5 ms·

Eh, everyone but the set theorists ignores the set theorists at this point. I think some exciting things happened in the 60's with forcing, but a lot of the sub

Eh, everyone but the set theorists ignores the set theorists at this point. I think some exciting things happened in the 60's with forcing, but a lot of the subsequent developments seem pretty contrived.

cgmg 9y ago
To an outsider, practically any area of advanced, pure mathematics looks contrived. That doesn't mean it's not beautiful and interesting, particularly to those who study it. Also, the claim that 'everybody ignores set theorists' is patently false.
- KGIII 9y ago
  They are a bunch of damned liars. 0.999... = 1 Not only can I prove it, I can tell you the excuses they use to justify it. I'm pretty sure that they just realized that it was 'proven' and had to then come up with a reason. I really do try to not think about it too much, though I may sometimes ponder it a bit while smoking. It's all horrible. I don't know what they hell they were thinking. Half of them probably belong in asylums.
  - Simon_says 9y ago
    People who get confused about that equality are in actuality confused about what a real number is. In particular they probably think that real numbers are digit expansions, and they have probably never seen a construction of the reals. You can show this person a proof, and you might get them to shut up about it, but they’ll never believe it on an intuitive level until you show them what real numbers actually are and what their relationship to their (non-unique!) digital representation is.
    - enord 9y ago
      I believe the confusion is spread across several layers. For those in the know, infinite decimal expansion is syntactic sugar for a limit over a particular infinite series of sums, but this convention sidesteps how people are taught syntactic manipulation and how it's intetpteted in a particular algebra. 0.999... is presented as a term, that can be syntactically replaced with an actual infinite decimal expansion, upon which the normal syntactic opetations can be applied, specifically "=" as in "0.999...=1". This is ofcourse impossible, because even if the rules are valid you would be applying them forever. This is the apparent absudity of the equation. In defence of people who hold this intuition: it's a valid concern in many branches of math(discrete, constructive, infinitesimal etc.), just not in informal expositions of the properties of real numbers.
      - cgmg 9y ago
        Yes, '0.999...' formally stands for the sum from n = 1 to infinity of 9 / 10^n.
        KGIII 9y ago
        Heh... You kinda prove my point for me. If you could step up to the board and actually expand the ... for us, we'd be much obliged. You can keep writing 9 forever and you will never reach a whole number. But, it does reach a whole number and there is proof! So, they made a rule. The rule is, of course, silly. Infinity itself is silly. Not even atoms are infinite. Not even atoms can be infinity divisible. It's a horrible idea, really. (I have karma to burn.) By the way, it's a damned good thing infinity doesnt exist. If you had a tape measure that was infinitely long and folded it in half, we aren't really sure what would happen. Hell, I wasn't using that karma for anything better. While I'm here... When you're done finding an infinite tape measure, I'd like it if you could bring me -1 apple. No, not a slip of paper representing an owed apple, but -1 apple. Math and reality are only loosely coupled. Mathematics is a language. It can tell lies. 1 + 1 = 74 See? That's pretty obviously a lie. It can tell stories, physicists are pretty fond of using it for that. It can be so very descriptive and elegant. It can be so deceptive, as we see with the 'well, let's just let that be represented by 9/10' silliness. It's pretty horrible, really. Like I said, it's best not to think about it too much. It's kind of contrived in a few areas.
        cgmg 9y ago
        > If you could step up to the board and actually expand the ... for us, we'd be much obliged. It seems you don't know what a sequence is. A sequence is just a function whose codomain is the natural numbers, i.e. you give any natural number you choose, and I give you the corresponding digit. I don't need to 'expand' the ellipses to define that function, any more than I have to write down all perfect squares in order to define what a perfect square is. Seriously, this is basic mathematics. You should study it before publicly embarrassing yourself by calling mathematicians 'liars'.
        9y ago
    - KGIII 9y ago
      No, I'm not confused about it. It's just a horrible, horrible lie. ;-) If it goes on forever, it can't reach the whole number. But, I can prove it is equal to one. Horrible.

cgmg 9y ago

To an outsider, practically any area of advanced, pure mathematics looks contrived. That doesn't mean it's not beautiful and interesting, particularly to those who study it. Also, the claim that 'everybody ignores set theorists' is patently false.

