantum, then, contains within itself the moment of the One. One commentator goes so far to suggest that the first three chapters of the Logic are entirely dedicated to establishing this one proposition. Michael John Petry. (1933–2003). This one is thus the principle of quantum. It is a self-relating, enclosing and other-excluding limit.
‘This one is therefore the principle of quantum, but as the one of quantity. For this reason it is, first, continuous, it is a unity; second, it is discrete, a plurality (implicit in continuous magnitude or posited in discrete magnitude) of ones that have equality with one another, the said continuity, the same unity. Third, this one is also the negation of the many ones as a simple limit, an excluding of its otherness from itself, a determination of itself in opposition to other quanta. The one is thus (a) self-referring, (b) enclosing, and © other-excluding limit’.
- ‘The Science of Logic’
When posited in all these three determinations, Quantum is Number. The Understanding sees Quantum as a continuous discrete magnitude, limit as a plurality.
‘Thus completely posited in these determinations, quantum is number. The complete positedness lies in the existence of the limit as a plurality and so in its being distinguished from the unity. Number appears for this reason as a discrete magnitude, but in unity it has continuity as well. It is, therefore, also quantum in complete determinateness, for in it the limit is the determinate plurality that has the one, the absolutely determined, for its principle. Continuity, in which the one is only implicitly present as a sublated moment — posited as unity — is the form of the indeterminateness’.
- ‘The Science of Logic’
It isolates this plurality as Amount. In Amount, Quantum contains the Many Ones. Across this plurality, Number is continuous. In Amount, Quantum determines itself as unique from other pluralities. Ten Amount proudly boasts that it is uniquely ten and not some other number. Amount is a plurality — of what? Units! ‘Any whole number is the ‘discerning’ of a sum within a continuous multiplicity of self-equal units, within an endless flow in which the unit endlessly repeats itself’, explains Geoffrey Reginald Gilchrist Mure, (1893–1979). Hence, ten is really ten units or 10 = 10 x 1. Hence, we immediately derive Unit. Amount and unit constitute the moments of number.
‘Quantum, only as such, is limited in general; its limit is its abstract, simple determinateness. But as number, this limit is posited as in itself manifold. It contains the many ones that make up its existence, but does not contain them in an indeterminate manner, for the determinateness of the limit falls rather in it; the limit excludes the existence of other ones, that is, of other pluralities, and those which it encloses are a determinate aggregate: they are the amount or the how many times with respect to which, taken as discreteness in the way it is in number, the other is the unit, the continuity of the amount. Amount and unit constitute the moments of number’.
- ‘The Science of Logic’
This brings us to Number in Number. Number, Hegel says, is a complete positedness. Positedness represents a self-erasing True Infinite that presupposes there is an other that constitutes its content. Thus, ten is simply not nine or eleven; it posits its being in all the Unit other numbers. The existence of these numbers Unit is presupposed.
Amount
It is also a complete determinateness, for in it the limit is present as a specific plurality which has for its principle the one, the absolutely determinate. Determinateness represents a cruder stage — Being which admits a unity with non-being but which refuses to self-erase: In the sphere of determinate Being, the relation of the limit to determinate Being was primarily such that the determinate being persisted as the affirmative on this side of its limit, while the limit, the negation, was found outside of the border of the determinate Being.
‘Regarding amount, we must examine yet more closely how the many ones in which it consists are in the limit. It is rightly said of amount that it consists of the many, for the ones are not in it as sublated but are rather present in it, only posited with the excluding limit to which they are indifferent. But the limit is not indifferent to them. In the sphere of existence, the limit was at first so placed in relation to existence that the latter was left on this side of its limit, standing there as the affirmative, while the limit, the negation, stood outside on the border of existence; similarly, with respect to the many ones, their being truncated and the exclusion of the remaining ones appears in them as a determination that falls outside the enclosed ones. But it was found in that sphere of existence that the limit pervades existence, that it extends so far as existence does, and that the something is for this reason limited by its very determination, that is, is finite’.
- ‘The Science of Logic’
Putting these points together, Number is a True Infinite. It becomes something other (positing); its being is determined externally, by all the numbers it is not. Yet it also stays what it is (Determinate Being); for this very reason, ten does not change into nine or eleven. ‘The concept of quantum … is not merely quantitative. Indeed, if number can show itself as qualitative, then it is because every quantitative difference (numerical e-quality and inequality) is always also a qualitative difference’, explains Andrew Haas.
