Cantor function, a.k.a. devil's staircase
en.wikipedia.org
en.wikipedia.org
lol that's not what it means at all. almost sure/almost everywhere/complement has measure 0 means exactly what it says: the set can be covered by a countable cover whose measure is zero. that's it and no more. no one ever says something like "the normal distribution 100% of the time is not 0" even though { x = 0 } has measure zero.
In this case, yes, it is true almost-everywhere, meaning that there the set of points where it is untrue has measure 0. And yet intuitively the function is increasing and so we ought to define "having a derivative" in such a way that shows that it is increasing. All this definition has done is crammed all of the increasing-ness into points that are vanishingly rare, so you can't find them. But they still exist, per _my_ definition of increasing!
I would instead choose to define "having a derivative != 0" in a different way, that captures the idea that, over any interval I choose, the function changes. Nevermind that the function becomes constant as I zoom in to a point. All that shows is that evaluating at points must not be a great way of modeling this.
Of course I'm not sure if that works exactly. But this is the reason I can't get enthusiastic about analysis: it seems like a bunch of silly definitions with silly consequences, with little relation to the actual mathematics of our universe that I care about.
There is always f(b) > f(a) <=> f increases in [a, b] or f(b) /= f(a) <=> f changes in [a,b].
But how useful is that?
I agree that it's not astounding. I think it's surprising because the definitions are well chosen, and intuitive in most cases, but this mathematical object just falls through the cracks.
So why? Real & Complex analysis are by far the best way humans have yet come up with to model certain things, particularly modeling a large subset of physical interactions in the universe.
So to some degree we are stuck with this. Mostly it's the fact that fields like R an C are actually pretty strange objects. Strange in ways that are hard to get your head around, and that leak out into any mathematics that you build on top of it.
So what to do? There is beauty there as well. And although we have tricks to hide most of this, most of the time, it's not a bad thing really that when you look closely the weirdness is apparent.
Do complex numbers have anything to do with complexity?
I agree there are sometimes unfortunate consequences of transliterating mathematical terms (e.g. "field", "smooth", "increasing") this potential for confusion with a lay audience (cf the famous (true?) story of a senator objecting the funding of "complex" analysis - was this your reference?). However, the alternative is worse, and peppers any conversation with intractable jargon.
A shockingly large amount of pure mathematics is asserted (with either no or little argument) as the best foundational groundwork for numbers, infinities, measures, etc. Digging a little deeper shows much of the dogma to be a dubious foundation for mathematics, leading to absurdities like Cantor's function (which 99.9% of math students marvel at, instead of more appropriately wondering "wait a minute -- maybe this means our foundations are messed up?").
[1] https://www.amazon.com/Approahttps://www.amazon.com/Approach...
There's as many variants of a derivative.
There are also many kinds of measure.
They all have limitations. Usually, the major limitation is handling sparse fields, because most integrals and derivatives do not hail from discrete mathematics.
Cantor's devil staircase is exactly one of those things that bridge discrete mathematics, (mostly discrete) number theory and (mostly continuous) analysis.
Or the one that still bothers me, ten years after I learned it. Is take an enumeration of the rationals and take the union of balls of with radius epsilon/2^n over this enumeration. And you can create dense open subsets of the real numbers, that miss basically every real number. Despite containing an open interval around every rational. The conclusion I’ve come to is the real numbers don’t make any sense.
Oh they definitely don't.
Almost all real numbers are uncomputable, which means they cannot be written down or communicated in any way.
Imaginary numbers aren't imaginary.
Infinities are unphysical, so you can understand them by analogy to anything that exists in the real Universe.
I haven't read your book specifically, but many people have thought about the problematic nature of infinity in analysis. There is a whole reformulation of analysis designed to fix that called nonstandard analysis [1]. I don't know much about it myself so can't comment on the content of these links, but it seems that Cantor's set still exists there [2] [3]
[1] https://en.wikipedia.org/wiki/Nonstandard_analysis [2] https://books.google.co.uk/books?id=hBHP5foeXCsC&pg=PA142&lp... [3] https://math.stackexchange.com/questions/2906390/what-does-a...
