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The Dialectics of the Infinitesimal Calculus

 

 

   The purpose of this article is to show that the general dialectical laws of the motion of the material world, and of human society and human thought, as formulated by Engels, and upon which the philosophical outlook of Marxism is based, are manifested by the more particular laws of mathematics, and most concretely by the branch of mathematics know as the Infinitesimal Calculus.  We may formally state the three laws upon which the dialectical materialist outlook rests as follows:-

 

1. All progress takes place through the unity and conflict, interpenetration

and transformation of opposites.

 

2. The law of the transformation of quantity into quality and vice versa.

 

3. The law of the negation of the negation.

 

   In a book which contains works written by Engels between 1873 and 1886, entitled The Dialectics of Nature, first published in the Soviet Union in 1925, Engels says the following:-

 

   “The so-called axioms of mathematics are the few thought determinations which mathematics needs for its point of departure.  Mathematics is the science of magnitudes; its point of departure is the concept of magnitude.  It defines this lamely and then adds the other complementary determinations of magnitude, not contained in the definition, from outside as axioms, so that they appear as unproved, and naturally also as mathematically unprovable.  The analysis of magnitude would yield all these axiom determinations as necessary determinations of magnitude.  Spencer is right in as much as what thus appears to us to be the self-evidence of the axioms is inherited.  They are provable dialectically, in so far as they are not pure tautologies.”  (Page 257)

 

    Engels goes on to tell us briefly what the axioms are, and they are surprisingly few and simple.

 

   “Nothing appears more solidly based than the difference between the four species of arithmetical operations, the elements of all mathematics.  Yet right at the outset multiplication is seen to be an abbreviated addition, and division an abbreviated subtraction of a definite number of equal numerical magnitudes … In algebraic calculation the thing is carried much further.  Every subtraction (a-b) can be represented as an addition (-b+a) and every division a/b as a multiplication a×1/b. A power can be put as a root (x²=√x4),a root as a power, (√x=x½)"     

 

   Engels goes further but this is sufficient to show that the two basic axioms of mathematics are addition and subtraction, and that the most complicated mathematical operations are no more than sophisticated expressions of these.  How are these axioms to be proved dialectically?  Elsewhere Engels shows that these axioms are indeed, as Spencer says, thought determinations inherited from the physical world of matter.

 

   “The whole of nature accessible to us forms a system, an interconnected totality of bodies, and by bodies we understand here all material existences extending from stars to atoms, indeed right to ether particles, in so far as one grants the existence of the last named.  In the fact that these bodies are interconnected already includes that they react on one another, and it is precisely this mutual reaction that constitutes motion.  It already becomes evident here that matter is unthinkable without motion.  And if, in addition, matter confronts us as something given, equally uncreatable as indestructible, it follows that motion also is as uncreatable as indestructible.  It became impossible to reject this conclusion as soon as it was recognised that the universe is a system, an inter-connection of bodies.  And since this recognition had been reached by philosophy long before it gained effective currency in natural science, one can understand why philosophy, fully two hundred years before natural science, drew the conclusion of the uncreatability and indestructibility of motion. (Op. Cit., page 70).

 

   We must follow Engels a little further to complete the picture.  Further down the same page he says:-

 

   “When two bodies act on each other so that a change of place of one or both of them results, this change of place can consist only in an approximation or a separation.  They either attract each other or repel each other. Or, as mechanics expresses it, the forces operating between them are central, acting along the line joining their centres.  That this happens, that it is the case throughout the universe without exception, however complicated many movements may appear to be, is nowadays accepted as a matter of course.  It would seem nonsensical to us to assume, when two bodies act on each other, and their mutual interaction is not opposed by any obstacle or the influence of a third body, that this action should be effected otherwise than along the shortest and most direct path, i.e., along the straight line joining their centres.  It is well known, moreover, that Helmholtz has provided mathematical proof that the central action and unalterability of the amount of motion are reciprocally conditioned and that the assumption of other than central actions leads to results in which motion could be either created or destroyed.  Hence the basic form of all motion is approximation and separation, contraction and expansion – in short, the old polar opposites of attraction and repulsion.”

