Someone told me that each equation I included in the book would halve the sales. I therefore resolved not to have any equations at all.
— Stephen Hawking, in his book A Brief History of Time
Whoever gave this advice to Mr. Hawking is probably partly responsible for the success, financial and otherwise, of his book. But as Paul K. Feyerabend said: "All methodologies have their limitations and the only ‘rule’ that survives is ‘anything goes’."
Open access online communities interested in high quality and productivity can benefit from a high degree of technical formality, including large amounts of mathematics.
When confronted with an economic problem, first translate into mathematics, then solve the problem, then translate back into English and burn the mathematics.
— Alfred Marshall
Another valuable advice in and of itself. But expressing technical issues in a natural language can be a mixed blessing.
Everyone can understand natural language, or so they think. Often any understanding solely arrived at by natural language is vague and can lead people to mistakenly believe that they understand the issue in question sufficiently.
A little bit of knowledge is a dangerous thing. It can convince you that an argument this idiotic and this sloppy is actually profound. It can convince you to publicly make a raging jackass out of yourself, by rambling on and on, based on a stupid misunderstanding of a simplified, informal, intuitive description of something complex.
— Mark Chu-Carroll, The Danger When You Don’t Know What You Don’t Know
The introduction of formal language and a focus on hard problems filters out a large subgroup of people who might otherwise believe that they are eligible for membership, able contribute something useful or that the community is interested in their critique.
More generally, many of the objects demonstrated to be impossible in the previous posts in this series can appear possible as long as there is enough vagueness. For instance, one can certainly imagine an omnipotent being provided that there is enough vagueness in the concept of what “omnipotence” means; but if one tries to nail this concept down precisely, one gets hit by the omnipotence paradox. Similarly, one can imagine a foolproof strategy for beating the stock market (or some other zero sum game), as long as the strategy is vague enough that one cannot analyse what happens when that strategy ends up being used against itself. Or, one can imagine the possibility of time travel as long as it is left vague what would happen if one tried to trigger the grandfather paradox. And so forth. The “self-defeating” aspect of these impossibility results relies heavily on precision and definiteness, which is why they can seem so strange from the perspective of vague intuition.
Terence Tao thinks that real insight only kicks in "once one tries to clear away the fog of vagueness and nail down all the definitions and mathematical statements precisely." This is another advantage of being technical as it introduces a high degree of focus by tabooing colloquial language and thereby reducing ambiguity.
To be technical means to be more formal. Formalizing problems is itself a problem that needs to be taken seriously as it is part of the eventual solution of each problem. Formalizing a problem is increasing the productivity of people who work on that problem by being more specific about what exactly it is that needs to be solved and by serving as a measure of progress.
Measure of Progress
Everyone involved has to be sure what is being asked for and what progress has been made so far. But to measure progress one needs to know what constitutes progress.
Progress ultimately means to approach a solution. But to approach a solution, first and foremost, one needs to formalize the problem, which in turn means to be highly specific about what would constitute a solution. And only by being strictly technical and by constantly trying to reduce any vagueness one can effectively approach the formalization of a problem and its eventual solution.