For formal logic, or elementary mathematics, the mere use of its laws is held sufficient to be proof, as with the proofs in geometry that the internal angles of a triangle equal 180 degrees. If the application of the laws of formal logic are applied to an argument, and result in a reductio ad absurdum, where A is, simultaneously, -A, this, alone, is held to be proof of the argument being false. Yet, as indicated earlier, in the real world, the internal angles of a triangle – for example drawn on a sphere – do not equal 180 degrees, and A is frequently, also, -A. It shows the limitations of formal logic, and elementary mathematics.
“Even formal logic is primarily a method of discovering new results, of advancing from the known to the unknown — and the same holds, only much more eminently so for dialectics, which by breaking through the narrow horizon of formal logic, also contains the germ of a more comprehensive world outlook. The same relation exists in mathematics. Elementary mathematics, the mathematics of constant quantities, moves by and large at least, within the confines of formal logic; the mathematics of variables, the most important part of which is the infinitesimal calculus, is essentially nothing but the application of dialectics to mathematical relations. Here, mere proof is decidedly pushed into the background, as compared with the manifold application of the method to new spheres of research. But almost all the proofs of higher mathematics, from the first proofs of the differential calculus on, are strictly speaking wrong from the standpoint of elementary mathematics. This is necessarily so, when, as in this case, an attempt is made to prove by formal logic results obtained in the field of dialectics.” (p 171-2)
Some years ago, one of my next door neighbours was a Professor of Mathematics at Keele University. One Summer afternoon, as we sat with other villagers in someone's garden, we, along with another neighbour, got on to a discussion of Philosophy. The other neighbour was Irish, and so, the ideas of his countryman, Bishop Berkeley, was included. As I recall, it was raised by my other neighbour, the Professor of Maths, in relation to the question of the Differential Calculus. Now, I am the first to admit that I am not a mathematician. I did not do 'A' level Maths, which meant that, when I first studied Economics at university, I was at somewhat of a disadvantage when everything was presented in mathematical terms, and understanding of the calculus was assumed. I had done 'O' level Maths, and had a perfect score in the Arithmetic paper, however. I had to quickly learn calculus in addition to everything else.
So, it was fascinating to me that, however we got on to this question of the differential calculus, and the inherent contradiction it involves, of A being -A, of a curve being simultaneously a straight line, the Professor was unhappy, even today, at the idea of accepting this contradiction, and hence his resort to Berkeley, and his comment, when I raised his name, that, in his youth, anyone who referenced Hegel was dangerous to be around. Yet, when I asked him how it could not be a real contradiction, he had no answer.
“To attempt to prove anything by dialectics alone to a crass metaphysician like Herr Dühring would be as much a waste of time as was the attempt made by Leibniz and his pupils to prove the principles of the infinitesimal calculus to the mathematicians of their time. The differential gave them the same convulsions as Herr Dühring gets from the negation of the negation, in which the differential also plays a certain role, as we shall see. Finally these gentlemen — or such as had not died in the interval — grudgingly gave way, not because they were convinced, but because it always came out right. As he himself tells us, Herr Dühring, is only in his forties, and if he attains old age, as we hope he will, perhaps his experience will be the same.” (p 172)
In fact, if we move to the 20th century, even Einstein was uncomfortable with the contradictions involved in Quantum Theory. Yet, in the same way, the results obtained by Quantum Theory have repeatedly been found to be correct, despite defying the laws of formal logic.
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