On Intuition in the Life and Work of Ramanujan

Portrait photo of Srinivasa Ramanujan in England

These past few weeks I have been engrossed in reading about Srinivasa Ramanujan (1887-1920), the brilliant mathematician from South India whose prolific work in number theory lit up the academic world from the early 20th century onwards.

In my studies in this subject I had heard the name, seen references, but didn't use his work.  Even though he's famous in the mathematical world and in India, I wasn't minded to give much attention until I happened to chat over lunch with Thomas Bewley, who described his experiences of playing Prof. H. F. Baker in the film, The Man Who Knew Infinity.  I promptly ordered a DVD and enjoyed watching the film, so I then ordered the substantial biography of the same name by Robert Kanigel.

I'm writing this having just finished Kanigel's book.  It's extensively researched and very detailed, covering Ramanujan's life and work; whilst aimed at the general reader, quite a lot of mathematical material has been presented, in a generally convincing way – Kanigel is numerate (he has a degree in mechanical engineering) and he has evidently spent considerable time grappling with the material in conversation with scholars.

I particularly like the way he shows how the relevancy of Ramanujan's work and various applications has ebbed and flowed; in his early years he struggled to make known his findings, but gradually some friends and associates tried to help promote his cause until they were able to tap into the British colonial networks.  Eventually there arose opportunity to write to the scholars at Trinity College, Cambridge, Britains' foremost centre for mathematical research.  Even then, Ramanujan had to keep persevering until he succeeded at the third attempt when his genius was recognised by Hardy, who nurtured Ramanujan's talent by instilling the rigours of proof and dissemination of the various results.  After Ramanujan's passing, Hardy continued to promote his cause through papers and continued reference.  Subsequent decades saw changing foci, but recently his work has become of great significance – his “mock theta functions” have been integral to the development of mock modular forms, which are now used in astrophysics, even to model singularities (black holes, etc.).

Personally, I might like to explore his work in partition functions (having a natural interest in combinatorics).  However, I am mainly interested at the moment in Ramanujan's spirituality, what we might learn about intuition.  Kanigel attempts to explore this area, knowing full well that members of the public like especially to know about how a human being can navigate the vicarious aspects of life and its innumerable obstacles, triumphing over adversity – the indomitable spirit.  Kanigel dutifully delves into this with cultural sensitivity – undertaking fieldwork in the foreign lands and cultures of the British Isles and India.  Through the information he gathers from interviewees, Kanigel recreates at some length daily scenes in which Ramanujan lived and breathed mathematics - in Kumbakonam, his home town, and in various other places such as Triplicane (now Tiruvallikkeni) (with its historical sites such as the Arulmigu Sri Parthasarathyswamy Temple), and other areas in the then Madras Presidency (now Tamil Nadu).

Even in a volume as extensively researched as this, the whys and wherefores as to Ramanujan's  mathematical discoveries can't be fully explained when it seeks to explicate an entire life story and indicate especially its mathematical import today.  Even so it's noticeable that whilst Kanigel appears comfortable explaining material facts, he finds it exceedingly difficult to fathom Ramanujan's spiritual inspiration.  Hence overall he writes sympathetically, but when it comes to religious aspects, he doesn't have much to say, and even occasionally strikes an incredulous tone.

For example, referring to a gathering that developed into philosophical discussion he writes (pp. 31-2):

Another time, when he was twenty-one, he showed up at the house of a teacher, got drawn into conversation, and soon was expatiating on the ties he saw between God, zero, and infinity - keeping everyone spellbound till two in the morning. It was that way often for Ramanujan.  Losing himself in philosophical and mystical monologues, he'd make bizarre, fanciful leaps of the imagination that his friends did not understand but found fascinating anyway. So absorbed would they become that later all they could recall was the penetrating set of his eyes.

I don't suppose Ramanujan felt lost; if anything, he was finding deeper relationships in what the author describes as "bizarre" and "fanciful".  Whilst it might have been tantalising to his audience, at the same time the culture readily accepted this kind of expression.

In another chapter Kanigel writes (p.66):

Later, in England, Ramanujan would build a theory of reality around Zero and Infinity, though his friends never quite figured out what he was getting at. Zero, it seemed, represented Absolute Reality. Infinity, or ∞, was the myriad manifestations of that Reality. Their mathematical product, ∞ x 0, was not one number, but all numbers, each of which corresponded to individual acts of creation. 

Kanigel's tone conveys shades of incredulity, but these kinds of views are taken seriously in many parts of the world.  At least in recent centuries, they seem to be more naturally appealing to Asians – from all over, whether the South, South-East, Far East, or the North.  So I'm interested to read accounts from their perspective, particularly Indian interpretations – how do they interpret Ramanujan today?