 

   Here then we have the derivation of the mathematical axioms of addition and subtraction.  Attraction brings bodies into unity, (addition), repulsion separates them, and if we concern ourselves only with what is left on one side after the separation of this unity, we have the familiar result of subtraction.  There can be no doubt that in referring to “the old polar opposites of attraction and repulsion”, Engels had in mind Hegel’s theoretical deduction and subsequent analysis of the concept of quantity.

 

   “Repulsion is, although negative, still essentially relation;  the mutual repulsion and flight is not a liberation from what is repelled and fled from, the one as excluding still remains related to what it excludes.  But this moment of relation is attraction and thus is in repulsion itself; it is the negating of that abstract repulsion according to which the ones would be only self-related affirmative beings not excluding one another … This Being, in the determination it has now reached, is Quantity.” (The Science of Logic, F. Hegel, ISBN 0 85950 0926, pp.175-177)

 

   Quantity is the unity of attraction and repulsion, of the one, (unity through attraction), and of the many, (separation through repulsion).  This is easily deduced from the language of every day usage.  If we speak of “a” quantity, it is always one quantity of many ones.  A heap of stones is one heap but many stones; a foot of space in one foot but twelve inches. Hegel’s analysis of the concept of quantity is important but wordy and obscure, so to make life easier we shall paraphrase.  The reader may wish to study Hegel directly for confirmation.

 

   Quantity, says Hegel, is both continuous and discrete.  A foot of space is continuous in that it is one continuous thing, one foot, since although it consists of discrete inches the inches are identical and therefore flow into one-another.  Nonetheless the inches are discrete things, and it is easy to see that quantity is wholly continuous and wholly discrete.  If we think of quantity from the side of continuity we have no idea where it ends, it is not necessarily infinite, but it has no limit. If we think of quantity from the side of discreteness then it must necessarily have a limit, because each inch is a finite limited thing and no matter how many finite things we add up the result must be finite.  So here we have quantity with a limit and we have arrived at a new concept, quantum, the idea of limited quantity, but without any specific limit being present to consciousness. 

 

   Quantum limits, and therefore negates, Quantity, and this negation is itself negated through the synthesis of Quantity and Quantum into the concept of Number.  A discrete quantity or magnitude is a plurality of ones, and a continuous magnitude is a unity. A number is precisely a plurality in a unity, the number 10 is one number containing 10 units. Having derived these basic axioms of mathematics, addition, subtraction, and number, we are ready to analyse a few mathematical operations and to abstract from them the general laws of dialectics given above.

 

   Let us begin with the simplest and most familiar of algebraic expressions:-

 

y  =  mx + b

 

   The left hand side, (LHS), of the equation expresses the philosophical category of Being, y is … The law of the relationship of y to x is expressed in the co-efficient m.  For every y there are so many Xs, that is to say,  m expresses the ratio of y to x, and  b is a constant that affects the value of y but is contingent, that is, nothing to do with the law of the relation of y to x.

 

   The right hand side, (RHS), expresses the inner content of y, the truth of y, which can only be revealed through analysis, the resolution of the equation.

 

   The equal sign expresses the philosophical category of Identity.  The LHS must necessarily be identical in magnitude to the RHS, and we notice that all the magnitudes are fixed and unchanging.  It is true that we can assign different values to m, x, and b, but in every case we begin and end with fixed magnitudes.  The limitations of this kind of procedure become apparent when we have to deal with practical problems such as the motion of bodies through space and time.  Consider the expression:-

 

v  =  s/t

 

   Here v is velocity, s is distance in miles, and t is time in hours.  If the distance is 60 miles and the time taken to move through this distance is 2 hours, then we have

 

v  =  60/2  =  30 miles per hour

 