Unit
Hegel has said that Number is a complete determinateness because of continuity. How so? Because, just as Attraction fused the Many into One, Attraction, so Continuity fuses the plurality into One. Hence, Number d — g is made into One by Number Continuity. Yet this One refers both to itself and all the other Units within it. Equally, this One’s being might be viewed as continuous plurality d, e, f or the negative unity g that holds it together. Either way, because it is complex, Number is a determinateness. Quantum is now beginning to recapture some of the content that Being-for-self shed from itself in Repulsion.
With regard to Amount, Hegel asks how the Many Ones (of which Amount consists) are present in Number. In effect, Amount assumes an external counter, who breaks off for his or her own purposes and isolates it from the many other Amounts that could have been isolated. The breaking off in the counting of the many ones and the exclusion of other ones appears as a determination falling outside the enclosed ones. For example, the counter, for reasons of his or her own, counts to 100. This amount is thus isolated from 99 or 101 by some external counting force. Of counting to 100, Hegel writes in the quantitative sphere a number, say a hundred, is conceived in such a manner that the hundredth one alone limits the many to make them a hundred but none of the hundred ones has precedence over any other for they are only equal — each is equally the hundredth; thus [the units] all belong to the limit which makes the number a hundred and the number cannot dispense with any of them for its determinateness.
‘Now, in the quantitative sphere, a number, say a hundred, is so represented that only the hundredth unit brings to the many the limit that makes them a hundred. In one respect this is correct; but, in another respect, none of the ones in the hundred has precedence over any other, for they are only equal; each is just as much the hundredth; they all thus belong to the limit that makes the number a hundred; this number cannot dispense with any of them for its determinateness; with respect to the hundredth, therefore, the rest do not constitute a determinate existence which is in any way different from it whether inside or outside the limit. Consequently, the number is not a plurality over against the limiting one that encloses it, but itself constitutes this delimitation which is a determinate quantum; the many constitute a number, a one, a two, a ten, a hundred, etc.’
- ‘The Science of Logic’
Number
Unit, then, is Limit to Amount. 100 is simultaneously one Unit, but it also implies 100 equal units contained therein, each one of which lays equal claim to being the 100th. A related point was made by Hegel earlier with regard to Attraction. In chapter three, Hegel stated that the Many Ones were fused into One by Attraction. We were not, however, to assume that, amidst the Many Ones, a single Caesar had risen to become the imperial One. Rather, each of the Many Ones had an equal claim to the laurel crown of One. So it is with the Units in Number.
Number has a limiting Unit — the 100th Unit. By this, 100 differs from 99 or 101. The distinction, however, is not qualitative. Qualitative distinctions are self-generated. Quantitative distinction is externally imposed. The units do not count themselves to 100. They require comparing external reflection — a mathematician — to do the counting. 100 is thus externally derived. Once this is accomplished, the 100th one remains returned into itself and indifferent to others.
‘Now, the limiting one is a discriminating determinateness, the distinguishing of a number from other numbers. But this distinguishing does not become a qualitative determinateness; it remains rather quantitative, falling only within the compass of comparing, of external reflection; as one, number remains turned back onto itself and indifferent to others. This indifference of number to others is its essential determination; it constitutes its being-determined-in-itself, but at the same time also its own exteriority. — Number is thus a numerical one that is absolutely determined but which has at the same time the form of simple immediacy, and to which, therefore, the connecting reference to an other remains completely external. Further, as numerical, the one possesses the determinateness (such as consists in the reference to other) as a moment in it, in its distinction of unit and amount; and amount is itself the plurality of the ones, that is, this absolute exteriority is in the one itself. — This intrinsic contradiction of number or of quantum in general is the quality of quantum, and the contradiction will develop in the further determinations of this quality’.
- ‘The Science of Logic’
All of the Units within the Amount are equal and mutually self repelling. Hegel emphasizes that Number is an absolutely determinate Unit,which at the same time has the form of simple immediacy and for which, therefore, the relation to other is completely external. In 5+7=12, nothing inherent in 5 demands that it be brought into relation with 7. Besides being this immediacy, Number is also a determinateness — a mediation. Its moments are Amount and Unit.