As Ramanujan is a national hero, there's no shortage of material, a fair amount being helpfully referenced in Kanigel's book.  Arriving as a newcomer, I try to find, where possible, sources with first-hand accounts, ideally published by authorities who have some historical connection.  My starting point has been a broad selection made available by The Institute of Mathematical Sciences in Tamil Nadu, a national research centre.

A section on books lists five volumes, including Kanigel's.  Among the others, I've been looking at  'Ramanujan - The Man And The Mathematician' by S. R. Ranganathan, part of Great Thinkers of India Series, published by Asia Publishing House in 1967.  The publisher is still registered, based in Mumbai, but I can find no website for it.  Dr Ranganathan was a mathematician and library information professional in India; an endowment in his name is associated with another publishing company, Ess Ess Publications limited, and copies of the book are readily available from them.  (But it's also not hard to find a free version online.)

This book records some accounts by those who knew and met with Ramanujan.  One of the respondents is Dr. Mahalanobis, who was there in the late night discussion that Kanigel refers to.  He recalls:

He sometimes spoke of “zero” as the symbol of the Absolute (Nirguna-Brahmam) of the extreme monistic school of Hindu philosophy, that is, the reality to which no qualities can be attributed, which cannot be defined or described by words, and which is completely beyond the reach of the human mind. According to Ramanujan, the appropriate symbol was the number “zero”, which is the absolute negation of all attributes.  He looked on the number “infinity” as the totality of all possibilities, which was capable of becoming manifest in reality and which was inexhaustible. 

(MN Reminiscences of Dr P C  Mahalanobis FRS,
Member of the Planning Commission of India: MP1, p.82)

Ramanujan starts with that philosophical position and then gives it mathematical expression, based on numerical properties that can exhibit the transcendent qualities.  These are not just made up fancifully, but rather there are references to a philosophical school.   Nirguna-Brahmam (or Para Brahman) is described in Hindu texts as the highest spiritual state, the formless Brahman, specifically in the sense of being absent of Maya, illusion.  It's a core belief in the Advaita Vedanta tradition.

Mahalanobis continued:

According to Ramanujan, the product of infinity and zero would supply the whole set of finite numbers.  Each act of creation, as far as I could understand, could be symbolised as a particular product of infinity and zero, and from each such product would emerge a particular individual of which the appropriate symbol was a particular finite number. I have put down what I remember of his views. I do not know the exact implication. 

Whilst Mahalanobis lacked understanding of the finer points, he could gain a general sense of what lay behind Ramanujan's words – there was valid and useful communication.  If they had been completely incomprehensible, then Ramanujan probably would not have sustained interest for so long.  Perhaps more significant still as an indication of the importance of this spiritual view, was the following reflection:

He seemed to have been perhaps emotionally more interested in his philosophical ideas than in his mathematical work. He spoke with such enthusiasm about the philosophical questions that sometimes I felt he would have been better pleased to have succeeded in establishing his philosophical theories than in supplying rigorous proofs of his mathematical conjectures.

This is a significant passage as it points to how important to him was his underlying spirituality of which mathematics was an expressions.  I think we see the deleterious effects of denying him support for this spirituality when Kanigel describes how to many Ramanujan appeared a much-changed man on his return to India in 1919.  Mentally and emotionally he was a different person: whereas previously he was full of fun and sociable in small groups, on his return he appeared withdrawn and angry.  It seems England was able to support his mathematics, but it came at the price of his Brahmin caste (at least for those who did not allow any exemptions to Samudrolanghana, the offence of crossing the sea) and his wellbeing.  There are areas that the book perhaps understates this sacrifice – which was more than the decline in his physical health.  Yet Ramanujan foresaw his own death (“I won't reach 35 years of age”), so the speculation around what might have been regarding alternative life paths and treatments of his tuberculosis should be set against that.

In modern times, we can still find views from India, especially religious teachers, who can give some indications of Ramanujan's spirituality.  Even though they might not have any formal background in mathematics and may lack rigorous language, they can express the 'inner voice', as it were.  For example, in his talk at SRCC College, available in a YouTube video, entitled The Secret of Ramanujan's Genius,  Sadhguru likens deities or, more specifically, murtis (forms) to energetic machines that are able to enhance particular faculties; unlike mechanical devices, such machines don't have moving parts, are easy to maintain, and are available all day and every day.  Ramanujan knew how to use the murti known as the goddess or deity Namagiri to receive mathematical insights and he seemed to be working continuously.  In the short excerpt, it's not explained how one cultivates practice of utilising these murtis, but in India it is typically through yogic or meditative training, and, as for most yogis, Sadhguru gives instruction in these, such as Isha Kriya.

