I'd written a blog for the INFORMS conference earlier this month based on my practice perspective, which emphasized the importance of contextual optimization rather than despairing over the 'not infallible' theoretical worst-case nature of certain mathematical problems. This is something well-internalized by those in the large-scale and non-convex optimization community, where 'NP-Hardness' is often the start, rather than the end point of R&D.
The wonderful 'Philip McCord Morse Lecture' at the recently concluded INFORMS conference in San Francisco by Prof. Dimitris Bertsimas of MIT touched upon this point, and the 'conclusions' slide in the talk explained this idea really well. To paraphrase, 'tractability of a problem class is tied to the context - whether the instances you actually encounter can be solved well enough'. I mentioned the Sulba Sutras in that blog - a well known body of work that epitomizes the Indian approach to mathematics as Ganitha - the science of computation. The genius, Srinivasa Ramanujan, was a relatively recent and famous example of a mathematician hailing from this tradition. The Indian approach is often algorithmic and more about rule generation than infallible theorem proving. Not that Indians shied away from proof ('Pramaana'. For example, see 'Yuktibhasa'). As I understand it, this sequential process of discovery and refinement does not lose sleep over theoretical fallibility, and consists of:
a) in-depth empirical observation of, and a deep meditation on facts,
b) insightful rule generation,
c) iterative, data-driven refinement of rules.
This quintessential Indian approach is applied not just to math, but to practically every field of human activity, including economics, commerce, art, medicine, law, ethics, and the diverse dharmic religions of India, including Hinduism and Buddhism. Panini's Sanskrit is a great example of this approach.
Panini, the famous Sanskrit grammarian (along with Patanjali) is perhaps the most influential human that much of the world does not know much about. His fundamental contributions to linguistics more than 2000 years ago continues to transform the world in many ways even today. Noted Indian commentator, Rajeev Srinivasan, has recently penned a wonderful article on Panini and Sanskrit. You can learn more about Panini's works by reading Dr. Subhash Kak's (OK State Univ) research papers (samples are here and here). This blog was in part, triggered by this article, and talks about Sanskrit and its contextual optimizations.
Abstract: Sanskrit recognizes the importance of context. Two examples that show how Sanskrit is optimized depending on the context, in two entirely opposite directions, is shown below.
Optimization-1. The grammar is designed to be entirely context-free as Rajeev Srinivasan's article explains, and anticipated the 'grammar' of today's high-level computing language by more than 2000 years: precise with zero room for ambiguity of nominal meaning. To the best of my knowledge, punctuation marks are not required, and order of the words can be switched without breaking down, although there may be personal preferences for some orders over the others, and the sentence remains unambiguously correct. An optimization goal here therefore is to generate a minimum (necessary and sufficient) number of rules that result in an maximally error-free production and management of a maximal number possible variations of Sanskrit text. In this case, Panini appears to have achieved the ultimate goal of generating a minimal set of rules that will produce error-free text, forever. There are other well-known optimizations hidden in the structure and order of the Sanskrit alphabet - more on that later.
Optimization-2. The final interpretation of many keywords in Sanskrit ARE contextual. Which means there are multiple, related interpretations for some words that have a nominal/generic meaning, but you have to optimize the final interpretation at run-time by examining the context of usage, to recover the most suitable specific choice. If the first optimization helped eliminate fallibility, this second optimization in a sense re-introduces a limited fallibility and a degree of uncertainty and freedom by design! This feature has encouraged me to reflect (recall Ganesha and Veda Vyasa), develop a situational awareness while reading, pay attention to the vibrations of the words, and grasp the context of keywords employed, rather than mechanically parse words and process sentences in isolation. A thoughtful Sanskrit reader who recognizes this subtle optimization comes away with a deeper understanding. For example, Rajiv Malhotra, in his book 'Being Different' (now in the top-10 list of Amazon's books on metaphysics) gives us the example of 'Lingam'. This can mean 'sign', 'spot', 'token', 'emblem', 'badge', etc, depending on the context. Apparently, there are at least 16 alternatives usages for 'Lingam' of which one best suits a given context is picked, and not simply selected at random. And of course, the thousand contextual names ('Sahasranamam') for Vishnu is well known in India. Some well-known western and Indian 'indologists' have ended up producing erroneous, and often tragic translations of Sanskrit text either because they failed to recognize this second optimization, or because they misused this scope for optimization to choose a silly interpretation, leading to comic or tragic conclusions.
Again, this contextual optimization approach by the ancient Indians is not just restricted to Sanskrit, but is employed gainfully in many areas, including classical arts, management, healthcare, ethics, etc., and of course dharmic religion. This contextual dharmic optimization has perhaps helped India in getting the best out of its diverse society, as well as keep its Sanskriti refreshed and refined over time. For example, the contextual ethics of dharma (ref: Rajiv Malhotra's book) has a universal pole as well as a contextual pole that allows the decision maker faced with a dilemma, to not blindly follow some hard-wired ideological copybook, but contemplate and wisely optimize his/her 'run-time' choice based on the context, such that himsa is minimized (dharma is maximally satisfied). Some posts in this space has tried to explore the applications of this idea'.
An earlier blog discussed a related example of seemingly opposite goals for contextual optimizations. When it came to mathematical algorithms, data, and linguistic rules in Sanskrit, a goal was to be brief and dense, minimize redundancy, and maximize data compression, so that for example, an entire Sine-value table or generating the first N decimals of Pi can be both encoded and decompressed elegantly using terse verse. Panini's 'Iko Yan Aci' in the Siva Sutras is a famous example of a super-terse linguistic rule. On the other hand, when it comes to preserving long-term recall and accuracy of transmission of Sanskrit word meanings as well as the precise vibrations of mantras (e.g. Vedic chants) that are critically linked to the 'embodied knowing' tradition of India, the aim appears to be one of re-introducing controlled data redundancy to maximize recall-ability, and error-reduction. This optimization enabled Sanskrit mantras to be accurately transmitted orally over thousands of years.
To summarize, contextual optimization is a powerful and universal dharmic approach that has been employed wisely by our Rishis, Acharyas, Gurus, and thinkers over centuries to help us communicate better, be more productive, healthier, creative, empathetic, scientific, ethical, and interact harmoniously with mutual respect.
update 11/22/2014: 'optimize the final interpretation at parse-time / read-time' is the intent for optimization-2, rather than the computer-science notion of 'interpretation at run time'.