   But this gives only an average velocity, and if the velocity varies during the course of the motion we have no way of knowing what the velocity of the body is at any one moment during its motion.  This was the problem Isaac Newton encountered when he found it necessary to describe the motion of bodies falling under the influence of gravity which accelerate uniformly at the rate of 32.2 feet/sec².  It will be seen from above that we express velocity as a ratio of distance to time, (distance divided by time), and if we want to know the velocity of the body at any one point in its path then we must identify this ratio at a single point in time, or in other words, we must be able to find the magnitude of s for any magnitude we assign to t. But to consider the velocity of a moving body at a single point in time and space is to consider both as having no magnitude at all, taking both as equal to zero, and to solve the problem of expressing this mathematically Newton developed the method we know as the infinitesimal calculus. The importance of this method is that it enables us to make calculations involving varying velocities  or varying magnitudes of any kind, and it is this method which most perfectly manifests the dialectical laws of motion given above.  The method of the infinitesimal calculus falls apart into two sides which are mirror image opposites, differential calculus which is analysis, and integral calculus which is synthesis.  For the moment we are concerned with the former.

 

  Generally if two variable quantities s and t are so related that, when any value is assigned to t there is thus determined a corresponding value of s, then s is termed a “function” of t.  The value of t may vary by any amount quite independently of s, but the value of s depends on the value of t, so we say that t is the independent variable and s is the dependant variable. We express this by taking the first letter of “function”, f, and placing t in parenthesis, s = f(t).  In the case of a body falling due to gravity, (ignoring the effect of air resistance),  the expression would be:-

 

f  ( t ) =   ½ gt²

 

where g is the rate of acceleration due to gravity and t  is the time through which the body falls: Or, since the result of the operation is the distance fallen, s :-

 

s  =   ½ gt²

  

   s is the dependant variable, t is the independent variable, and as explained above, the rest of the terms express the law of the relation between the two. The method Newton adopted to find the velocity of a body at a single moment in time, say after falling for 10 seconds, begins by assigning a small increase or increment to the time of motion of the body, t.  We express the small increase in time with the term δt, δ being the Greek letter delta. Since s varies in dependence on t, then s will increase by a small amount also, although by a different amount according to the law of their relation, and this we express with δs. We call these small increments differentials, and having now assigned the value 10 to t, we can express the last equation with the small increases applied like this:-

 

s+δs = ½g(10 +δt)²  feet

 

    Taking the acceleration due to gravity as 32 feet/sec.² and expanding the RHS,

 

s+δs = ½ × 32(10 + δt)²   feet

 

s+δs = 16(100 + 20δt + δt²)   feet

 

 s+δs = 1600 + 320δt + 16δt²  feet

 

   If we wish to know the value of the small increase in s, δs, we must subtract the value of s from both sides, but we can only subtract like from like, and since we already know that s = ½gt², and we have assigned the value 10 to t, we can do it like this:-

 

 

δs = 1600 + 320.δt + 16δt² - 16 × 100  feet

 

δs = 1600 + 320δt +16δt² - 1600  feet

 

δs = 320δt + 16δt²   feet

 

    The velocity of the body is the ratio of the differential of distance to the differential of time, and to express this we must divide δs by δt, observing the rule of treating both sides of the equation the same.

 

                                                                       δs  =  320δt + 16δt²  feet/second
                                                                       δt               δt

                                                                      δs =  320 + 16δt  feet/second
                                                                      δt           

 

   But for so long as the differentials δs and δt represent actual magnitudes we still have under consideration a distance in space and a duration in time; we have still not arrived at a single point in space and time.  To determine the velocity at a single point we must limit these differentials to zero so that the equation looks like this:-

 

 

                                                                      0   =  320 + 16δt  feet/second

                                                                      0

 

 

                                                                     0   =  320 + 16  ×  0  feet/second

                                                                     0

 

                                                                     0  = 320 feet/second     

                                                                     0

 

 