And so to geometry and arithmetic. Hegel distinguishes and relates geometry and arithmetic. Geometry, the science of spatial magnitude, has continuous Magnitude as its subject matter. Arithmetic trades in discrete Magnitude. Perhaps this can be seen in the Cartesian plane, [René Descartes, (1596–1650)], defined as a two-dimensional coordinate plane, which is formed by the intersection of the x-axis and y-axis. The x-axis and y-axis intersect perpendicular to each other at the point called the origin.
Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2, 3) in green, (−3, 1) in red, (−1.5, −2.5) in blue, and the origin (0, 0) in purple
On the Cartesian plane, xy = lOO is a rectangle and so is continuous through its allotted space and the arithmetical 100 is so to speak contributory to a certain discreteness in that it is simply neither 99, 101, nor any other Number. Spatial geometry nevertheless implies and continues into arithmetic and Hegel returns to the point that from the geometric point springs the line of its own accord, when the point is Limit of the line. Limit is a correlative term; when point is designated as Limit, we must think of what point limits — the line. Hegel believes that this demonstration indicates that spatial magnitude, that is, geometry free and clear of Number, generates numerical magnitude. Spatial magnitude (the point, which is also a numerical one) immediately sublates itself and continues on to become the line of many Ones. Accordingly, geometry is never entirely isolated from arithmetic, just as Continuity is never entirely isolated from Discreteness. Arithmetic trades in Number without contemplating what Number is. To arithmetic, Number is the determinateness which is indifferent, inert; it must be actuated from without and so brought into a relation.
‘Arithmetic treats number and its figures, or, more accurately, it does not treat them but rather operates with them. For number is indifferent, inert determinateness, such as must therefore be activated and brought into connection from outside. The modes of connection are the species of calculation. These are performed in arithmetic one after the other, and it is evident that each depends on the other even though the thread guiding their progression is not brought out in arithmetic. However, the systematization to which the textbooks are rightly entitled can easily be drawn from the conceptual determination of number itself’.
- ‘The Science of Logic’
Arithmetic is the tool of an outside will. Numbers do not add themselves. Arithmetic has various modes of relation — addition, multiplication, etc. Arithmetic not being a speculative enterprise, the transition from one of these modes to another is not made prominent. These modes can, however, all be derived from the very concept of Number. Number has for its principle the one and is, therefore, simply an aggregate externally put together, a purely analytic figure devoid of any inner connectedness.
‘Because of its principle, which is the one, number is in all instances an external aggregate, simply an analytical figure without any inner connectedness. And because it is thus produced only externally, all calculation is a generation of numbers, a counting or, more precisely, a summing up. Any diversity in this external production that always simply repeats itself can rest only on a difference in the numbers that are to be summed up; but any such difference must itself be imported from elsewhere as an external determination’.
- ‘The Science of Logic’
Thus, an external counter breaks off the counting at, say, 100, thereby isolating this Number from the infinite others the counter may have preferred. All calculation is essentially counting, indeed Hegel regards mathematical operations as narrating a story regarding numbers.
‘It is well known that Pythagoras philosophised with numbers, and conceived number to be the basic determination of things. To the ordinary mind this interpretation must at first sight appear to be thoroughly paradoxical, and indeed quite mad. So the question arises, what we are to make of it. To answer this question we must first remember that the task of philosophy consists just in tracing things back to thoughts, and to determinate thoughts at that. Now, number is certainly a thought, and indeed it is the thought which stands closest to the sensible world; more precisely, it expresses the thought of the sense-world itself, because we understand generally by that what is mutually external and what is many. a So we can recognise in the attempt to interpret the universe as Number the first step toward metaphysics’.
- ‘The Encyclopedia Logic’
Suppose we have two numbers chosen by the counter. Whatever relation these two numbers have must also be supplied by the counter. The counter must decide whether to subtract or divide these numbers. Number has qualitative difference within it — Unit and Amount. But the identity or difference between these Numbers is entirely external. Numbers can be produced in two ways. We can count up the units and produce a number. Or we can subdivide from an aggregate already given. That is, given 100, we can negate 70 of the Units and isolate 30. In both cases, counting is implicated. One is positive counting. The other is negative counting. In counting, the Amount of the Unit is set arbitrarily. We can count five single Units. Then we can decide to count some more — seven more units are added, or 7+5=12. In this expression, the relation of 7 and 5 is a complete contingency. These two Numbers are quite indifferent to each other. They were simply put together by the mathematician for her own private purposes — an arranged, not a romantic, marriage. In analyzing analysis, Hegel will summarize this point by announcing that arithmetic is basically one — magnitude as such. If this one is rendered plural, or unified into a sum, this is done externally. How numbers are further combined and separated depends solely on the positing activity of the cognizing subject.
