The 38th ISSA issued a call for participation in open debate on the philosophy of mathematics.1 The current idealistic philosophy is called formalism; and currently the only comprehensive alternative to it is my realistic philosophy of zeroism.
A very distinguished formal mathematician (FM) attended, and the following are the minutes of an informal conversation.
1. CKR: Why are real numbers needed for calculus?
FM: That is how the calculus developed in the West. It was used successfully for applications to physics.
2. CKR: So the issue is practical applications? Most practical applications of physics are done today on computers using floating point numbers. So, we should teach floating point numbers.
FM: No, understanding is also important.
[3.1. Newton failed to understand the calculus. As I have argued, Newton's physics failed because of the resulting intrinsic conceptual confusion.2 Where Newtonian physics worked, it involved only the Indian method of numerical solution of ordinary differential equations (as done on a computer today, using floating point numbers). ]
3.2. There are alternative ways to understand calculus using Indian non-Archimedean arithmetic and zeroism. Why not use them instead of reals and formalism? A critical comparison is needed to demonstrate that one is superior to the other. Who did it?
3.3. Real-number calculus fails in numerous situations in contemporary physics (e.g. discontinuities and shock waves, S-matrix expansion).3 This failure cannot be resolved by replacing calculus with Schwartz distributions or even using non-standard analysis.4 Non-Archimedean arithmetic works where both fail. But a new philosophy (zeroism) is needed to handle the situation where there are an infinity of infinitesimally different entities. This philosophy is like the natural language practice of using one name pi to denote the multiplicity of distinct entities 3.14, 3.1415, etc.
3.2. I intuitively prefer real numbers.
3.3. I don't know physics.
CKR: Zeroism uses empirical proof, as Indians did. For practical applications, empirical proof is certainly adequate. Why not use it instead of deductive proof?
FM: Empirical proof is fallible.
CKR: Yes, this is accepted. A rope may be mistaken for a snake, and vice versa. But what makes you think deductive proof is infallible? Mathematical proof uses 2-valued logic. Why choose it? On cultural grounds? Just because the church said that a particular logic is universal since it binds God? That is not true; there are many logics such as Buddhist and Jain logic. Mathematical theorems are relative to both postulates and logic. So how is logic decided? If logic is decided culturally, theorems are only cultural truths. If logic is decided empirically then the choice of logic is itself fallible, by your argument, so theorems based on that are even more fallible. So, in either case, deductive proof is more fallible than empirical proof.
FM: You are obsessed by the church.
(CKR: I have already explained in this very meeting the theoretical importance of studying the church in understanding the West and its superstitions which plague mathematics and science to this day.5 )
CKR: Anyway, when you choose real numbers on grounds of intuition that is a cultivated intuition. In other words that is indoctrination. Why should this indoctrination be imposed on millions of students? What if Dinanath Batra were to say the same and impose his intuitive preferences on millions? Are you not making that equally legitimate?
FM: You are ignoring the aspect of aesthetics.
CKR: Where is the aesthetics in math? I left formal math since I found it ugly and repulsive. Millions of students hate math, and find it hard. They would not do so if there really was beauty in it. On the contrary, I have demonstrated an alternative way to teach calculus which makes it easy. Students like it.
FM: Your sample size is very small.
CKR: But the sample on the other side is very large as clear from my lifelong personal experience or the responses to my articles in Dainik Bhaskar which went to 15 million people.
FM: Whatever way you teach mathematics students will hate it, because the basic problem is that teachers are bad.
[CKR: That is a counter-factual assertion being used to maintain status quo. A bad volleyball coach does not make students hate volleyball.]
At this point the conversation was discontinued, since FM was finding it very stressful.
(Minutes prepared by C. K. Raju (CKR).)
Summary of round 1 and further questions for round 2.
Background: Colonialism was con-all-ism. It just made stupid claims of superiority, like the stupid racist claims for which the West is famous. Macaulay said the West is immeasurably superior in science, based on racist history. Colonial education taught blind imitation, and rewarded it with some bribes.
Q 1. You are basically advocating the “ape the West” superstition, which we should eliminate, along with supporting bogus myths like “Euclid”. If it is not aping, you need a critical examination of alternatives. Have you done so? Did any one else do so in two centuries?
2. Most people understand numbers empirically. They manage arithmetic and its everyday applications to commerce lifelong without ever hearing of Peano's axioms. I support that everyday practice. But you say that most people lack understanding. As proof you only repeat the racist myth “West is superior”. Why? Why is an empirical understanding of numbers as in natural language inferior? (Religious connection: Peano's axioms bring in infinity by the backdoor. That metaphysics of infinity is tied to the key church theology of eternity and is actually INFERIOR to other concepts of infinity/eternity.)
