École Normale Supérieure, Paris Sciences et Lettes University
Challenging the relevance of mathematical counterfactuals in biology
Recently, the issue of whether there are mathematical explanations outside pure mathematics has received substantial attention. There are some accounts of mathematical explanatoriness outside mathematics. For instance, Baker (2005) argues that distinctively mathematical explanations are characterized by the indispensability of mathematical objects. Lyon and Colyvan (2008) appeal to the notion of inference to the best explanation and consider that mathematical explanatoriness relies on whether mathematics constitute the best explanation available.
Lange (2013) argues that some explanations are distinctively mathematical because mathematical necessity, and not causes, has most of the explanatory burden (hence, mathematical explanations are of a modal kind). Baron, Colyvan and Ripley (2017) bring to this debate recent ideas from counterfactual reasoning and difference making. They suggest that there is a mathematical explanation outside mathematics when, should a given mathematical fact be otherwise, the fact outside mathematics would be different as well.
I will challenge the idea that counterfactual changes in pure mathematics ought to translate to changes in biology. I will discuss the case of the hexagonal shape of honeycomb cells and argue that it is not feasible to translate geometrical facts into biological shapes even from a neo-Aristotelian perspective (which acknowledges a relationship between abstract mathematical facts and empirical facts). According to my arguments, if pure mathematical fact about geometry were different, biological shapes would not be necessarily different. One can study the potential relevance of mathematical counterfactuals, what a mathematical counterfactual is a counterfactual about, if one explores the historical origins of the area of pure mathematics in question (for Euclidian
geometry: rope stretchers in Egypt and Ancient Greece; for number
theory: pebble accountants in Ancient Greece). If pure mathematical facts about geometry do not make a difference on biological shapes, then, a fortiriori, the rest of accounts do not hold either:
mathematical facts are not indispensable in the explanation of the biological shape, they do not constitute the best available explanation, and do not determine biological shapes by mathematical necessity.
However, I will also argue that, because this disconnect is not complete, mathematics does make an epistemic difference acting as heuristics in biology. To this end, I will discuss another case of “hexagons” in biology hitherto untreated by the philosophical literature on mathematical explanations: the hexagonal periodicity of grid cell activity. Because grid cells were discovered in 2004 but their hexagonal periodicity was observed only after 2005, one can isolate the impact of such an epistemic difference in the explanations of grid cell activity.
To do this, I will reconstruct the “mathematical” explanation featuring the intuitions of biologists/neuroscientists.
References:
-Baker, A. (2005). Are there Genuine Mathematical Explanations of Physical Phenomena?. Mind 114, 223–238.
-Baron, S., Colyvan, M., & Ripley, D. (2017). How mathematics can make a difference. Philosophers’ imprint, 1-19.
-Lange, M. (2013). What makes a scientific explanation distinctively mathematical?. The British Journal for the Philosophy of Science 64(3), 485-511.
-Lyon, A. & Colyvan, M. (2008). The Explanatory Power of Phase Spaces. Philosophia Mathematica 16, 1–17.