‘It is well known that ‘analytical science’ and ‘analysis’ are the names of preference of arithmetic and the sciences of discrete magnitude in general. And in fact their typical method of cognition is most immanently analytical and we must now briefly consider why this is so. — Any other analytical cognition begins from a concrete material that has an accidental manifoldness within; every distinction of content and every advance to further content depend on this material. The material of arithmetic and of algebra is, on the contrary, an already totally abstract and indeterminate product from which every peculiarity of relation has been eliminated, and to which, therefore, every determination and every joining is something external. This product is the principle of discrete magnitude, the one. This relationless atom can be increased to a plurality and externally determined and unified into a sum; the increasing and the limiting are an empty progression and an empty determining that never gets past the same principle of the abstract one. How the numbers are further combined and separated depends solely on the positing activity of the knowing subject’.
- ‘The Science of Logic’
We can also count six Units of two (multiplication). So multiplication also counting. What counts as a Unit (one, two, etc.) is externally decided by the mathematician. Counting, however, is tedious and so, to save time, we learn by rote what the products of two numbers are. The sum 7+5=12 is chosen by Hegel because Immanuel Kant, (1724–1804), used this very sum to demonstrate that arithmetic is a synthetic proposition:
‘We might, indeed at first suppose that the proposition 7 + 5 = 12 is a merely analytical proposition, following (according to the principle of contradiction) from the conception of a sum of seven and five. But if we regard it more narrowly, we find that our conception of the sum of seven and five contains nothing more than the uniting of both sums into one, whereby it cannot at all be cogitated what this single number is which embraces both. The conception of twelve is by no means obtained by merely cogitating the union of seven and five; and we may analyse our conception of such a possible sum as long as we will, still we shall never discover in it the notion of twelve. We must go beyond these conceptions, and have recourse to an intuition which corresponds to one of the two — our five fingers, for example, or like Segner in his Arithmetic five points, and so by degrees, add the units contained in the five given in the intuition, to the conception of seven. For I first take the number 7, and, for the conception of 5 calling in the aid of the fingers of my hand as objects of intuition, I add the units, which I before took together to make up the number 5, gradually now by means of the material image my hand, to the number 7, and by this process, I at length see the number 12 arise. That 7 should be added to 5, I have certainly cogitated in my conception of a sum = 7 + 5, but not that this sum was equal to 12. Arithmetical propositions are therefore always synthetical, of which we may become more clearly convinced by trying large numbers. For it will thus become quite evident that, turn and twist our conceptions as we may, it is impossible, without having recourse to intuition, to arrive at the sum total or product by means of the mere analysis of our conceptions. Just as little is any principle of pure geometry analytical. ‘A straight line between two points is the shortest’, is a synthetical proposition. For my conception of straight contains no notion of quantity, but is merely qualitative. The conception of the shortest is therefore fore wholly an addition, and by no analysis can it be extracted from our conception of a straight line. Intuition must therefore here lend its aid, by means of which, and thus only, our synthesis is possible’.
- ‘Critique of Pure Reason’
Hegel, however, thinks it is analytic: The sum of 5 and 7 means the mechanical. conjunction of the two numbers, and the counting from seven onwards thus mechanically continued until the five units are exhausted can be called a putting together, a synthesis, just like counting from one onwards; but it is a synthesis wholly analytical in nature, for the connection is quite artificial, there is nothing in it or put into it which is not quite externally given.
‘Yes, five is given in intuition, that is, it is an entity put together in an entirely external fashion by the arbitrary repetition of the thought one; but seven is equally not a concept: there are no concepts here to go outside of. The sum of 5 and 7 is the conceptually unconstrained joining of the two numbers; one can call such a mechanical process of counting from seven onwards until the five ones have been exhausted an adding, a synthesizing, exactly like the counting from one onwards to five — a synthesizing, however, which is wholly analytical in nature, for the combination is only an artifact in which there is nothing, or to which nothing is added, which was not previously there in external fashion. The postulate that 5 be added to 7 stands to the postulate of counting in general in the same way as the postulate that a straight line be extended stands to the postulate of drawing a line’.
