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Math is fake. The realm of perfect platonic mathematical objects and proofs is a self-consistent fictional narrative that society made up, kind of like the pantheon of Greek gods. The main difference between math and mythological lore is that you use them in different situations for different purposes.
A less trollish way of making this point is that math is a collection of interesting tautologies. If you do the work to make sure there is a sufficiently tight correspondence between mathematical objects and objects in the real world, these tautologies can sometimes tell you useful things about the real world. But those things will only ever be statistical regularities. If you mistake the map for the territory, and make extremely confident and lopsided gambles based on the mathematical tautologies, you will lose badly:
An Interesting Wager
For example, consider the statement 2+2=4. When dealing with platonic mathematical objects, it is always tautologically true.
However, if you think, as a matter of religious faith, that two plus two always equals four in all contexts, then I propose the following wager:
Using a fair random number generator, take two samples from the set of numbers that round to 2, i.e. U[1.5,2.5), and add them together. If the result rounds to 4, I will give you a dollar, and if the result rounds to 5, you will give me $100.
This is a very bad gamble, because it can be shown with simple probability theory that the numbers will round to 5 1/8th of the time. (the sum is a triangular distribution with min 3, peak 4, and max 5, and 1/8th of the area of the triangle is above 4.5)
Clearly the statement ‘2+2=4’ is true more often than the statement ‘2+2=5’, so that it is more reasonable to say ‘2+2=4’, but there is a huge difference between ‘a reasonable prediction that is true more often’ and ‘always axiomatically true’.
When doing any kind of cost-benefit analysis or welfare estimation, always keep this in mind. Always question the links or assumptions you have made between your math and the things in the real world. Sometimes 2 and 2 will make 5 because of random error, and sometimes your data-gathering process is flawed and biased enough that two of the things you call 2 will routinely add up to the thing you call 5.
That wager is more just a toy example to bootstrap your intuitions. It is a guide for who is and is not lying to you. A cost-benefit analysis that never has numbers that don’t add up is lying to you. Any real analysis by someone who is competent and numerate, and trusts you to be sensible and numerate, will have something that pattern-matches to ‘2+2=5’ or ‘2+2=3’ in roughly one fourth of the calculations.
This is because of Significant Figure Arithmetic. In any situation where there is inherent measurement error, you should not report every decimal place. You should only report the ones that are larger than the margin of error of the measurement tool. However, you should keep unreported ‘guard digits’ in the intermediate calculation to avoid round-off errors accumulating in the calculation. Because you are not reporting these, the underlying calculation may be something like ‘2.3 + 2.4 = 4.7’ but you report it as ‘2+2=5’.
Aggressively rounding numbers is much more honest than false precision. It sends the message to the reader that this result really is quite uncertain, and should be treated as such. Most of the analyses we will be doing will have 90% confidence intervals that are very wide, often spanning an order of magnitude or more. This is real knowledge, and often allows us to be much more certain of a number than we were before, but we should never pretend that the estimate is more precise than that. When dealing with such large confidence intervals, it is silly to report either bound to more than one significant figure.
If you see an analysis of any real-world thing that presents a conclusion or bottom-line number with more than three significant figures, you can be quite certain that the author is either innumerate or forced to comply with an innumerate style guide. Most of the time, you should only use one or two significant figures, unless your measurements are extraordinarily precise.
