Some interesting reflections by Jason Smith.
Instead of talking about math and "mathiness," perhaps we should speak about formalization of informal arguments and conceptual models in the interest of rigor.
Humans think and communicate using ordinary language, which is product of evolution and consists of many alleys and byways as well as highways. As Ludwig Wittgenstein sought to show, ordinary language can be something of a minefield. Meaning is context-dependent. Failure to appreciate the use of signs in context leads to misunderstanding signs as symbols.
Wittgenstein explored rigorous expression before turning to ordinary language. The Tractatus was a response to Russell and Whitehead's Principia and also to Frege. Russell was so impressed he invited Wittgenstein to Cambridge.
The Tractatus as an exposition of the logic of description can be seen as a work not only about logic but also the philosophy of science. Wittgenstein was an engineer by training. His Tractatus was developed out of the introduction to Hertz's Principles of Mechanics.
Wittgenstein's work turned the focus of Anglo-American philosophy in the direction of analysis. There are arguably three periods in the history of Western philosophy. Ancient and medieval philosophy were chiefly absorbed in the question, What is there? In founding modern philosophy, Descartes pointed out that prior to answering this question, it is necessary to answer the question, What can we know about what is? Wittgenstein shifted the focus from thought to language, asking the question, What can we say meaningfully?
Formalization of language is an attempt to increase rigor in expression. However, it is not the only way, since logic is broader than formal logic and mathematics. Clear and precise expression is a challenge because the ordinary language that grounds our thinking process contains a lot of noise, and it is a temptation to confuse some of that noise with a signal.
The history of Western thought is generally traced to Socrates as his student Plato set forth the dialectical method in the Dialogues. It is clear that the method that Socrates employed was intended to clarify expression by introducing rigor through debate. This was the beginning of the scientific method, as it were. If one insisted instead in talking about gods, they were sent to the temple.
Science is usually thought as a being chiefly empirical, so Aristotle is credited with founding the scientific approach. But prior to observation is expression, and expression involves the use of signs as symbols. Getting clear on how language works is a prerequisite to gaining knowledge as the confluence of experience and understanding. Aristotle agreed. His works begin with the logical investigations, Prior and Posterior Analytics. Aristotle's logic can be viewed as a proto-formalization of logic implicit in the Dialogues.
Tradition has it that the lintel of Plato's Academy was inscribed with, "Let no one ignorant of geometry enter." The term "geometry" may have meant "mathematically literate" in context. Aristotle wrote in the Analytics that one should not talk "geometry" with those who are not geometrical, that is, not so inclined. This could be extended to mean that one should not attempt to be rigorous with those who are non-rigorous and who cannot deal with abstraction.
Descartes initiated the second great step in formalization in his Discourse on Method, the full title of which is Discourse on the Method of Rightly Conducting one’s Reason and Seeking Truth in the Sciences. "Clear and distinct ideas" are fundamental to correct method.
Descartes was a mathematician as well as a philosopher. Descartes developed analytic geometry and Leibniz initially developed calculus, although it was Newton that usually credited. Formalization begins to incorporate mathematics, and mathematics is developed to support greater formalization.
Since then, the overwhelming view is that formalization in science is chiefly mathematical. This quite obviously followed since science is about measurement of quantity and changes in quantity. But basic logic still applies. You can't add apples and oranges, and the meaning of terms cannot shift within the same context.
Occam's razor also applies: Let expression be as simple as possible for the task. This is where over-formalization and "mathiness" can come in. Is the exposition as simple as it could be to get the job done? Or is it being complicated as a matter of jargon that protects the trade?
I don't think any of this is controversial. Say what you mean as simply as possible and as rigorously as need be, and don't get pedantic. To the degree that a subject is amenable to quantification, the preferred method is to use math appropriately. Is there any argument about this?
Alfred Marshall advised economists to use math intelligently to clarify their thought and make their point.
I had a growing feeling in the later years of my work at the subject that a good mathematical theorem dealing with economic hypothesis was very well unlikely to be good economics: and I went more and more on the rules - (1) Use mathematics as shorthand language, rather than as an engine of inquiry. (2) Keep to them till you have done. (3) Translate into English. (4) Then illustrate by examples that are important in real life (5) Burn the mathematics. (6) If you can’t succeed in 4, burn 3. This last I do often. — Letter to A.L. Bowley, 27 February 1906, cited in: David L. Sills, Robert King Merton, Social Science Quotations: Who Said What, When, and Where Transaction Publishers, 2000. p. 151.
Keynes said of Marshall:
The study of economics does not seem to require any specialized gifts of an unusually high order. Is it not, intellectually regarded, a very easy subject compared with the higher branches of philosophy and pure science? Yet good, or even competent, economists are the rarest of birds. An easy subject, at which very few excel! The paradox finds its explanation, perhaps, in that the master-economist must possess a rare combination of gifts. He must reach a high standard in several different directions and must combine talents not often found together. He must be mathematician, historian, statesman, philosopher – in some degree. He must understand symbols and speak in words. He must contemplate the particular in terms of the general, and touch abstract and concrete in the same flight of thought. He must study the present in the light of the past for the purposes of the future. No part of man's nature or his institutions must lie entirely outside his regard. He must be purposeful and disinterested in a simultaneous mood; as aloof and incorruptible as an artist, yet sometimes as near the earth as a politician. Much, but not all, of this many-sidedness Marshall possessed. But chiefly his mixed training and divided nature furnished him with the most essential and fundamental of the economist's necessary gifts – he was conspicuously historian and mathematician, a dealer in the particular and the general, the temporal and the eternal, at the same time. — John Maynard Keynes, Essays In Biography (1933), p. 170.Keynes thought of economists as wearing many hats. This accords with the ancient conception of the philosopher as lover of wisdom. A philosopher studies the whole, regarding nothing as alien. Robert Heilbroner called the great economists "worldly philosophers." They are not only social and political scientists but also social and political philosophers.
I would say that the placing math on a pedestal in economics may tempt some economists to forget that that they are doing something other than natural science, which the conception of "market forces" is apt to convey if not designed to do so. Math is not a end in itself other than for those doing pure mathematics. Those who use math for other than pure mathematics are doing applied math. What they are doing may involve much more than math and even more than is quantitive, since human life is qualitative as well.
There is also pure science and applied science. Theoretician are dong pure science, while engineers are going applied science or technology. Engineering uses science as theoretical knowledge to make stuff. So it is an art. Some theoretical physicists are exploring the boundary of science and philosophy.
Economics also has theoretical and applied aspects. Theoretical economists explore model space. Applied economists attempt to make economics useful in explaining the world. Some applied economists also use economic to affect the affairs, e.g., through policy formulation.
Conflating pure and applied results in confusion. What holds in a model does not necessarily hold in reality. Exploring model space is interesting and important, but it is not the whole of the field.
Moreover, and most importantly, theoreticians may not be well qualified in areas of application, such as policy formulation. A noble prize in theory doesn't necessarily quality a person as a policy adviser.
It is also useful to remember that Ronald Coase received The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel for "the Coase theorem," which is not expressed mathematically.
There is also a place for people like John Kenneth Galbraith that translated economics and politics economy into a level of expression that non-economists could understand and use in approaching policy.
Generally speaking, math cannot be used to sell an idea politically without appealing to argument from authority since most of the audience won't understand the math. Voters need to understand the rationale for policy and have criteria for evaluating it. Otherwise it is a matter of proliferating post hoc ergo propter hoc (before therefore because) fallacies.
There's a difference between buying a piece of technology that depends on complicated math and intricate science that very few have the capacity to understand and buying into to policy prescriptions based on "experts."
In summary, there is a need for rigor in expression. Math is one tool in the toolbox for achieving rigor. But just as tools can be applied to uses for which they are not fitted, so too can logic and math.
When this is unintentional, it is called ignorance. When it is intentional, it is called sophistry. So while rigor is a plus, user beware. Rigor is a only prerequisite — a necessary condition but not a sufficient one.