The Theory of Everything

During the holiday break, I read Stephen Hawking’s The Theory of Everything. Reading this book was excruciatingly agonising. If not for the awesome Christmas dinner my friends Marilena and Kostas cooked for our big group of Greek friends, and the wonderful Yorkshire Bettys tea and Christmas cake supplied, along with plates, teacups, classical music and a comfy sofa, by dear friends Angela and John, and many miles of hiking, running and biking, I might have lost sanity a little bit. This book stirred up so many questions in me about black holes, the universe, God(s?), our existence and so on. You could almost insert a why question after every single sentence in this book. That is how I pleasantly suffered through reading it.

Can there really be a unified theory of everything? Or are we just chasing a mirage? There seem to be three possibilities:

  • There really is a complete unified theory, which we will someday discover if we are smart enough.
  • There is no ultimate theory of the universe, just an infinite sequence of theories that describe the universe more and more accurately.
  • There is no theory of the universe. Events cannot be predicted beyond a certain extent but occur in a random and arbitrary manner.

I am most inclined to argue for the second and third possibilities. Perhaps this has something to do with my own resignation that if the first possibility is true, there may be no meaning of my own work and existence if not towards that ultimate quest of finding that unified theory and I am neither a physicist nor an mathematician. I had to repeatedly seek comfort in hiking in the woods to think and to reconcile what I could do with my minuscule amount of knowledge and time on the earth (relatively speaking, totally negligible at the grand scheme of the universe).

This is how 2017 ended and 2018 started: with a very long hike in the Santa Cruz Mountains, and the thoughts provoked by two great books The Very Hungry Caterpillar and The Theory of Everything, and the question what challenges to set for myself next.

How To Solve It: A New Aspect of Mathematical Method

My book of this week is How to Solve It: A New Aspect of Mathematical Method by eminent mathematician George Polya. Polya was one of the most influential mathematicians of the 20th century. O’Connor and E F Robertson wrote a short biography of Polya, giving us a glimpse of this extraordinary scientist and teacher.

In the book Mathematical People: Profiles and Interviews, there is a quote from Polya: “I came very late to mathematics. … as I came to mathematics and learned something of it, I thought: Well it is so, I see, the proof seems to be conclusive, but how can people find such results? My difficulty in understanding mathematics: How was it discovered?” This might be one motivation behind Polya’s writing of a book like How to Solve It. I like this short review of the book:

How to Solve It will show anyone in any field how to think straight. In lucid and appealing prose, Polya reveals how the mathematical method of demonstrating a proof or finding an unknown can be of help in attacking any problem that can be “reasoned” out – from building a bridge to winning a game of anagrams. Generations of readers have relished Polya’s deft – indeed, brilliant – instructions on stripping away irrelevancies and going straight to the heart of the problem.

How to Solve It was first published in 1945. While reading this book, I could not help questioning myself: how did I not come across and read it in my early teens? Damn me for the hefty loss of not having done so! It would have enlightened and saved me from many struggles of searching for ways to solve problems not only in maths, but also other subjects. As I had not read it back then, I cannot tell what a young student would think of this book. I am indeed curious to know your thoughts, if you had. As a trained computer scientist with a lot of passion for mathematics, many articles in this book brought me moments of epiphany. Polya’s words brought me the realisation of what we have learned to do through our scientific education and research, with the shortcoming that we have done so with less reflection and less articulation of what that set of thought processes are.  

How did I come across How to Solve It eventually? I owe this discovery to two gentlemen: Nuwan Jayasena and Charles Simonyi. Many thanks to Nuwan that I attended a Stanford Engineering Hero Lecture, given by Charles Simonyi in 2015. In this lecture, Charles noted how much he valued the book How to Solve It and recommended everyone to read it. How fascinating that it took one Stanford PhD (Nuwan) to invite me to a talk given by another Stanford PhD (Charles), in which I was connected with a book by a Stanford Professor (George Polya)! I am very grateful.  

You might start to wonder: woman, are you ever going to tell me what this book is about?

There are four parts: In the Classroom; How to Solve It: A Dialogue; Short Dictionary of Heuristics; Problems, Hints and Solutions. Here I focus on the “How to Solve It” list and some of the heuristics discussed in the book.

In the “How to Solve It” list, Polya distilled the problem solving processes to the following four steps. These suggestions and questions are quoted directly from the book.

  1. Understanding the problem
    • You have to understand the problem.
    • What is the unknown? What are the data? What is the condition?
    • Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory?
    • Draw a figure. Introduce suitable notation.
    • Separate the various parts of the condition. Can you write them down?
  2. Devising a plan
    • Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution.
    • Have you seen it before? Or have you seen the same problem in a slightly different form? Do you know a related problem? Do you know a theorem that could be useful? Look at the unknown! And try to think of a familiar problem having the same or a similar unknown.
    • Here is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible?
    • Could you re-state the problem? Could you restate it still differently? Go back to definitions.
    • If you cannot solve the proposed problem try to solve first some related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem? Could you solve a part of the problem? 
    • Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary?
    • Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown?
    • Could you change the unknown or the data, or both if necessary, so that the new unknown and the new data are nearer to each other?
    • Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem?
  3. Carrying out the plan
    • Carrying out your plan to get to the  solution: check each step.
    • Can you see clearly that the step is correct? Can you prove that it is correct?
  4. Looking back
    • Examine the solution obtained.
    • Can you check the result? Can you check the argument? Can you derive the result differently? Can you see it at a glance?
    • Can you use the result, or the method, for some other problem?