Another perspective is shared in a presentation on teachings by Sri Aurobindo & The Mother: the quality of beauty is highlighted in a post where Sandeep, the author, asks: Where does Mathematics come from?   According to teachings in this tradition, having some correlation to the energetic machines, it as though humans have inner beacons of light that can be directed towards specific arts; an agile mind can shine the light in different directions.  But here, this longer article also emphasizes  development (I'd choose the word bhavana) of the capacity of attention and concentration.  Other posts on that informative site, including one that considers some views of Roger Penrose, describe how a prerequisite is knowing how to bring the mind into stillness (once the mind is at a standstill one can move easily in any direction); bringing the mind to a standstill is  key to allowing novel ideas to arise.

From my own Buddhist perspective, I would highlight that Ramanujan's superlative ability can only come through sustained kusala karma (skilful intentional actions), usually over many lifetimes.  In this way he would have generated puñña (merit), a kind of energetic fuel that with continued cultivation crystallizes as paramis (perfections) – puñña gives you the capacity to achieve, paramis enable that capacity to be readily and instantly available.  Perhaps Ramanujan refrained strictly from intoxicants leading to great clarity and receptivity of mind – certainly even in such a foreign environment he practised strictly as a Brahmin, so he retained that quality of mind seeking perfection.

Sharing some research in number theory

I have been exploring the life of Srinivasa Ramanujan, the brilliant Indian mathematician whose efforts in number theory from about a century ago are still captivating mathematicians today.  Reading about him in The Man Who Knew Infinity, Robert Kanigel's detailed biography, has prompted me to revisit the little bit of research in algebraic number theory that I undertook at the University of Glasgow. 

Under the direction of Professor Robert (“Bob”) Odoni, I submitted a thesis, Norms of Ideals in Direct Sums of Number Fields and Applications to the Circulants Problem of Olga Taussky-Todd, in April 1992.  Its originality was due to my supervisor; I barely understood the material and for various reasons felt that I couldn't continue towards a doctorate, but I wrote up what I had for a Master's and in September I duly underwent the viva with the external examiner, Dr. Stephen Wilson from Durham University.   Years later, I feel that I should in some way honour the memory of Prof. Odoni  by making the thesis available as a PDF file on Academia.

Taussky-Todd, who originally posed the problem about circulant matrices, was a versatile mathematician who explored many branches; in a long and productive life her output was considerable.  She initially studied chemistry because that her father was director of a vinegar factory, but then she switched to mathematics.  There is a parallel with Prof. Odoni, whose initial undergraduate studies were in chemistry, but he was not satisfied with the kinds of questions the subject addressed and so he turned to mathematics and never looked back.  I'm pleased to know that his contributions have been largely preserved in various papers, collated in a ResearchGate profile.

My thesis includes half a dozen references to Prof. Odoni's work (I have PDF versions that I'm happy to share).  He encouraged collaboration in research in many ways; when speaking of publications, I remember him praising in particular the Institute of Mathematics of the Polish Academy of Sciences, which is reflected in several of the listed publications.  All the papers he submitted there have been digitised and are being shared freely, reflecting this kind of openness. 

On a technical note, I prepared the thesis using (plain) TeX on an Atari ST, using GNOME   as a text editor.  I stored the files on a PC-compatible floppy and eventually transferred them off this obsolescent storage medium.   The sources I found a few weeks ago were from the time of the original submission, so they did not include any corrections.  Fortunately I still had these in the form of handwritten notes, which I could duly incorporate into the TeX sources.  And, wonderfully, on installing TeX years later I could still compile them to generate a DVI file and thence the PDF!  The final task was to modify the PDF to draw some straight lines to complete a couple of diagrams, which I achieved with PDFill.

As to why I'm reading about life of Ramanujan, I'm currently investigating the role of intuition in maths and computing for a little book I'm writing.

A class in Game Theory for Management students

My host, Prof. Yukari Shirota, had arranged that whilst at Gakushuin University I would give an undergraduate class in some topic of mathematics to students in the Department of Management. For such a one-off, I had a fairly free hand as to subject matter; what seemed to be considered most valuable for the students was the (rare) opportunity to hear a native speaker of English (though actually I'm ethnically less than half English).

I chose Game Theory, the subject made famous by John von Neumann and Oskar Morgenstern in their landmark work, Theory of Games and Economic Behavior. I first came across Game Theory whilst at secondary school. In fact I became quite engrossed, venturing into Birmingham Central library to conduct research for an extended essay on a typewriter [yes, it was a long time ago]! I was fascinated by its confluence of mathematics, economics and psychology; the last of these particularly intrigues me. On this occasion my main reference was Games, Theory and Applications (1st edition) by Lyn Thomas, which I used during my own undergraduate studies at Southampton University. For anyone wanting to further their study, I would strongly recommend a textbook like this.

