5 ms·

> 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 con

> 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.
        samus 3y ago
        It's also perfectly possible to draw maps of places that don't exist in reality, i.e., that only exist in somebody's brain.
        xanderlewis 3y ago
        A good point. So the real numbers exist in the same sense that anything else they might describe does. My original point was really a comment on the way that the subject is often taught: the real number system is not introduced as some arbitrary set of rules that is as good as any other (though of course it is); it is introduced as if it were as incontrovertible an idea as, say… integer addition. I don’t think it is. It can perfectly well be argued that it makes sense and is useful, but its naturality is another matter. I personally find the leap from R to C less controversial than that from Q to R.
        samus 3y ago
        Yes, I agree. Real numbers are a quite nebulous construct for which the need is a bit hard to see. Complex numbers are basically a hack to solve quadratic equations in all cases.
  - danbruc 3y ago
    I agree with what you have said but have an additional worry. It seems to me that the standard logic we use might be heavily rooted in the real world. It either rains or doesn't rain seems to closely reflect how the classical world works. But would we intuitively accept the same laws of logic if we lived in an universe that was obviously some kind of many worlds universe, where it always rains and doesn't rain?
    - pdonis 3y ago
      > would we intuitively accept the same laws of logic if we lived in an universe that was obviously some kind of many worlds universe, where it always rains and doesn't rain? You don't have to change the laws of logic that you use to do unitary QM (which, taken to extremes, is what gives you a "many worlds universe"). You can describe it just fine using mathematical equations based on the same laws of logic you're familiar with. In other words, what you are calling "standard logic" actually contains many more possibilities than just the simple Boolean "rains or doesn't rain".
      - danbruc 3y ago
        Sure, if the system is expressive enough, you can emulate other systems in it, you can build fuzzy logic on top of classical logic. Many worlds was just a random example which is not too far from the classical world. But can we not imagine an universe that is very different from ours, that outright contradicts our logic? After all logic is just build on top of axioms that make sense to us, I don't see that this necessarily grants them some universal truth.
        pdonis 3y ago
        > can we not imagine an universe that is very different from ours, that outright contradicts our logic? Can you give an example? If you can't, doesn't that in itself answer your question?
        danbruc 3y ago
        Not necessarily. How and what we can think might be constraint by our brains or maybe even by the universe itself. I don't think any human can make sense of two time dimensions but it seem premature to me to conclude that no universe could have more than one time dimension. Anything I could suggest would by definition not make any sense to us as it defies our logic but could it really not exist? It might of course just be a stupid idea. How would you even think about something like this if you throw out the foundation of our thinking?
- deleted 3y ago
  [deleted]

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.
      - samus 3y ago
        It's also perfectly possible to draw maps of places that don't exist in reality, i.e., that only exist in somebody's brain.
        xanderlewis 3y ago
        A good point. So the real numbers exist in the same sense that anything else they might describe does. My original point was really a comment on the way that the subject is often taught: the real number system is not introduced as some arbitrary set of rules that is as good as any other (though of course it is); it is introduced as if it were as incontrovertible an idea as, say… integer addition. I don’t think it is. It can perfectly well be argued that it makes sense and is useful, but its naturality is another matter. I personally find the leap from R to C less controversial than that from Q to R.
        samus 3y ago
        Yes, I agree. Real numbers are a quite nebulous construct for which the need is a bit hard to see. Complex numbers are basically a hack to solve quadratic equations in all cases.
- danbruc 3y ago
  I agree with what you have said but have an additional worry. It seems to me that the standard logic we use might be heavily rooted in the real world. It either rains or doesn't rain seems to closely reflect how the classical world works. But would we intuitively accept the same laws of logic if we lived in an universe that was obviously some kind of many worlds universe, where it always rains and doesn't rain?
  - pdonis 3y ago
    > would we intuitively accept the same laws of logic if we lived in an universe that was obviously some kind of many worlds universe, where it always rains and doesn't rain? You don't have to change the laws of logic that you use to do unitary QM (which, taken to extremes, is what gives you a "many worlds universe"). You can describe it just fine using mathematical equations based on the same laws of logic you're familiar with. In other words, what you are calling "standard logic" actually contains many more possibilities than just the simple Boolean "rains or doesn't rain".
    - danbruc 3y ago
      Sure, if the system is expressive enough, you can emulate other systems in it, you can build fuzzy logic on top of classical logic. Many worlds was just a random example which is not too far from the classical world. But can we not imagine an universe that is very different from ours, that outright contradicts our logic? After all logic is just build on top of axioms that make sense to us, I don't see that this necessarily grants them some universal truth.
      - pdonis 3y ago
        > can we not imagine an universe that is very different from ours, that outright contradicts our logic? Can you give an example? If you can't, doesn't that in itself answer your question?
        danbruc 3y ago
        Not necessarily. How and what we can think might be constraint by our brains or maybe even by the universe itself. I don't think any human can make sense of two time dimensions but it seem premature to me to conclude that no universe could have more than one time dimension. Anything I could suggest would by definition not make any sense to us as it defies our logic but could it really not exist? It might of course just be a stupid idea. How would you even think about something like this if you throw out the foundation of our thinking?
deleted 3y ago
[deleted]