At my IIT (BHU) lecture (see also previous post), I emphasized Bertrand Russell’s remark that there is neither meaning nor truth in (formal) mathematics. Hence, any nonsense proposition one desires (such as “All rabbits have two horns”) can be proved as a formal mathematical theorem from appropriate postulates: Russell’s sole criterion being that the postulates should be “amusing”.
To drive the point home, I pointed out how, long ago, when I still believed in formal math, I used to teach a course (A) on Real Analysis while also teaching a more advanced course (B) on Advanced Functional Analysis, in the math department of Pune University. In the elementary course (A) I taught
Theorem: A differentiable function must be continuous. (Therefore, a discontinuous function cannot be differentiated.).
In the more advanced course (B) I taught
Theorem: Any (Lebesgue) integrable function can be differentiated infinitely often. (Therefore, a function with simple discontinuities can be differentiated infinitely often.)
I have made exactly this point earlier in this blog.
“Now, for several years I taught real analysis to students and mathematically proved in class that a discontinuous function cannot be differentiated. I also taught advanced functional analysis (and topological vector spaces and the Schwartz theory according to which every Lebesgue integrable function can be differentiated). In the advanced class, I mathematically proved the exact opposite that a function with a simple discontinuity can be differentiated infinitely often (and the first derivative is the Dirac δ).”
The question is which definition of the derivative should one use for the differential equations of physics? As pointed out in Cultural Foundations of Mathematics (or see this paper) the issue can only be decided empirically, unless the aim, like that of Stephen Hawking and G. F. R. Ellis, is to spread Christian superstitions about creation using bad mathematics.
Superstitions go naturally with ignorance. One such ignorant professor from the IIT mathematics department was present during my lecture. His knowledge was limited to the first of the theorems above, and he ignorantly believed that it was some kind of absolute truth, which everyone was obliged to believe. He objected to my claim that a discontinuous function can, of course, be differentiated, and walked out to show his contempt of my claim.
Even the students had heard of the Dirac δ, and agreed with me. The next day during the workshop, I explained that I had engaged with this question since my PhD thesis. But the professor remained absent, though his ignorance was exposed before the students. He is welcome to respond by email; I will post it publicly since it is sure to further expose his ignorance.
Oliver Heaviside applied first applied this to problems of electrical engineering over a century ago, and Dirac, formerly an electrical engineer, then applied the Dirac δ to physics. It remains very useful because it is the Fourier transform of white noise (flat spectrum or the unit function), and used even in the formal mathematical theory of Brownian motion.
Earlier in the lecture, the same professor, contested my claim that probability was invented in ancient India, and taken from India in the 16th c., where credit for it was later falsely given to people like Pascal and Poisson. (more…)