Anti-Duhring, Part I, Philosophy, III - Classification. Apriorism - Part 7 of 7

Mathematical axioms, Engels says, can be reduced to two, borrowed from logic. First, the whole is greater than the part, and second that, if two magnitudes are equal to a third, they are equal to each other, i.e. if A = C, and B = C, A is also equal to B.

“The remaining axioms relating to equality and inequality are merely logical extensions of this conclusion.

These meagre principles could not cut much ice, either in mathematics or anywhere else. In order to get any further, we are obliged to import real relations, relations and spatial forms which are taken from real bodies. The ideas of lines, planes, angles, polygons, cubes, spheres, etc., are all taken from reality, and it requires a pretty good portion of naïve ideology to believe the mathematicians - that the first line came into existence through the movement of a point in space, the first plane through the movement of a line, the first solid through the movement of a plane, and so on. Even language rebels against such a conception. A mathematical figure of three dimensions is called a solid body, corpus solidum, hence, even in Latin, a tangible object; it therefore has a name derived from sturdy reality and not at all from the free imagination of the mind.” (p 49-50)

Duhring notes that, although mathematical elements such as number, magnitude, time, space and motion are empirically observed, in relation to things in the real world, for example, I might count 10 sheep, or 10 Rolls Royces, the concept of number, i.e. of 10, exists on its own, abstracted from whether it is sheep or Rolls Royces. But, as Engels says, this is true, more or less of every abstraction. Marx makes the point in The Poverty of Philosophy.

“In the world schematism pure mathematics arose out of pure thought — in the philosophy of nature it is something completely empirical, taken from the external world and then divorced from it. Which are we to believe?” (p 50)

Back To Part 6

Back To Table of Contents

Michael Roberts' Fundamental Errors, IV - The Transformation of Values Into Prices - Part 2 of 2

As Marx notes,

“The foregoing statements have at any rate modified the original assumption concerning the determination of the cost-price of commodities. We had originally assumed that the cost-price of a commodity equalled the value of the commodities consumed in its production. But for the buyer the price of production of a specific commodity is its cost-price, and may thus pass as cost-price into the prices of other commodities. Since the price of production may differ from the value of a commodity, it follows that the cost-price of a commodity containing this price of production of another commodity may also stand above or below that portion of its total value derived from the value of the means of production consumed by it. It is necessary to remember this modified significance of the cost-price, and to bear in mind that there is always the possibility of an error if the cost-price of a commodity in any particular sphere is identified with the value of the means of production consumed by it. Our present analysis does not necessitate a closer examination of this point.”

(Capital III, Chapter 9)

In other words, when Marx first set out his calculations of the transformation of values into prices of production, he only carried that as far as the transformation of the output prices. However, as he sets out, here, many outputs are simultaneously inputs for other spheres of production. If the value of coal is transformed into a price of production, for example, and is changed from £10 per ton, to £15 per ton, this coal is also, simultaneously, an input into the production of, for example steel. But, in calculating the price of production of steel, the former, untransformed value of £10 per ton was used, whereas, in this capitalist economy, in which commodities sell at prices of production, rather than values, the steel producer must actually pay £15 per ton for it, and this consequently, changes the calculation of their own organic composition of capital, and rate of profit, and consequently, their price of production. It is the point that Marx makes above, when he says,

“Suppose, the average composition is 80c + 20v. Now, it is possible that in the actual capitals of this composition 80c may be greater or smaller than the value of c, i.e., the constant capital, because this c may be made up of commodities whose price of production differs from their value.”

Back To Part 1

Anti-Duhring, Part I, Philosophy, III - Classification. Apriorism - Part 6 of 7

Early humans had to observe the world around them in order to begin to understand it, and, thereby, to survive in it. To go from observation to understanding requires categorisation, sticking labels on things, which becomes manifest, and takes a great leap forward with language. But, alongside it goes the need not only to categorise, but to count, which is why humans use as the basic means of such counting the fingers of their hands. Similarly, to count distances, they also use parts of the body – a foot, a stride, the length of a forearm, and so on. Only later do these things become merely names for those lengths, whilst an abstract, common unit of measurement takes their place.

The need to measure becomes vital to human survival. When settled agriculture begins, not only is it necessary to measure areas for cultivation, but measurement of time, of the seasons, and so on is required. None of it is possible without mathematics, but the mathematics itself could not develop without observation of the real world.

“... as in every department of thought, at a certain stage of development the laws abstracted from the real world, become divorced from the real world, and are set over against it as something independent, as laws coming from outside, to which the world has to conform. This is how things happened in society and in the state, and in this way, and not otherwise, pure mathematics is subsequently applied to the world, although it is borrowed from this same world and represents only one part of its forms of interconnection — and it is precisely only because of this that it can be applied at all.” (p 48)

One reason that mathematics develops concepts such as the point, which do not, and cannot exist in the real world, is precisely because it tries to model reality on the basis of a formal logic, of the syllogism, whereas reality is dialectical, i.e. it is inherently contradictory, because its essence is flux. The syllogism denies that something can simultaneously be something else. A cannot be -A. So, for example, a straight line cannot be a curved line. But, how then to explain a tangent? At the point of tangency, the angle of the straight line is equal to the angle of the curved line. In other words, at that point it is both curved and straight! How to resolve that? If the line is reduced to a point, which has no extension, i.e. is reduced to something that does not exist, and cannot exist.

Take another example, but, now, in relation to time. I have discussed these issues previously in relation to the arguments of supporters of the TSSI. If an object is in motion, say a ball that is dropped, this motion necessarily involves contradiction. It means that the object is in two places simultaneously, it is both at A and -A (not A). The syllogism cannot accept such a conclusion. But, the reality shows that, as well as starting at A, and ending at B, with no stops along the way, this very fact means that when I measure the position of the object, at any other moment in time, it must, similarly, start at one position and end at another. Each moment of time has a duration, a start and end, and, consequently, in this moment, the object is at two different positions. That is true no matter how short that duration might be. The only way around that is to theorise a zero in time, a point in time that has no duration. But, that, again, is something that does not, and cannot exist in the real world.

At the macro level, the arrow of time, points in one direction, and whilst this may not be true at the quantum level, it is still not true that what exists or can exist is a zero of time, but that there is uncertainty, with the arrow of time pointing in different directions, with different degrees of probability. Its only when the consequence of these backward and forward movements is viewed in their totality that we see the arrow of time moving in one direction. If you look at a mountain side, from a distance, it appears as a more or less straight upward line. But, look at it from a closer distance, and the straight upward line becomes a line with periodic changes in angle. Look even more closely, and you will see that, at some places on this line, it does not slope upwards at all, but downwards.

Back To Part 5

Forward To Part 7