This subject called Mathematics, in fact it’s larger family, the complex of Analytics that fall under the banner of Logico-Mathematical Method, is erected on multiple pillars. If they shake an inch the entire structure will swing by a mile. So it’s a good idea to step down with a flashlight and take a look at these foundations every now and then.
The first and most important is the pillar labeled with the little word ‘ Is’ [we will get to it in the Posts, by-and-by]. The second of the pillars is the Principle of Contradiction. And the third, often ignored, is the concept of the Tautology.
Here is Professor Bertrand Russell:
‘It is clear that the definition of ‘Logic’ or ‘Mathematics’ must be [newly] sought..[we must] no longer be satisfied [defining] logical propositions as those that follow from the Law of Contradiction..[but] must admit a wholly different class of propositions..[that] all have the characteristic which we
agree..to call Tautology.
For the moment, I do not know how to define ‘Tautology’..I know of none that I feel to be satisfactory, in spite of feeling thoroughly familiar with the characteristic of which a definition is wanted.
At this point, therefore, for the moment, we reach the frontier of knowledge on our backward journey into the logical foundations of Mathematics.’
Honest and finely crafted lines from the co-author of Principia Mathematica. In English: ‘The ‘[Foundational] Principles of Mathematics’.
In the early part of the last century, Cambridge Analytic Philosophers, lead by Bertrand Russell [Wittgenstein wrote his Doctoral Dissertation under him, if I recall] and bewitched by the Rational, had tried hard to anoint it as the only true God. And they had thought long and hard about the meaning of words like ‘Tautology’. Kurt Godel’s work, among others, ended their ambitions for good.
As a teenager growing up in Madras [now, Chennai], I spent my free afternoons at the British Council Library mainly because it was free, air-conditioned and always had pretty girls visiting from the neighboring college.
And the library carried all his books [he won a Nobel for Literature]. And I must have read everything the good professor ever wrote. [I admired his courage: ‘Traitor to Class and Country’]. He was a hero to my youth.