Would you like to explore this topic in this slightly extended post?

Good! So on a Wednesday morning I gave a class in the Multimedia Mathematics series to about 25 4th year undergraduates, kindly assisted by Prof. Shirota who gave explanations (sometimes in English, sometimes in Japanese) and encouragement. Normally there would be hands-on for students, but things were kept simple so that I only had to give a presentation with software demonstrations. I read somewhere some statistic indicating that the Japanese have more slides per presentation than any other nation, so I armed myself with 70+ slides. That's excessive, probably more suitable for 2-3 classes, but I had only one at my disposal and I wanted to share sufficient material to give a reasonable feel of the mathematical methods involved, including inductive reasoning and aspects of probability. I also included quite a few pictures and came up with simple examples to show how many situations in life can be treated as a game - such as growing tomatoes or commuting.

At the same time it was important for me not to rush delivery - speaking more slowly than usual, there was no way I would cover all this material in one go, so I just used a selection of slides, starting with the main concepts and proceeding via a few hops to a couple of famous non-zero-sum case studies, the Prisoner's [really prisoners'] Dilemma and The Battle of the Sexes.

I also had another prop with me - some reasonably authentic-looking notes in Pounds Sterling (Casdon PlayCash that I bought from the local Boswells store). I used it as an ice-breaker:

An introduction to Game Theory

View more presentations from Paul Trafford

(Thanks to the American Mathematics Society for this idea, which I first saw on 'Who wants to be a mathematician?' roadshow).

Yes, a £1,000 giveaway! Except it was a bit credit-crunched: as only £937 cash was in the bank, I tried to explain that the British banks are struggling at the moment and a cheque could make up the remainder, £63.

The students were divided into 6 teams, A-F, one team per island, each with a representative. Each team was a 'player'. Teams had to choose a number N>=1; a team that picked the highest number would receive a share of £1000/N. This game had two rounds as follows:

• Round 1 [slide 3 above]: no communication
  We should have collected pieces of paper; as it was, numbers went something like: 20,50,80,100,50,52.
  Team D won £10.
• Round 2 [see slide 4]: communication allowed
  Teams chatted about this and quite quickly came to a decision, yielding the optimal result: 1,1,1,1,1,1, so each team gained £1000/6!

I was struck by the smoothness in reaching this outcome (and lack of betrayal among the teams) and think this may reflect a general culture in Japan of collective action and perhaps conformity, something that has helped the country to become such a productive and powerful economy. Quite different from an individualistic view, where it would be seen as problematic. It was later related to me that when someone says, "Ne...?" ("Isn't that so?"), there is often a feeling of obligation to say, "Ne!" ("Yes, that's so."). This would tend to support a culture of opinion leaders and followers. Ne?

In terms of software demonstration, I used mainly two tools, both released under open source licenses. The first was Gambit, which is a dedicated Game Theory suite that provides for the analysis of non-zero-sum games in both normal and extensive form. It has options to carry out computation, particularly of Nash Equilibrium, though one of its current limitations is that it restricts itself to games where players have to choose their moves independently. The other tool was Maxima, a Computer Algebra System, which I used for the graphical visualisation of payoff regions. Maxima by itself has only a command line interface, but it can invoke gnuplot to render graphical output and there is a choice of graphical interfaces: in my case I ran wxMaxima. All these are bundled together in the distribution.

I find the topics of communication and cooperation to be of philosophical interest. A standard definition of a cooperative game is couched in terms of business contracts (in the UK we can think of the Co-op supermarket) and so in such games players are said to enter binding agreements. It is used accordingly as a basic binary categorization and its importance is evident in e.g. providing assurance for the mathematical calculations. However, it means communication becomes secondary and I'd argue that [human] communication is more fundamental - it's what made the huge difference in the giveaway of slides 3 and 4 and to my lay-person's thinking, co-operation was established through a collectively agreed strategy before it became binding. No ties are needed to work together! Communication doesn't imply cooperation, but it usually precedes it.

So, I should issue the caveat that my slides exhibit a natural personal bias to this voluntary sense of cooperation, illustrated, for example in the Battle of the Sexes, in which a young married couple have free time at the weekend for an outing. The only issue is that the husband prefers a sporting venue, whilst his wife prefers a concert (so the story goes), but the bottom line is that they'd both prefer to be together than go there separate ways - see slides 65-71.

