Source: Social Democrat, Vol.10 no.3, March 1906, pp.137, 145.
Transcribed: Ted Crawford. for the Marxists’ Internet Archive.
Will you afford me a little space in which to refute an assertion of Bax? Not that I wish to substitute myself for Rothstein, who needs no one to answer for him, nor do I desire to discuss the value “in itself” of the Marxist method; I avoid the scholastic discussions beloved of Bax’s metaphysical mind.
Bax claims to demonstrate the imperfection of the method in affirming that the natural and mathematical sciences are not attached by any connection to the economic conditions, or if such connections exist they are of insignificant importance. This is a rather bold assertion!
Bax says, with perfect justice, “that man attained to natural knowledge essentially through observation of fact (supplemented later on by experiment), and reasoning from fact.” But are not the every-day experiences and observations made constantly by men in procuring the means of existence a thousand times more numerous and varied than those made by savants in their petty scientific laboratories? Are those observations which are made in the gigantic economic laboratory not susceptible of forcing man to reason and to seek out general laws?
“The doctrine of Natural Selection” which Bax has cited in support of his thesis is on the contrary an excellent example of the superiority of the method that he condemns. In effect, Darwin gathered in the economic world the observations and the experiences which he needed to complete the observations which he and the naturalists had made in the natural world, and to conceive his doctrine. Let Bax open the “Origin of Species” and he will read that Darwin says that the first suggestion of his theory was afforded him by Malthus’s “Law of Population,” which placed to the account of Divine Providence the miseries of the workers engendered by capitalist production, just as Aristotle made nature responsible for slavery. It was by starting from the social struggles of man that Darwin conceived the idea of the natural struggles of animals. But industrial and commercial competition, which, on the one hand, deteriorated the producer by poverty and excessive toil, and, on the other, transformed the capitalist into a social parasite, could not furnish him with the idea of progressive evolution; that was suggested to him by economic phenomena of another order. He saw and admired the farmers and breeders around him, who experimented upon the various animals, long before the naturalists had dreamt of doing so, and who, by “artificial selection,” perfected horses and other animals in order to increase their exchange value. Darwin is, perhaps, the naturalist who has devoted the most attention to the variations of domestic animals. He was led to think that Nature did unconsciously what the farmers did intentionally for the sake of profit. It may, therefore, be advanced that the doctrine of natural selection could only have been produced in an epoch of ferocious commercial competition, and in a country in which methodical and intelligent breeding was carried on.  It is necessary to add that the works of constructing the railways, and of mining coal, which have given birth to a new science, geology, by drawing from the bowels of the earth the remains of extinct plants and animals, have incontestably prepared the scientific mind for the idea of the progressive evolution of the organic world.
“The history of mathematics is a crucial refutation of the one-sided Marxian view,” says Bax. Let us see. He is obliged to recognise “that geometry had its origin in land measurement may be perfectly true. But it is the correctness of the formulation of the space-relations involved in it that is the crucial point for the science as such. The practical necessities which led men’s attention to these relations is the mere superficial and proximate cause.” Quite too metaphysical is this disdain for the “practical necessities” which have caused men to count and to calculate, and which have furnished the axioms of mathematics. It is not because the mathematical sciences make abstraction of the properties of things and consider only a few of them – number in arithmetic and algebra, the point and the line in geometry – it is not because in the abstract sciences observation and experimentation are consequently useless and are replaced by speculation, that one ought, as philosopher of “the thing in itself,” to declare that they owe nothing to experience. They are groups of speculative theorems rigorously deduced from a small number of axioms of an incontestable and incontested truth; the axioms, then, are of capital importance. If they are not found mathematical sciences cannot exist, and if they are erroneous the rigorous speculative deductions are false. But the axioms – (two and two make four; a straight line is the shortest way from one point to another; from a given point we can only draw one line parallel to another straight line, etc. – this third and important axiom bears the name of the postulatum of Euclid) – are undemonstrable. Leibnitz has vainly endeavoured to demonstrate that 2 and 2 make 4. They have been given to us not by reasoning but by experience, and, I would add, by economic experience.