- ‘The Science of Logic’
[Side note: when I was doing my first philosophy course at the University of Leeds some thirty years ago, included was a module on Kant in which the then tutor of the course Roger M. White whose name I can mention because he is never going to read this article explained how Kant thought 7 + 5 = 12 was a synthetic proposition but it wasn’t till much later that mathematician, logician, and philosopher Friedrich Ludwig Gottlob Frege, (1848–1925), showed it to be analytic. If only I had known then what I know now, I could have told him, hold on a minute, Hegel got there first!]
Maybe the issues are too subtle. David Gray Carlson complains: It is not clear to me why Hegel is so heated in denouncing Kant’s arithmetic’s synthetic nature. Was Kant not simply saying that 5 and 7 do not add themselves — that mathematical knowledge is obtained not through definition but through intuition and construction? And is not Hegel in complete agreement that addition is a matter for the external counter? In short, ‘synthesis’ to Kant is what ‘externality of content’ is for Hegel’. Hegel believes that 5 + 7 already contains the command to count 7 more beyond 5. The result contains nothing more than what was in 5 + 7 — the command to keep counting. So arithmetic is analytic only. The difference between Kant and Hegel could be put this way. Kant focuses on the fact that a mathematical sentence must be constructed. Hegel focuses upon what the sentence requires after it is constructed.
Hegel also objects to Kant’s view that arithmetic is a priori, that is to say, not derived from experience. If we synthesize experience, then knowledge is merely empirical, contingent, and a posteriori. Hegel attacks the very distinction of a priori and a posteriori: every sensation has in it the a priori moment, just as much as space and time, in the shape of spatial and temporal existence, is determined a posteriori. (Anyone who has studied Philosophy 101 will have been taught about the a priori/a posterior distinction. A priori, relating to or denoting reasoning or knowledge which proceeds from theoretical deduction rather than from observation or experience. A posteriori, relating to or denoting reasoning or knowledge which proceeds from observations or experiences to the deduction of probable causes. But it is a distinction that warrants being subjected to a certain amount of scrutiny).
‘Just as vacuous as the expression ‘to synthesize’, is to say that this synthesizing takes place a priori. Counting is of course not a determination of the senses, which, according to Kant’s definition of intuition, is all that is left over for the a posteriori, and it certainly is an affair conducted on the basis of abstract intuition, that is, one which is determined by the category of the one and where abstraction is made from all other sense determinations and no less so also from concepts. The a priori is something altogether all too vague; feeling, determined as drive, sense, and so on, has in it the moment of the a priori, just as much as space and time, in the concrete shapes of temporal and spatial existence, is determined a posteriori’.
- ‘The Science of Logic’
This plaint is related to Hegel’s criticism of the unknowable thing-in-itself. For Hegel, our knowledge of objects is always a unity of our perception (a posteriori) and the authentic integrity of the object (a priori).These demonstrations are made in the early chapters of the ‘Phenomenology of Spirit’. Hegel praises, after a fashion, Kant’s notion of the synthetic a priori judgment as belonging to what is great and imperishable in his philosophy.
‘Kant’s concept of synthetic a priori judgments — the concept of terms that are distinct and yet equally inseparable; of an identity which is within it an inseparable difference — belongs to what is great and imperishable in his philosophy. To be sure, this concept is also equally present in intuition, for it is the concept as such and everything is the concept implicitly; but the determinations selected in those examples do not exhibit it. Number is an identity, and counting the production of it, which is absolutely external, a merely superficial synthesis, a unity of ones that are rather posited as not inherently identical with each other, but external, each separate for itself; the determination of the straight line of being the shortest between two points is based on a moment of abstract, internally undifferentiated identity’.
- ‘The Science of Logic’
‘As Kant recognized, philosophy could obtain new knowledge of what was necessarily and universally the case only insofar as concepts could be set in binding relation to what was not their immediate identity’, explains Richard Dien Winfield, (1950 -). But what he likes about it is the speculative content Kant never brought to light. In the synthetic a priori judgment, something differentiated equally is inseparable. Identity is in its own self an inseparable difference. If arithmetic is a priori synthetic, then 7 + 5 can be kept apart and also not kept apart simultaneously. Difference and identity each have their moments in 7 + 5=12. But this identity of identity and difference (the identity of identity and difference — a key Hegelian slogan — is expressly considered in the ‘Doctrine of Reflection;), is no mere property of the a priori synthetic judgment. It is just as much present in intuition -a posteriori judgment. Hence, the compliment to Kant is, at best, ironically tendered.