In Gambit (using this source file), we can compute the Nash equilibrium points. If we assume x is the probability with which the husband choose the first venue and y is similarly the probability that the wife chooses the first venue, then the expected returns e1 and e2 are given respectively by:

• e1(x,y)=5xy -4x -4y +4 - (eq1)
• e2(x,y)=5xy -4x -4y +4 -(eq2)
• where 0 <=x <=1, 0<= y <=1.

For minimax we set both of these equations to equal the value of the game. Gambit can do the calculation for us. The following screenshot shows the matrix used and underneath three equilibrium points.

The first of the equilibrium points are the respective the minimax strategies. But a value of 4/5 seems rather poor and would suggest - if the payoff matrix is a true reflection - that both 'battlers' will reason that settling on any venue would be better. Indeed, underneath are two other equilibrium points that return expected returns of 1 and 4 and vice versa.

However, the computation of individual points doesn't give a full picture. Just a few lines of Maxima instructions enables us to compute the region covered by all mixed strategies. It generates a 3D parametric plot, and we can initially set the z-axis to be constant, so with a bit of dexterity, you can rotate it to show the following:

The x and y-axes denote the respective expectations for husband and wife. I'm fascinated by the shape: the attentuation to the corners (1,4) and (4,1) - this particular graph reminds me of someone sitting in a hammock! Note that the point (4/5,4/5), which is the expected value of the game under minimax, lies a long way from those corners and it's also nested deeply in the region. It's certainly not on the boundary since in equations 1 and 2 above, if we set x=y=0.5, we get e1(x,y)=e2(x,y)=1.25.

It's a graph that assumes no cooperation, which is not a very optimistic view of a newly wed couple. We'd expect them to work something out in the form of a cooperative strategy, pure or mixed, so that whenever they have an outing they will go to one of these attractions together. If that's the case, then we can simplify the equations so that the (0,0) outcomes are factored out. The resulting graph is a line, the convex closure of the original region:

(Incidentally, I wonder if there is some metric indicating how far one is from cooperation in choosing minimax, perhaps defined in terms of the angle created by the expected returns with pairs of 'pure cooperative' vertices - the smaller the angle, the greater the missed opportunity for cooperation...?)

In the case of the Prisoner's Dilemma there is no minimax strategy. Geometrically, if you plot that graph you get a triangle, i.e. the set of points in Euclidean space is already convex.

Student Response and Feedback

In the event the class listened attentively and concentrated well. The opening game helped to stimulate interest, which they seemed to sustain for the duration. I was informed that they could understand most of what I said, which was a relief since preparing this class felt a bit like navigating in the dark. Certainly a few of the students gave responses that indicated they understood particular concepts. Although I didn't receive questions at the end (same kind of traditional response as Thai and other oriental students), facial expressions were not blank or bemused. This may have been helped in no small measure by Prof. Shirota, who produced (in one evening/night!) a translation into Japanese of some (possibly all?) of the slides. This would also encourage students in further reading and assist them in an assignment - a write-up about the Prisoner's Dilemma.

Whilst at Gakushuin, I also met Prof. Jun WAKO, who is a specialist in Game Theory. I hope he would approve of my presentation, but at least he may now have a few more enquiries from interested students...

Research Visit to Gakushuin University

Located towards the northwest of central Tokyo, Gakushuin is a private academic corporation that comprises schools and colleges as well as a university that currently has 9000 students. It is famous in Japan because of its historical connections with the Imperial Court. When I arrived a couple of weeks ago, I found the Mejiro campus spacious and verdent, surprisingly so given its centrality; I was informed that among the universities inside the JR Yamanote loop, it's the second largest after the University of Tokyo.

The present campus, like much of Tokyo, has seen many changes, but among the modern blocks there still remain a few of the older buildings, a little over 100 years old:



The modern multi-storey block on the right (East building no. 1?) obviously offers more capacity, but in the distance are some well-maintained old classrooms, cherished by staff and students even today. To the left is the former library building, now housing the Museum of History, graced by a venerable old tree in front of a small pool and foundation:


However, I wasn't here to be a tourist (although during my week's stay I did have a chance to wander), but rather to explore aspects of e-learning. The opportunity had arisen following earlier exchanges of ideas in the UK: in 2005, Oxford University Computing Services (OUCS) received a visit from Professor Yukari Shirota of the Department of Management, Faculty Economics, Gakushuin University. I arranged for her to give a presentation on some interactive software she had developed that guided students through the study of some topics in mathematics. The system's architecture was based on solution plans to word problems and delivered using an intelligent agent (animated by the Microsoft wizard).

