Zeroism vs formalism in math

Jay Jolly asked:

“Could you expand more on the the concept of “Zeroism”as applied in Mathematics?What changes do you foresee in pedagogy of Mathematics if the philosophy is changed from formalism to zeroism?”

First I have a whole chapter on zeroism in my book Cultural Foundations of Mathematics (chp. 8 with the subtitle: sunyavada vs formalism). (The editor of a Cambridge journal solicited the book for review, and the reviewer then lied that the book did not contain any philosophy beyond chp. 2 to justify the extraordinary procedure of reviewing only 2 chapters of the book. This is in the long Western tradition of suppressing and defaming opposing viewpoints when unable to counter them. Even today, Western universities don’t teach non-Western philosophy presumably just because they are afraid this would damage their own philosophy built on the weak foundations of theology.)

What is zeroism?

There are three aspects of zeroism which are relevant. First, zeroism regards the empirical (though fallible) as an appropriate means of proof, superior to any metaphysics. (This is contrary to formalism which is wholly metaphysical, and regards its particular metaphysics as superior to both physics and all other systems of metaphysics.)  Zeroism accepts fallibility, as in the fallibility of science.

Second zeroism accepts the impossibility of “perfect” or precise represenation of anything. This is the practice in natural language that one name refers to a multiplicity of entities. When applied to persons, this means that we use one name to refer to a “person” who is, in reality, a constantly changing entity from childhood to old age.  (We cannot even specify, “this person at x instant” because the instant cannot be precisely specified, and by the time of the specification, even that specious instant is gone forever beyond observation.) When applied to numbers this means that the number pi can variously refer to 3.14 or 3.1415 etc, as needed for a particular practical problem, with the rest of the infinite series being “zeroed”. This is similar to the common practice of rounding, though built on a different epistemological foundation.

This was the point of Western confusion about the calculus, as in Descartes’ confusion: he thought the infinite series for pi must be naively summed. He was right that physically carrying out this infinite sum would require an infinite amount of time, which was not available to humans. He was wrong in thinking that stopping at any stage was not perfect, hence not mathematics: for any practical application we will and must stop at some finite stage of precision. So, it is the demand for infinite precision which is mistaken.  This persistent confusion about math as perfect and infinitely precise is what eventually led to the  formalist solution just to postulate that the series for pi could be summed metaphysically, which metaphysics is never needed for any practical purpose.

Third, zeroism brings out the variations in logic, so that the deductive proofs do not constitute certain knowledge, and are of little practical value. The theorems would obviously vary with the logic used. (Even intuitionsists recognized that math would change if proofs by contradiction were abandoned.) Accordingly, the church vision of mathematics as  concerned with proof, and not calculation has to be abandoned. The church focus was on using math to teach reason as a “universal means of persuasion”, since after the failure of the Crusades it tried to convert Muslims using “reason”.  However, for others, the objective of math was (and remains) commerce, science and technology.

What are the advantages of zeroism over formalism?

I have some papers and a monograph forthcoming on the changed pedagogy and its advantage. For the time being, you could take a look at my article “Decolonising math and science education” at the recent conference on “Decolonising our universities”, at http://multiworldindia.org/wp-content/uploads/2009/12/C.K.-Raju1.pdf.  The proceedings are scheduled to the published by next month. Here is a sample quote from that paper on the pedagogical disadvantages of formalism.

“The disadvantages of the existing (calculus with limits) curriculum are the following. First, limits cannot really be taught correctly even according to formalism, since that would involve teaching formal real numbers and axiomatic set theory. These are regarded as advanced topics, and most students of science and engineering are not exposed to them. While formal real numbers are taught in advanced courses on mathematical analysis, axiomatic set theory is not taught even to mathematicians, except to specialists in logic. Hence, even professors of mathematics in IITs are not familiar with it. The assumption is that some confused and naïve ideas about set theory and real numbers are adequate for most people. On this same assumption, naïve set theory is taught to school children with all sorts of wrong definitions, and that confusion persists with most people for the rest of their lives. One completely fails to see why students should be taught limits in a naïve and confused way when the objective of limits is purportedly to eliminate naivette and confusion! Apparently, this is intended to teach some sort of ritual obeisance to the confusion implanted into the calculus by the West.”

Basically the pedagogical difficulties with formal math arise because formalism attempts to grasp infinity metaphysically. This metaphysics of infinity gets pushed into math at every level, including the natural numbers, which are to be understood thorugh Peano’s axioms, and not computer arithmetic, say. This metaphysics involves specific beliefs about eternity/infinity, useful for church rule, but very confusing for most people who have a different metaphysics and correlate numbers with empirical observations.  This metaphysics would have to go.

Zeroism makes math easy because it does away with the idea that that manifest correlation (as in relating the number 2 to two stones) must be mediated by a biased metaphysics of infinity. Briefly, zeroism is a realistic epistemology, hence easier.

Accordingly, with zeroism, one can teach natural numbers the natural way they are taught to children, by reference to physical objects. One does not have to unlearn this later. One can teach that the length of a curved line (which Descartes regarded as beyong the human mind) can be easily measured with a flexible string. And one can teach calculus without limits, as I am advocating.

Then there is the more advanced issue about differentiating discontinuous functions in a way which makes physical sense. This means that instead of teaching that physicists and engineers should be obedient to mathematicians, we teach mathematicians that their theories are but auxiliary physical theories.

Teaching of math become easy because reference to the empirical is permitted (which is what formalism eliminated, even from the Elements), and the focus is on practical value, not an illusory “perfection”  based on the erroneousbelief that this “perfection” is possible through metaphysics. (Proponents of sunyavada vigorously argued about the errors of such idealism.)

Then the pedagogy of math must change focus to calculation (which is of practical value) as opposed to theorem-proving.  From this perspective, theorem-proving only indoctrinates people into a particular method of reasoning  (deriving from Naiyayikas, whose syllogisms are known as “Aristotelian logic” in the West).   I call this indoctrination, because India also had other methods of reasoning, and because the underlying logics were different, Naiyayikas and Buddhists could hence not agree despite a thousand years of debate.

At the same time Zeroism cannot be challenged on the grounds of lack of rigor. It is an advantage that it does not agree with the religiously-biased Western metaphysics which is sought to be made into a universal norm. Zeroism also does not suffer from the same problem of religious bias. 

 

 

 

 

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