In any case, Hegel argues against Kant’s assertion that Euclidean geometry is synthetic. Kant conceded that some of its axioms are analytic. As Douglas Richard Hofstadter, (1945 — ), elucidated, ‘Euclid gave these four postulates upon which all geometry is based: (1) a straight line segment can be drawn joining any two points. (2) Any straight line segment can be extended indefinitely in a straight line. (3) Given any straight line segment, a circle can be drawn having the segment as radius and one end point as center. (4) All right angles are congruent’. A fifth was added, but it turned out to be subjective, not objective: (5) If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. The suspension of the fifth postulate leads to non-Euclidean geometry.
But Kant held as synthetic the proposition that the shortest way between two points is a straight line (see above passage). In contrast, Hegel has held that, at least if point is Limit, the line generates itself. This self-generated line is inherently simple. Its extension does not involve any alteration in its determination, or reference to another point or line outside itself.
‘Here too, however, we are not dealing with the concept of straight in general but with the straight line, and this latter is already something spatial, intuited. The determination of the straight line (or, if one prefers, its concept) is none other than that it is an absolutely simple line; that is, that in coming outside of itself (the so-called movement of the point), it refers simply to itself; that in being extended, no diversity of determination, no reference to some other point or line outside it, is posited; it is simple, absolutely internally determined direction. This simplicity is indeed its quality, and if the straight line seems difficult to define analytically, this is so precisely because of its simplicity and self-referential character, whereas reflection looks for determination first and foremost in a plurality, in a determining through something else. But there is absolutely nothing inherently difficult in grasping this determination of simple internal extension, this absence of determination by another; Euclid’s definition contains nothing else than this simplicity. — But now the transition of this quality to the quantitative determination (‘the shortest’) that allegedly constitutes the synthetic factor is in fact entirely analytical. As spatial, the line is quantity in general; when said of the quantum, the simplest means the least, and when said of line, it means the shortest. Geometry can accept these determinations as a corollary to the definition; but Archimedes in his books on the sphere and the cylinder took the most advisable course by introducing this determination of the straight line as an axiom, in just as correct a sense as Euclid included the determination concerning parallel lines among the axioms, for the development of this determination into a definition would have also required determinations that do not belong to spatiality immediately but are rather of a more abstract qualitative character, like the simplicity and sameness of direction just mentioned. These ancients gave even to their sciences a plastic character, rigorously confining their exposition to the distinctive properties of their material and thus excluding from it anything heterogeneous to it’.
- ‘The Science of Logic’
Euclid was therefore correct to list amongst his postulates the analytical proposition that the shortest way between two points is a straight line. Because this definition includes nothing heterogeneous to geometry, Euclid’s proposition is analytic.
Which brings us to subtraction and division. Subtraction and division are negative counting. In subtraction (that is, 12–5 = 7) the Numbers are indifferent or generally unequal to each other. That is, given a line segment of 12 units, we could have subdivided the line as 7 and 5, or 9 and 3, or 11 and 1, etc. The two Numbers into which a line of 12 units is subdivided bear no relation to each other. If we make the two Numbers (qualitatively) equal, then we have entered the province of division. Suppose we isolate a Unit — say, 6. The Number 12 now has a Unit of 6 and an Amount of two.
Division is different from multiplication, however. In multiplication, where 6–2 = 12, it was a matter of indifference whether 6 counted as Amount or Unit. This is the commutative property of multiplication, ab = ba. Division would seem to operate on another principle. 12/6 is not the same as 6/12. But, remembering that negative counting takes 12 as given, it is likewise immaterial whether the divisor (6) or quotient (2) is Unit or Amount. If we say 6 is Unit, we ask how often 6 is contained in 12. If we say that the quotient (2) is Unit, then the problem is to divide a number [12] into a given amount of equal parts here, 6 and to find the magnitude of such part.
‘Division is the negative species of calculation with this same determination of difference. It is equally immaterial which of two factors, the divisor or the quotient, is taken as unit or as amount. The divisor is determined as unit and the quotient as amount whenever the stated task of the division is to see how many times (the amount) a number (the unit) is contained in a given number; conversely, the divisor is taken as amount and the quotient as unit whenever the stated intent is to partition a number into a given amount of equal parts and to find the magnitude of the part (of the unit)’.
