Mathematically semiliterate scientists?

Galileo famously wrote, “The laws of Nature are written in the language of mathematics.” Somewhat less known is his admission “If I were again beginning my studies, I would follow the advice of Plato and start with mathematics.”

This does not seem to be the view of Edward O. Wilson —the well known Harvard biologist (myrmecologist) and author of many books including those on consilience and sociobiology—who claimed in an article earlier last year:

For many young people who aspire to be scientists …I have a professional secret to share: Many of the most successful scientists in the world today are mathematically no more than semiliterate

and

exceptional mathematical fluency is required in only a few disciplines …Far more important throughout the rest of science is the ability to form concepts, during which the researcher conjures images and processes by intuition. …Pioneers in science only rarely make discoveries by extracting ideas from pure mathematics.

The mathematician Edward Frenkel responded, and there is no reason to question his observation that “(Wilson) does not understand what mathematics is and how it is used in science”.

Indeed, EO Wilson admits that he had difficulties following calculus, a first year course usually taught to all science and engineering students, the understanding of which hardly requires “exceptional mathematical fluency” although it is essential to, for instance, mathematical statistics, with applications also in biology. 

The reason for teaching future scientists and engineers mathematics — besides introducing them to mathematical concepts, facts and techniques (of calculus etc that, anyhow, are gradually being left to computers) — is to make them understand what mathematics, mathematical models, can and cannot do. Wilson speaks of his cooperation with a “mathematical theorist” who could not have solved his biological problem without input from him, the biologist, as if this was not something rather obvious. Less obvious is the fact that the scientist or engineer consulting a professional mathematician has to know what to ask, how to mathematically formulate his/her problem. This can be helped by a mathematician but cannot be delegated to him/her entirely.

Once I spent an hour or so with an engineer who came to me with what he thought was a mathematical problem until I convinced him that it was not so, that there were a number of mathematical formulations of his problem, and that it was a matter of engineering (rather than mathematical) expertise to decide which one of them properly modelled his problem.

As I used to tell my students, the question of what is the probability of all TattsLotto numbers drawn next week being even is a mathematical question, but not very interesting. The question of which numbers will be drawn is much more interesting but it is not a mathematical question.

A good point against Wilson’s contention that scientists can get away with being “mathematically no more than semiliterate” was provided recently when a graduate student with little math education demolished an influential book on psychology by showing that the mathematics it was based on was wrong or inappropriate. The author of the book had to admit that “she didn’t really understand the mathematics behind it” that she used to build her theory on.

Mathematics is important not only for results and techniques immediately applicable in science or engineering (which is still more than “number crunching”), but  mathematics also plays a fundamental role in our attempts to model, and hence better understand, physical reality at its most basic level. It could be seen as the solid point required by Archimedes because, “while our perception of the physical world can always be distorted, our perception of the mathematical truths can’t be” as Frenkel puts it. Einstein expressed the exceptional position of mathematics thus: “As far as the laws of mathematics refer to (physical) reality, they are not certain; as far as they are certain, they do not refer to (physical) reality”. It is this certainty combined with its  "unreasonable effectiveness", as famously expressed by Eugene Wigner, that makes mathematics indispensable for science and scientists.

The “ability to form concepts, during which the researcher conjures images and processes by intuition” that Wilson favours, is often enhanced by the scientist’s ability to “think mathematically”, to know to what extent can scientific problems (and perhaps also philosophical ones) be illuminated by insights coming from mathematics. Not just to “extract ideas from pure mathematics” as he thinks.

For instance, concepts like infinity, boundedness, dimension, curvature, singularity, etc, can lead the “conjuror’s” intuition towards a scientific theory only if he/she properly understands their mathematical meaning. There are many philosophical problems associated with quantum physics, its relation to a “common sense” understanding of our observations, or even to classical, Newtonian physics, but the mathematics behind it is clear, problem-free, indeed Archimedes’ solid point. Similarly for still speculative models, like superstring theory. 

Even for a non-specialist, familiarity with some mathematical concepts and relations (not necessarily Wilson’s “exceptional mathematical fluency”) helps to better understand popular expositions on those abstract matters. This is true of anybody wanting to have a science-informed worldview, not just professional scientists. Of course, on the level of a critical understanding of contemporary theories of the nature of physical reality “exceptional mathematical fluency” is an absolute necessity. And also conversely, without a vey good background in mathematical statistics one could not debunk one of the strongest advocates of Intelligent Design Wiliam A. Dembski, who bases his arguments on advanced statistics.

Scientists’ insights often work through metaphors. For instance, the software-hardware metaphor for the mind-brain relation (whatever it’s worth) provides some insight into the not yet satisfactorily understood relation, provided one is familiar with the software-hardware relation.

Mathematics, because of its seemingly purely mental constructions, offers an abundant source of insights for a scientist familiar with them, even when the insight is only a metaphor helping to better understand a given situation, without necessarily leading to a full-fledged theory.

As a trivial example, a sequence whose terms are getting closer and closer to a limit, without necessarily reaching it might be seen as a metaphor for physical theories coming closer and closer to “the truth” about physical reality, without necessarily reaching it. Conversely, the concept of observer in relativity theory does not come from any philosophy, it is just a metaphor for the coordinate system in which the physical laws are mathematically expressed.

I even believe that mathematical literacy (at whatever level), which carries with it a sense for logical rigour and formal coherence, helps one to better understand and express the rational framework of worldviews, especially one’s own, be they of a theist, atheist or whatever orientation.