It is probable that animals have bequeathed us many axioms. For example, ducks in going to the water follow a straight line as being the shortest way; pigeons know that 1 and 1 make 2, since they do not sit until they have laid two eggs, etc. Economic experience has given a value to the axioms inherited from the animals and has caused others to be discovered of equal importance, as, for instance, the postulatum of Euclid.
We know that numeration is very limited among savages, that many of them can only count up to 20 and that the first figures have, in their language, the names of the fingers because they count by naming and touching the fingers one after the other. The savage must extend his numeration in proportion as the number of animals and other objects he may possess increases. When they are too numerous to be counted on the fingers, he makes use of pebbles, as is shown by the word “calculate,” which comes from the Latin calculus, signifying pebble; in order to obtain an account of their augmentation, he is obliged to invent addition, the beginning of arithmetic, and algebra, the operations of which are only additions transformed, complicated by unknown and imaginary quantities, and simplified; and in order to state any decrease in their numbers he must invent subtraction, which is only the addition of that which remains to that which has disappeared – the unknown quantity to be found. The Romans performed these two operations with pebbles, as is evidenced by the expressions calculum ponere, to place the pebble, and calculum subducere, to withdraw the pebble, which indicated that they added and subtracted, by adding or taking away pebbles. As exchanges multiplied it became necessary to calculate the number of objects to be given in order to obtain some other; in order for anyone to estimate his wealth in animals and other objects, he had to invent multiplication, which is only a long addition simplified. The traders of the maritime cities of Asia Minor and of Greece made multiplications long before Pythagoras had erected the table which bore his name, and which, perhaps, they had invented. When they had to share out the gains of a commercial expedition according to the number of participants they discovered division, which is a complicated operation of multiplications and subtractions. Many centuries after economic necessity had compelled men to find the four rules of arithmetic, the mathematicians made their theoretical demonstration.
If the possession of flocks and herds and other objects developed numeration, and brought forth the invention of the rules of arithmetic, the manufacture of baskets and of receptacles for liquids engendered the idea of capacity; and the production of precious liquids, such as wine and oil, taught the measurement of the capacity of vessels.
The savage, while he lived by fishing, the chase, and on the wild fruits of the earth, did not dream of measuring the land; but when he became a cultivator and had to divide the arable land among the different families, he had to learn to measure it. The Greek philosopher attributed to the Egyptians the invention of geometry, because after each inundation of the Nile it was necessary to redistribute the fields, the bounds of which had been swept away by the overflowing river. The men of all countries had no need to go to the school of the Egyptians; the agrarian divisions they made every year were the masters that taught them the first elements of geometry.
The savage cultivators, not knowing how to measure surfaces, solved the problem of equal division of land, by dividing the field to be shared out- generally a level ground more or less plane – by long and narrow bands, which, having the same length and the same breadth, were equal; these bands were quadrilaterals, the sides of which were parallels, as the straight furrows which bounded them were of an equal distance from each other. The obtaining of these straight furrows had such an importance that in many languages the word “straight” has come to signify that which is just. The equal length of the furrows was obtained by passing over each an equal number of tines a staff which served as measure. This measuring staff had in their eyes so august a character that in the Egyptian hieroglyphics it signified justice and Truth; while among the Russian peasants the staves used for measuring in the division of land are called sacred measures. Haxthausen, who, about 1846, assisted at one of these divisions in Russia, declared that the measurements are made as accurately by the illiterate peasants as they would be by scientific land-surveyors. This primitive land measurement, which may still be seen in India, gives birth, says Paul Tannery, the erudite historian of the Science Hellene, to “a collection of processes, but loosely related, serving for the solution of the usual problems of life, and the demonstration of which, when it was made, found its support on propositions regarded as evident, but which were rigorously proved very much later.” One of these propositions is the famous postulatum of Euclid, on which rests geometry. This “empirical” geometry, long before the creation of scientific geometry, enabled the Egyptians, the Greeks, and, in fact, all peoples, to construct monuments which, by their grandeur, their solidity, and their harmonious proportions awaken the wonder and admiration of modern engineers.
The primitive cultivators divided level lands which they regarded as plane. The geometry of Euclid starts from the hypothesis that space is absolutely plane. Consequently, two straight lines at equal distance from each other are parallel and can never meet, as well in the level lands of the primitive cultivators as in Euclidian space.
