… Of Great Mathematicians….

Albert Einstein was not a mathematician. In his scientific biography, “Subtle is the Lord”, his friend and colleague Abraham Pais identifies, especially in his Appendix, over thirty mathematicians with whom Albert collaborated: in his quest to interpret Physical Reality. As you know, he never achieved this goal. It was not his fault. He simply lacked the appropriate mathematical framework.

In the Technology section of the September 2018 issue of Electronics World, brief interview comments appeared under the heading of :“University of Southampton develops electronics technology that can learn “biologically””. It was the word “biologically” which caught my attention and encouraged me to look up the original paper:“ Seamlessly fused digital-analogue reconfigurable computing using memristors”. It relates to their work, funded by the Royal Society, EPSRC and EU PF/7 Ramp Project, on Memristors for mixed signal technology i.e. the analogue-discrete technology boundary.

Therein their phrase “essentially multiplicative interactions”[p4] chimed with a phrase of the mathematician, John Conway who stated that “there is something about multiplication we do not understand”. Conway's context related to the Complex Number System. In signal processing the CNS is only required for Doppler Processing wherein the concept of quadrature needs retaining. On further examination of these Southampton researchers body of work, another issue, the Linearity verses Non-Linearity aspect in Mathematics emerged. Clearly these Southampton researchers are also suffering from a lack of an appropriate mathematical framework. But do they appreciate and do they acknowledge this simple fact: that all Western Science and Technology is build on the mathematical concepts of Addition, Subtraction, Multiplication, Division and extracting the Roots of Unity?

Consider that numerous great mathematicians have all sought the removal of the complex number system. Cauchy proposed to carry out all and more that complex numbers do by operating with congruences whereas Kronecker sought to use the concept of a placeholder. What did Boole propose?

When problems involving these manipulations become hard to solve, we turn to Transforms to aid with obtaining a suitable mathematical format. To clarify the concept of “transforms” and their relationship with mathematics, it is worth pointing out that the fundamental transform is Multiplication. True Multiplication is Repeated Addition. But it is normally achieved via the multiplication tables we are all taught in school. “Russian Multiplication” is an alternative systematic approach which avoids the requirement of memorising multiplication tables (other than the first 2x table) by reducing multiplication to repeated doublings and halvings. In the binary number system, even the need of memorisation of the first multiplication table is avoided; as division by two is easily accomplished in this system.

Previously, I have brought readers attention to a Nature paper from 2013:”Synthetic analog computation in living cells” by R. Daniel, J.R Runins, R. Sarpeshkar and T. Lu. All from MIT.

In a related 2014 paper “Threshold Logic Computing: Memristors – CMOS Circuits for FFT and Vedic Multiplication.” by Alex Pappachen James, Dinesh S. Kumar, and Arun Ajayan, these authors tackle similarly cast mathematical modelling issues.

But are all these researchers merely clutching at straws? I believe so.

The foundational model is the Number System. Next comes the applied Time Domain model. In that order. Then the transform itself. In Fourier land the time domain is where we devise and apply the models of Physical Reality. Essentially sets of simultaneous equation models and differential equation models. Are they the same? In the Transfomed Space we are merely dealing with abstract parameter manipulation, beginning with periodicity, at the cost of sacrificing the time dimension. As it has been stated “we are merely pushing things around to see what pops up”. In essence we are building models-on-top-of-models. Alas, not without a definite purpose. Fourier's achievement was to advance solutions to Newton's boundary-value problems concerning the fitting of solutions of differential equations to proscribed initial conditions: The central problem of Mathematical Physics. As I have stated elsewhere physical laws must be independent of coordinate systems used in describing them, if they are to be valid.

But what of Biology? How does this Western twin approach of Analytic Geometry (c1637) and the Calculus (c1666) translate to BioPhysics, Bio-electronics and Synthetic Biology? Simple answer: it doesn't. What has nature stll to teach us? Could it be that extra mathematical process(s) need inventing to augment division and the extraction of roots? So, as with Einstein, I believe that these researchers are all in good company.

Barry McKeown
October 2018