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To be more precise, it has derivative zero almost everywhere, so title is slightly misleading. To fully grasp this, the article links topics from measure theory

To be more precise, it has derivative zero almost everywhere, so title is slightly misleading. To fully grasp this, the article links topics from measure theory.

Dylan16807 6y ago
To be clear, this is the measure/probability theory definition of almost everywhere, which means that while there are exceptions, it's true 100% of the time.
- throwgeorge 6y ago
  >it's true 100% of the time. 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.
- 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.
    - 6y ago
- p0llard 6y ago
  > it's true 100% of the time. 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.
conistonwater 6y ago
I'd say the title is correct almost surely.
- dhosek 6y ago
  Reminds me of something my complex analysis prof said with respect to some constant that it was unknown whether it was rational or irrational: “There’s no reason to think it’s rational, and statistically, the probability of any real number being rational is 0.”
  - alberto_ol 6y ago
    He could have even said that that constant was almost certainly transcendental. It always surprises me that almost all real are transcendental, but that there are only 2 transcendental real numbers I am familiar with: pi and e.
    - dhosek 6y ago
      I forget the constant, but as I recall, that was likely true as well.
    - sukilot 6y ago
      That's because almost all transcendentals are not computable or nameable, whereas all algebraics are. So when you limit to the space of nameable computable numbers you get a different statistical intuition.
dang 6y ago
Ok, we'll move "increasing function with 0 derivative" to almost nowhere in the title above.

To be more precise, it has derivative zero almost everywhere, so title is slightly misleading. To fully grasp this, the article links topics from measure theory.

Dylan16807 6y ago

To be clear, this is the measure/probability theory definition of almost everywhere, which means that while there are exceptions, it's true 100% of the time.

throwgeorge 6y ago
>it's true 100% of the time. 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.
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.
  - 6y ago
p0llard 6y ago
> it's true 100% of the time. 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.

conistonwater 6y ago

I'd say the title is correct almost surely.

dhosek 6y ago
Reminds me of something my complex analysis prof said with respect to some constant that it was unknown whether it was rational or irrational: “There’s no reason to think it’s rational, and statistically, the probability of any real number being rational is 0.”
- alberto_ol 6y ago
  He could have even said that that constant was almost certainly transcendental. It always surprises me that almost all real are transcendental, but that there are only 2 transcendental real numbers I am familiar with: pi and e.
  - dhosek 6y ago
    I forget the constant, but as I recall, that was likely true as well.
  - sukilot 6y ago
    That's because almost all transcendentals are not computable or nameable, whereas all algebraics are. So when you limit to the space of nameable computable numbers you get a different statistical intuition.

dang 6y ago

Ok, we'll move "increasing function with 0 derivative" to almost nowhere in the title above.