Sunday, April 30, 2006

A Research Genealogy Project?

The Mathematics Genealogy project provides a field to categories 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.

Saturday, April 29, 2006

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...

Tuesday, April 25, 2006

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.

Saturday, April 22, 2006

Structure and Flow - an example in XML editing

Carrying on with the balance of structure and flow, although I left formal methods research a long time ago, I still come across it as a recurring theme in IT work.

At my present workplace, as a software developer I found myself with the task of extending a web-based system to allow anyone to use the web to edit some data encoded in XML. XML (short for 'eXtensible Markup Language) is a hot topic that promises the recording of meaningful information, its long term preservation and wonderful exchange and interoperability among software systems (e.g., because it's stored in a text file, so you can read an XML file in Notepad). An XML file is a data file, basically a hierarchical structure of tags and content. It's structure and data in one.

So where's the flow? That comes in the editing, because to edit the documents I devised a system that used a functional programming language called XSLT [well, it looks like it should be functional, though proof "by example" doesn't look like proof!]. Every change to an XML document is carried out in terms of XSLT, i.e. suppose we have XMLDOC1, then apply an XSLT stylesheet xslt1 to get XMLDOC2; and then apply xslt2 to get to XMLDOC2 and so on. In practice, each stylesheet defines a slight change in the document, with all else remaining the same. The operative verbs are simply: add, delete, and update. Perhaps you could use the word 'perturbation' for this?

As it happens, an XSLT stylesheet is actually an XML document, so again it has all those nice qualities described above, which means that using this system not only allows you to have a convenient text-based history of the documents, but also of the transformations and I can not only share data, but the transformations needed to carry out changes. There's a walkthrough illustrating what I mean through a number of screenshots.

Now there's an irony in using XSLT for change because as instances are XML documents it means that the transformations are themselves data and structure. So in one sense we have a sequence of data structures - where data and the way it changes is in the same format. But what I haven't addressed is how you actually generate and carry out the transformations. This requires a processor!

I've grown to appreciate this system for it gives me a sense of holism - a stream of documents and transformations in one flow. It certainly intrigued one of my colleagues, for whom XML and XSLT are very much his bread and butter!

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

A copy of Bohm's book (Routledge Classics 2002) arrived last week, conveniently just before I set off for a few days' holiday, staying at my father's house. It looks fascinating, so I'll jot down some responses, though at this stage I don't know how far I'll take this. In any case, I should say I can be a very slow reader!

The introduction develops some rationale for Bohm's new perspective, which appears to have emerged from deep personal observation, a state of absorption, as well as his considerable experience as a physicist.

When I thought of 'wholeness' and 'reality,' what came first to mind were the elements, especially depicted in the dhammakaya meditation tradition as a sphere - the four elements of earth, fire, air and water at cardinal points surrounding the space element at the centre and within that the element of consciousness.

Things can be observed at different levels, on different scales. My impression is that at any given scale, science is familiar with progress/movement through stages and has developed laws of motion that model this accurately. However, what laws or models are there for movements between scales? What about the flow between levels of abstraction? I raise this because in the bit of literature I encounter, there seem to be different models for macro and micro, so what is happening on the journey from macro to micro?

The tensions between/balance of structure and flow can be found in many disciplines. I came across it whilst doing research in the field of [concurrent] formal methods in computer science, in which mathematical techniques are used to specify and analyse software systems. You can make a crude division in terms of orientation: one is 'structure' based, viz the so-called 'axiomatic' techniques of VDM, Z etc.that are oriented around sets; the other is 'flow'-based, which is the emphasis in process algebras - how systems are defined in terms of the actions that can be carried out from state to state rather than descriptions of the states per se and hence action-based or operational semantics. This was brought home to me by a very valuable survey of formal methods by Jonathan Ostroff [Formal Methods for the Specification and Design of Real-Time Safety Critical Systems", The Journal of Systems and Software, Vol. 18, No. 1, pp 33-60, Elsevier Scienc Publishing Co. Inc., New-York, April 1992.]

[p. xii] The content of consciousness to be 'reality as a whole'? It's quite an assumption that there can be consciousness of whole reality - is that possible? I'm glad Bohm emphasises the importance of view - it affects everything!

In the introductory class on Buddhist texts that I attended in Spring, Richard Gombrich explained how the Buddha always taught about consciousness of... [and the teachings state that viññana (translated as consciousness) is one of the 5 heaps that are not part of deathless nibbana].

[p. xiii] I can see that this work is very much contraflow vs prevailing views that have become entrenched since the so-called 'Age of Reason.' A process-oriented view was something the Buddha expounded 2500 years ago, expressed succinctly in Pali as sabbe sankhara anicca... - "all conditioned formations are impermanent." The growing interest in the Buddha's teachings presents a veritable challenge to those who separate subject from object and take a materialistic view, which seems the predominant characteristic of European thought during the past few hundred years.

[p. xiv] A language with verbal emphasis. Again, the Buddha focused teachings a great deal on processes of mind: indeed the path to Enlightenment, the Eightfold Noble Path is expressed in terms of verbs, starting with 'Right View' and detailed modes of practice themselves as expressed in e.g. the Satipatthana Sutta describe exercises through the four modes of mindfuless (body, feelings, mind, mental qualities) - that are always working with change; magga is a flow/process of going through stages and something that may be worth noting is that what also occurs is a subtle progression in the nature of observation.

