Not So Distinctively Mathematical Explanations: Topology and Dynamical Systems

2022, Synthese [Springer]

(Jha, A., Campbell, D., Wilson, P. & Montelle, C.)

https://link.springer.com/article/10.1007/s11229-022-03697-9

This paper argues distinctively mathematical explanations, which are presupposed to be non-causal explanations, are actually causal explanations in disguise.

A Mathematical Model of Dignaga’s Hetu-cakra

2020, Journal of Indian Council of Philosophical Research [Springer]

https://doi.org/10.1007/s40961-020-00217-3

This paper provides a formulation to deconstruct styles of analogical reasoning in Indian philosophy.

Modal Modelling: Why Micro-physics Matters

(With Wilson, P. & Campbell, D.)

This paper denies the modal claim that antipodal weather patterns must exist and instead argues for an empiricist reading of these modal claims. It demonstrates this by showing why temperature is a variable that is much more complicated to mathematically model than usually presumed. One way to demonstrate the argument is to showcase the the microphysics of fluid dynamics by studying (a) temperature discontinuities at water vapour interfaces in phase transitions, and (b) the breakdown of the continuum hypothesis and Navier-strokes equation in micro-fluids with high Knudsen numbers.

A Buddhist Take on Mathematical Modelling

(Proceedings of the University of Cambridge Postgraduate Conference: Dynamical Encounters Between Buddhism and the West)

The Tale of a Medieval Scroll: Mathematical Discoveries

(Part of a University of Canterbury project on the history and mathematics of medieval astronomical scroll.

I investigate three broad ways in which mathematical models and explanations can unify disparate physical phenomena:

(1) Unification via non-causal distinctively mathematical explanations (DMEs)

As for (1), I examine modal claims concerning rich and complex systems, such as n-tuple pendulums, antipodal weather patterns, and micro-fluids with my supervisors Phillip Wilson, Douglas Campbell, and Clemency Montelle at Canterbury, New Zealand. Many non-causal explanations, such as topological explanations of complex physical phenomena, omit perturbations in such systems and such omissions lead to a more inflated robustness claim associated with such modal explanations. My research models these perturbations mathematically and investigates the effect of such perturbations on these complex systems. In several cases, the modal claims break down when the richness of the system is evaluated with mathematical counterfactuals that are physically permissible. This has interesting implications for philosophical arguments concerning modal metaphysics, the indispensability of mathematics, and the unifying capabilities of such non-causal explanations in general.

(2) Unification via analogical models

As for this, I will be working with Hasok Chang at Cambridge HPS where I will examine accounts of analogical reasoning in sciences as coherent epistemic activities. This is motivated by a pragmatist view of mathematical modeling and formal analogies in Physics that employ analogical reasoning. If mathematical models are coherent only with reference to a certain epistemic context, then general arguments that extrapolate structural claims analogously from one model to another are philosophically challenging. And I argue that we need a lot more than mere structural isomorphisms and universality class arguments that have been recently put forth to support analogical reasoning in analogue experiments.

(3) Unification with the partial structures approach

As for (3), I will be working with Roman Frigg at LSE where I will explore if the partial structures approach is a promising way to unify disparate physical phenomena. Any account of partial structures must philosophically address the parsing out of partial structures from rich and complex systems and show whether the structural claims made with regards to such structures can be held up fundamentally. Here, I advance an argument against the existence of fundamental structures given how mathematical models suppress and often conceal important micro-physical details in target systems, which might be irrelevant for a certain explanandum, but relevant for others.