Prof. Shirota is a computer scientist of long-standing - for instance, she co-authored an introduction to UNIX in 1984. During the past decade, Prof. Shirota has been developing e-learning systems to aid in the teaching of mathematics to her Management students. Inspired by George Pólya, her research has been focused on problem solving, invoking techniques in A.I. and especially visualisation, to help make sense of how the formulae and equations are used in word problems in Economics. I was particularly struck by her idea that A.I. might be able replicate the rhythm of instruction from teacher to pupil, quite similar perhaps to the rhythm of communication between a mother and baby. I hadn't come across anything like this thinking in the UK.

More recently, Prof. Shirota's research has concerned the provision of integrated tools that enable academic staff to create a range of online materials to direct students step by step in tackling certain types of questions, particularly in the field of bond mathematics. These systems are typically Web applications with scripts that invoke computer algebra systems such as Maple and Maxima to deliver step by step instructions. Some recent examples of this work are evident in overview of activities for 2011.

With regard to the financial mathematics, Prof. Shirota and her colleagues have used a conceptual approach based on entity-relationship diagrams to relate variables to formulae and equations. So the tasks of understanding may be characterised by being able to visualise and understand this map and its relations. How may that process be best aided online? In its entirety, the complete diagram is too extensive and detailed to show all at once, so any online implementation will need navigation - to focus on particular formulae and relations, but also to 'zoom out' and see the model as a whole. So does this suggest navigating it like, say, Google Earth or some other way based more closely on the relationships?

It's in addressing these considerations that I have been offering some input with my background in mathematics, somewhat distant now, and more recently e-learning and Web development, gained mainly whilst in the Learning Technologies Group at OUCS. Determining effective solutions is a multi-faceted task: its design, especially in terms of user interface, should be soundly rooted in principles of cognition. As I am not trained in educational psychology, I have to deal more with the nuts and bolts of the learning context itself, primarily in terms of the current and emerging technologies. In this regard, I'd say that whilst the emphases in pedagogy vary from country to country, the predominance of personally own computing devices has presented new variables to factor into the education at all levels. It's particularly this phenomenon, which I tend to call mobile and ubiquitous computing, that I was encouraged to explore ahead of my visit in the form of a survey paper on e-learning systems for mathematics, with particular reference to business and economics.

To give some structure in this rather broad landscape, I initially made reference to some recent features of e-learning in the UK. There's a strong focus on learner-centred education; within that I'm particularly interested in processes of deep reflection, stemming from the RAMBLE project I led in mobile blogging and learning environments. I'm not sure about the merits of focusing so much on individual predispositions, but I felt that the work on open educational resources and podcasting as a delivery mechanism was a useful vehicle to illustrate how learning has emerged from being concentrated in a classroom or workshop and flowed out into less formal environments. It was with this image in mind that I developed a thread to show how computer assisted learning for mathematics has similarly emerged from the laboratory into the open and is now squarely aiming at handheld tablet and multi-touch devices.

The paper, whose full title is 'Ubiquitous e-Learning: Designing Web Systems for Economics and Business Mathematics', has been published in Gakushuin Economic papers and is now available online - in HTML and PDF formats.

A Research Genealogy Project?

The Mathematics Genealogy project provides a field to categorise dissertations according to the Math Subject Class. Seeing how the selection is very broad, e.g. covering computer science, I was prompted to wonder what about genealogy projects for other subjects? There appear to be a few ideas and initiatives, including Thomas Witten's proposal for a Physics PhD Genealogy project, the High Energy Physics directory, the Software Engineering Academic Genealogy, the Theoretical Computer Science Genealogy and the Notre Dame University academic genealogy, that covers current members of its departments of Chemistry & Biochemistry and Physics.

It's a very fragmented picture, with independently developed systems, very partial coverage of researchers and yet already some duplication. It will become even more so as subject disciplines keep growing...

So it makes sense to me to take a fundamentally more integrated view that incorporates research in any field, one that can also have a richer model, taking into account different kinds of research qualifications, not just PhDs; and different kinds of relationships, not just formal supervisor-student; thereby responding to issues raised in the Mathematics PhD in the United Kingdom.

The findings yielded on this broader base will be fascinating, showing among other things how disciplines evolve over the generations, shedding light on questions such as: What happened to descendants of those who studied classics? What did the ancestors of computer scientists research? Many trends can be observed. There's a lot of talk in the UK about lifelong learning, so how about considering lifelong and generational research?

Another aspect that needs attention is the quality of entries. It's a tall order for just one central team responsible for verifying information received and compiling the database, which is the current arrangement at the Mathematics Genealogy Project. It would be better to distribute the workload and make use wherever possible of local expert knowledge, suitably authorised to update data in the areas with which they are familiar, whilst allowing for as wide public participation as possible.

So what's the solution?

