John H. Conway’s Recipe for Success

John H. Conway - A Recipe for Success

I did have a recipe for success, which was always keeping six balls in the air. What I mean is: Always be thinking about six things at once. Not at the same time exactly, but you have one problem, you don’t make any progress on it, and you have another problem to change to. I had a set mix: one of the problems could be a crossword puzzle in the newspaper or something like that. Nowadays it might be a Sudoku puzzle. One of them might be a problem that would instantly make me famous if I solved it, and I don’t expect to solve it, but don’t give up on it; it’s worth trying. There must also be one problem where you can definitely make progress just by working hard enough. Then when the guilt level rises sufficiently—and I felt guilty in Cambridge for not doing any work—you could make progress. It is sort of a routine problem that is not completely dead and might be useful. So that is my recipe for success.

Source: http://www.ams.org/notices/201305/rnoti-p567.pdf

P.S – I discovered after this was written Tanya Khovanova shared something similar here.


The End

Alice laughed. “There’s no use trying.” she said. “One can’t believe impossible things.”

“I daresay you haven’t had much practice,” said the Queen. “When I was your age, I always did it for half-an-hour a day. Why, sometimes I’ve believed as many as six impossible things before breakfast.”

– Lewis Carroll Through the Looking-Glass

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50 Weeks of Mathematics

You know that I write slowly. This is chiefly because I am never satisfied until I have said as much as possible in a few words, and writing briefly takes far more time than writing at length.

– Karl Friedrich Gauss

Completely disregarding Gauss’ sage advice, I commit to “sketching” (see my previous blog post: The Importance of Sketching) 50 Mathematical Topics in 50 weeks for your reading pleasure.

A sketch of the proposed topics are as follows:

Sunday 29 March 2015 Week 1 Set Theory
Sunday 5 April 2015 Week 2 Mathematical Logic
Sunday 12 April 2015 Week 3 How to Prove It – Mathematical Proof
Sunday 19 April 2015 Week 4 Single Variable Calculus
Sunday 26 April 2015 Week 5 Linear Algebra I
Sunday 3 May 2015 Week 6 Linear Algebra II
Sunday 10 May 2015 Week 7 Multi variable Calculus I
Sunday 17 May 2015 Week 8 Multi variable Calculus II
Sunday 24 May 2015 Week 9 An Introduction to Probability
Sunday 31 May 2015 Week 10 Computability Theory
Sunday 7 June 2015 Week 11 Group Theory
Sunday 14 June 2015 Week 12 Graph Theory
Sunday 21 June 2015 Week 13 Euclidean Geometry
Sunday 28 June 2015 Week 14 Analysis I
Sunday 5 July 2015 Week 15 Analysis II
Sunday 12 July 2015 Week 16 Analysis III
Sunday 19 July 2015 Week 17 Elementary Number Theory I
Sunday 26 July 2015 Week 18 Elementary Number Theory II
Sunday 2 August 2015 Week 19 Abstract Algebra – Rings and Fields
Sunday 9 August 2015 Week 20 Abstract Algebra – Vector Spaces and Modules
Sunday 16 August 2015 Week 21 Abstract Algebra – Galois Theory
Sunday 23 August 2015 Week 22 Topology I
Sunday 30 August 2015 Week 23 Topology II
Sunday 6 September 2015 Week 24 Complex Analysis I
Sunday 13 September 2015 Week 25 Complex Analysis II
Sunday 20 September 2015 Week 26 Ordinary Differential Equations
Sunday 27 September 2015 Week 27 Partial Differential Equations
Sunday 4 October 2015 Week 28 Measure Theory
Sunday 11 October 2015 Week 29 Statistics: A Crash Course I
Sunday 18 October 2015 Week 30 Statistics: A Crash Course II
Sunday 25 October 2015 Week 31 Electrodynamics
Sunday 1 November 2015 Week 32 An Introduction to Quantum Mechanics
Sunday 8 November 2015 Week 33 Special Relativity
Sunday 15 November 2015 Week 34 General Relativity
Sunday 22 November 2015 Week 35 Combinatorics
Sunday 29 November 2015 Week 36 Gödel’s Theorems
Sunday 6 December 2015 Week 37 Functional Analysis I
Sunday 13 December 2015 Week 38 Functional Analysis II
Sunday 20 December 2015 Week 39 Algebraic Geometry
Sunday 27 December 2015 Week 40 Algebraic Topology
Sunday 3 January 2016 Week 41 Algorithmics
Sunday 10 January 2016 Week 42 Calculus of Variations
Sunday 17 January 2016 Week 43 Elementary Fluid Dynamics
Sunday 24 January 2016 Week 44 Computational Complexity I
Sunday 31 January 2016 Week 45 Computational Complexity II
Sunday 7 February 2016 Week 46 Machine Learning I
Sunday 14 February 2016 Week 47 Machine Learning II
Sunday 21 February 2016 Week 48 Stochastic Processes
Sunday 28 February 2016 Week 49 Elementary Numerical Analysis
Sunday 6 March 2016 Week 50 Looking Back and Looking Forward

