6 ms·

This is one of those things where a surprising fact isn't astounding, but just makes me think the definitions are badly chosen. After all -- of course if you pi

by ajkjk 6y ago

This is one of those things where a surprising fact isn't astounding, but just makes me think the definitions are badly chosen. After all -- of course if you pick definitions in weird ways, you get weird results. 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.

WmyEE0UsWAwC2i 6y ago
That sounds like something taken from the book: Burn Math Class. 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?
ubercow13 6y ago
But can you build a fruitful system of analysis from your different definitions that doesn't have any equally surprising facts once you dig any deeper? The proof would be in the pudding. It's not like the choice of definitions in mathematics is arbitrary, they stick because they turned out to be useful.
cjohnson318 6y ago
> This is one of those things where a surprising fact isn't astounding, but just makes me think the definitions are badly chosen. After all -- of course if you pick definitions in weird ways, you get weird results. 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.
ska 6y ago
The problem with this way of thinking is that "the actual mathematics of our universe that I care about" doesn't (yet?) exist. The definitions you find badly chosen have been thought about, hard, by many generations of very smart people. Some with the same basic reaction to all this you have. And yet, after all that, the definitions remain. And the alternatives proposed, at best are not compelling. 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.
- heavenlyblue 6y ago
  You are mixing up definitions and names. Do complex numbers have anything to do with complexity?
  - ska 6y ago
    I'm not quite sure where this comment comes from - what I am saying has nothing to do with the names; I use the conventional names for the natural reasons. Perhaps you meant the potential ambiguity of terms? 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.
georgewsinger 6y ago
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.
biddlesby 6y ago
I can sympathise with you on that. The whole of pure mathematics seems to be about making definitions and seeing what weird places and surprising connections you get. This isn't as exciting as seeing how mathematics can describe and predict phenomena in the real world.
- ubercow13 6y ago
  Analysis is all about explaining calculus, one of the most useful and practical real-world mathematical fields. So you should find it pretty exciting.
- sukilot 6y ago
  Almost none of pure mathematics is like that. 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.
all-fakes 6y ago
> 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. 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?

by ajkjk 6y ago

WmyEE0UsWAwC2i 6y ago

That sounds like something taken from the book: Burn Math Class. 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?

ubercow13 6y ago

But can you build a fruitful system of analysis from your different definitions that doesn't have any equally surprising facts once you dig any deeper? The proof would be in the pudding. It's not like the choice of definitions in mathematics is arbitrary, they stick because they turned out to be useful.

cjohnson318 6y ago

> This is one of those things where a surprising fact isn't astounding, but just makes me think the definitions are badly chosen. After all -- of course if you pick definitions in weird ways, you get weird results. 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.

ska 6y ago

The problem with this way of thinking is that "the actual mathematics of our universe that I care about" doesn't (yet?) exist. The definitions you find badly chosen have been thought about, hard, by many generations of very smart people. Some with the same basic reaction to all this you have. And yet, after all that, the definitions remain. And the alternatives proposed, at best are not compelling. 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.

heavenlyblue 6y ago
You are mixing up definitions and names. Do complex numbers have anything to do with complexity?
- ska 6y ago
  I'm not quite sure where this comment comes from - what I am saying has nothing to do with the names; I use the conventional names for the natural reasons. Perhaps you meant the potential ambiguity of terms? 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.

georgewsinger 6y ago

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.

biddlesby 6y ago

I can sympathise with you on that. The whole of pure mathematics seems to be about making definitions and seeing what weird places and surprising connections you get. This isn't as exciting as seeing how mathematics can describe and predict phenomena in the real world.

ubercow13 6y ago
Analysis is all about explaining calculus, one of the most useful and practical real-world mathematical fields. So you should find it pretty exciting.
sukilot 6y ago
Almost none of pure mathematics is like that. 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.

all-fakes 6y ago

> 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. 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?