Mathemorphism

Quantifying stuff is a bedrock of the natural sciences. By saying this, I’m intentionally attempting to juxtapose some precise measurement (“quantifying”) of something vague (“stuff”) while also communicating its foundational importance via a metaphor (“bedrock”). I have to work hard to notice that I’m doing so. Perhaps I am so steeped in my practice as a scientist, that I’m well-conditioned to think the way I do, perhaps unreflectively.

 

As a computational scientist and educator, using the language of mathematics as a formalism to discuss scientific concepts and theories is a key part of my job – especially when I’m teaching physical chemistry. As a chemist, attempting to connect visible macroscopic properties to nanoscopic stuff we cannot directly observe, I equally use the language of metaphor with a good dose of pictorial representation. Essentially, I’m in the business of helping students build abstract conceptual models to help them think chemically – as a chemist would, until it (hopefully) becomes second nature.

 

From one viewpoint, mathematics has been extraordinarily successful in helping scientists think about and represent what goes on in the natural world. You’ve perhaps heard the phrase “the unreasonable effectiveness of mathematics”. The very notion seems to defy explanation. But its power is apparent – in that we scientists often eventually make an appeal to mathematics when we go down a rabbit hole trying to explain something and eventually reach the end of our personal knowledge. If we can represent something mathematically, we think we know something fundamental about it, even if this turns out to be illusory.

 

I’m enjoying reading John Ziman’s Real Science; I would classify it as a practical philosophy of science. It tackles the subject matter broadly and thoughtfully and I’ve found it to clarify ideas that were fuzzy in my own mind about the nature of science and its practice. Ziman perceptively reminded me that much of what I do is trying to set up a tractable mathematical model to represent some complex system, and that by doing so I am likely making over-simplified assumptions. He would also say that this isn’t wrong (nor is this about right and wrong) but reminds me to be thoughtful about the limitations and assumptions inherent in the process. Here’s a quote from him that I found very helpful when he brings in syntax and semantics – ideas I have been wrestling with.

 

“On the one hand, mathematical formalisms have the advantage that they are semantically wide open – that is, the terms that occur in them can be designated to mean whatever we want them to represent theoretically in each case. On the other hand, mathematical formalisms are syntactically very restricted – the relationships symbolized by mathematical operations on these terms are highly specialized and are very often meaningless.”

 

This discussion on the use of mathematics and science is embedded in Chapter 6 (“Universalism and Unification”) of Ziman’s book. It’s a rich chapter that has gotten my mind buzzing as he discusses the role of classification in science – it is a bedrock of what we do as scientists and what we perceive as thinking scientifically. Ziman also emphasizes the widespread use of schemas and the notion of a system that’s inherent to the scientific endeavor. We “schematize the seamless web of the real world by representing it as a more or less closed and coherent set of relationships between potentially separable entities.” Furthermore, what differentiates the ‘soft’ sciences from their ‘hard’ cousins isn’t necessarily that they are more complex, but that classification becomes extremely challenging. Reductionism depends heavily on classification. Ziman writes that the “notion of a model defies formal definition… that a theoretical model is an abstract system used to represent a real system, both descriptively and dynamically.” I’m inclined to agree after steeping myself in Rosen’s work. Ziman also describes computational modeling as building an increasingly important bridge between the twin poles of theory and experiment.

 

Ziman also reminded me of how much metaphor is used in scientific reasoning. I’ve discussed this before in a previous blog post, but lately it has made me think about whether I should ease up on decrying anthropomorphic explanations in my chemistry classes. That’s the subject for a future post. After all, what I’m doing in my daily practice as a computational chemist is what Ziman refers to as “arithmomorphic”. Since this doesn’t always involve arithmetic, I will generalize it to “mathemorphism”, perhaps a close cousin of metamorphism. Not to be confused either with metaphorism. It’s hard not to get all tangled up just thinking about all of this.