Encouraging students to see value in things that can't be measured

Question

It's very tempting for a student who is overly excited about mathematics to discount intellectual work in other fields, particularly the humanities, where the nature of knowledge and knowledge formation is very different: hardly ever will someone in the humanities have the last word on a specific question.

There are a wide variety of real world problems that require a combination of technical and human perspectives to solve, and so I believe it is our responsibility as educators to cultivate a willingness towards this sort of cooperation in our technical-minded students. I've seen mathy types far too often propose technical solutions to what are ultimately social problems.

Do you have thoughts on how we can help our students in advanced math courses see the value in "softer" fields? Do you do anything for this in your classroom? Do you know of any resources I could look into for this?

Addenda:

• Perhaps a more useful perspective on this question is "how can we teach pure math in a way that best leaves students open to the value of other fields?"
• This perspective also has the advantage that it means this question isn't an exact duplicate of How can I teach my students that other disciplines are important too?
• For context, my personal interest in this question is that I have a small youtube channel where I post lecture videos in pure math, and would like to gently steer my audience in a positive direction.

Answer 1

Take this advice with a grain of salt, as I have not implemented anything like it yet.

I have been thinking a lot about the responsibility that we have, as mathematics educators, to not only give students technical tools but to help them to integrate mathematical thinking into how they view the world. Mathematics is one way that we gain and organize knowledge about the world, but it is not the only way.

One great example of how this could look is Tom Murphy's textbook "Energy and Human Ambition on a Finite Planet": https://escholarship.org/uc/energy_ambitions

I think that this book could be suitable for a "quantitative literacy" course at most Universities. The course would help students to build mathematical skills such as unit conversion, modeling with linear, polynomial, and exponential functions, setting up and solving linear, polynomial, and exponential equations, understanding rates of change, etc.

However, these skills are being developed in the context of grappling with big questions. A student might have to grapple with questions like "If we try and store one day of solar energy globally using lead-acid batteries, how much lead will be required? How does that compare with the global supply of lead? What are the implications of this for society?".

So while mathematics is a part of the toolset we need for understanding our predicament, it is clear that it is not sufficient. We need to understand history (how did we end up as reliant on fossil fuels as we are?), our philosophical values (what makes a life "good"? Are we able to live good lives with significantly less energy expenditures?), sociology (we see the need for a change: how do human beings make these kinds of transitions?), etc.

Answer 2

I think the most straightforward way of showing the value of "softer" fields is to point out that the softer fields are what mathematics is based on. I always tell my students, "step 1 of math is philosophy." If you have not philosophically determined that the entities in question obey the relevant formulas, then the mathematics is useless. Thus, the ability to use mathematics requires the softer fields, such as philosophy, to even get started.

But it even goes deeper than that. The verification of mathematics requires philosophy. How do we know that the law of non-contradiction is true? This is a philosophical question. Godel incompleteness shows that there are an infinite number of proofs which require philosophical intuition - i.e., deductive mathematics will be unable to reach them.

Philosophy also determines what sort of functions should be allowed in the descriptions of reality. Some people philosophize that reality is computable, and therefore any full description of reality should only include computable functions. Others have a more expansive view, and can include other sorts of functions in their descriptions of reality.

Even for the sciences, before you can assign numbers to anything, you have to determine what the quantifiable entities are! Thus, we have to philosophize about nature before we can even apply numbers to the question. Polanyi covers a lot of this in his book, "Science, Faith, and Society." Another book in this vein is Clouser's "Myth of Religious Neutrality," where he shows that one's starting metaphysics influences what sort of mathematics they use.

Additionally, there are questions about what classifies as a proper explanation. This is, again, a philosophical question. For information on this topic, see Keas' paper, "Systematizing the Theoretical Virtues."

Other fields do not have the certainty of mathematics, and that is often what students are searching for. However, in order to gain certainty, one must structure their thinking in such a way that makes it possible. Making sure one is doing that correctly is a philosophical notion.

Additionally, there are many things which are not quantitative, or at least not in the usual sense (usually, there are aspects of anything that can be quantified, but the core reality may be non-computational). For instance, beauty is non-quantifiable. Beauty, however, is actually more primary for living a joyful life than productivity, while quantity usually contributes more to productivity than joy.

A great quote from Niebuhr from his book, "The Irony of American History":

Yet we cannot deny the indictment that we seek a solution for practically every problem of life in quantitative terms; and are not fully aware of the limits of this approach. The constant multiplication of our high school and college enrollments has not had the effect of making us the most "intelligent" nation, whether we measure intelligence in terms of social wisdom, aesthetic discrimination, spritual serenity or any other basic human achievement. It may have mad us technically the most proficient nation, thereby proving that technical efficiency is more easily achieved in purely quantitative terms than any other value of culture....No national culture has been as assiduous as our own in trying to press the wisdom of the social and political sciences, indeed of all the humanities, into the limits of the natural sciences...the result is frequently a preoccupation with the minutiae which obscures the grand and tragic outlines of contemporary history, and offers vapid solutions for profound problems.

Answer 3

I think the way to go after this is tangentially, not directly. Encourage them into looking at fields like Bayesian predictions, where decisions are made on imperfect information, but have real consequences (e.g. betting prediction markets). Or to doing analysis of science or business problems. In particular, if you can gently put them into situations where they don't immediately succeed, but they see how semiquantitative process work, they may be incentivized to learn them.

This is not to say that there is no value in literature, art, etc. (there is huge value, they concern the human condition, of living/dying/striving). But you have to use the indirect approach. Maybe occasional career/life stories or the like. And realistically, you don't have all the time in the world and need to get the basic job done, first. Life will eventually teach these kids who live in an abstraction. So, I would set the bar low (for accomplishing your objective) and be gentle. Encourage them to do coops and clubs and the like also. Something to get life experience from.

It's still going to be hard though. Every year you get another crop of wet behind the ear kids. Heck, even on the math stuff, the next year rolls around and the incoming froshes don't know calculus again. And you taught it to them last year. That's teaching for ya. ;)

FWIW, I'm routinely flabergasted by commenters and questioners here (a very high IQ group by the way) making logical fallacies. E.g. evaluating teaching techniques or curriculum design (which are probabalistic situations, full of confounding input variables, economic factor limitations, multiple output variables to optimize, and incomplete information) based on single counter/pro examples. As if they were divided by zero math gotchas. Total wrong analytical frame of reference.

So...I do think there is something to this idea of a blind spot for math types. But fixing it...that's a hard task.

P.s. https://www.youtube.com/watch?v=Rf6BhxuxSR8 (math v engineering mindset)