
How a destructive idea paved the way for modern math - dnetesn
http://nautil.us/issue/53/monsters/maths-beautiful-monsters-rp
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rocqua
I suppose the inaccuracies caused by the 'prose' of Ampère's false proof that
all continuous functions are differentiabe might be what inspired Hilbert's
demand for rigor. In turn, this lead to Godel's incompleteness.

We are then left in the curious situation of despising any hand waving, whilst
knowing deep down that at some point we need to accept things without rigorous
proof. Of course, that other reason to hand-wave remains. Often, things are
`obvious' and yet very very tedious to proof. It is an odd balance to strike.

~~~
whatshisface
Unprovable statements weaken provable ones no further than the extent to which
the undecidability of the halting problem makes my toaster burn my Eggos.

~~~
visarga
Your toaster has a halting problem?

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whatshisface
Yes, it has the halting problem for a machine much less powerful than a Turing
machine, where the answer is always yes unless the thermostat fails. The idea
is that just like there are programs less powerful than the halting oracle
there are statements less powerful than the completeness of arithmetic.

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xelxebar
This reminds me of the Devil's Staircase [0], a continuous function that is
flat almost everywhere and yet still spans the range [0,1].

[0]
[https://en.wikipedia.org/wiki/Cantor_function](https://en.wikipedia.org/wiki/Cantor_function)

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danharaj
Synthetic differential geometry formalizes calculus in a way that precludes
most monsters. Quite nicely, it allows you to use infinitesimal reasoning. The
price to pay is the law of excluded middle. If you're a programmer, you're
probably fine giving that up.

The wiki page has good references:
[https://en.m.wikipedia.org/wiki/Synthetic_differential_geome...](https://en.m.wikipedia.org/wiki/Synthetic_differential_geometry)

~~~
adrianratnapala
Only if you are that exceedingly queer kind of programmer who thinks that
which is not within reach of Coq can never be truth.

I am a programmer, but I like C better, and _assert(A && !A)_ looks like a
monster to me.

~~~
irishsultan
Well, you probably meant _assert(A || !A)_ , but even so that will of course
never be an issue if _A_ is a value, it becomes only a problem if _A_ is
replaced by something like _a()_ (which I assume to be a pure function for the
purpose of this comment). Of course _a() || !a()_ is never going to be false
either, so _assert(a() || !a())_ will never abort your program, but that
doesn't mean _a() || !a()_ will ever be true (if _a()_ never finishes
calculating then neither will _a() || !a()_ ).

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Bromskloss
The monster in question is the Weierstraß function [0], noteworthy for being
continuous everywhere yet differentiable nowhere.

I don't if the article talks about other "monsters" too. I didn't look further
than to this one.

[0]
[https://en.wikipedia.org/wiki/Weierstrass_function](https://en.wikipedia.org/wiki/Weierstrass_function)

~~~
jhkim
It seems like the equation shown in the article doesn't satisfy the condition
listed on the wikipedia page. Did the author make a mistake?

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supergarfield
The Wikipedia article gives Weierstrass's original condition, but it was later
improved to just ab ≥ 1 (as noted, not very visibly, at the end of the
article).

However, the Nautilus article also says that "Conventional wisdom held that
for any continuous curve, it was possible to find the gradient at all but a
finite number of points", which is clearly not true, so I'd be cautious about
its technical correctness.

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alimw
On what basis are you calling this assertion "clearly not true"?

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bonzini
You can define a periodic function that goes from 0 to 1 as a straight line
and then from 1 to 0 as another straight line, basically a triangle wave. It
is not differentiable on an infinite (and countable) number of points:

[https://en.m.wikipedia.org/wiki/Triangle_wave](https://en.m.wikipedia.org/wiki/Triangle_wave)

~~~
lmm
I imagine they would have been talking about functions on the interval rather
than our modern sense of functions on the reals. (Of course there are plenty
of examples on the interval as well, but no obvious ones)

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bonzini
Take something like arcsin sin pi/2/x(x-1), where "arcsin sin pi*x/2" is
"pointy" like a triangle wave. There you have it.

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WallWextra
Analysis in general is notorious for these kinds of things. There's a neat
little Dover book called Counterexamples in Analysis full of them.

~~~
mathperson
There is also "counterexamples in topology"!

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graycat
Beauty and the Beast: Smooth meets the Monster.

Let n be a positive integer and let R be the set of real numbers. Let R^n be
Euclidean n-dimensional space. E.g., R^1 is the real line; R^2 is the plane;
and R^3 is (approximately) the space we live in.

Let C be a closed subset of R^n.

In R^2, examples of closed subsets include a sample path of Brownian motion
(as in the OP) and the Mandelbrot set. An example in R^1 is a Cantor set of
positive measure.

Then there exists a function f: R^n --> R so that (1) f is infinitely
differentiable, (2) f(x) = 0 for all x in C, and (3) f(x) > 0 for all x not in
C.

So, smooth beauty f meets the monster beast set C.

Of course, the level set of any differentiable function is closed. But now we
know that any closed set can be the level set of an infinitely differentiable
function. So, we also know that there can be an infinitely differentiable
function positive outside of C, 0 on the boundary of C, and negative on the
interior of C. So, we know that for any closed set, there is an infinitely
differentiable function that has the boundary of the closed set as a level
set. Since a level set of a differentiable function is closed, we also have
the the boundary of any closed set is closed.

"I have found a truly wonderful proof," but the mathematical notation is too
difficult to type into a blog post! But, no worries: I published it in JOTA.

The Brownian motion example was noticed by A. Karr.

~~~
xelxebar
Interesting. Or put more tersely,

Let O be an open set in R^n. There exists a smooth function f: R^n -> R with
support cl(O).

Off the cuff, I'd try proving this by covering O with bump functions and
trying to glue those together. I'll try thinking about this more when I've got
time. :)

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Myrmornis
Really nice essay. I think it might be an enjoyable read even for the majority
for whom real analysis is somewhere between excruciatingly boring and
excruciatingly painful.

~~~
alfredallan
Really nice indeed, but I think it'd have been much nicer with actual
examples/equations.

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notduncansmith
This video has more information about these "monsters":
[https://www.youtube.com/watch?v=56gzV0od6DU](https://www.youtube.com/watch?v=56gzV0od6DU)

Fractals are one of the most important epistemological tools of our time.

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mar77i
It's as if the universe aligned for Nautilus (which is also a snail producing
a natural fractal as its house) to publish such a fascinating article about
how calculus and geometry partially unite to produce such a fascinating
article.

Wait... what?

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gogopuppygogo
I love stories about anti-authoritarians. While they don't specifically state
that Karl Weierstrass was one, it seems clear to me from his actions that he
was.

~~~
thenewwazoo
But was he a true Scotsman?

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dogruck
This essay’s conclusion is it’s best part. I like how it points out how
radical abstract concepts eventually reunited with natural phenomena.