Well I am committed now. Watch this space …


Some useful Resources in compiling this sketch:

http://www.maths.cam.ac.uk/undergrad/course/

https://www.math.ucla.edu/ugrad/courses

http://www.ox.ac.uk/admissions/undergraduate/courses-listing/mathematics

http://en.wikipedia.org/wiki/Areas_of_mathematics


The End

Throughout my whole life as a mathematician, the possibility of making explicit, elegant computations has always come out by itself, as a byproduct of a thorough conceptual understanding of what was going on. Thus I never bothered about whether what would come out would be suitable for this or that, but just tried to understand – and it always turned out that understanding was all that mattered.

– Alexander Grothendieck

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The importance of sketching. The best way to learn is to write.

“Those who know, do. Those that understand, teach.”
― Aristotle

I have recently discovered Terence Tao‘s utterly brilliant blog, and write this post to comment on his advice surrounding time management and the importance of writing things down.


I currently have close to 12 posts at various stages of completion in draft form awaiting inspiration, requiring polishing, or just plain not working; destined for a twilight like existence in some type of blog post purgatory.

What is important to note is that these posts all have a sketch of the intended content, and looking over some I am surprised at how well formed they actually are, and it is a great resource for ideas and thoughts that would take me a long time to reconstruct without these sketches. The sketching of arguments is a fantastic time management tool, in fact it is probably the single best time management advice anyone could ever receive. Tao gives valuable advice in this post about batching low intensity tasks, and other gems, but if you sketch all those subtle insights that you stumble upon, the snowballing effect of the increased comprehension and ready stock of ideas is hard to describe. It is the best productivity boost I know of.

Terence Tao is considered one of the greatest mathematicians of his generation:

If you’re stuck on a problem, then one way out is to interest Terence Tao,”

– Charles Fefferman, Mathematician at Princeton University

He is not only an exceptional mathematician, but he seems to be incredibly generous with his time in posting expository articles on his work, and more general advice based on his undeniable successes. I came across his blog recently and have devoured multiple articles, and even purchased a few books based on his blog; I found Compactness and Contradiction particularly good. [His two volume treatment of Analysis is incredible]

Terence Tao comments on the effectiveness of writing down sketches of your arguments and insights on a particular topic, in that it clarifies your thinking, which in and of itself is invaluable, but more importantly it saves time, and literally saves the ideas themselves. To quote:

There were many occasions early in my career when I read, heard about, or stumbled upon some neat mathematical trick or argument, and thought I understood it well enough that I didn’t need to write it down; and then, say six months later, when I actually needed to recall that trick, I couldn’t reconstruct it at all. Eventually I resolved to write down (preferably on a computer) a sketch of any interesting argument I came across – not necessarily at a publication level of quality, but detailed enough that I could then safely forget about the details, and readily recover the argument from the sketch whenever the need arises.

I have presented this advice to others, but with a slightly different emphasis. The absolute best way to learn a topic, is to write an expository article.


If you think you understand something, even something not particularly complex, just spend 15 minutes sketching an “Idiots Guide to …” article about that topic. I will be willing to bet a full English pound you will either hesitate, stumble, or discover something novel, a new perspective; every time I do this (and I have started to do this for anything of note) I find new connections that were only dimly realised previously, and it can spark creativity! The most important outcome is that you will have clarified your understanding of the topic in question, and have a readily available sketch, that will aid future comprehension and consolidate your learning on that topic. It is very satisfying browsing my /sketches folder (with many sub-folders organised by topic) on my trusty Ubuntu laptop to see all my recent learning. [Note: I write my notes in a standard TeX AMSMath template. For a very good introduction to TeX see this youtube series – no really this is the best first exposure to TeX I have ever found! I cannot recommend  familiarity with TeX highly enough, I personally find it such a pleasure to write in that it is no longer even mildly a chore to write these expositions, and the output is beautiful. There is also a TeX to WordPress converter that I will most likely be trialling in my next post.]

I recently sketched an overview of asymptotic notation for example, using Knuth‘s TAOCP Volume 1, and his collaborative work (with Graham and PatashnikConcrete Mathematics that I will turn into my next blog post to show you how these sketches look. I am consistently surprised that this simple exercise has such a palpable impact on my understanding of a topic, and provides a physical artefact that somehow makes everything infinitely more memorable.

To see some of Tao’s expositions, visit https://www.math.ucla.edu/~tao/preprints/Expository/


My best advice for time management – write down what you know.

My best advice for effective learning – write down what you think you know.


