Leonardo da Vinci’s ability to embrace uncertainty, ambiguity, and paradox was a critical characteristic of his genius. —J Michael Gelb
Say you want to use a mathematical metaphor, but you don’t want to be really precise. Here are some ways to do that:
- Tack a +ε onto the end of an equation.
- Use bounds (“I expect to make less than a trillion dollars over my lifetime and more than $0.”)
- Speak about a general class without specifying which member of the class you’re talking about. (The members all share some property like, being feminists, without necessarily having other properties like, being women or being angry.)
- Use fuzzy logic (the ∈ membership relation gets a percent attached to it: “I 30%-belong-to the class of feminists | vegetarians | successful people.”).
- Use a specific probability distribution like Gaussian, Cauchy, Weibull.
- Use a tempered distribution a.k.a. a Schwartz function.
Tempered distributions are my current favourite way of thinking mathematically imprecisely, thanks to this book: Theory of Distributions, a non-technical introduction.
Tempered distributions have exact upper and lower bounds but an inexact mean and variance. T.D.’s also shoot down very fast (like exp −x², the gaussian) which makes them tractable.
For example I can talk about the temperature in the room (there is not just one temperature since there are several moles of air molecules in the room), the position of a quantum particle, my fuzzy inclusion in the set of vegetarians, my confidence level in a business forecast, ….. with a definite, imprecise meaning.
Classroom mathematics usually involves precise formulas but the level of generality achieved by 20th century mathematicians allows us to talk about a cobordism between two things without knowing precisely everything about them.
It’s funny, the more “advanced” and general the mathematics, the more casual it can become. Even post calc 1, I can speak about “a concave function" without saying whether it’s log, sqrt, or some non-famous power series.
Our knowledge of the world is not only piecemeal, but also vague and imprecise. To link mathematics to our conceptions of the real world, therefore, requires imprecision.
I want the option of thinking about my life, commerce, the natural world, art, social networks, and ideas using manifolds, metrics, groups, functors, topological connections, lattices, orthogonality, linear spans, categories, geometry, and any other metaphor, if I wish.