- ‘The Science of Logic’
Which brings us to exponents. In multiplication and division, the two Numbers are related to each other as Unit and Amount. Yet Unit and Amount are still immediate with respect to each other and therefore simply unequal. If we insist that Unit and Amount be equal, we will complete the determinations immanent within Number. This last mode of counting is the raising of Number to a power. Take 6 to the power of 2 = 36. Here, the several numbers to be added are the same.
‘The two numbers that are determined with respect to each other as unit and amount still are, as numbers, immediate to each other and are therefore unequal in general. The further equality is that of unit and amount themselves, and, with this, the progression to the equality of the determinations inherent in the determination of number is completed. On the basis of this complete equality, counting is the raising to a power (the negative counterpart of this calculation is the extraction of a root), and this raising to a power constitutes — in the first instance, as the squaring of a number — the complete inherent determinateness of counting where (1) the many numbers to be added are the same, and (2) their plurality or amount is itself the same as the number which is posited a plurality of times, or the unit’.
- ‘The Science of Logic’
Should not Hegel have said the two numbers (6 and 6) to be multiplied are the same? No. Hegel has said that multiplication is counting, like addition. Hence, we shall count six units. Each unit has six in it. In short, we count from 1 to 6. Next we count from 7 to 12, and so forth. Eventually we reach 36. The point is that in squaring 6, Amount equals Unit. The square is in principle those determinations of amount and unit which, as the essential difference of the Notion, have to be equalized before number as a going-out-of-itself has completely returned into self. The arithmetical square alone contains an immanent absolute determinedness.
‘It is clear from what has been said that the arithmetical square alone contains a determinateness which is inherent to it and absolute; for this reason the equations of higher formal powers must be reduced back to it, just as in geometry the right-angled triangle contains a determinateness absolutely inherent in it which is expounded in the Pythagorean theorem, and for this reason all other geometrical figures must also be reduced to it for their total determination’.
- ‘The Science of Logic’
Here we have a preview of what, in chapter six, will be called the Ratio of Powers. The premise is that if we insist that Unit equals Amount, Number resists outside manipulation and thereby shows its quality. If x to the power of 2 = 36, then x must be {6, -6}. The Ratio of Powers is the last stage of Quantity. Here is where Quantum recaptures its integrity and wins independence from the counters who have controlled and dominated it prior to that point. ‘However, a square is precisely not a quantum, not a part, not something externally limited’, explains Hegel in the ‘Jena System’. Hegel concludes this section with this observation that it is an essential requirement when philosophizing about real objects to distinguish those spheres to which a specific form of the Notion belongs. Otherwise the peculiar nature of a subject matter which is external and contingent will be distorted by Ideas, and similarly these Ideas will be distorted and made into something merely formal.
‘The step-by-step determination of the species of calculation that we have just given cannot be said to be a philosophical treatment of them, not an exposition of their inner meaning as it were, for it is not in fact an immanent development of the concept. However, philosophy must know how to distinguish what is by nature a self-external material; it must know that, so far as this material goes, the concept can make its way forward only externally, and its moments also can be only in the form peculiar to their externality, as here equality and inequality. To distinguish the spheres to which a specific form of the concept belongs, that is, in which the concept is present in concrete existence, is an essential requirement when philosophizing about real objects. This is to prevent ideas from interfering with the peculiar nature of externality and accidentality, and the ideas themselves, because of the disproportionateness of the material, from being distorted and reduced to a formalism. Here, however, the externality in which the moments of the concepts appear in this external material, number, is the appropriate form; since these moments display the subject matter in the conceptual form of the understanding which is appropriate to it, and also, since they contain no demand for speculative thought and therefore have the semblance of being easy, they deserve to be employed in elementary textbooks’.
- ‘The Science of Logic’
Such a caveat means that speculative philosophy and higher mathematics each has its sphere and each should be wary of allowing the other to interfere unduly with its project.
Number is the absolute determinateness of quantity, and its element is the difference which has become indifferent. The indifference of Number implies that Number finds its content imposed upon it from the outside. Thus, arithmetic is an analytical science. It does not contain the Notion. Arithmetical combinations are not intrinsic to the concept of Number but are effected on it in a wholly external manner.
