If we take mathematics, astronomy, mechanics, physics and chemistry, Engels says,
“... it can be asserted that certain results obtained by these sciences are eternal truths, final and ultimate truths, for which reason these sciences are called the exact sciences. But this is very far from being the case for all their results. With the introduction of variable magnitudes and the extension of their variability to the infinitely small and infinitely large, mathematics, which was so strictly moral in other respects, fell from grace; it ate of the tree of knowledge, which opened up to it a path of most colossal achievements, but at the same time a path of error, too. The virgin state of absolute validity and irrefutable proof of everything mathematical was gone for ever; the realm of controversy was inaugurated, and we have reached the point where most people differentiate and integrate not because they understand what they are doing but from pure faith, because up to now it has always come out right.” (p 109-10)
In other words, the differential calculus, without which most modern maths and science would be impossible, itself breaches the rules of the syllogism. In orthodox economics, for example, based on marginalist theories, the point of tangency is fundamental. If we take an indifference curve, at the point of tangency with the budget constraint line, it is where the consumer is indifferent between good A or B. Either side of this point, they can improve their welfare by exchanging one for the other, and so will do so up to this point. But, how do they, then, choose one or the other as any choice implies inequality?
Similarly, firms employ factors of production up to the point at which the marginal revenue product from them is equal to their price, and, similarly, they expand output to the point where the marginal cost of production of a good is equal to its price. But, as described earlier, all of these points of tangency required for the determination of optimality conditions, require the existence of such a point, which, when considered more closely, has to be a point as defined in mathematics, as being of zero size. It depends on something that can only exist abstractly, and not in the real world.
Engels; argument was strengthened within a couple of decades of his death, both in the work of Einstein, on Relativity, but also, the work of Planck, Bohr, Schrodinger and others in relation to Quantum Theory. Again, without the latter, none of modern electronics would be possible.
Engels argues that, with many of these sciences, it would be impossible to verify the theories, because it would be impossible to observe the reality. In fact, we have been able to do that, to a far greater extent too, and, thereby, to confirm the theories. Einstein's theory of relativity, which showed that Newton's laws were themselves only partial truths, argued that gravity is not a force, but represents a distortion of the fabric of space-time, due to mass, and this distortion also causes light to be bent, so disproving the “truth” that it travels in straight lines. That was proved by observing the light from distant stars being bent around the sun - gravitational lensing – during an eclipse.
Engels doubted that we would be able to observe the movement of molecules, but the development of the electron microscope has made that possible, and now have the ability even to manipulate individual atoms via nano-technology. At the other extreme, in the realm of the very large, the Hubble, and, now, James Webb telescopes have enabled us to see to the outer reaches of the known universe, and so also, back in time to within around 300 million years after the Big Bang. That, plus spacecraft sent in orbit around, and landing on other planets, means that we can observe the same processes of geology and meteorology.
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