4 ms·

I mean, real numbers certainly do exist mathematically. They're not very difficult to rigorously construct, so I can see why your tutor would be bewildered by t

I mean, real numbers certainly do exist mathematically. They're not very difficult to rigorously construct, so I can see why your tutor would be bewildered by the statement. Whether or not they're a good description of coordinates in spacetime is a pretty separate matter.

xanderlewis 3y ago
I know how they can be constructed. But mathematical ideas don’t exist in a vacuum — they’re supposed to reflect some intuition or describe some (possibly very abstract) reality. My worry came from an inability to reconcile the theory with any such intuition. You can very fruitfully study the real numbers as has been done for centuries; that’s not up for debate.
- 4death4 3y ago
  > they’re supposed to reflect some intuition or describe some (possibly very abstract) reality. Says who? Math is the study of finding all possible logical conclusions from a given set of assumptions. Reality has nothing to do with it.
  - xanderlewis 3y ago
    > all possible logical conclusions I don’t think this is accurate. All logical conclusions would be a horrible mess of symbols because we don’t care about most of them (because they’re considered trivial) and the list would be infinite and filled with statements that are very similar to each other. The subject isn’t about brute force in that way. It is more true to say that mathematics is about finding all meaningful / interesting logical conclusions from a given set of assumptions. What those two words mean is then the interesting question (and, of course, a matter of personal taste). Reality certainly has something to do with that choice, which is why I don’t think you can escape it. It’s fundamental to the subject, no matter how seemingly abstract you get.
    - pdonis 3y ago
      > It is more true to say that mathematics is about finding all meaningful / interesting logical conclusions from a given set of assumptions. But that depends on what particular people find meaningful or interesting. Lots of people--or at least lots of mathematicians :)--appear to find many things interesting that have nothing at all to do with "reality". Also, it's important not to confuse "reality" with our models of it. Real numbers are unquestionably very useful in constructing models of reality. That will still be true even if it turns out that actual "reality" is not a continuum.
      - xanderlewis 3y ago
        > But that depends on what particular people find meaningful or interesting Yes. That’s why I said it’s a matter of taste. My claim is that what makes things meaningful is how they relate to reality, and by reality I mean things like the three dimensional world and the passing of time we perceive. Whilst most of pure mathematics seems ridiculously abstract on first glance, I don’t believe there is a single topic that truly has no connection to reality. > it's important not to confuse "reality" with our models of it This is a frequently repeated point and I’m inclined to agree, but any such model also exists in reality and so it’s hard to know where to draw the line.
        pdonis 3y ago
        > any such model also exists in reality Yes, but there's still a distinction between the model and the thing modeled. A map of your city is not the same as the city, even though the map is a real object in the same reality as the city. Similarly, real numbers are a model that is instantiated in the brains of mathematicians, papers in the literature, etc. Those things are of course real objects. But real numbers are used to model lots of other real things.
        xanderlewis 3y ago
        Of course there’s a distinction between a map and the thing it’s a map of. But they’re both parts of reality.
- chongli 3y ago
  The subject you’re looking for is philosophy of mathematics. Here’s a good place to start [1]. Unfortunately, if you’re looking for certainty you won’t find it there. Philosophy of mathematics is fascinating but studying it will give you more questions but few answers. One advantage of studying that I will vouch for is that you’ll learn that most of the arguments you had were already thought of many years ago. [1] https://plato.stanford.edu/entries/philosophy-mathematics/ https://plato.stanford.edu/entries/philosophy-mathematics/
  - xanderlewis 3y ago
    Yes! I am aware of PoM, just surprised so many mathematicians are seemingly disinterested. This is also a lovely, very readable, book on the subject: https://mitpress.mit.edu/9780262542234/lectures-on-the-philosophy-of-mathematics/ https://mitpress.mit.edu/9780262542234/lectures-on-the-philo...
    - chongli 3y ago
      I think most mathematicians are uninterested in PoM because their interest in math comes from an aesthetic impulse and a problem-solving drive. In short, they like math because it’s beautiful and challenging, and challenging problems are fun for them to solve. They may also be motivated by prestige, to varying degrees, though I think mathematicians are fairly low on the prestige-valuation scale (viz. Grigori Perelman). Though I enjoy PoM myself, it actually feels a bit weird to wonder why mathematicians don’t care about it. To me, it’s like wondering why chess grandmasters don’t seem to care that chess is a fictional game and not real warfare.
      - xanderlewis 3y ago
        > it actually feels a bit weird I won’t argue with that. :-) > it’s like wondering why chess grandmasters don’t seem to care that chess is a fictional game and not real warfare You’re exactly right. That is why I will never be a chess grandmaster!
- derbOac 3y ago
  But haven't real numbers served us pretty well in application as well, for most of the time? Modern physics is anything but intuitive in my opinion. Maybe reality isn't well described by real numbers, or maybe it is; but my sense is the reason why physicists have used real numbers is because they've worked pretty well up until a certain relatively recent point, when other intuitions about reality have started to fall apart as well.
  - samus 3y ago
    Part of the problem is that they maybe work too well. A lot of particle theory is model fitting without probiding us new insights into a possible underlying reality. Or something that would force a paradigm shift. Real evidence of something completely contradictive to our existing theories would be a dream come true.

xanderlewis 3y ago

I know how they can be constructed. But mathematical ideas don’t exist in a vacuum — they’re supposed to reflect some intuition or describe some (possibly very abstract) reality. My worry came from an inability to reconcile the theory with any such intuition. You can very fruitfully study the real numbers as has been done for centuries; that’s not up for debate.

4death4 3y ago
> they’re supposed to reflect some intuition or describe some (possibly very abstract) reality. Says who? Math is the study of finding all possible logical conclusions from a given set of assumptions. Reality has nothing to do with it.
- xanderlewis 3y ago
  > all possible logical conclusions I don’t think this is accurate. All logical conclusions would be a horrible mess of symbols because we don’t care about most of them (because they’re considered trivial) and the list would be infinite and filled with statements that are very similar to each other. The subject isn’t about brute force in that way. It is more true to say that mathematics is about finding all meaningful / interesting logical conclusions from a given set of assumptions. What those two words mean is then the interesting question (and, of course, a matter of personal taste). Reality certainly has something to do with that choice, which is why I don’t think you can escape it. It’s fundamental to the subject, no matter how seemingly abstract you get.
  - pdonis 3y ago
    > It is more true to say that mathematics is about finding all meaningful / interesting logical conclusions from a given set of assumptions. But that depends on what particular people find meaningful or interesting. Lots of people--or at least lots of mathematicians :)--appear to find many things interesting that have nothing at all to do with "reality". Also, it's important not to confuse "reality" with our models of it. Real numbers are unquestionably very useful in constructing models of reality. That will still be true even if it turns out that actual "reality" is not a continuum.
    - xanderlewis 3y ago
      > But that depends on what particular people find meaningful or interesting Yes. That’s why I said it’s a matter of taste. My claim is that what makes things meaningful is how they relate to reality, and by reality I mean things like the three dimensional world and the passing of time we perceive. Whilst most of pure mathematics seems ridiculously abstract on first glance, I don’t believe there is a single topic that truly has no connection to reality. > it's important not to confuse "reality" with our models of it This is a frequently repeated point and I’m inclined to agree, but any such model also exists in reality and so it’s hard to know where to draw the line.
      - pdonis 3y ago
        > any such model also exists in reality Yes, but there's still a distinction between the model and the thing modeled. A map of your city is not the same as the city, even though the map is a real object in the same reality as the city. Similarly, real numbers are a model that is instantiated in the brains of mathematicians, papers in the literature, etc. Those things are of course real objects. But real numbers are used to model lots of other real things.
        xanderlewis 3y ago
        Of course there’s a distinction between a map and the thing it’s a map of. But they’re both parts of reality.
chongli 3y ago
The subject you’re looking for is philosophy of mathematics. Here’s a good place to start [1]. Unfortunately, if you’re looking for certainty you won’t find it there. Philosophy of mathematics is fascinating but studying it will give you more questions but few answers. One advantage of studying that I will vouch for is that you’ll learn that most of the arguments you had were already thought of many years ago. [1] https://plato.stanford.edu/entries/philosophy-mathematics/ https://plato.stanford.edu/entries/philosophy-mathematics/
- xanderlewis 3y ago
  Yes! I am aware of PoM, just surprised so many mathematicians are seemingly disinterested. This is also a lovely, very readable, book on the subject: https://mitpress.mit.edu/9780262542234/lectures-on-the-philosophy-of-mathematics/ https://mitpress.mit.edu/9780262542234/lectures-on-the-philo...
  - chongli 3y ago
    I think most mathematicians are uninterested in PoM because their interest in math comes from an aesthetic impulse and a problem-solving drive. In short, they like math because it’s beautiful and challenging, and challenging problems are fun for them to solve. They may also be motivated by prestige, to varying degrees, though I think mathematicians are fairly low on the prestige-valuation scale (viz. Grigori Perelman). Though I enjoy PoM myself, it actually feels a bit weird to wonder why mathematicians don’t care about it. To me, it’s like wondering why chess grandmasters don’t seem to care that chess is a fictional game and not real warfare.
    - xanderlewis 3y ago
      > it actually feels a bit weird I won’t argue with that. :-) > it’s like wondering why chess grandmasters don’t seem to care that chess is a fictional game and not real warfare You’re exactly right. That is why I will never be a chess grandmaster!
derbOac 3y ago
But haven't real numbers served us pretty well in application as well, for most of the time? Modern physics is anything but intuitive in my opinion. Maybe reality isn't well described by real numbers, or maybe it is; but my sense is the reason why physicists have used real numbers is because they've worked pretty well up until a certain relatively recent point, when other intuitions about reality have started to fall apart as well.
- samus 3y ago
  Part of the problem is that they maybe work too well. A lot of particle theory is model fitting without probiding us new insights into a possible underlying reality. Or something that would force a paradigm shift. Real evidence of something completely contradictive to our existing theories would be a dream come true.