Mathematically semiliterate 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.