   Mathematicians who base themselves on the law of formal identity are dumbfounded by this.  How can nothing equal something?  They accept that it does and operate on this basis, without understanding the underlying truth of the matter.  They will sometimes describe this awkward circumstance as a “blind spot in mathematics”. So as to gloss over the problem the convention of substituting the English letter d for the Greek δ has been adopted to express the moment of differentiation, so that we proceed from

 

                                                                    δs = 320 + 16δt  feet/second
                                                                    δt

                                                                                       to

 

                                                                   ds  = 320 feet/second
                                                                   dt

 

   Another wheeze formal thinking mathematicians resort to in order to gloss over this shortcoming in the language of mathematics is to say that “we catch the differentials at the moment of their disappearance”, but still does the thing exist or not? Formal logic is based on the axiom that “A = A, A does not equal not A”.  But this Parmenidian logic is seen to be false when we realise that if this is true then all motion and change is impossible.  Hegel explains this in his Science of Logic. First he sets out the Kantian antinomy concerning the formal opposition of the finite and the infinite in time and space:-

 

   “It is impossible for anything to begin, either in so far as it is, or in so far as it is not; for in so far as it is, it is not just a beginning, and in so far as it is not, then also it does not begin.  If the world, or any thing, is supposed to have begun, then it must have begun in nothing, but in nothing - or nothing – is no beginning; for beginning includes within itself a being, but nothing does not contain any being.  Nothing is only nothing.  In a ground, a cause, and so on, if nothing is so determined, there is contained an affirmation, a being. For the same reason, too, something cannot cease to be; for then Being would have to contain Nothing, but Being is only Being, not the contrary of itself.”  (Op. Cit., page 104)

 

   Hegel’s criticism of this false logic follows:-

 

   “It is obvious that in this proof nothing is brought forward against becoming, or beginning and ceasing, against this unity of being and nothing, except an assertoric denial of them and an ascription of truth to being and nothing, each in separation from the other.” (Ibid.)

 

   Right at the beginning of the Science of Logic Hegel explains that this unity of Being and Nothing that we have discovered in the differential calculus is the truth, and the metaphysical separation of them is quite unfounded.

 

   “Being, pure being, without any further determination.  In its indeterminate immediacy it is equal only to itself.  It is also not unequal relative to another; it has no diversity within itself nor any with a reference outwards.  It would not be held fast in its purity if it contained any determination or content which could be distinguished in it or by which it could be distinguished from an other.  It is pure indeterminateness and emptiness.  There is nothing to be intuited in it, if one can speak here of intuiting; or, it is only this pure intuiting itself.  Just as little is anything to be thought in it, or it is equally only this empty thinking.  Being, the indeterminate immediate, is in fact nothing, and neither more nor less than nothing”  (P.82)

 

     Being and Nothing are of course opposites, but we see above how they are also identical.  Being is both itself and its own opposite, Nothing.  Hegel continues by giving us the equation in reverse:-

 

   “Nothing, pure nothing:  It is simply equality with itself, complete emptiness, absence of all determination and content - undifferentiatedness in itself.  In so far as intuiting or thinking can be mentioned here, it counts as a distinction whether something or nothing is intuited or thought.  To intuit or think nothing has, therefore, a meaning; both are distinguished and thus nothing is (exists) in our intuiting or thinking; or rather it is empty intuiting and thought itself, and the same empty intuition or thought as pure being.  Nothing is, therefore, the same determination, or rather absence of determination, and thus altogether the same as pure Being.”