But the idea of curved space was introduced in science towards the middle of the nineteenth century. Lowachevsky, Ricman, Helmholtz, Sophus Lee, and other mathematicians, rejecting the postulatum of Euclid, created what has been called the non-Euclidian geometry, of which the rigorously deduced theorems are, however, in complete contradiction to the theorems of the Euclidian geometry taught for two thousand years as the absolute truth. The illustrious mathematician Gauss, who, at the end of the eighteenth century, already foresaw the possibility of a non-Euclidian geometry, dared only to speak of it in private letters, which have been published recently, for fear of arousing “the clamours of the Boeotians.” The solutions of the new geometry, which overthrow all accepted ideas, are, according to the mathematicians, more simple from the purely mathematical point of view, than the solutions of the old geometry; which, however, retains its practical utility because surveyors, engineers and architects, operating on surfaces of small extent, neglect, like the primitive cultivators, all their unimportant curves. The creators of the new geometry, nn the contrary, take account of every curvature in space, however slight it may be; and they also think there will be as many non-Euclidian geometries as there are places on the terrestrial globe.
From whence came this idea of the curve of space?
The savage cultivators regarded the level lands they divided among themselves as plane. When men conceived an idea of the earth, they imagined it to be flat, like a disc, said Archelaus. But their voyages having shown to the traders of the Mediterranean cities that different places of the earth were lighted one after the other by the sun, they represented the earth as a hollow half-sphere, the border of which was lighted before the bottom, But as a result of astronomical observations, the Greeks, towards the fifth century BC. regarded the earth as a solid sphere. But the idea of the sphericity of the earth remained barren Practically and theoretically. It led to no practical result until the fifteenth century, when Columbus, misled by art error of calculation of Ptolemy, discovered America instead of the maritime route which he sought for commerce with the Indies and which the Venetians monopolised. It was necessary still to wait some centuries before the sphericity of the earth, demonstrated every day merchant ships, determined the mathematicians to deduce from it the theoretical consequences. The geometricians, after having taken account of the observations collected by sailors, merchants, travellers and savants, conceived the earth as a sphere, flattened at the two poles, and enveloped in an atmosphere corresponding to its solid form. All the plans constructed on the earth or in space would, therefore be necessarily curved; all the lines traced on these plans would, perforce, be curved; the line which describes the flight of a cannon ball, whatever may its initial velocity, is a curved line. The curvature these plans and these lines must vary as the place which they are traced is more or less distant from the Equator. The postulatum of Euclid, on which geometry rests, and which cannot be demonstrated by reasoning, is then false experimentally. The non-Euclidian geometries, which appeared to be erroneous, opposed to reason, because they contradicted the truths to which men had been accustomed for thousands of years, are then a superior approximation to the truth. Absolutely plane space, which the necessities of agrarian divisions and of architectural constructions had introduced into the heads of the mathematicians, began to be elbowed out by the idea of curved space only after commercial voyages and expeditions on land and sea had popularised the idea of the sphericity of the earth and of its atmospheric envelope.
Bax, therefore, cannot say that “the history of mathematics is a crucial refutation” of the Marxist method.
I may remark, in conclusion, that Marx did not present economic determinism as a doctrine, but as a tool for historical research, valuable only according to the ability of him who uses it. In his hands it has given us the theory of the class struggle, which explains the political history of human society. If after an essay with economic determinism Bax finds it defective, it is because, like all metaphysicians, he has been unskilful in applying it, and, like the bad workman, he ascribes his own want of skill to the tools.
1. All plants and animals cultivated and bred for the market have for centuries been transformed by persistent experimentation; It is only since the last forty years that naturalists, who had confined themselves to observation, are attempting some few timid experiments. If, like Darwin, they had begun by making themselves acquainted with the experiments on plants and animals made by cultivators and breeders of all countries, and if they had studied their method of work, they would be surprised by their numberless experiments, quite as interesting as that of De Vries on the Oenotheries, of which they had retained only the practical results, without drawing the theoretical consequences, as the Dutch botanist had done from his one experiment.
2. The French word “droit” is the equivalent of both “right” and “straight.” – TRANS.
Last updated on 14.9.2008