In the belief that it might help the reader to appreciate the enormity of what has been done if he is shown, very much in outline and without going into detail, the mechanics of how it was done, I reproduce the following very informative passage from a book from which I have already quoted more than once in this chapter, Evert Beth’s The Foundations of Mathematics – A Study in the Philosophy of Science. As with some others that I have used, this book is useful to my purpose because the author does not take a position against what happened and therefore cannot be accused of bias in favour of my conclusions. He is merely reporting the facts that occurred. I shall interpose occasional comments. Pages 38 and 47 of the book (my emphases added throughout):
“Aristotle’s theory of science postulates, as we have seen, every science to have a deductive structure, to start from principles accepted as self-evident, and to have an empirical foundation.”
In other words, the principles set out by Aristotle, which by the turn of the seventeenth century had survived examination over some two thousand years, were that truth could be arrived at by a combination of (1) what one could see or otherwise experience with the senses, (2) what every reasonable person would agree to be self-evidently true, (3) what could be deduced by correct use of logic, and (4) what could be tested by experiment.
“About 1600, it became more and more clear from scientific practice that science could hardly hope to satisfy all three of these postulates at the same time.
“In mentioning the date 1600, I do not mean to imply that it was only then that the development of modern, ‘non-Aristotelian’, science had its beginning. It is known, from the studies of scholars such as P. Duhem, Dijksterhuis, P. Rucker, and A.C. Crombie, that the roots of modern science reach far back into the Middle Ages, and there may even be much truth in the opinion of such authors as R. Eisler and A. Frenkian, who place the origin of certain number of fundamental conceptions of modern science and philosophy in times far beyond Greek antiquity. It is not until 1600, however, that non- and anti-Aristotelian conceptions took scientific forms which could successfully rival and even supersede the solid edifice of peripatetic [i.e. Aristotelian] science.
“From then onward, it became customary to recognise two different types of science, one of which conforms to the postulates of deductivity and evidence, whereas the other answers to the requirement of an empirical foundation. Rationalism, as defended by Descartes, has a preference for the first type of science which I shall call rational science, whereas empiricism, typically represented by Locke, fosters empirical science, as the second type is called. The opposition between the two schools of rationalism and empiricism should not, however, be overrated, as these schools have their origin in the same historical situation and present quite a number of common features in their doctrines.”
Indeed the two opposing errors would have the same origin. The important thing is not the particular error that is propagated but that there should be errors. And once an error has been proposed, which will inevitably be attacked as demonstrably untrue, it is vital to set up another error in opposition to it, because otherwise the first error will be opposed only by the truth - leaving little doubt about which the victor will be. Provided two ideas in opposition to each other are both wrong, those responsible for propagating the errors do not mind which side we join; and the fact that two ideas are in opposition to each other carries the insidiously seductive implication that one of them must surely be right. In other words each confers a spurious legitimacy on the other.
“Leaving Kant aside, we may sum up the outcome of the development of the theory of science during the l7th century as follows. There are two types of science:
“(a) Rational science, which starts from principles, accepted as self-evident, and proceeds by rigorous logical deduction and so conforms to Aristotle’s postulates of deductivity and evidence, but not necess-arily to his reality postulate.
“(b) Empirical science, which starts from experimental data and proceeds by analysis; it conforms to the reality postulate, but not necessarily to the postulates of deductivity and evidence.
“Consequently, speculative philosophy has to make a choice between being either a rational or an empirical science, and it accordingly splits up into the currents of rationalism and empiricism. Kant, by bringing together rational and empirical science, made an attempt to restore, as far as possible, Aristotle’s unitarian theory of science; in my opinion, however, he was not successful.”
He certainly was not. Far from restoring Aristotle, which he could have done very easily simply by returning to Aristotle, he was advancing yet another error; but that is another story, outside the scope of this appendix.
“On the contrary, rational science turned farther from Aristotle’s ideal, by dropping his evidence postulate also. The development of non-Euclidean geometry constituted the first move in this direction; the decisive step was taken as a result of contemporary research into the foundations of logic and mathematics, Each of the modern schools in this field - logicism, cantorism, formalism, and intuitionism - attempted, initially, to maintain the postulates of deductivity and of evidence; they were all forced to drop the one or the other of these postulates.
“An equally significant development can be observed in empirical science. Here, in spite of Mach’s phenomenalism, the reality postulate had to be attenuated in order to preserve the ability to construct suitable deductive theories. Modern physical theories do not conform to the evidence postulate, and, recently, the transition to quantum logic has even necessitated a revision of the postulate of deductivity.