‘We saw that number is the absolute determinateness of quantity, and that its element is the difference which has become indifferent: implicit determinateness which is posited at the same time as only entirely external. Arithmetic is an analytical science, since all the combinations and all the differences that occur in its subject matter do not originate in it but are imported into it entirely from outside’.
- ‘The Science of Logic’
It is therefore no problem for speculative thought, but is the antithesis of the Notion and when thought engages in arithmetic, it is involved in activity which is the extreme externalization of itself, an activity in which it is forced to move in a realm of thoughtlessness and to combine elements which are incapable of any necessary relationships.
‘Arithmetic does not have any object that might harbor within it such inner relations as would be concealed to knowledge at first, because they are not given in its immediate representation, but are elicited only through the effort of cognition. Not only does arithmetic not contain the concept and the intellectual task of conceptualization that goes with it: it is the very opposite of the concept. Here, because of the indifference of the combined to the combining — a combining that lacks necessity — thought finds itself engaged in an activity which is at the same time the utter externalization of itself, a tour de force in which it moves in an element void of thought, drawing relations where there is no capacity for necessary relations. The subject matter is the abstract thought of externality itself’.
- ‘The Science of Logic’
Numbers are supposed to be educational yet Hegel believes this to be overstated for occupation with numbers is an unthinking, mechanical one. The effort consists mainly in holding fast what is devoid of the Notion and in combining it purely mechanically.
‘As for any supposed use that number and calculation might have for basic pedagogical formation, it follows by itself from what has been said so far. Number is not an object of the senses, and to be occupied with number and numerical combinations is not the business of the senses; such an occupation, therefore, encourages spirit to engage in reflection and the inner work of abstraction, and this is of great, though one-sided, importance. For, on the other hand, since the basis of number is only an external, thoughtless difference, the occupation proceeds without a concept, mechanically. The effort consists above all in holding on to something non-conceptual, and in combining it non-conceptually. The content is the empty ‘one’.’
- ‘The Science of Logic’
Calculation dulls the mind and empties it of substance. It is so thoroughly debased Hegel observes that it has been possible to construct machines which perform arithmetical operations with complete accuracy.
‘So the solid nourishment of moral and spiritual life in its individual shapes on which, as the noblest aliment, education should nurture the young spirit, is to be ousted by this ‘one’ which is void of content; when those exercises are made the main subject and the main occupation, the only possible outcome must be to dull the spirit and to empty it of both form and content. Since calculation is so much of an external and therefore mechanical business, it has been possible to manufacture machines that perform arithmetical operations with complete accuracy. It is enough to know this fact alone about the nature of calculation to decide on the merit of the idea of making it the main instrument of the education of spirit, of stretching spirit on the rack in order to perfect it as a machine’.
— ‘The Science of Logic’
‘Soldier’s farewell’
by Johanna Kinkel, (1810–1858), née Maria Johanna Mockel
How can I bear to leave thee,
One parting kiss I give thee;
And then what e’er befalls me,
I go where honour calls me:
Farewell, farewell, my own true love!
Ne’er more may I behold thee
Or to this heart enfold thee,
With spear and pennon glancing
I see the foe advancing:
Farewell, farewell, my own true love!
I think of thee with longing,
Think thou, when tears are thronging;
That with my last faint sighing,
I’ll whisper soft while dying:
Farewell, farewell, my own true love!
‘Ritters Abschied’
Weh’, daß wir scheiden müßen, laß mich noch einmal küßen.
Ich muß an Kaiser’s Seiten, in’s falsche Welschland reiten.
Fahr’ wohl, fahr’ wohl, mein traues Lieb.
Ich werd’ auf Fernen auen, dich niemals wieder schauen.
Der Feinde grimme Schaaren, sind kommen angefahren.
Fahr’ wohl, fahr’ wohl, mein traues Lieb.
Ich denk’ an dich mit Sehnen, gedenk an mich mit Tränen.
Wenn meine Augen brechen, will ich zuletzt noch sprechen:
Fahr’ wohl, fahr’ wohl, mein traues Lieb.
‘How precious also are thy thoughts unto me, O God! how great is the sum of them! If I should count them, they are more in number than the sand: when I awake, I am still with thee’. — Psalm 139:17–18. And so to beloved dedicatee, for whom everything I do is done for thee, my thoughts are of you, and you are with me always albeit so far away.
Coming up next:
Intensive Quantum.
To be continued….
