A Certain Ambiguity
A Mathematical Novel
Gaurav Suri & Hartosh Singh Bal
I am nearing the end of reading A Certain Ambiguity by these two Indian mathematicians, after I had been attracted to it by its advert, where it purported to be about Infinity. As my current researches are philosophical investigations in both Mathematics and Science, I immediately ordered the book.
But, on commencing to read it, I was surprised by the realisation that such a book could only have been written in the USA, and by someone who subscribes to the present day consensus there. Yet, by their names, these authors seemed to be from Sikh heritage, while the book delivers a debate between "American" Christianity and Atheism. Before any content is dealt with, this surprising situation seems to say quite a bit about the motives of the authors.
So, my excitement at finding a source to expand my own work, was immediately undermined by speculation about these motives. For, if they were dealing with the problems and capacities of Mathematics, the inclusion of a dogmatic christian judge (no less) seemed to betray "inclusion" motives... What do you think?
The actual content is contained within two discussions eighty years apart.
The first details one between a christian judge and an Indian mathematician, after the latter had questioned and disagreed with some christian "speaker" in a public park. In spite of this being supposedly a "forum" for the populace to "air their views", the hostile reaction to Sahni's comments was immediately amplified into a major attack, and Sahni was arrested and charged with blasphemy, under some ancient and forgotten law.
The second, much later debate, is set in the present day, and contains a long series of discussions between a mathematics lecturer and a small group of his students (one of whom is the grandson of the man Sahni).
The kernel in both these discussions, as the authors see it, is whether there is any certainty in this world, or, to be more accurate, in the vastly different sub-worlds defined by the judge's christianity and the Indian's view of mathematics.
The judge sees everything as based on his belief in God, while the mathematician sees everything as based on his favoured axiomatic method, as used in mathematical proofs. Sahni keeps to the example of Euclid's Elements, which established this methodology in Alexandria 2000 years ago, where the subject was greek Geometry. Euclid managed to integrate ALL known geometrical ideas of his time into a self-consistent system based on a handful of axioms. Our mathematician considered that this was the most powerful method known to man, and could see that it could be extended in use to the world at large.
As a mathematician myself, I must say that I was somewhat surprised at the childishness of BOTH their positions. Did they really live in the 20th Century?
I had personally dumped religion at the age of 13, as it offered NO understanding of anything. Indeed, quite the reverse, it was clearly an alternative to any attempt to understand how the world works. And though I too fell in love with Euclid at about the same age, and could prove all the theorems without any trouble, I could not see how such an artificial system could ever be used generally to deal with everything in the world.
My respect for theorems soon matured into a respect for the 'included' logic. But even then it was clear what this method was all about. Logic was able to generate a whole panoply of consequences from a given set of assumptions and assertions. It could amplify a set of such things into an extensive penumbra of dependent, though not immediately evident extensions.
But, to reduce this general method to the axiomatic version contained in Euclid, was, even then, obviously juvenile, as logic was the basis of discussions about everything in the world. Axiomatic structures are only a special, limited, subset of this 'universality'. The idea of a complete, logical system of Absolute Truth was, to say the least, "a bit much". From the outset, it was clear that the assumptions on which Geometry was based were merely simplifications of what actually existed in reality. For in the real world there are no dimensionless points, no lines of zero thickness, and no infinite planes. These were clearly invented to serve as a basis for a model of the forms of reality. And, it must be emphasized that, by the time you reached the end of Elements, you did not know reality, you only knew the model, and, in addition, you perhaps grasped some idea of the power of Formal Logic. As a pupil in school, I soon moved my allegiance from Mathematics to Science, because the latter did at least attempt to tackle more of reality than Mathematics ever did or could!
I hoped for understanding via Science.
But I was an able mathematician, and have used this remarkable tool all my life. My attitude to it was always pragmatic, rather than for revealing the truth of reality. I solved real world problems with it.
