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Oh, so human diseases in the form of bacteria/viruses! And humans working on gain-of-function research.

Reminds me of a discussion I've had recently about whether humans solve complex systems of [mechanical] differential equations while moving. The counter-argument was "do you think that a mercury thermometer solves differential equations [while 'calculating' the temperature]?"

This one is a classic, so I can just copy-paste the solution from Google. The more interesting point is that this is one of those cases where math doesn't correspond to reality.

In the spirit of "trying to offer concrete models and predictions" I propose you a challenge: write a bot which would consistently beat my Rob implementation over the long run enough that it would show on numerical experiments. I need some time to work on implementing it (and might disappear for a while, in which case consider this one forfeited by me).

One of the rules I propose that neither of us are allowed to use details of others' implementation, in order to uphold the spirit of the original task.

It does work for negative bases. Representation of a number in any base is in essence a sum of base powers multiplied by coefficients. The geometric series just has all coefficients equal to 1 after the radix point (and a 1 before it, if we start addition from the 0th power).

Oh, thanks, I did not think about that! Now everything makes much more sense.

Those probabilities are multiplied by s, which makes it more complicated.
If I try running it with s being the real numbers (which is probably the most popular choice for utility measurement), the proof breaks down. If I, for example, allow negative utilities, I can rearrange the series from a divergent one into a convergent one and vice versa, trivially leading to a contradiction just from the fact that I am allowed to do weird things with infinite series, and not because of proposed axioms being contradictory.
EDIT: concisely, your axioms do not imply that the rearrangement should result in the same utility.

The correct condition for real numbers would be absolute convergence (otherwise the sum after rearrangement might become different and/or infinite) but you are right: the series rearrangement is definitely illegal here.

It actually would, as long as you reject a candidate password with probability proportional to it's relative frequency. "password" in the above example would be almost certainly rejected as it's wildly more common that one of those 1000-character passwords.

Stamp collecting (e.g. "history" and "English literature") does not count.

Interesting to see your perspective change from this post and it's comments, which suggested that history is a useful source of world models. Or am I misinterpreting past/current you?

You cannot falsify mathematics by experiment (except in the subjective Bayesian sense).

Actually, that's technically false. The statements mathematical axioms make about reality are bizarre, but they exist and are actually falsifiable.

One of the fundamental properties we want from our axiomatic systems is consistency — the fact that it does not lead to a logical contradiction. We would certainly reject our current axiomatic foundations in case we found them inconsistent.

Turns out it's possible to write a program which would halt if and only if ZFC is consistent. I would not recommend running this one as it's a Turing machine and thus not really optimized (and in any case, ZFC being inconsistent is unlikely, and it's even more unlikely that the proof of it's inconsistency would be easy to be found with current technology), but in theory one might run one of such machines long enough to produce a contradiction, which would basically physically falsify the axioms.

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