Are you excited and motivated enough to read the book yourself by these questions now? I love reading this book, if not for anything else other than the questions. Polya’s writing style of this book reassured me that I am not too crazy of asking many questions. But then, I never care at all whether others think less of my intellectual capability, because I would admit I am an idiot forthright. I am an idiot who is trying very hard to shed off each layer of stupidity every day through learning and working. Is it not wonderful to imagine that you are losing weight that way metaphorically, without sacrificing your awesome appetite for a giant neapolitan pizza?

A final point: although many examples in this book are maths related, the book is much more fundamental than prescribing recipes for solving maths problems. To me, many suggestions for investigation, thought processes, heuristics, planning and evaluation are very applicable for any domain, such as litigation, economics, social science and civil engineering.

The Selfish Gene

Richard Dawkins mentioned his concern about people’s misjudgement of the book because of its title. Unfortunately, I was one such plonker. This book sat on my bookshelf for many years. I did not want to touch it because I shallowly inferred from the title that the book is about finding justification in biology for selfishness and moral degradation (if I may, as a far-fetched extension). Reading it might shatter my strongly held view that we human should promote altruism and not be selfish. I opened it one day recently, not remembering when and why exactly. It has proved to be a fascinating read and has demonstrated my pre-assumption to be completely wrong. Dawkins mentioned in the book that, retrospectively speaking, he should have named the book The Immortal Gene instead. I would have benefited from this book many years earlier if that were the title. The fault is all mine though. I learned the lesson to not make the verdict based on the title alone.

A few thoughts occupied my mind while reading this book. There is no answer applicable to us as a group. I think each individual would have one’s own opinion. Also I would like to point out that I am not a biologist at all, merely one who is interested in the subject.

We are the manifestation of our genes to certain extent. I understand gene survival theory as that we maximise the chance of prolonging the existence of any given gene by producing and bringing up offsprings, or helping to increase the chance of survival of that gene as carried by our family members. Last year, when my father was in a coma and in imminent danger for a lengthy period, I, against all my scientific and atheist mind, was praying madly with uttermost sincerity that God would let me give up 30 years of my lifespan in exchange for 30 more years for him, or an even better deal if God is cruel and unjust, 30 years only for 10 years. It was not an attempt to preserve my genes, because it does not propagate back to prior generations at certain ages. It was not altruism either. Did I want him to live because I could not cope with the possibility of losing him? Or, did I want him to live because I truly think he could enjoy many more years of life that would be more relaxing and peaceful than one at working age, despite the prospect of being severely disabled? Looking back, I dismissed the second question fiercely and wanted father to be with us regardless of the medical conditions. I could not cope with the future of living in endless regrets that I have hardly looked after him. Damn me that I have not taken him travel around the world as I wished! Curse me that I have not done X, Y, Z for him! A very long list of unfulfilled wishes. To cope with the fear of not having a chance to give back, I had a belief that he must live. That was a strong and very selfish belief. On one hand, I find this book very convincing. On the other hand, we are so miniscule in the biology evolution, our thoughts and behaviors are greatly influenced by other factors besides the genes. This kind of non-gene-survival related influence appeared to be more dominating than the survival theory in the relatively constrained timeframe.

I am particularly drawn to Dawkins’ meme theory and more keen to find out more about further research on memes. Meme is the name given by Dawkins for a new kind of replicator. “It conveys the idea of a cultural transmission, or, a unit of imitation. Examples of memes are tunes, ideas, catch-phrases, clothes fashions, ways of making pots or of building arches. Just as genes propagate themselves in the gene pool by leaping from body to body via sperms or eggs, so memes propagate themselves in the meme pool by leaping from brain to brain via a process which, in the broad sense, can be called imitation.”

The following paragraph from the book gives us great hope of the legacy we could create in our very trivial and limited lifetime in the long evolution river:

When we die there are two things we can leave behind us: genes and memes. We were built as gene machines, created to pass on our genes. But that aspect of us will be forgotten in three generations. Your child, even your grandchild, may bear a resemblance to you, perhaps in facial features, in a talent for music, in the colour of her hair. But as each generation passes, the contribution of your genes is halved. It does not take long to reach negligible proportions. Our genes may be immortal but the collection of genes that is any one of us is bound to crumble away. Elizabeth II is a direct descendant of William the Conqueror. Yet it is quite probable that she bears not a single one of the old king’s genes. We should not seek immortality in reproduction. But if you contribute to the world’s culture, if you have a good idea, compose a tune, invent a sparking plug, write a poem, it may live on, intact, long after your genes have dissolved in the common pool. Socrates may or may not have a gene or two alive in the world today, as G. C. Williams has remarked, but who cares? The meme complexes of Socrates, Leonardo, Copernicus and Marconi are still going strong.

The end of the first edition was very upbeat and positive: We are built as gene machines and cultured as meme machines, but we have the power to turn against our creators. We, alone on earth, can rebel against the tyranny of the selfish replicators.

It is a book well worth reading. Do not let the title stop you.