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.
Unprovable statements weaken provable ones no further than the extent to which the undecidability of the halting problem makes my toaster burn my Eggos.
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.
There seems to me an enormous mistake there. ... Suppose I
convince [someone] of the paradox of the Liar, and he says,
'I lie, therefore I do not lie, therefore I lie and I do not
lie, therefore we have a contradiction, therefore 2x2 = 369.'
Gödel's incompleteness theorem doesn't immediately imply that we have to accept some things without rigorous proof. It just shows the limits of what you can prove within one axiomatic system.
His second incompleteness theorem implies we can never rigorously know our axioms are consistent. Thus, no proof is 'truly' rigorous. We have to depend on the prose of 'these axioms are so self evident that it is obvious they are consistent'.
This is what originally made people suspicious of the axiom of choice. It is much less self evident than the other axioms of ZF.
It's better to think of axioms as definitions. If the axioms are inconsistent, it just means nothing satisfies the definition, which isn't the end of the world. People have spent time studying hypothetical properties of polynomial-time factorization algorithms, knowing full well there may be no such things if P<>NP.
There are models of ZF where the axiom of choice holds, and there are models of ZF where the axiom of choice doesn't hold. To "assume" the axiom of choice, is nothing more than to restrict attention to the former. No more controversial than, say, deciding to write a paper specifically about golden retrievers, rather than a paper about all dogs.
If your axioms are inconsistent, you can use reductio ad absurdum to prove any statement. I'd say that it is rather problematic when I can prove any statement and it's converse.
Of course, just because the axiom of choice isn't obvious doesn't mean that ZFC is inconsistent. It just means that the argument for ZFC being inconsistent is more hand-wavy.
Of course you can prove any statement. The statements only hold for things which satisfy the axioms. "If X is a model of ZF where 1=0, then X also satisfies anything else you want." (Fine, there is no such X so who cares.) You're getting confused because of the shorthand where statements within the model are stated as if they are statements about the world we live in. That's just a shorthand.
Right, but Hilbert wanted to axiomatize arithmetics and all of mathematics in order to make proofs rigorous. It was one of his 23 problems, and Godel solved it (negatively).
It wasn't so much to make proofs rigorous, as to determine whether or not there would always exist "true" statements which could not be proved. Hilbert had already proposed a set axioms for second-order arithmetic - he just didn't know whether they were both complete and consistent.
I've heard that propositional logic helps with this because you don't assume everything is either true or false, but rather look at what can be implied.
You can then decide if you want the law of the excluded middle or not. I'm paraphrasing a talk I watched but would love to hear from someone knowledgeable on this subject...
If you have heard of the Curry-Howard equivalence, that programs and proofs are the same thing, the proofs in question are constructive, intuitionistic proofs.
I've been leaning towards intuitionism for exactly this reason.
Sadly, it makes a lot of results harder to proof (and iirc it even makes some results impossible). I've been toying with the idea that these results are simply 'not useful'.
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.
For anyone interested, Anders Kock has a nice slim-but-dense book on the subject entitled "Synthetic Differential Geometry" [0]. Kock even offers a PDF of an older version [1] on his site.
"The first is that most of the analytic data for describing the class of smooth manifolds can be encoded into certain fibre bundles on manifolds: namely bundles of jets (see also jet bundle)."
This is a technically precise, compact way of saying "most of what we care about of smooth manifolds is how differential equations on them behave locally and how that local information can be glued together".
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()).
"The Weierstrass function could perhaps be described as one of the very first fractals studied, although this term was not used until much later. The function has detail at every level, so zooming in on a piece of the curve does not show it getting progressively closer and closer to a straight line."
Does anybody know any even earlier "fractal studied" before they were named so? I guess the author probably meant "Bolzano function" (~1830, published in 1922, mentioned in the Notes section).
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.
Mathematicians didn't state this as theorem or axiom, so it's hard to pin down exactly what they might have believed; the people answering the question talk about "except at isolated points" rather than "except at a finite number of points".
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:
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)
It can also be integrated, and can be used as the basis for doing Fourier transforms. Completely useless as far as I've researched, but curious nonetheless. Here's an example paper: https://arxiv.org/abs/1502.07734
I take issue with this. The function is an infinite series of cosine functions which can be shown to be convergent everywhere. It's also easy to write a series for the derivative of it, but that is probably divergent and hence not well defined anywhere. It's a nice trick. But if one truncates the series at any finite number of terms, the problem goes away and it is differentiable everywhere.
I see parallels to the countability of the reals, and a similar issue with vague notions of infinity.
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.
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. :)
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.
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.
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.
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.