 

   Hegel goes on to explain how these opposites, Being and Nothing, find unity and identity in the moment of Becomimg:-

 

   “Pure being and pure nothing are, therefore, the same.  What is the truth is neither Being nor Nothing, but that Being – does not pass over but has passed over- into Nothing, and Nothing into Being.  But it is equally true that they are not undistinguished from each other, that, on the contrary, they are not the same, that they are absolutely distinct, and yet are unseparated and inseparable and that each immediately  vanishes in its opposite.  Their truth is, therefore, this movement of immediate vanishing of the one in the other:  Becoming, a movement in which both are distinguished, but by a difference which has equally immediately resolved itself.” (P.82) And further,  “The unity, whose moments, Being and Nothing, are inseparable, is at the same time different from them and is thus a third to them; this third in its own characteristic form is Becoming.” (P.93)  

 

   Here we have the truth of the relation of Being and Nothing at the universal, abstract level, and this truth is just as real at the particular and individual levels.  Dealing with this question in The Dialectics of Nature, Engels quotes Hegel as saying, “Nothing that is opposed to something, the nothing of any something, is a determinate nothing” … and again “the self-contradictory resolves itself into nullity, into abstract Nothingness, but essentially only into the negation of its particular content.” (Page 221).

 

   It is this truth that surfaces in the equation that says that nothing equals something, to the eternal irritation of the formal, non-dialectical thinker.  Now that we have completed the process of differentiation let us see how things stand with respect to the three laws of dialectics given above. For convenience we shall re-state the laws:-

 

1. All progress takes place through the unity and conflict, interpenetration

and transformation of opposites.

 

  

   The opposites, distance s and time t, are held in unity by the Equals sign. They are in conflict since the independent variable t is causing the dependant variable s to change; t penetrates s by imparting to it the law of its motion.  The form which results is a part of the motion, the velocity at one point in time, and the content of this part is the whole motion, since the body had necessarily to fall for 10 seconds to bring this form into being. Hence form is the part and the content is the whole. As yet there is penetration in one sense only, the penetration of s by t, so that the first law of dialectics is at this stage only half complete.

 

2. The transformation of quantity into quality and vice versa.

 

   Although we are dealing here with quantity, the question of quality now arises, because since all quantity has gone, been negated, we are left with zero, nothing, but while nothing is bereft of quantity it is rich in quality. The content is a quantity of distance and time which is qualitatively determined as a form of motion, that is, motion which manifests a law.  As this quantity is negated, reduced to nothing, we find that while this nothing is devoid of quantity it is nonetheless a determinate nothing, the determination being the definite relation between space and time, the law which it manifests. We see that quantity has been transformed into quality, but the reverse has not happened as yet, so that this second law is also at this stage only half complete.

 

3. The law of the negation of the negation.

 

   This law is a unity of the first and second law. The first negation is that which is described in the first law, the quantitative negation of the original quantities of both s and t  into the new quantities which result from the differential increase. There is no doubt that the original quantities have been negated - they definitely no longer exist – only the new quantities exist.  At the same time it is the negation of quantity into quality as given by the second law. Once again this third law in incomplete because the second negation has not yet taken place. The significance of this second negation is explained below.

 

   As mentioned above, the mathematical method of the Infinitesimal Calculus falls into two sides.  So far we have dealt with one side only, Differentiation, which is a process of analysis, that is, a method of starting from a whole, breaking it down into its constituent parts, and considering the parts in isolation. In the example given we began with the movement of a body of 10 seconds duration which manifested a certain law of motion.  Knowing this law we were able to abstract one of the parts of the motion of the body, one moment in the 10 second duration, (since we are dealing with the completion of the 10 second duration then this moment the the end of the 10th. and last second), and to discover the velocity of the body at that moment, 320 feet per second. Further, we noticed that, having completed the analysis, the dialectical laws of motion remained in a half completed state. We can demonstrate the completion of these laws by tracing through the other side of the Infinitesimal Calculus method, Integration, which is synthesis, the opposite of analysis, the placing together of parts into a whole.

