What makes a discipline ‘mathematical’?
January 19, 2015 4 Comments
While walking to work on Friday, I was catching up on one of my favorite podcasts: The History of Philosophy without any Gaps. To celebrate the podcast’s 200th episode, Peter Adamson was interviewing Jill Kraye and John Marenbon on medieval philosophy. The podcasts was largely concerned with where we should define the temporal boundaries of medieval philosophy, especially on the side that bleeds into the Renaissance. A non-trivial, although rather esoteric question — even compared to some of the obscure things I go into on this blog, and almost definitely offtopic for TheEGG — but it is not what motivated me to open today’s post with this anecdote. Instead, I was caught by Jill Kraye’s passing remark:
I’ve never heard of the Merton school before — which a quick search revealed to be also known as the Oxford Calculators; named after Richard Swinehead‘s Book of Calculations — but it seems that they introduced much more sophisticated mathematical reasoning into the secundum imaginationem — philosophical thought experiments or intuition pumps — that were in vogue among their contemporaries. They even beat Galileo to fundamental insights that we usually attribute to him, like the mean speed theorem. Unfortunately, I wasn’t able to find sources on the connection to medicine, although Peter Adamson and Jill Kraye have pointed me to a couple of books.
Do you have pointers, dear reader?
But this serendipitous encounter, did prompt an interesting lunchtime discussion with Arturo Araujo, Jill Gallaher, and David Basanta. I asked them what they thought the earliest work in mathematical medicine was, but as my interlocutors offered suggestion, I kept moving the goalposts and the conversation quickly metamorphosed from history to philosophy. The question became: What makes a discipline ‘mathematical’?
My views on this question are confused, and I don’t think it has a simple answer. So for this post I am going to try something new, and present my musings as a dialogue between three characters. Image a nice sunny day near your local mathematical institute, meandering through the park are the essentialist siblings Alice and Bob Ee, and the relativist Remi. They are discussing what makes a discipline mathematical in hopes of having a reasonable definition from which to find the genesis of mathematical medicine.
Alice Ee: Methodology is where we should start, I can offer a sufficient condition there. If the discipline follows the definition-theorem-proof style while discussion abstract objects or structures then it is mathematical.
Bob Ee: This seems to be a very strong condition, is there anything other than mathematics that satisfies it? The vast majority of results in mathematical oncology — or mathematical medicine more generally — do not follow the definition-theorem-proof style, so from this perspective if we asked when mathematical medicine started, the answer would be “not yet”.
AE: It is very restrictive that is why it is a sufficient condition and not necessary; but it isn’t simply equivalent to mathematics. For example, consider mathematical physics: it isn’t mathematics, but the thing that distinguishes it from theoretical physics is the more rigorous definition-theorem-proof engagement with its material.
Remi: I’m not sure if it is even sufficient, at least if you want to reflect the word ‘mathematical’ as it is commonly used. Consider theoretical computer science, it isn’t called ‘mathematical computer science’, but it proceeds almost exclusively by definition-theorem-proof, maybe with a few more widely believed but not yet proven conjectures than is typical in pure math.
BE: Well, let’s take another lesson from theoretical computer science and look at the other direction: some lower bounds or necessary conditions. If a field is mathematical then it quantifies properties of interest.
R: This seems as loose as Alice’s condition was restrictive. Almost any science, experimental or theoretical, will classify as mathematical under these definitions. If the ancient Greeks said that health is maintained through the balance of the four humors then were they quantifying at that point simply by saying four? Further, I don’t think it is reasonable to call mathematical a field that thinks they’ve ‘invented’ (and tried to name after themselves), in 1994, a method that has actually been in introductory calculus classes for over a century. There needs to be some continuity with mathematics, at least at the level of basic knowledge.
We should heed the role of ‘mathematical’ as a qualifier in everyday use. As such, I would suggest that mathematical is a relative word, that just means ‘more continuous’ with mathematics in any given domain. Whenever, we have some science (or other similar pursuit), it tends to break down into a theoretical and experimental branch. Part of that theoretical branch then becomes a mathematical branch, distinguished from its parent by a higher resemblance to the practice of mathematics.
AE: How does this account for domains where there is not a branch commonly known as ‘theoretical’. Consider the case of psychology, for instance, where we have experimental psychology and mathematical psychology, but people seldom talk of a ‘theoretical psychology’. Something similar happens in economics, except where the experimental branch is known as ‘behavioral’. On the other hand, you also have domains, like computer science, where a clearly mathematical discipline — as you mentioned in your response — is simply known as ‘theoretical’.
BE: Also, viewing ‘mathematical’ as a purely relative property renders moot our original intent of finding out when mathematical medicine originated. At any point in history, we could view some branch as ‘more continuous’ with mathematics. It seems obvious that there has to be at least some basic threshold of ‘mathiness’ that a discipline has to cross in order to become mathematical, but then you are back at the essentilist position that Alice and I were advocating. In particular, on the importance of a lower bound on ‘mathematical’.
So, dear reader, can you help Alice, Bob, and Remi to resolve their debate? What makes a field mathematical, especially from a historical perspective? Is mathematical medicine, or mathematical oncology more specifically, a proper use of the qualifier ‘mathematical’?
Sylla, Edith D. (2011). Oxford Calculators Encyclopedia of Medieval Philosophy, 903-908 DOI: 10.1007/SpringerReference_187789