“It is easily understood how these developments have alarmed the representatives of the various schools of speculative philosophy, which, as we have seen, derives its origin, and even its right of existence, from Aristotle’s theory of science. This accounts for the violent protestations of speculative philosophers against the developments in modern science which gave rise to the establishment of such theories as non-euclidean geometry, mathematical logic, the theory of relativity, and quantum mechanics, each of which, in one respect or another, implies an infringement of the postulates underlying Aristotle’s theory of science; the unanimity of these protestations is, indeed, in a peculiar contrast to the common discord among speculative philosophers.”
In other words, putting it into nice simple language, the scientific system of Aristotle, which required equal weight to be given to both logic and what one could experience through the senses, was supplemented by either logic without the need to test it against experience or experience without logic. Rational science, on the one hand, denied the need for evidence, and empirical science, on the other, denied the need for reason. Thus, to point out only the most fundamental limitations: rational science said that if a thing could be shown to be mathematically true it was true even if common sense showed it to be false, examples of which we saw earlier on in the main body of this essay;* while empirical science limited its discoveries to what could be directly perceived, and no matter how demonstrably true something that was outside the range of the senses might be, it could not be accepted.
And, funnily enough, once sufficiently unhinged from reality, both erroneous systems of science eventually ended up by denying their own foundations; so that rational science, as Dr. Beth has just stated and as we had already seen, ceased even to adhere to the rules of logic and empirical science ignored empirical evidence. The wheel has turned full circle and unity has been restored; but this time it is not Aristotelian unity but a unity of universally self-contradictory madness.
* And when mathematicians are let loose on such distortions of reality, the results, surely not surprisingly, can be bizarre indeed. I offer an example.
We take a column of figures; we add 0 to it; in doing this we have not affected the result. So far so good; and similarly if we subtract 0 from the column of figures. But the inductive conclusion that has been too swiftly drawn from this and similar facts is that it is equally safe and mathematically valid to multiply or divide by zero. And it is not equally safe and mathematically valid; emphatically it is not.
It is obviously not good enough for me simply to assert this; I shall have to show that it is true. And in order to do that I shall have to suspend briefly my resolve to avoid any mathematics in this chapter and include a sequence of equations which not everyone will be able to follow. But courage, please! – even those readers who have not attempted to get their minds into training by seeing if they could make more sense of Eddington’s helpful dissertation on the displacement of the Frauenhofer lines than Dr. Lynch managed to make!
Unless you have never learnt any algebra or have forgotten all you knew, it will not be difficult to follow it. And even everybody else need not panic; for all they need to be aware of is that, although the following sequence of equations ought to be valid, since it obeys all the rules of mathematical manipulations (a fact they can confirm with any friend who knows enough mathematics), nevertheless it leads to the patently aberrant conclusion stated in the last line:
Suppose a = b
therefore ab = a2
therefore ab - b2 = a2 - b2
therefore b(a-b) = (a + b)(a - b)
therefore b = a + b
therefore b = 2b
therefore 1 = 2
Yes, one is equal to two, according to the rules of mathematics. Did you spot the flaw? It is that, on the right-hand side of the fourth line of the equation, the multiplicand “(a - b)” is equal to zero.95 And thereafter the equations are erroneous because the zero has been used to divide. And whereas even most mathematicians would acknowledge that this is not allowed in the context given, it does not stop them doing similar things themselves. And when they start treating infinity in the same way, the results are even more disastrous; and, of course, the more complicated the mathematics, the less easy it is for the layman to spot where he is being hoodwinked.
A whole chapter or book would need to be devoted to showing exactly why; but suffice it for the moment to say that these illegitimate manipulations were involved from the start in the integral and differential calculi invented and developed from the sixteenth century onwards, and underlie the whole of the branch of pseudo-mathematics founded by Georg Cantor to which reference will shortly be made in footnote 97.
All that now remains for me to say about the need for mathematics to be valid is that it should certainly not be supposed that, provided scientists get their sums right, their practical conclusions will definitely be correct. As I have pointed out earlier, the other main problem lies in the fact that mathematics sometimes supplies several different possible answers to the same calculation, between which, from the mathematical point of view, there is nothing to choose. (...)
EINSTEIN AND MODERN PHYSICS
by N. Martin Gwynne