The End

I am always doing that which I cannot do, in order that I may learn how to do it.
– Pablo Picasso

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It was the best of times … An Introduction

It was the best of times, it was the worst of times, it was the age of wisdom, it was the age of foolishness, it was the epoch of belief, it was the epoch of incredulity, it was the season of Light, it was the season of Darkness, it was the spring of hope, it was the winter of despair, we had everything before us, we had nothing before us, we were all going direct to Heaven, we were all going direct the other way – in short, the period was so far like the present period, that some of its noisiest authorities insisted on its being received, for good or for evil, in the superlative degree of comparison only.

A Tale of two Cities – Charles Dickens (1859)

There are not many better introductory passages than Dickens’ A Tale of two Cities. The introduction to this blog will not hope to meet that high standard (I was tempted to crowbar in “great expectations” but though better of it); it will only briefly outline what type of content can be expected in future posts, with an aside about human memory.


There is a wonderful anecdote about John Von Neumann in relation to A Tale of Two Cities. It is recounted in Herman Goldstein‘s The Computer from Pascal to von Neumann (source – retrieved December 28th 2014):

As far as I could tell, von Neumann was able on once reading a book or article to quote it back verbatim; moreover he could do it years later without hesitation. He could also translate it at no diminution in speed from its original language into English. On one occasion I tested his ability by asking him to tell me how the ‘Tale of Two Cities’ started. Whereupon, without pause, he immediately began to recite the first chapter and continued until asked to stop after about ten or fifteen minutes.

This left a strong impression (and the linked article’s contents in general) in my teens and instilled a deep interest (that I still enjoy at 27) in the possibilities of a trained mind. I was also fascinated with stories of equally impressive feats: memorisation of pi to hundreds and thousands of digits, memorisation of dictionaries, human calculators, hyperthymesia etc. These feats seemed alien and far beyond my reach. I had what could be described as a good memory, and a flash of intelligence, but 1000 digits of pi?

I was always rather sceptical about the truth of the Von Neumann anecdote in the hyperbolic sense claimed by Goldstein: photographic memories do not exist, and I doubted Von Neumann was able to recite any book he had read. (There are very interesting stories that I will no doubt remark upon in future posts, not about the famous Solomon Shereshevsky but Eug­enia Alexeyenko, Kim Peek, Stephen Wiltshire and others who may possess something as close as is possible to a photographic memory).

I later came across Norman Macrae’s Biography of Von Neumann where the myth was dispelled. Macrae makes pains to state that this ability was restricted to texts Von Neumann had “fearsomely concentrated” on:

His powers of memory were awe-inspiring, but only about matters on which he had fearsomely concentrated his mind. He could recite verbatim pages and pages from books such as Dickens’ A Tale of Two Cities which he had read fifteen years before … This was because he had concentrated hard when first reading them … [in the case of A Tale of Two Cities] he was trying to get a feel for proper English syntax before emigrating to the United States.

This is something that I, or anyone in fact, can do with a little technique and effort. (Things of this kind will no doubt be discussed in this blog). Inspired by Von Neumann, I can recite the introductory passage from A Tale of Two Cities, and it was achieved with surprisingly little concentration; it could not be described as fearsome anyway. This is along with many more impressive feats that are equally as achievable.


What will this blog contain?

In this blog I will post musings mathematical, physical, computational and any other topic(s) of interest (book reviews, recent research etc.).

I currently work in IT and will, from time to time, post interesting cases that may provide some useful insight for others working on similar problems.

A lot of my free time is spent self-studying higher-mathematics with a side interest in physics and the theory of computation.

I hold many other varied interests: history, literature, poetry, chess, psychology, film, memory, magic, language, politics and art to name a few. All I am sure all will feature at some point.

I don’t expect the blog to be widely read, it is more for myself. If you do find anything interesting, take issue with an exposition, or feel compelled to comment, all feedback is welcomed.

In each post, as an addendum, I will always try to include a quote or quotes, be they pithy, profound or frivolous, that in some way relate to its content. The reason for this is not pretension, it is that during my reading I always find that a thought I have had (recently, rarely or often) is so consistently expressed more concisely, cogently and skilfully by another mind, that I tend to collect them, or their impression leaves an indelible mark.

Put another way I often encounter …

What oft was thought, but ne’er so well express’d

An Essay on Criticism – Alexander Pope

I hope to post at least weekly, and that at least one person will find it useful.


The End

A book is made from a tree. It is an assemblage of flat, flexible parts (still called “leaves”) imprinted with dark pigmented squiggles. One glance at it and you hear the voice of another person, perhaps someone dead for thousands of years. Across the millennia, the author is speaking, clearly and silently, inside your head, directly to you. Writing is perhaps the greatest of human inventions, binding together people, citizens of distant epochs, who never knew one another. Books break the shackles of time ― proof that humans can work magic.

Cosmos – Carl Sagan

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