1. Recognize the History-centric (HC) aspects of mainstream Western mathematics from a reading of Dr. C. K. Raju's essay on "Zeroism and Calculus without Limits"
2. Attempt an original extension of the ideas and conclusions in this paper in terms of our HC thought system (HCTS) Model (History-Centrism is a term coined by Rajiv Malhotra in a recent book. See prior blog posts for complete details).
Part 1: History-Centrism in Mainstream Western Matha. Dr. Raju notes that not much is known about Euclid to this day although the work 'Elements' has attributed to him, and he has become well-recognized via the popular term "Euclidean geometry". Dr. Raju uses this example as symptomatic of the general pattern of attempts by the church to seek non-Indian and non-Arabic foundations for Western science to make his point that science and Abrahamic religion in the west has been closely linked for a long time and says "the difficulty in understanding quantum mechanics is primarily due to the excess baggage of theology in science".
b. Dr. Raju then takes on the 'question of time'. Here he cites his book 'The Eleven Pictures of Time' that shows how truth-claims with respect to time are fundamental to Abrahamic religions, value systems, and science, and in particular, the religious aspects of the truth claim and its assertion of universality has created a few problems (I haven't read the book yet, so I can't comment).
c. A case is made for the 'history-laundering' of Calculus by the Church where Dr. Raju notes "the fact is that the calculus developed in India, over a thousand years, in response to the clear economic need of monsoon-driven agriculture. Just as “Arabic numerals” and related Indian algorithms were imported in Europe by Florentine merchants, the calculus was imported in Europe due to the clear economic need of an economy driven by overseas trade, and the difficulties of (and specific to) European navigation".
To summarize the introduction of the paper, the Western truth claims over the nature of time and the history-centrism injected into science (Calculus is shown as a prominent example where much of the complexity of learning and teaching it is really an artifact of such an intervention) has created a few problems. This situation is next contrasted with the approach taken in Dharmic Thought Systems (DTS, specifically Buddhism):
d. Buddhist logic or Catuskoti is that of four alternatives: This, Not This, Both This And Not This, Neither This Or Not This. Obviously this is incompatible with the two-valued logic that is exclusively used in the west. India also has another alternative in the Jaina school, Syadavada that allows seven alternatives. Dr. Raju notes that the two-valued logic was also known in India for a very long time, so calling it 'Aristotelian' may be inaccurate. Thus, we find in India a diversity of logical traditions based on empirical evidence in contrast with "this is the only way" approach that characterizes mainstream Western math. Dr. Raju summarizes this situation thus: "one cannot appeal to empirical experience to support the current use of 2-valued logic in formal mathematics. Thus, formal mathematics prides itself on being entirely metaphysical".
e. Thus we have a clash of approaches. A diverse, empirical based logic schools of India, and the metaphysics-justified two-valued logic of mainstream West. What is the impact of this choice on the science that comes out the respective schools of thought? There is an argument to be made that “deduction (West) is less certain than induction (Indic)”.
f. The coup-de-grace of this paper is delivered via a brilliant and useful interpretation of the Buddhist notion of Sunyavada as 'Zeroism (rather than null or zero) as follows "The understanding,
therefore, is that sunya does not simply mean emptiness or void, sunya refers to the nonrepresentable part which is ignored or zeroed in a representation. The meaningful
statement that every representation leaves out some non-representable elements, should not be converted into the meaningless statement that everything is void".
g. The author then goes into an in-depth discussion (that is beyond the scope of this post for the time being) of the practically beneficial properties of Zeroism, in particular noting that Calculus taught without limits and supertasks, and based on first order differential equations is a better alternative, both practically and from the point of view of pedagogy, and in this age of the computers and internet, 'there is no doubt' that the latter way is the superior and rational approach.
Part-2: History-Centrism in Western MathematicsDr. Raju's paper is able to make a reasonable case that mainstream Western Math over-emphasizes deduction over induction and abstract theorem-proving over insightful empirical verification. Furthermore, the root of such deductions are often buried in axioms that are essentially tied to mathematical truth claims. Certain prophet-like historical characters such as Euclid and Newton, on the basis of their reputation (perhaps obtained via genuinely insightful work in modeling and solving real-life problems) reinforce the notion of 'perfect science' and its theological root, thereby discouraging questions. Consequently, students and future practitioners blindly accept truth claims as reality simply based on "Euclidean Geometry" and "Newtonian Physics" despite the fact that both these areas have been proven to be non-universal in application, and that the more reliable way of verifying an idea is via the empirical method based on 'Zeroism' that is the cornerstone of DTS. Furthermore, it is also important for HCTS to establish the historicity of people like Euclid as well as provide evidence that they were always 'church-friendly (Newton's criticism of the church in particular has been downplayed considerably, per the paper). On the other hand, the DTS approach allows a student to develop a process of empirical verification of the practical viability of a postulated idea from first principles, without blind dependence on truth claims solidified via historical reputation. We see these major differences between HCTS and DTS approach to math modeling:
1. Pluralism: There are multiple ways of analyzing and interpreting a given physical problem. No claim on universality of a "historic law" and standardization needs to be made with respect to one particular approach, although each student may have their own favorite approach. Thus a DTS-based society was able to come up with 2, 4, and 7 valued logical outcome models. The practical value of such alternative non-western approaches may well become critical in the future as the speed of computer systems (e.g. quantum computing) goes up and the quality of artificial intelligence techniques increase.
2. Unlike mainstream western deduction which relies on abstraction, i.e. 'one-time perfect deductive proof given by reputed expert, then blindly invoke this theorem forever', DTS encourages empirical verification every time we are confronted with a new problem context, thereby requiring the student or practitioner to understand the intricacies in the posed problem in-depth, and from first principles. More recently in the west, the practice of geometrical construction-based proofs for theorems has regained traction and a few complex proofs in computer science and math are 'computer-assisted' nowadays, bypassing the need for a reputed deduction-centric human expert.
3. DTS allows for and accounts for a realistic level of uncertainty and probability in mathematical modeling and allows for more logical outcomes than the perfectly deterministic "Aristotelian law of the excluded middle" abstraction.
4. DTS encourages a student to re-experience and reproduce a first scientific discovery by empirical verification from scratch taking Zeroism into account, rather than just dogmatically relying on the 'expert guarantee' of the first (western-legitimized) discoverer.
The western quest for perfection via an absolute metaphysical truth in a math representation and based on supertasks has never really worked in practice. Approximation, round-off error, and neglecting that which is not useful in a context is reality and therefore there is often a considerable gap between a 'perfect math model' on paper and its computer translation that renders the former to be often worthless in terms of providing any practical insight. Futhermore, we observe an unbridgeable duality of "theoretical math" versus "applied math" in the west, which to the best of my knowledge was not present in ancient Indian DTS society where the need for such a binary categorization of theory versus practice never really arose. Among the most famous and important quotes as far as mathematical models is "All models are wrong, but some are useful". In other words, there may well be no perfect representation of some processes (Sunyavada), and it is more important to focus on the usefulness and practical value of such representations in the given context. In fact, this approach is the cornerstone of Analytics, the ongoing revolution in the world of data-driven applied math modeling, computerized decision optimization, and prediction. Students and engineers who can recognize, and more importantly, apply these Indian and Dharmic thoughts and concepts stand to gain a competitive edge over their competitors in the global technical workplace.