For example infinitesimal numbers make no sense, in the sense that there is no such thing (in any possible world) as a positive number that is smaller than every positive real number. Of course, you can definitionally augment `(R+, <)` with formal objects `u` s.t. `u < x` for any `x ∈ R+` (where `R+` designates the positive reals), and then close these under the field operations `(R, +, *)` (I assume this is how infinitesimals are constructed). But that doesn't mean you have written down a model of actual numbers (which include the so-called "infinitesimal" numbers), since there's no such things as infinitesimal numbers.
You could argue "well, maybe infinitesimals aren't actually numbers (and perhaps more strongly: the theory of infinitesimals fails to have any metaphysically possible model), but they are still useful to us". For example: calculus is surely useful to us, and calculus can be built over the "infinitesimals". I don't have anything against this. But many people studying non-standard analysis incorrectly think that because infinitesimal numbers are being formally studied, that must mean there are good arguments out there for their actual existence.
Another incorrect dogma from traditional mathematics has to do with infinities being treated as numbers. Infinities are not numbers, for there is no such thing as a number which is greater than every natural number. There are countable (and even uncountable) collections (that is perfectly coherent). But that doesn't mean infinity is literally a number. Yet in traditional mathematics, it is merely asserted that infinities are numbers (and that indeed there are an infinity of these infinities, that can be ordered by size).
Even the ordering relation is asserted without argument. For example, let `N` be the naturals and `P(N)` be its power set. In traditional mathematics, we say that `|N| < |P(N)|` since every any injective function from `N` to `P(N)` fails to be surjective. Sure, you can define `<` to satisfy this. But that doesn't mean, in reality, the "less than" relation does satisfy this! It is merely asserted, without argument, that `|N| < |P(N)|`, when it would seem to most smart, reflective people (before they are corrupted by years of traditional mathematics training) that `|N|` and `|P(N)|` aren't orderable in the first place, since they are both infinite collections (the former being countable and the latter being uncountable), and infinite collections can't be ordered by their size (in the "less than", "greater than" sense of ordering we use with actual numbers). Of course you can define an ordering on infinite collections using the surjective relation, but that doesn't mean it's mapping onto the actual "numbers ordered by size" relation that we had in mind before we started studying infinite collections. Perhaps this is why beginning students studying infinities in analysis find it so mind-boggling: they're being taught a bunch of forced definitions/constructions which don't actually map onto any real (or even metaphysically possible) models.
When you take the traditional mathematics route literally, you run into all sorts of absurdities. For example, if you treat ∞ as a literal number, you have to patch up the field operations on it. You have to start saying: "∞ + 1 = ∞", which implies it's not the case that "∞ < ∞ + 1". You can find several other examples in that book I linked to.
The original Cantor's function (a function that is continuous everywhere, but has zero derivative almost everywhere) that OP linked to strikes me as something that fails to have a real model. I don't doubt that such a function is logically possible, as constructed. But it likely means our definitions of "continuous", "derivative", "function" (or other such mathematical constructions) are failing to map onto the abstract objects we set out to when we started studying them in the first place.
Sort of like how, in probability theory, you can have events which are possible but nevertheless have probability zero (classic example: throwing a dart at the real number line means that the probability it will land on any particular real number is zero). It's totally logically coherent to set things up this way, but that it is logically possible for this to happen means our foundations/definitions/models are all screwed up (though, I admit, still pretty useful nevertheless).
>that it is logically possible for this to happen means our foundations/definitions/models are all screwed up
Why? Simply because it feels unintuitive to you? Where is the useful alternative proposed by this theory? Is there any sound definition of what is 'metaphysically real' and why there is any reason to dismiss things which are not?
For example consider FLT - the proof surely uses fields and concepts with countless 'absurdities' inside them. And yet it proved something quite real and tangible. Consider that there may be no alternative proof of FLT. In that case, doesn't that validate those absurd concepts and give them a reality of their own that doesn't require any 'metaphysical' form? They have revealed a truth about the physical natural numbers we can all understand. To me that gives them a very compelling 'realness'.
Of course it could be that you can throw out all the absurd definitions and still build a useful enough system to prove such things - I am sure people have tried. Maybe the point is that not enough people have tried?
I should say also the only reason I read this book was because I liked the author's other books which have nothing to do with mathematics. Before reading it I thought "man who gives a shit about philosophy of mathematics? can we just do some real math please?", but this book changed my mind about that.