KGIII 9y ago
They are a bunch of damned liars. 0.999... = 1 Not only can I prove it, I can tell you the excuses they use to justify it. I'm pretty sure that they just realized that it was 'proven' and had to then come up with a reason. I really do try to not think about it too much, though I may sometimes ponder it a bit while smoking. It's all horrible. I don't know what they hell they were thinking. Half of them probably belong in asylums.
- Simon_says 9y ago
  People who get confused about that equality are in actuality confused about what a real number is. In particular they probably think that real numbers are digit expansions, and they have probably never seen a construction of the reals. You can show this person a proof, and you might get them to shut up about it, but they’ll never believe it on an intuitive level until you show them what real numbers actually are and what their relationship to their (non-unique!) digital representation is.
  - enord 9y ago
    I believe the confusion is spread across several layers. For those in the know, infinite decimal expansion is syntactic sugar for a limit over a particular infinite series of sums, but this convention sidesteps how people are taught syntactic manipulation and how it's intetpteted in a particular algebra. 0.999... is presented as a term, that can be syntactically replaced with an actual infinite decimal expansion, upon which the normal syntactic opetations can be applied, specifically "=" as in "0.999...=1". This is ofcourse impossible, because even if the rules are valid you would be applying them forever. This is the apparent absudity of the equation. In defence of people who hold this intuition: it's a valid concern in many branches of math(discrete, constructive, infinitesimal etc.), just not in informal expositions of the properties of real numbers.
    - cgmg 9y ago
      Yes, '0.999...' formally stands for the sum from n = 1 to infinity of 9 / 10^n.
      - KGIII 9y ago
        Heh... You kinda prove my point for me. If you could step up to the board and actually expand the ... for us, we'd be much obliged. You can keep writing 9 forever and you will never reach a whole number. But, it does reach a whole number and there is proof! So, they made a rule. The rule is, of course, silly. Infinity itself is silly. Not even atoms are infinite. Not even atoms can be infinity divisible. It's a horrible idea, really. (I have karma to burn.) By the way, it's a damned good thing infinity doesnt exist. If you had a tape measure that was infinitely long and folded it in half, we aren't really sure what would happen. Hell, I wasn't using that karma for anything better. While I'm here... When you're done finding an infinite tape measure, I'd like it if you could bring me -1 apple. No, not a slip of paper representing an owed apple, but -1 apple. Math and reality are only loosely coupled. Mathematics is a language. It can tell lies. 1 + 1 = 74 See? That's pretty obviously a lie. It can tell stories, physicists are pretty fond of using it for that. It can be so very descriptive and elegant. It can be so deceptive, as we see with the 'well, let's just let that be represented by 9/10' silliness. It's pretty horrible, really. Like I said, it's best not to think about it too much. It's kind of contrived in a few areas.
        cgmg 9y ago
        > If you could step up to the board and actually expand the ... for us, we'd be much obliged. It seems you don't know what a sequence is. A sequence is just a function whose codomain is the natural numbers, i.e. you give any natural number you choose, and I give you the corresponding digit. I don't need to 'expand' the ellipses to define that function, any more than I have to write down all perfect squares in order to define what a perfect square is. Seriously, this is basic mathematics. You should study it before publicly embarrassing yourself by calling mathematicians 'liars'.
        9y ago
  - KGIII 9y ago
    No, I'm not confused about it. It's just a horrible, horrible lie. ;-) If it goes on forever, it can't reach the whole number. But, I can prove it is equal to one. Horrible.