We can go further with emphasising verbs and one of the most striking example can be found in the Buddha's instruction to Bahiya (see previous entry), "in the seeing, just the seen; in the hearing, just the heard, ..." But this is for a very very advanced practitioner, on the brink of full final attainment. So conventionally the subject-object paradigm is often more practical ... I wonder what Bohm's 'rheomodes' is all about and how far this language can be taken...?

Tuesday, April 18, 2006

Tuning in to Dhamma with the 3D Crystal Radio

The Dhamma has been made clear in many ways by Master Gotama, as though he were righting the overthrown, revealing the hidden, showing the way to one who is lost, or holding up a lamp in the dark for those with eyesight to see forms...

[MN. 7, Vatthupama Sutta]

I'm going to try to develop further the ideas expressed in the previous entry and have a feeling that there could emerge a few strands of research.

There's no known contemporary written account of the Buddha's teachings - it was very much an oral traditon. So when we say the Buddha "taught," what gets recorded in written form as his "teachings" certainly loses a great deal. So this term probably needs lots of qualification along the lines of, say, the Buddha "transmitted" and it is worth paying special attention to the commonly used term applied to his disciples of "Savakas," the "listeners" or "receivers".

So, I'm taking 'Savakas' as my cue or prompt. My previous entry introduced briefly an analogy with holography. Just to use a bit more of the terminology, I was comparing the teachings with the interference patterns (hologram) produced on a special film when a light (called a reference beam) is shone at and interferes with light from the object to be 'recorded' (called the object beam). Shining the right light (the reference beam) at the hologram generates the light from the original object (i.e. the object beam) thereby providing a faithful 3D semblance. At least, that's my beginner's understanding of the process, just paraphrasing a Wikipedia article.

The main points I wish to highlight are that there are two components necessary to reproduce a faithful reproduction of the original whole - the appropriate recording (on film) and the right light shone onto the film.

Now to take the comparison further, it is as thought the Buddha possessed the reference beam and for someone to understand they too need the reference beam to reconstruct the original 3D object, the Dhamma object, as it were.

How do they generate the reference beam? In considering how the 'right light' (or reference beam) is produced by the mind, it's easier for me to try working with another analogy in which we may liken the mind somewhat to a crystal radio set. The crystal lies at the heart of the set because it acts as the detector, converting radio waves into sound that is meaningful to us.

The ability to interpret a signal depends upon the kind of crystal and also its size and quality. So it seems to me that it can be likened to the inner treasure of paramis, perfections accrued through meritorious actions over many lives, specially as a crystal itself takes a very long time to form. In practice, for radios, it's relatively easy to find the right kind of crystal that can do a good enough job, so the analogy is partial. However, we may also say that a radio's ability to tune in to different stations is similar to the way people can tune in to different kinds of teachings.

So what's the significance of the holography analogy? At the moment, what I'm presenting are probably just a few pieces of a jigsaw. Even so, I think some research could analyse the Buddha's teachings using the latest findings in physics and psychology to explore new kinds of mental maps. It would mean putting to one side many of the assumptions currently used in linguistic and textual analysis so there is space to allow for aspects hitherto considered irrelevant or dull, such as repetition. I think it would be instructive to provide different ways of looking at the Tipitaka through a variety of visual representations and mappings - linear and nonlinear.

One particular interest is abstraction or, looking the other way, expansion or reification: which teachings expand on others? Are these teachings characteristics of interference patterns? Is there something analogous to concentric rings to be drawn, where the inner core is the teaching at its most abstract, as in the Bahiya Sutta, and where the outer circles containing the inner core are supporting details, as in the Malunkyaputta sutta?

I think holography could be useful in casting light on how the mind perceives and processes. Also pertinent are studies in physics - particularly quantum theory - and the implication that these studies have on the study of mind.

I know little about holography or physics, let alone how they may relate to mind, so have just ordered Wholeness and the implicate order by David Bohm, and The Holographic Universe by Michael Talbot, which was a bundle offered by Amazon.

[Quote at beginning of article added on 4 December 2008]

Sunday, April 02, 2006

Holographic teachings from the Buddha?

When you read the texts that relate the teachings of the Buddha, you find some very short passages. They can seem fragmentary and it can be tempting to doubt their authenticity. However, I've long felt that the Buddha taught very precisely to his audience knowing that they could tune in so effectively that a few choice words sufficed to prompt progress to Enlightenment, whereas if read conventionally out of context they would appear odd and make little impact. So I wrote a short essay [with a long title of]: Observations on how kamma affects listeners and its implication for interpreting the Buddha's teachings.

In that essay I thought intuitively of holography as a good analogy for how one can recover the whole from fragments: just as a certain beam of light shone against the fringe pattern on a photographic film can reconstruct a faithful 3D representation of the original object, so the Buddha knew that the listener could penetrate the specially recorded words of the Dhamma teachings and reconstruct the essence of Enlightenment by tuning in (or 'shining the right beam').

Sorry if this is expressed clumsily.