I'm quite sure that the biggest consideration is organisational, not technical. It's probably a workflow problem and perhaps can be addressed by appealing to other international networks, most likely business networks. The quality control needs to rest with academic departments and it seems sensible that they should deal with information relating first to their department, then their institution and then neighbouring institutions. So I envisage an international network of genealogy research nodes where public contributions would be submitted though their nearest research node rather like, "contact your nearest reseller."

A few days ago I attended a presentation by someone who has done work for the World Wide Web consortium and he re-iterated the point that if there's one technical issue affecting software above all others it's scalability. So any proposal probably ought to design and develop a system that distributes the processing (cpu and resources) as well as the administration, though the computing power need not be distibuted per site (big companies typically use a few data centres containing large numbers of rack-mounted PCs). This suggests an application for a parallel computing grid.

I don't know what the implementation itself should look like: it could well be underpinned by a relational database or might even be a special kind of wiki (thinking about how that can really grow rapidly). However, the data model should certainly be given careful consideration. How to deploy it on the Internet? How to authenticate and authorise? Lots of questions will pop up if one investigates further!

Mathematics Genealogy: Indexing

Exploring some of the entries in the Mathematics Genealogy project has led me learn about some interesting and unexpected connections, but it's also highlighted quite a number of limitations regarding accuracy and promptness of updates. Saying this is really just an indication that if you offer something good, then people will be looking for more!

One particular issue is that the total number of descendants requires a separate process to run as explained by the FAQ, which says:

Because of the time required to run the descendant counting program, it is only run once per week (early morning US Central Time on Sundays), while our data is updated nightly.

That surprises me somewhat as with around 100,000 people with not very many details stored per user and few relations, it's not a big complex database. The issue here is probably that it's a relational database and the advisor-student relationship is hierarchical, somewhat like a tree structure. However, it's not a tree because of having multiple parents (multiple advisors), but rather a directed graph, where the nodes represent the mathematicians and the edges correspond to the advisory relationship. [I'm taking definitions from MathWorld, an encyclopaedia that provides clear and nicely formatted explanations with diagrams]. Further, I think there is a fair chance that it would be more general than a simple directed graph from the scenario of the same supervisor supervising a candidate in more than one thesis - although it might sound unlikely today, it is quite plausible a few centuries ago, when a researcher could be at the forefront in a number of fields. I'd also expect it to be an oriented graph, in that supervision is expected to go in one direction, but it's not inconceivable that a student produces a thesis separately in two fields under two supervisors and then shares the knowledge back across.

Returning to the problem of counting, hierarchical relationships are easy to model in a relational database, but retrieving even summary counts may mean a lot of spidering through the hierarchy, which can be very slow. The key consideration is how to index the database. I'm not a database expert, but have seen this issue in the daily work I undertake as an administrator of WebLearn an e-learning system based on software called bodington, which is essentially a web database application. The system contains various resources, arranged hierarchically, in trees, so more specific than the genealogy case.

Jon Maber, the original developer, had started work on Bodington in the mid 90s and had thought about the issue of efficent queries about resources within a given branch; he reviewed approaches to indexing and decided to adopt the tree visitation model devised by Joe Coelko. Celko had given consideration to this graph theory problem and came up with SQL for Smarties: A Look at SQL Trees, an article that appeared in DBMS, March 1996 . Basically, each node or vertex has two indices - left and right - that are numbered according to a complete tour of all the nodes, visiting each twice. It means that selecting the number of descendants of a resource a simple SQL statement that subtracts one index from another at the given node. However, there is a trade-off in that every time you update the database you need to update the index, so if lots of changes are being made it can be a major performance issue.

Celko's solution may not be appropriate in this case, but it looks like the of approach that may lead to a suitable index that will allow real-time queries of how many descendants. The article was published more than 10 years ago, so I expect research has progressed a fair bit since then.

Mathematics Genealogy

I recently came across the Mathematics Genealogy project, which offers a fascinating view of how guidance in doctoral research in maths is passed down from supervisor to student. It's a service currently hosted by North Dakota State University and supported by the Clay Foundation. It's quite a simple idea that can reveal fascinating details about the evolution of subjects studied and 'lineages' of famous mathematicians.

I have an entry, but it's wrong! I did indeed submit a thesis with that long title and study under the guidance of the late Prof. Robert Odoni, but only obtained an M.Sc., having decided after about a year not to continue for the doctorate. The Prof's record is far from complete - he was previously at Exeter University for quite a few years, so I expect he would have had doctoral students there. Also not there yet: he was supervised mainly by Harold Davenport and then towards the end by Alan Baker, both distinguished number theorists.

