which make the representation problem explicit, for both integers and real numbers.
IV
The Contemporary Relevance of the Revised History
9 Math Wars and the Epistemic Divide in Mathematics
411
European historical difficulties with Indian mathematics and present-day learning
difficulties in mathematics
Using the principle that phylogeny is ontogeny, the historical European
difficulties in understanding the algorismus and the calculus are here re- lated to difficulties that students today have in understanding elementary mathematics. Historically, both algorismus and calculus greatly enhanced the ability to calculate, but only in a way regarded as epistemologically insecure in Europe for periods extending to several centuries. Since, in fact, the formalist epistemology of mathematics is too complex to be taught at the elementary level, the same situation persists in "fast forward" mode today in the classroom.
This epistemic divide has been exacerbated by
computers which have again greatly enhanced the ability to calculate, albeit in a way regarded as epistemologically insecure. In view of the preceding considerations, it is proposed to accept mathematics-as-calculation as epis- temically secure, and to teach mathematics for its practical value, along with the related notion of number, despite Plato and assorted footnotes to him.
A Distributions, Renormalization, and Shocks
425
Difficulties with the continuum approach to the calculus and an example of how
The belief that the calculus found a final and satisfactory solution with the
formalisation of real numbers is not valid. The formalisation of real numbers only side-stepped the central problem of representation, which persists even in to the present-day formal mathematical extensions of the calculus in the Schwartz theory of distributions. The differences between the two philosophies of mathematics--(a) formalism vs (b)
śūnyavāda [empiricism +
acceptance of non-representability]--though subtle, are here demonstrated to have practical applications also to areas other than computing and math education, particularly to physics and engineering. Thus, the alternative philosophy of mathematics is here related to suggested improvements in (a) the current renormalization procedure used to tackle the problem of infinities in quantum field theory, to allow use of any polynomial Lagrangian, and (b) the theory of shock waves, to make it more accurate in real fluids like air, water etc. The suggested improvements, however, require empirical inputs to finalize the mathematical derivation. Thus, the other key idea, like that of
Śrīharşa, is to bring out the limitations of formal mathematics
also from within formal mathematics--namely, to demonstrate that formal mathematics, without empirical inputs, quickly reaches a sterile end.