What A Certain Ambiguity never even mentioned were the uses of mathematics. All of its discussions were clearly limited to Pure Mathematics, Applied Mathematics didn't even deserve a mention, which when you think about it, is remarkable! But, I am clear that the reason it wasn't mentioned is that Applied Mathematics is not an isolated pure system. It is involved, everyday, in the attempted application of pure, abstract forms back into reality - and when you attempt to do this, all that purity and elegance vanishes. Multiple 'frigs', approximations, fittings and downright invented models are found to be essential. The abstractions, isolated, extracted and perfected by Pure Mathematics simply do not fit with real world situations as they stand. They have lost too much in that processing to have the necessary content for a perfect fit. Obviously, discussions about Essence and Eternal Truth do not gel very well with such things, and so Applied Mathematics is dumped for being ugly, as if it didn't exist at all!
Much later in life, I was able to categorize the situations of the role of Mathematics much more soundly, when I embarked upon a period of research into the Processes and Productions of Abstraction, which quickly grew into a major area of study. These studies have since extended into a much wider area, which at present occupies all of my time, so that it has now inflated to be accurately termed Philosophy.
The Processes and Productions of Abstraction
A short film explaining these ideas
Mathematics, as it is discussed in this book, is quite clearly the study of Pure Form in isolation from Reality. Yet, its centre is its universality and generality - things that can only be established in Reality. These properties arise from the fact that its abstract equations can be used in many different areas. How else could it be termed "general" and "universal"?
Now, all these questions are much too important to be tidily confined into limited areas in order to make the arguments easier. At the present time in modern sub-atomic physics, the majority of scientists would have us believe that the Essence of their subject is Mathematics. They have abandoned Science as it has been developed over many centuries to replace it by formulae alone. Now, to move to such a position does also abandon Reality as the supreme arbiter, and instead moves into a non-scientific position which is clearly a branch of Idealism.
The book, surprisingly applauded by many world famous academics, is, as I hope that I have been able to demonstrate, decidedly infantile in its dealing with the important Big Questions, that are clearly untreatable by the limited disciplines employed. It is also surprisingly old-fashioned! The Religion/Science debate has been stone dead for many years. Why on earth is it now resuscitated in this book? It can only be because these old, answered questions have not yet been resolved in the States. Modern Christian Fundamentalism is clearly alive and well in that country, and even, we are told, in the Oval Office. The debate in this book reflects the bigger debate in American society at this point in history, but, as is usual in such things, it does not reflect the real issues or address the crucial questions. It is closer to what close advisers to the White House put into their president's mouth, when untenable things are proposed as the Truth.
Mathematics is neither the opposite of Religion, nor can they ever be reconciled. What utter nonsense.
Indeed, my researches show that Mathematics is actually extremely Idealist, and the degeneration of Modern Physics is characterized by its abandonment of Science for the thin gruel of Mathematics as Essence. Such trends get more like Religion every day.
Why should such an easy target as Mathematics be chosen as the "enemy" in these discussions? I think the answer is transparently clear! It was chosen because it is an easy target. Mathematics is no general philosophical method. It is essentially Formal Logic, but in such a limited way that I have characterized its area of application as being only in Ideality - the world of pure form alone. It is easy meat.
I know because I have spent time doing a very different task - the criticism of Mathematics compared with Science - the weaknesses of description as compared to explanation - the whole trajectory of Mankind's methods in attempting to understand Reality. So, the easier target was purposely chosen in order to allow the "correct" general conclusions at the end of the book.
I had wondered whether I ought to write a full conspectus of this book, but not for long! It is simply not worth the effort. Dawkins might feel that he has a job combating American Fundamentalism, but not me! I have better things to do with my time.
I am sure that a such a book as this will become very popular in the Mid-West of the USA, and much discussion (as in this book) will ensue among privileged college students. But such a level of treatment only reveals inadequacies and scarcely requires a full treatment.
After all, I dumped these ideas and questions before I was 14 years old, and when you become a man, you put away childish things...