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For most of my life I would have unthinkingly attacked any attacker of the traditional foundations of mathematics (including the standard definitions in Analysi

For most of my life I would have unthinkingly attacked any attacker of the traditional foundations of mathematics (including the standard definitions in Analysis, Measure Theory, etc). But after reading Michael Huemer's "Approaching Infinity"[1], I am now sympathetic with your skepticism. 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/Approaching-Infinity-M-Huemer/dp/1137560851ching-Infinity-M-Huemer/dp/1137560851 https://www.amazon.com/Approahttps://www.amazon.com/Approach...

AstralStorm 6y ago
Are they messed up though? There are many variants of integration. 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.
hilbertseries 6y ago
There are a couple constructions like this in mathematics that... really make you think. Like Banach Tarski doesn’t make any sense. 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.
- Dylan16807 6y ago
  > 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.
- sukilot 6y ago
  Real numbers aren't real. Imaginary numbers aren't imaginary. Infinities are unphysical, so you can understand them by analogy to anything that exists in the real Universe.
ubercow13 6y ago
Just because it is unintuitive doesn't make it absurd. It is still well defined and its behaviour is understood and can be described. It's one thing to think 'this is so messed up that maybe the foundations are messed up' and it's another to actually find a new set of foundations which don't produce any 'absurd' results at all but are still useful. 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 https://en.wikipedia.org/wiki/Nonstandard_analysis [2] https://books.google.co.uk/books?id=hBHP5foeXCsC&pg=PA142&lpg=PA142&dq=nonstandard+analysis+cantor+set&source=bl&ots=E-v1mVyEZ4&sig=ACfU3U1D-yqabnvCA3gFLCZuCfqUwgosFg&hl=en&sa=X&ved=2ahUKEwirkb-z8vvpAhV0oXEKHfp4BikQ6AEwDnoECAoQAQ#v=onepage&q=nonstandard%20analysis%20cantor%20set&f=false https://books.google.co.uk/books?id=hBHP5foeXCsC&pg=PA142&lp... [3] https://math.stackexchange.com/questions/2906390/what-does-a-hyperreal-version-of-the-cantor-set-look-like https://math.stackexchange.com/questions/2906390/what-does-a...
- georgewsinger 6y ago
  Robinson's non-standard analysis is also discussed (and rejected) in the book I linked to. My book argues (among other things) that while all of the constructions of traditional mathematics are logically possible (not very interesting), many of them are metaphysically impossible for any object to instantiate. 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).
dimes 6y ago
I think you can derive enough real analysis to define the Cantor function using only ZFC. You can read the axioms in [1], but they are very simple and “obvious” so to speak. What’s more interesting is where ZFC is known to be incomplete. Cantor’s continuum hypothesis is known to be unprovable in ZFC, which is fun the think about. [1] https://en.m.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory https://en.m.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory

AstralStorm 6y ago

Are they messed up though? There are many variants of integration. 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.

hilbertseries 6y ago

There are a couple constructions like this in mathematics that... really make you think. Like Banach Tarski doesn’t make any sense. 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.

Dylan16807 6y ago
> 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.
sukilot 6y ago
Real numbers aren't real. Imaginary numbers aren't imaginary. Infinities are unphysical, so you can understand them by analogy to anything that exists in the real Universe.

ubercow13 6y ago

Just because it is unintuitive doesn't make it absurd. It is still well defined and its behaviour is understood and can be described. It's one thing to think 'this is so messed up that maybe the foundations are messed up' and it's another to actually find a new set of foundations which don't produce any 'absurd' results at all but are still useful. 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 https://en.wikipedia.org/wiki/Nonstandard_analysis [2] https://books.google.co.uk/books?id=hBHP5foeXCsC&pg=PA142&lpg=PA142&dq=nonstandard+analysis+cantor+set&source=bl&ots=E-v1mVyEZ4&sig=ACfU3U1D-yqabnvCA3gFLCZuCfqUwgosFg&hl=en&sa=X&ved=2ahUKEwirkb-z8vvpAhV0oXEKHfp4BikQ6AEwDnoECAoQAQ#v=onepage&q=nonstandard%20analysis%20cantor%20set&f=false https://books.google.co.uk/books?id=hBHP5foeXCsC&pg=PA142&lp... [3] https://math.stackexchange.com/questions/2906390/what-does-a-hyperreal-version-of-the-cantor-set-look-like https://math.stackexchange.com/questions/2906390/what-does-a...

georgewsinger 6y ago
Robinson's non-standard analysis is also discussed (and rejected) in the book I linked to. My book argues (among other things) that while all of the constructions of traditional mathematics are logically possible (not very interesting), many of them are metaphysically impossible for any object to instantiate. 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).

dimes 6y ago

I think you can derive enough real analysis to define the Cantor function using only ZFC. You can read the axioms in [1], but they are very simple and “obvious” so to speak. What’s more interesting is where ZFC is known to be incomplete. Cantor’s continuum hypothesis is known to be unprovable in ZFC, which is fun the think about. [1] https://en.m.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory https://en.m.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory