Vague theories: a defence

Most scientists believe that vagueness is bad in a scientific theory. In fact, if it’s vague, it cannot really be a theory at all. At the hard end of the social sciences, economists insist that theories should be formalized in the crisp language of mathematics. Even in softer-edged disciplines, which often get accused of vagueness and waffle, very few people defend vagueness explicitly. Instead, they use arguments about the meaning-ladenness of human action, or the holistic interconnectedness of the social world, to argue that social science should aim for understanding rather than explanation. But understanding is still meant to be clear.

Actually, mathematics is not enough to avoid vagueness, as Paul Romer has pointed out in a controversial article on “mathiness” in the theory of economic growth. Mathiness is a word coined on the analogy of the Daily Show’s “truthiness”: when something “feels true in your gut” rather than actually being true. Mathy economic theories have a valid mathematical argument in their model, but the connections between the maths and reality are vague or ill-defined:

McGrattan and Prescott (2010) establish loose links between a word with no meaning and new mathematical results.

Boldrin and Levine (2008) make broad verbal claims that are disconnected from any formal analysis.

This suggests a puzzle. If the words and verbal claims in a theory must be connected via tight links to a formal analysis and mathematical results, why aren’t they part of the mathematics themselves? If economists build models because ordinary language is imprecise, then why is any ordinary language allowed?

Having sown these seeds of doubt, I will present three examples, and a bonus fourth, of theories that are vague but useful. I am not arguing that vagueness is a theoretical virtue. Instead, it is a disadvantage that must be traded off against competing values, such as relevance. Also, perhaps, the tradeoff approaches infinity as vagueness goes to zero: a perfectly defined theory would be a theorem in pure mathematics, with no relationship to the world; insofar as it connects to the world, it has to be a bit vague.

Theory 1: watch out for that truck!

 

This is a very simple theory. Its usefulness, if you are about to step into the road, should be obvious. But it is vague, because a key term is not precisely defined. What exactly is a truck? When does a van become a truck? Even people from Texas aren’t sure.

Van? Truck? Vruck?

Theory 2: Western societies are threatened by Islamic terrorism

 

This theory could be true or false. It is perfectly legitimate to argue that terrorism’s threat to the West is manufactured, or hyped, by the warmongers in power. But the theory is at least worth arguing over. After the Twin Towers attack, Western societies needed to decide what to do, which involved making judgments about this theory. Yet the theory is clearly vague. There is no clear uncontested definition of terrorism, or of “Islamic” terror. (Is it Islamic terror when a small-time drug dealer or mentally unstable loser decides to ram a van into some people, after posting some Islamic-tinged rants on Facebook?) Or of “threat”. Or of “societies”. Or of “Western”. A lot of the debate about this theory consists in making terms more precise. But you have to start somewhere.

Theory 3: fluxions

 

Modern science, with its mathematical basis, was kickstarted by the enormous prestige of Isaac Newton’s theory of gravity. Under that theory lay his advanced mathematics, in particular his development of the mathematical derivative – he famously fought Leibniz over priority on the topic.

The derivative can be loosely defined as the slope of a function. Awkwardly, its precise definition was not clarified until 150 years after Newton wrote. In particular, the idea of a slope at a point involves the idea of a “limit”, which was finally explained by Bolzano and Weierstrass in the 19th century.

For the first half of its history, Newton’s theory, the key theory underpinning the development of modern science, was vague, meaning not that it had an uncertain real world referent à  la “truck”, but that we could not even explain clearly what a key term meant. Newton recognized this explicitly and was uncomfortable enough to write a book about it.

Bonus Theory 4: the truth shall set you free

 

This is a saying attributed to Jesus. The actual quote, from John 8:32 in the King James Version, is: “And ye shall know the truth, and the truth shall make you free.”

The theory is frankly obscure. But it would be hard to say that it has not been influential. It may be that with respect to it, we are in the position of eighteenth-century mathematicians with respect to the calculus. It is a theory that seems important, but we cannot explain why.

I'll finish by quoting Edmund Burke's catchily-named A Philosophical Enquiry into the Origin of Our Ideas of the Sublime and Beautiful. The argument in this remark seems superficial, but I think a deeper point is lurking there.

But let it be considered that hardly anything can strike the mind with its greatness, which does not make some sort of approach towards infinity; which nothing can do whilst we are able to perceive its bounds; but to see an object distinctly, and to perceive its bounds, is one and the same thing. A clear idea is therefore another name for a little idea.