Also: I agree that traditional mathematics is useful. I trust that FLT true (not to mention it's an awesome achievement). And I even trust that proofs which make heavy use of concepts I find philosophically dubious will almost always turn out to still hold when placed under the "right" sort of foundations.
There are some exceptions to this though. For example, I absolutely don't trust stuff like this: https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
By "don't trust", I mean I seriously doubt there is anything at all out there in the universe (or in the metaphysically possible universe) that behaves like this. Instead, it's just a logical result which leans heavily on some problematic mathematical assumptions which were never actually justified. It should be taken as a reductio ad absurdem that something went wrong with the inputs (or at least evidence as such). 100+ years ago, people spent time wrestling with the foundations of math, but now it's looked down upon as a waste of time/not "serious" math, etc.
How is that even remotely relevant?
[1] https://en.m.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory
But the normies obsess over the pathological cases, often because they are described in misleading mathematically incorrect terms (such as Banach Tarski).
It's kind of similar to law, where people love misreading judicial opinions to "prove" that law is crazy.
The Cantor function is not differentiable at the points of "increase", so there is no problem with it growing at those points. I'm not sure what your complaint is.
> it seems like a bunch of silly definitions with silly consequences, with little relation to the actual mathematics of our universe that I care about.
That's honestly really silly. A lot of analysis is extremely practical, and is fundamental to the study of this universe. Yes, the language allows you to express non-physical situations, but you know you can just ignore those situations, right?
I don't think this is a great way of expressing it really since it's not very clear what "100% of the time" means formally; to most people (myself included), saying "100% of the time" in the context of whether a property holds over a set is to say that the property holds for all elements of the set.
This is completely different to saying it holds almost everywhere, which is to say it holds on the complement of a measure zero set.
https://en.wikipedia.org/wiki/Minkowski%27s_question-mark_fu...
> Georg Cantor (1884) introduced the Cantor function and mentioned that Scheeffer pointed out that it was a counterexample to an extension of the fundamental theorem of calculus claimed by Harnack.
I don't know if it has any real world applications other than as a reminder to be careful about assumptions and definitions. Most people think that increasing/monotone functions must have positive derivatives but the Cantor function shows this is not the case.
That's because Cantor's function does not have a derivative that's defined everywhere. Of course a function without derivative is not the integral of that non-existent derivative.
If most people think that increasing/monotone functions must have positive derivatives, they're apparently forgetting about non-differentiable functions.
> The Cantor function, also known as the Cantor Staircase, is a bizarre function that is continuous and has a derivative of 0 at every point where it is differentiable. In fact, it is differentiable at every point other than on the Cantor set, which is a set of measure zero.
So from a measure theoretic perspective the derivative of the Cantor function is well defined and it is equal to 0 almost everywhere. Almost everywhere equality is an equivalence relation and the derivative of the Cantor function is in the same equivalence class as the 0 function.
And a function in that equivalence class will have an integral that is equal to a constant function on all intervals where that function is defined. If you also integrate over intervals where the function is only defined almost everywhere, all bets are off.
I guess that is one way to think about it, but if that is the case you can just differentiate a constant function and you the fundamental theorem of calculus won't return it to you (if you don't like that example just take the Heaviside function). In finite dimensional linear spaces, the derivative operator has a rank 1 kernel, but the integral does not. From that perspective, the derivative destroys information (specifically it destroys constants), the integral operator cannot restore it automatically and the whole + C is meant to identify an equivalence class for the indefinite anti-derivative.
There are concrete application examples, but in general one guy that was particularly interested in fractals in the real world was Mandelbrot. He published several influential books on fractals in nature (and in finance). Another famous name is Taleb, although imho he's much more a populariser than a researcher.
The results before/after that result have other niceties: a one sided fundamental theorem of calculus for monotone functions (even though the Cantor function has 0 derivative!) or the fact that monotone functions can also be decomposed as the sum of continuous monotone function and a jump function such as Cantor's function.
He was treated horrifically by the mathematics establishment. There is some justice in learning his history all these years later.
https://en.wikipedia.org/wiki/Cantor_set
This and other wacky analysis things is what inspired me to study pure math in undergrad.