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JoshuaFox comments on What math is essential to the art of rationality? - Less Wrong
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An understanding of the insights behind math is essential. But I wonder: To improve your rationality, how often do you really solve an equation, arithmetically (as opposed to just going by feel) calculate probabilities from Bayes Rule, or derive a formal proof?
The concrete practice is an indispensable way of arriving at the insight. ("No royal road to geometry" etc.)
Achieving facility with the concrete work is evidence that you have the insight. Evidence to yourself, the one person you need to prove it to.
To be avoided is gaining a mere feeling of understandishness. Anyone can learn to say "light travels along geodesics in curved space", but if you can't calculate the precession of Mercury, you don't know general relativity.
Yes, concrete practice may be indispensable to the insight. But once you have the insight, do you ever need to calculate to help you with a practical problem? Almost never, I think.
When you know things, you discover uses for them. Knowing arithmetic, you can easily decide whether the supereconomy giant size really is a good deal. Knowing prob/stats/causality, you can dismiss a lot of reporting as junk, and be able to say exactly why. Quadratic equations are often used as an example of useless knowledge, and yet I find myself solving those from time to time, and not just at work (in the narrow sense of what people pay me to do).
Yes, arithmetic does come in useful, for example in those cases.
Can you give an example of when you have used actual arithmetical calculations to explain why some prob/stats/causality were junk, or where you solved a quadratic equation?
It's not some minor trick, like how to fold a t-shirt, it's useful everywhere.
It's common enough that I don't even notice it as a thing. But for example, a political survey shows a 2% advantage for one party. The sample size is given and I know at once that the result is noise. (sigma = sqrt(pqN).) Knowing how correlation and causality relate to each other disposes of a lot of bad reporting, and some bad research. Or I want to generate random numbers with a certain distribution; that easily leads to pages of algebra and trigonomentry.
For a more extensive illustration of how knowing all this stuff enables you to see the world, see gwern's web site.
Certainly, it is useful everywhere to understand. But very few people actually run calculations (other than basic arithmetic). Gwern and you are very rare exceptions. I think the world could use more of that.
I am greatly flattered to be mentioned in the same breath as Gwern. The world could indeed use a lot more Gwerns.
But it's like what lionhearted just posted about history: when you know this sort of thing, you see its use. And by seeing its use, you can do things that would not previously have come to your attention as possibilities.
If you get diagnosed with an illness and are given the sensitivity and specificity of the test, being able to calculate your risk is valuable and many doctors get this wrong.
The trouble is that knowing the formula and being able to use it in daily life are two different things. On one of the LW censuses a significant portion got a question intended to test knowledge of Bayes rule wrong.
So other than learning the formula, how do you suggest that we learn to apply it, other than actually applying over and over, until one stops getting problems wrong? It seems that it's just a function of practice.
I don't have a good answer to that question.
There are studies of mental biases that suggest that a lot of people who can manage to apply the formula to textbook problem fails to apply it when you give them a political scenario. Most people fail to think clearly and motivated reasoning if the question becomes meaningful for them.
As far as I understand CFAR tries to teach bayes theorem in a way that people will actually use it. I however neither know their exact curriculum nor know the success rate of their approach.
How often have you received this info in practice? How often have you done the calculations?