 

   In order to explain the method of integration we must go back and extend our knowledge of Differentiation a little.  It is now the general practice to skip over the differentiation process described above by operating to a formal rule which has been discovered to work well enough.  If we consider our original equation

 

s =  ½ gt²

 

we can differentiate s  with respect to t, that is, proceed directly to the differential equation, by adopting a rule which is expressed generally in mathematical language thus:-


                                                                                         dy  =  na.x(n-1)

                                                           dx     


 Referring back to our own equation, n refers to the index attached to t, in our case 2, and a refers to the co-efficient of  t, in our case ½g, (and g = 32).  So to differentiate by rule, we multiply the co-efficient by the index, then subtract 1 from the index, the result in our case being:- 

 
                                                                                    ds = 32t feet/second
                                                                                    dt

 

 

  It has been necessary to explain how to differentiate by rule because the method of integration is simply to reverse the process, that is, to add 1 to the index and then to divide by the new index, which returns us to the original equation. We use the elongated letter “s” to indicate the integration process, and we attach the symbol dt to indicate that we are integrating with respect to t, so the next equation means “the integral of 32t with respect to t.

 

32t.dt  = 16t²

 

   We are starting with the part, the velocity reached and the end of the motion, and we wish to re-connect this with the whole, the distance fallen during the whole 10 seconds.  We therefore assign the value 10 to t. 

 

 16 × 10² = 1600 feet

 

   The resulting value, 1600 feet, is the distance the object falls in the whole 10 seconds of its motion. We have placed the part back in the whole and this is synthesis.  How do things now stand with respect to the three laws of dialectics?

 

1. All progress takes place through the unity and conflict, interpenetration

and transformation of opposites.

 

  

   We saw in the differential process that t penetrated s causing it to change, the resulting form being a part of the motion, its velocity after 10 seconds, the content of which form was the whole of the motion, since the body had necessarily to fall for 10 seconds to bring this form into being. In the integration process a penetration in the opposite sense, inter-penetration, takes place, completing the first law of dialectics which, we remember, was only half completed by the differentiation. The part, the velocity after 10 seconds, is now penetrated by the whole, the time taken for the whole motion, and this gives rise to a leap, and the opposites, form and content, are transformed into each other.  The form is now the whole, the whole distance through which the body moves, 1600 feet, and the content of this form is the part, the velocity of the body after 10 seconds.

 

2. The transformation of quantity into quality and vice versa

 

   In the process of differentiation we saw that the quantity of the differentials was negated, they became nothing, a purely qualitative, determinate nothing, the quality being the law of their relation. Quantity had been transformed into quality.  With the completion of the integration process this transformation of quantity into quality is negated by the opposite process, the transformation of quality into quantity, since the quality manifested in the nothing of the differentials, the law of motion of the falling body, enables us to calculate the quantity of distance contained in the whole 10 second motion of the body, 1600 feet. 

 

3. The law of the negation of the negation.

 

   The first negation is the quantitative negation of the original quantities of both s and t into the new quantities which result from the differential increase, which reaches a limit at the moment of differentiation, the velocity of 320 feet/second.  This is analysis, the part abstracted from to whole.

 

The process of integration negates this negation by synthesising the part, the velocity reached and the end of the motion, 320 feet/second, with the rest of the motion of the body.  The part is still there, but now contained in the 1600 feet and 10 seconds of the whole motion, although it is itself negated, has no existence, as its own individual self. Such a process is sublation, the annihilation of a thing and its simultaneous preservation in a new form of Being. The second negation is not a simple repeat of the first. Its significance lies in the development to new content which, in any natural process, results from the second negation. However, it must be explained that in order to clearly show the relation between the two sides of the infinitesimal calculus, differentiation and integration, and the associated completion of the three laws, we have carried out a process in which the second negation is a simple reversal of the first, and no new content can be expected to arise in such a case.

 

   Hopefully we have demonstrated that, if we train ourselves to think according to dialectical logic instead of the formal logic which presently predominates in the sciences, we can develop profoundly deeper insight into mathematics and with that into every science which relies upon it for its work, and it is difficult to imagine a science which does not. Even more importantly, or perhaps urgently, we have demonstrated that dialectical materialism , Marxism, is a soundly based science and the necessary guide to practice in social, economic and political science, and offers the best possibility of finding solutions to the problems currently besetting mankind.

 

Terry Button.

August 2008