I do have a Ph.D. in theoretical computer science on 'The Use of Formal Methods for Safety-critical Systems,' for which my supervisor was Dr Kate Norrie, but there's no link shown. That lineage goes through Frobenius, whose work was fundamental to my M.Sc. and before him Gauss, one of the most prolific mathematicians known, and as I write almost 1/3 of all people in the database are his descendants!

It's noticeable that via Odoni, one traces back a UK line of researchers that only goes back as far as the 20th Century, whereas the line via Norrie is mainly German and goes back to the 17th Century. This apparently reflects the fact that in the UK, PhDs were only introduced in the 20th Century. This means that British mathematicians are poorly represented in the genealogy project as it stands, a situation discussed at length in 'The Mathematics PhD in the United Kingdom'. (Incidentally nice to see T. M. Fred Smith mentioned - he kindly acted as my main personal tutor for my B.Sc. at Southampton, even after I later changed my registration and dropped stats in favour of pure maths!)

I've filled in update forms over a week ago, but as yet there have been no changes in the entries of my supervisors or myself. So I'm wondering about improvements...

Notes on reading 'Wholeness and the implicate order': Introduction (3).

Still more notes in response to the intro (with more baggage that I bring). Although these are presented as notes jotted as I read, in practice, I usually tap away and later on do some tidying up. Most entries are prepared offline, on a handheld computer (HP Jornada 720, as usual :-) It allows for me to sit on a comfy chair, edit to my heart's content, whilst using only modest amounts of electricity (or battery power).

[p. xi] Thought and reality: for the Buddha, the reality he was primarily concerned with was dukkha, typically translated as 'suffering' or 'unsatisfactoriness' concerning which he taught a lot about subtle processes (e.g. the dependent chain of contact, feeling, perception and so on, yet the essence is expressed in a simple connection, in the first two verses of the Dhammapada:

1. Mind precedes all mental states. Mind is their chief; they are all mind-wrought. If with an impure mind a person speaks or acts suffering follows him like the wheel that follows the foot of the ox.

2. Mind precedes all mental states. Mind is their chief; they are all mind-wrought. If with a pure mind a person speaks or acts happiness follows him like his never-departing shadow.

[p. xv - xviii] Bohm continues to summarise what lies in the chapters ahead, and comes to the later chapters. He is looking for a holistic theory that takes a wold view that includes consciousness and evidently is not content with the discontinuities at the sub-atomic level, in which results given are in terms of statistical aggregates. I find it interesting that research is oriented to concrete predictions, that are applicable: indeed even 25+ years later, even though physicists are well-versed in wave/particle duality, I tend to hear about funding for particle accelerators or measurements concerning sub-atomic particles, such as the MINOS project .

However, it may be that it's the level of aggregates where we need to work. Again, the Buddha gave many teachings on khandas, which translate as 'heaps' or 'aggregates', and the processes surrounding them. But, as expressed e.g. in the Parivatta Sutta, the key requirement is direct personal observation.

This is what I was trying to get at in my first foray in this area, when on the basis of little more than intuition and reading an article in Scientific American, I posted a perhaps overly bold (and, now it seems arrogant) message to Usenet, entitled 'Quantum Theory and Meditation,' especially as it was my first proper posting! I received a flame within 3 days and more vitriole followed, yet there also flowed some rich dialogue and friendship. The main point I was trying to make is that the most interesting results depend upon's one own observation and not that of any instruments set up to do the observations for you.

I touched on just special relativity at school, when I read and wrote an essay on some of Bertrand Russell's 'The ABC of Relativity,' but that's about 20 years ago and so I have very little detailed knowledge.

So that's my baggage, so I look forward to reading what Bohm presents concerning quantum theory and relativity, and his new approaches.

Already though the book conveys the sense that there's a lot of feeling one's way for research directions. There's a kind of sustained balance or tension between wholeness and division, to which I can relate to intuitively from the period I spent doing a bit of research in number theory, in that the object of my research was to elicit the integer values of the determinant of a certain kind of matrix, which is a problem worked mainly in the field of algebraic number theory, but actually the main result was in terms of densities, saying "most values of 'the right type' are integer values of the determinant," and thus a result of analytic number theory.

So what? Well, many mathematicians like simplicity, symmetry, wholeness and completeness, wherein they can find great beauty. For some, it evidently meant so much, among whom Kronecker is well known among mathematicians for his remark:

God created the integers, all else is the work of man.

But, on reading a summary of his life, it sounds that this strongly held belief led to immense friction.

This reminds me of the conflict in views dismissed by the Buddha in the Tittha Sutta in the Udana. All in all it's best that I have no expectation about any absolute answers concerning the cosmos; rather, my goal should remain to learn something that may improve my understanding of the composition of the Buddha's teachings.