
Mathematical Patterns That Eventually Fail - fanf2
https://johncarlosbaez.wordpress.com/2018/09/20/patterns-that-eventually-fail/
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weinzierl
“The man who has fed the chicken every day throughout its life at last wrings
its neck instead, showing that more refined views as to the uniformity of
nature would have been useful to the chicken.”

― Bertrand Russell, The Problems of Philosophy

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westoncb
That's a pretty interesting quote. Can anyone provide some more context for
it?

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unsung
He was talking about some of the limits of just inferring that repeated
behavior is induced ad infinitum. He talks about it more. [0]

[0] www.personal.kent.edu/~rmuhamma/Philosophy/RBwritings/ProbPhiloBook/chap-
VI.htm

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whatshisface
The decision to wring the chicken's neck was made by a brain that did not
repeat its state even a single time.

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dwaltrip
I took it to emphasize how all models are approximations and thus may fail in
surprising ways.

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Jeff_Brown
There's a very big list of these, including links to good papers, on
StackOverflow[1].

[1] [https://math.stackexchange.com/questions/111440/examples-
of-...](https://math.stackexchange.com/questions/111440/examples-of-patterns-
that-eventually-fail)

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orangecat
My favorite example of this sort is from the same blog:
[https://johncarlosbaez.wordpress.com/2016/05/06/shelves-
and-...](https://johncarlosbaez.wordpress.com/2016/05/06/shelves-and-the-
infinite/) . It's a sequence generated by completely normal integers, but to
prove that it diverges (after an astronomically large number of items) you
have to assume the existence of a specific large transfinite cardinal.

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jobigoud
I can't help but notice that Greg Egan is commenting on the post!

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skinner_
More than just commenting. John Baez's blogpost is just elaborating on a math
trick invented by Greg Egan. It's a variation on an idea by Hanspeter Schmid.

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amelius
I wonder how often a (widely expected-to-be-true) mathematical hypothesis
fails, versus how often it turns out to be actually true, versus how often it
remained unknown so far.

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enedil
If a hypothesis is widely expected-to-be-true, then there needs to be a good
evidence towards it, which in turn means that it stood unresolved for a long
time. That's why probably most of them are still unsolved.

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tirumaraiselvan
Empirical evidence clearly does not work in mathematics :)

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yeukhon
Wow hold on! The first is so widely taught I am surprised! Has there been
accidents because people/software not knowing about such failing point? This
is so dangerous. I titled “math that eventually fails to teach students the
real math”.

I’m not sure how many math and engineering professors are aware of #1. Do PhDs
learn this stuff?

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pishpash
Length of pattern means very little without knowing the significance of each
step. As the article shows, something contrived can be built out of a short
pattern to look like a long pattern.

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sk5t
This could use a better title, to spare those of us with somewhat average
levels of mathematics skill five minutes of bewilderment.

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codr4
Thank you for that, I need more bewilderment like I need another hole in the
head.

Next!

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coldtea
Original title on HN was "A mathematical pattern that fails after about 10^43
examples".

An even more impressive example:

10^1000 - x > 0 where x ∈ ℕ.

Fails after (10^1000)+1

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tomjakubowski
Could you say what you find impressive about that? I'm not seeing it.

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coldtea
It was tongue in cheek, based on the fact that the original HN submission
title was too generic...

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antidesitter
Presumably the pattern would have to be interesting to merit a submission.

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csomar
Since we don't have a way to apply "infinity" or infinitely big _n_ to math
equations/problems, isn't it more correct to say that we don't know if any of
the equations we have will either fail or prevail?

We know that some functions don't converge but that's only because we can't
try _n_ infinite numbers. How are we certain that these non-convergent
functions do not converge to some value and that math breaks at big scales (or
smaller scales). =much like our universe=

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gpm
We can absolutely say what happens to equations with big n, math has no
problem talking about large numbers. n > n * n for n > 1 for instance is
fundamentally true - by virtue of our definitions of "1", "integers", ">" and
"n * n", our logical system (generally zfc), and our definition of true.

That's not to say that the real world physics will continue to match our
models for big n, or anything like that. Just that our abstract mathematical
models continue to "work".

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schoen
I think you mean n < n * n rather than n > n * n here.

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gpm
Oops, yes.

