Mathematical Patterns That Eventually Fail | Hacker News Reader

Mathematical Patterns That Eventually Fail

johncarlosbaez.wordpress.com

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“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

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

He was a pretty smart guy who said some smart things.

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

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

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

"the trend is your friend until the end when it bends"

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-...

My favorite example of this sort is from the same blog: https://johncarlosbaez.wordpress.com/2016/05/06/shelves-and-... . 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.

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

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.

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.

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.

Finding a counter example is often easier (in those cases that fail) instead of a proof.

Empirical evidence clearly does not work in mathematics :)

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?

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.

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

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

Next!

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

The interesting facts are non obvious ones.

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

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

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

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=

How do we know that the universe isn't sitting on the back of a giant turtle?

Most of the math that we use is self-consistent, and our proof system is, as far as we can tell, relatively consistent with the known properties of the universe.

What I'm getting at is that no, we don't actually KNOW with 100% certainty that, for example, certain proven divergent series continue to diverge at infinity, if your standard for proof requires trial of an infinite span of numbers; but such speculation is so far out of the realm of what is known and "proven" according to our [mostly] self and externally consistent set of knowledge[1] that it is more or less in the realm of pseudoscience/metaphysics.

1. Excluding the Incompleteness Theorem

I'm about to go off-line so I can't enter a protracted discussion, but your comment leaves me completely baffled. For example, you say:

We know that some functions don't converge but that's only because we can't try n infinite numbers.

That's just patently false. For example, we know that this sequence doesn't converge:

1 + 1/2,

1 + 1/2, + 1/3,

1 + 1/2, + 1/3 + 1/4,

1 + 1/2, + 1/3 + 1/4 + 1/5,

... etc.

We know that won't converge, and not being able to try infinite numbers has nothing to do with it.

So what you say seems to me to be, on the surface, pretty much complete nonsense. To one who is trained in mathematics, what you're saying feels very similar to people who haven't travelled much saying that the Earth is obviously flat.

We know that won't converge, and not being able to try infinite numbers has nothing to do with it. So what you say seems to me to be, on the surface, pretty much complete nonsense. To one who is trained in mathematics, what you're saying feels very similar to people who haven't travelled much saying that the Earth is obviously flat.

With respect, this response seems needlessly harsh and supercilious. That’s the kind of stuff that discourages people from going deeper into a field. The question appears to have been asked in good faith ... why not be more charitable and constructive?

And as you probably know, it was discovered that many sequences that supposedly “won’t converge” indeed do converge under analytic continuation, there are reasonable and terribly subtle philosophical questions to ask about the validity of inductive reasoning, and there are working mathematicians who consider themselves finitists.

Usually those proofs start off by saying "pick an N as large as you wish", and then showing that the statement is true. Thus, you're certain that it doesn't break at big scales because you allowed any big scale you wished at the start.

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".

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

Oops, yes.

Your stated inequality fails for all n > 1

In ℤ₃, it is almost correct. There, “n ≥ n * n”.

No, for a quite fundamental reason: numbers are a thing we made up, and we made them to work a certain way, and they don't work that way. Part of the definition of an integer is that every integer has a successor, which is also an integer. And—presuming we have an analytical understanding of a function—a function of the integers won't suddenly "malfunction" if fed a sufficiently-large one. A function isn't a computer.

I mean, you could have your own axiomatic set-theory where mathematical induction doesn't work, but I'm not sure how to construct it without it being a very strange set-theory indeed.

Because we derive analysis from the counting numbers and fill in the gaps to get to the reals. These are philosophical and logical constructs. Whether I work with 1+2 or n^n^n^n^... doesn't matter, as the number still obeys the logical laws we created them with.

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