by Desmond Sander

Last night, watching an excellent SBS film about the mathematics genius Ramanujan, a thought occurred to me which might interest others who wonder about the effectiveness of Mathematics and Physics as ways of understanding what happens.

As the movie made plain, Ramanujan was most reluctant to supply the proofs of his theorems that Hardy and others demanded. I think it likely that, thanks to some fluke in the make-up of his unique brain/mind, Ramanujan was able to see his congruences directly, so to speak, and viewed them as not needing the proofs that are viewed as essential by Logicist Mathematicians, including the very distinguished Mathematician GH Hardy. As well as by Bertrand Russell, the inventor of modern logicism, who also featured in the film.

Ramanujan’s extraordinary insights are not so different in kind, it seems to me, from the insight of Pythagoras that led him to believe the Pythagoras Theorem long before it was proved. What those insights, separated by a couple of millennia, have in common is that they have to do with geometry, in particular the geometry of the circle.

In the Pythagoras case, all that was required to believe what might at that time have been called the Pythagoras conjecture was careful drawing (or imagining drawing), on a flat sheet, with straight-edge and compass and the kind of direct empirical insight that is nowadays disparaged as visual proof. Around a century later, Plato and Euclid between them had the idea that became logic, with its axioms and inference rules. So, I find myself wondering whether it might be that Ramanujan chose not to prove his results because he viewed those Logicist proofs as an easy waste of time.

I think that the easy proofs which currently pre-occupy the Logicist/Analytic Philosophers in leading Australian Universities nicely confirm the view that Logicist thinking is easy.

Mathematics/Physics, I would say, is much deeper than that.

Does anyone out there disagree?

Note (20/12/2022)

Ramanujan’s formula for Pi can be found here
or a pdf downloaded from here: Ramanujan’s Magnificent Formula for Pi.

6 Replies to “Ramanujan”

  1. It’s the same conflict I experience in my own thinking. In my case it is between a cumbersome, verbal mode groping towards being logical and an ~10 x more powerful mode I have come to call my “WIMP-ish wormhole activity” with WIMP being an allusion to its Windows Icon Mouse Pointer structure, what before WIMP was invented, I used to call an “Alice in Wonderland”/”visionary” structure . “Wormhole” is an un-PC allusion to what Ramanujan was on about, his getting it all from his God. It is the same with me, except that my God is what I believe to be a Universal Life-Form. My “WIMP-ish wormhole activity” has been mostly wasted career-wise, but one thing I get from it are some intense, everlasting intuitions about particular people. Also many omens, strings of coincidences happening in real life which seem to be too meaningful to be random.

  2. Another thing I loved about this article is that it is a great example of Less is More. Its concentration on the single point that our usual Western demand for rigorous mathematical proof may be wasting many important new discoveries, a point also clearly made in the film.
    Two points I can add a single illustration to :
    I should have concentrated my presentation, at an AGC Conference in Hobart ~20 years ago, of my Planetary Meta-Geology theory, on demonstrating and explaining fully the gestalt method that had convinced me that I was right. There really are symmetries, “Continental Drift Contradictions”. I should still do that. Indeed I shall. I intend doing this, working on it within a few weeks, thanks to this article. Thanks John and Desmond.
    I had shown telling symmetries in superpositions of rocky planet antipodal conjugacies, by showing them against obviously assymmetrical, much less beautiful, false superpositions showing false antipodal conjugacies, obtained by leaving out say the lateral inversion normally done, in combination with a vertical inversion, to obtain antipodal conjugacies.
    I had been lucky to be able to obtain antipodal maps and photos of the Moon, Mars, Venus, Earth freely off the Internet.
    Detail 1:20,000 such maps of Amazonia and Indonesia I obtained from the Library work the same way, show beautiful symmetries (obviously planetary shock wave fringes, what I was calling Super Huge Impact Tectonogenesis).
    These conjugacies included many telling alignments of “shock wave inscription manifestations” as rivers, mountain ranges and coastlines.

  3. Knowing nothing about geology, I would like to know the meaning of those mathematical-sounding “conjugacies” that Fang refers to. Anyhow, I find it interesting that Fang seems to accept that Nature provides the kind of direct access to Truths about Reality, whatever that may be, that I prefer to call Empirical Facts. An example of this is the intuition/belief of Pythagoras (and Euclid) that the area of a triangle in a plane does not change when it is moved in that plane.

    To call this “direct access” is to recognise that it is independent of both Logic and Rationality, which is not to say that what individuals say/think using that direct access is illogical or irrational. It is simply not covered by the Logicist way of thinking that was promoted over two millennia ago in the Athens of Aristotle and that is still promoted today in the Philosophy Departments that dominate that Professional Discipline in Australia.

    Personally, I think that an apparently non-teleological NATURAL Selection has generated some individual brain/minds that in their own limited ways are somewhat capable of that direct access which Ramanujan had in spades. Whether you choose to attribute this to Ramanujan’s God or Fang’s Universal Life-Form is neither here nor there. Call it NATURE if you like.

    1. I believe these “Truths about Reality” were called “synthetic a priori postulates” by Kant.

  4. According to Wikipedia, on Kant,

    “In the Critique of Pure Reason (1781) he countered Hume’s sceptical empiricism by arguing that any affirmation or denial regarding the ultimate nature of reality (‘noumenon’) makes no sense.”

    I am making a very specific claim about the ultimate nature of reality. It is not a claim that I can prove correct, but it certainly makes sense. To argue with that, I’m afraid you need to read the detail of that claim.

    The way of thinking that I call Strict Empiricism, is closer to Hume than to Kant. Nevertheless, I like to think that:


  5. I cannot help wondering what Fang might make of the following thought about Ramanujan’s extraordinary mind.

    Seeing a picture (with your eyes) is very different from hearing a claim (with your ears), because speaking/thinking a sentence takes TIME while a picture is directly grasped as a unity by whoever is able to see/imagine it. Hence, I am inclined to believe that what gave Ramanujan his extraordinary ability to “see” his congruences (theorems that take time to state, let alone to prove) was down to his imagining/seeing the curves (perhaps on a 2-dimensional plane like Euclid’s) which those congruences capture/describe.

    Something like this was foreseen by Heraclitus of Ephesus, around 500 BC, when he said:

    “The eyes are more exact witnesses than the ears.”.

    That is because a picture is what it is, a FACT that may be observed but cannot lie, though people can utter lies about facts, including pictures.

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