3. Deductive proofs are INFERIOR since they are MORE fallible than empirical proofs. That holds whichever way you decide logic: culturally or empirically. That knocks the bottom out of formal math. So why continue with formal math? (Religious connection: That superstition relates to Crusading church theology that logic binds God, but not facts, so that God cannot create an illogical world, but can create the facts of his choice. That is, the West put logic above God who was above facts. Hence, the Western superstition that 2-valued logic is universal, and proofs based on it are superior to empirical proofs.)
4. Why do we teach math? We should teach math for its practical value. That is why most students want to learn it; for its practical value. Calculation, not proof, is central to practical value. Hence, we should teach calculation. You evade the issue by shifting from practical value to understanding. In actual fact, most students of physics and engineering never learn about real numbers (since too complicated), and hence never understand real-number calculus. They learn the formula that (d/dx)ex = ex but do not understand what ex is. So, what you actually teach them is to accept your authority, WITHOUT understanding, based solely on your claim of “superiority” which was never established. (Religious connection: The church had little use for calculation except to calculate the date of Easter and everyday commerce. It wanted to persuade others, especially Muslims, hence valued a “universal” method of persuasion or proof. We should value practical calculation, not metaphysical proof.)
5. You speak of the value of math to physics. A cocktail of practical value and metaphysics is dangerous, because there is no link between the metaphysics of formal math and its value to physics. (E.g. physical time may be discrete, not like the real line.) Numerical calculations not deductive proof is good for most practical applications of physics. So if math has value because of applications to physics, the real stuff is the calculation which is what we should teach. Do you agree? Further, when I probe deeper into the physics, pointing out that the real-number calculus does NOT work in many situations, you evade it by saying you don't know physics. Well, in that case, don't talk of value to physics. Also tell that to the government. In future all formal mathematicians should ethically desist from masquerading as experts, and prescribing what sort of math should be taught to physicists and engineers (and social scientists). You offer neither practical value nor a critical basis for what you call “understanding”, but only insist that status quo should be maintained. (Religious connection: Colonial/church education is all about instilling imitative practices without understanding. That is what is actually taught.)
6. When I offer an alternative understanding of calculus through traditional Indian “non-Archimedean” arithmetic and zeroism, you evade it by appealing to your personal intuition. That is not intuition but indoctrination. Others too can appeal to their intuition and dismiss yours. Can we justify what we teach to millions on grounds of mere personal intuition? If so, whose intuition? (Religious connection: Western philosophers from Plato to Kant valued “intuition” on account of their superstition linking math to the soul and to its innate knowledge.)
7. You say math is concerned with aesthetics, when the fact is that millions of students hate it. I left formal math because I found it repulsively ugly. What is the basis of the claim of aesthetics in formal math? (Religious connection: this claim of aesthetics too is linked to Western superstitions about the soul. Plato clubbed math and music, as arousing the soul (hence linked aesthetics to virtue). But, today, most people still find music aesthetic, but find math as unaesthetic. Something happened to math. The church grabbed it. So, the claim of aesthetics survives only as a superstition contrary to facts.)
8. The claim that students would dislike any other way of understanding and teaching math is a facile counter-factual, and an attempt to preserve status quo. Are you afraid to allow alternatives to be tried?
2For further elaboration and references, see http://ckraju.net/papers/Calculus-story-abstract.html.
3See abstract cited above.
4Applying non-standard analysis to define products of Schwartz distributions is inadequate, since the ambiguity was only partly resolved by demanding that the final result should be standard. In the presence of discontinuities, different forms of differential equations become inequivalent. (E.g. Riemann's mistake was to suppose that even in the presence of shocks the fluid dynamical equations of conservation of (a) mass, momentum, and energy are equivalent to the equations of conservation of (b) mass, momentum, and entropy.) So the correct form of the equations (or the correct association of factors) must be decided empirically. This situation (of an infinity of infinitesimally different entities) inevitably arises even with Colombeau's product of distributions: where there are an infinity of different entities associated with δ2. However, with that inferior product, there is no way to fix things because a huge infinity of alternatives is involved.
5 Specifically, the church and its education system is an important determinant of what Marxists call "superstructure". Church education started during the Crusades, and developed for the church aim of imperial expansion through “soft power” or organized falsehood (after force failed against Muslims during the Crusades) Western universities were under church control from the 11th c. Univ of Bologna, and until the end of the 19th c. That education was designed to produce an army of indoctrinated missionaries who would spread superstitions. Contrary to Marxist theory, this education system developed independently of capital. But this superstructure was neglected by Marx and even Gramsci in his understanding of education. Macaulay 1847 proposed to use church education as the cheapest means of counter-revolution. This proposal was based on the observation that church education indoctrinates and enslaves minds. Colonial education was church education when it first came to India in a big way in 1847.