The ‘Foundations of Mathematics’

The Wittgenstein family in Vienna, summer 1917. From left, siblings Kurt, Paul, and Hermine Wittgenstein; their brother-in-law, Max Salzer; their mother, Leopoldine Wittgenstein; Helene Wittgenstein Salzer; and Ludwig Wittgenstein.

Here is Ludwig Wittgenstein from his Tractatus Logico Philosophicus, a seminal text on the Philosophy of Logic:

The Tautology is unconditionally true; the Contradiction is in no condition true…the Truth of Tautology is certain, of Proposition possible, of Contradiction impossible.

‘Tautology and Contradiction are without sense..Tautology leaves to Reality the whole infinite logical space; Contradiction fills the whole logical space and leaves no point to Reality.

Neither one of them therefore can in any way determine Reality..(They) are the limiting case of the combination of symbols, namely their dissolution.

O.K. So what in heaven’s name is a ‘Tautology’? I’m glad you asked. For strictly speaking, we don’t know.

What is a Contradiction? We are not too sure either.

But here are examples of what we think they mean:

‘It is raining’ is a proposition. You can verify its truth by looking out the window. ‘It is raining or not raining’ is a Tautology: it’s truth, a Logician would say, is certain. ‘It is both raining and not raining’ is a Contradiction: it’s truth, a Logician would say, is impossible.

‘It is neither raining nor not-raining’ however is Sweet Nonsense. The Logician does not see the need to dignify it with a comment.

To search within the limits of the familiar and the sensible is to look for your lost keys under the lamp-post, ‘because that’s where the light shines’.

Tautology and Contradiction are the Logico-linguistic limits of the legitimate expression. They mark the boundary of the sensible. Go past that boundary and you are in absurdist territory.

Respectable folks largely live in the zone between Tautology and Contradiction, the mapped terrain of ‘Conventional Understanding’ [vyavharasatya].

Worthwhile Teaching however [and there is not much of it around], begins at this border and moves outward into zones of ever-increasing Absurdity.

[The metaphor of the light under the lamp-post goes back to early Sufi literature but has been appropriated as the ‘Streetlight Effect’ by modern Academia.]

Ludwig Wittgenstein taught Logic and Language at Cambridge with Bertrand Russell [Principia Mathematica] and was a reluctant founder of Analytical Philosophy.

I always liked Professor Wittgenstein. He was the established star at Cambridge, a serious philosopher who also had a fan-following. [Stranger things happen. Paris shut-down for Jean-Paul Sartre’s funeral.]

And Wittgenstein just turned and walked away from it all once he stopped believing in what he was teaching. That’s intellectual honesty.

In his celebrated phrase of informed and abject capitulation: ‘Whereof one cannot speak, thereof one must be silent.’

Is the characterization: ‘It is just as it is; things are just as they are’ an empty Tautology? How about ‘True Nothing’ and ‘Not-True Nothing’?

What’s a Tautology?

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. A 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 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.

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 of conviction; he claimed he studied Mathematics only because it was the nearest thing he could find to ‘Certainty’. He was a hero to my youth.

One thought on “The ‘Foundations of Mathematics’”

Comments are closed.