What is to be Done: III
The essential tasks for the Marxists of today
When I was teaching “A” Level mathematics in the late 1960s, I had to teach what is called The Calculus. This was a series of techniques independently and more or less simultaneously invented by both Isaac Newton and Leibnitz to deal with the mathematical study of Rates of Change. These covered all such situations in Reality, but are most clearly and accurately encapsulated by the series of relations involving Distance, Time, Speed, and Acceleration.
These are a closely related set: the rate of change of Distance with Time being termed Speed, and the rate of change of Speed with Time giving us Acceleration. Now relations between pairs of these variables could be experimentally studied, and the results fitted to standard forms available from Mathematics. But, it was soon clear that scientists needed to move between these various variables at will, and some natural relations required different selections from the set to deliver what was required. For example air resistance is related to speed, and so MUST involve that Rate of Change in its descriptive formula. So though Scientists and Mathematicians could manipulate the separate equations, they couldn’t convert one form into another. Such a process was NOT mere manipulation. A whole new level of conversion was found to be involved, but these involved the plotting of the pair of related variables onto Graph and thereafter to construct lines on the graphs which could deliver (by calculation) the required rate-of-change values. Various frigs (or short cuts) were found, but these just had to be remembered. What was needed was general method of doing these tasks covering ALL possible equations. AND, most important it had to be the job of a teacher to explain why these tricks worked, and even to supplying a general way of doing it for ALL possible cases.
The classic route is to plot out the given equation as a graph. For pairs of variables this was very easy to do, and students could follow the processes as demonstrated, but for any number of variables above two, the graphical method was at first difficult, and then impossible.
Needless to say most students did NOT like this area of Mathematics.
Let us see what was involved for just two variables, and then see how Newton and Leibnitz then cracked it for ALL cases. What was required was to find out the rate of change of one of the variables with respect to the other.
For example we might require the change in Distance with Time - we were requiring the Speed!
Now such things were easy when speeds were constant, but if these were changing moment by moment they were seemingly unobtainable. It had long been the practice of Mathematicians to construct a straight line addition onto the graph at the point where we need the rate of change.
Obviously, such a construction has to try to match the “slope” of the graph accurately, and a right angled triangle constructed with this line as its hypotenuse. The Tangent of the angle of slope would give us our required Rate of Change.
Thus, the instantaneous Rate of Change, at the given moment, was extracted from the graph. Also, it is clear that for every point where we required this information, this whole process would have to be repeated. This is a tedious process and needed to be replaced with something more accurate and quick.
What Newton did was to correctly assume that this crucial slope could be directly derived from the original equation WITHOUT all this geometrical construction and trigonometry. His researches discovered a manipulative technique wherein the “slope” of (say) y = x3 was shown to be 3x2 at every point on the curve (i.e. for every situation covered by this equation). This led him (and Leibnitz quite separately of course) to a general form where for y = xn, the slope would be nxn-1 This general process was given the name Differentiation, and was expanded to cover all known formulae.
But why did this (and the following related techniques) work? Remembering such a trick was useful but NOT really informative.
Both inventors attempted to establish their manipulations by geometrical proof. It amounted to drawing a Chord between two points A and B, on the curve of the equation, and setting up a right angled triangle with these two points giving the hypotenuse of that figure.
The coordinates of the points at A and B could easily be used to find the lengths AC and BC and a very inaccurate estimate of the required slope (at A) could be obtained from that of the chord AB.
What happened next was philosophically very interesting. Both inventors assumed that perfect Continuity pertained for the given equation, and B could be brought closer and closer to A. As this happened, the calculated slope of AB would change, until it STOPPED at a final and accurate value just as the point B became coincident with A. At that precise moment, the slope of the chord AB would be identical with the slope of the curve at A.
But, there was a problem.
At this final situation the triangle used to calculate the slope completely vanished, so that the slope at A (the Tangent of the angle at A) became BC/AC which was 0/0 (A and B were the same point). But surprisingly the slope didn’t vanish too. It actually reached its correct finite value. But how could we find it from 0/0. Both inventors were able to establish, theoretically, the final form of each Differentiation from this construction, but it certainly posed important questions. Their modified versions after differentiation worked well, but they had only demonstrated rather than proved why this was the case. The whole idea of 0/0 being all sorts of different and finite things seemed WRONG. Our mathematicians had come up against the consequences of the assumption of Continuity.
Indeed, this anomaly was merely swept under the carpet, and students were told just to remember the frig to get the right answers. But, it was, of course, very unsatisfactory. The mathematicians had used an argument, which they had taken to The Limit, and without a clear justification conjured up the Right answer out of a simple Frig. The students naturally wondered why I had bothered to point out this situation. “Surely”, they insisted, “if the correct answer was obtained, then what could possibly be amiss?”
So, I asked them to consider the division of things in general into smaller and smaller pieces. “Could this go on forever?”, I asked. The answer was clearly, “No!” So I proposed that the assumption of Continuity was only OK, so long as we remained within the limits of Applicability of an equation. Indeed ALL equations had this limit, and would fail if it were transgressed. They began to see that Mathematics was NOT absolute but conditional and limited.
It was an important lesson. All philosophies that attempted to construct an absolute picture of Reality based on Mathematics were simply wrong. Mathematics was NOT the essence of Reality, but a pragmatic and limited bag of practical frigs.
Now, for many years a similar discussion had been going on in Science (or more precisely in Physics) as to whether everything studied was similarly perfectly Continuous. But researchers kept finding descrete particles, which could never be shoe-horned into a conception of total and universal continuity.
The assumptions of Newton and Leibnitz could be wrong! The assumption that perfectly continuous equations covered all of Reality came into question.
Now, this everyday pair of assumptions: “Reality is Continuous” and “Reality is made up of Descrete fundamental units”, had been first revealed 2,500 years ago by Zeno of Elea (in ancient Greece). Zeno was so taken aback by the consequences of these two assumptions that he designed a series of Logical Paradoxes to show that BOTH of these assumptions led to contradictions when used in the study of Motion. No-one was able to refute the towering logic of his case. Indeed, they didn’t even know what his purpose was! They thought that he was saying that Logic is impossible, whereas, of course, he was saying NOTHING about Logic as such (indeed he himself was using it to make his points). He was, on the contrary, attempting to draw attention to our underlying, everyday assumptions that were the basis for subsequent logical arguments and derivations. He proved that the Foundations for our Logic were both man-made and inaccurate.
Needless to say, I taught Zeno to my A Level students and Mathematics was set alight!
This post is the third in a new blog series entitled "What is to be done?" on the crises in both Marxism and Science, and how a revolution is necessary in both. This body of work is AVAILABLE NOW as a Special Issue. Read it all here!