5 ms·

Ah, I see you're a mathematical formalist. That's one philosophy of what math is, but it's not the only one. What's going on in my mind when I visualize 3-dime

by kmill 5y ago

Ah, I see you're a mathematical formalist. That's one philosophy of what math is, but it's not the only one. What's going on in my mind when I visualize 3-dimensional manifolds and manipulate them? Cutting pieces apart, reattaching them elsewhere, stereographically reprojecting objects embedded in a 3-sphere. I don't see where the symbols are exactly here -- it seems like I'm (at least trying to) manipulate "actual" mental objects, ones my visual cortex are able to help me get glimpses of. Everything I'm doing can be turned into symbolic reasoning, but it's a fairly painful process. It seems closer to the basis for the philosophy of intuitionism, that math derives from mental constructions. I think it's rather interesting that many creatures (including ourselves) have a natural ability to at a glance tell how many things there are, up to about five objects or so. Maybe it's useful evolutionarily for counting young, or for detecting when a berry has gone missing, etc. I think it's also rather interesting that many creatures have episodic memory. Sometimes it seems to me that the invention of symbolic number derives from these two capabilities: through our experience that things can happen in sequence, we extend our natural ability to count while believing these "numbers" are meaningful. Each number is a story that plays out for what's been accumulated.

lisper 5y ago
The boundary where abstract noodling around with mental images becomes math lies precisely at the point where you can render those images symbolically. That is the thing that distinguishes math from all other forms of human mental endeavor.
- WhitneyLand 5y ago
  So spending say 5 years, noodling around with abstract mental concepts and ending with symbols that turn out to be manipulatable and useful in a mathematical way means, you haven’t done math until year 6 starts? I’m trying to see the line as clearly drawn as your comment suggests. The exact boundary doesn’t seem as simple as, stops here / starts here.
  - lisper 5y ago
    > So spending say 5 years, noodling around with abstract mental concepts and ending with symbols that turn out to be manipulatable and useful in a mathematical way means, you haven’t done math until year 6 starts? Yes, that's right. Until you write down the symbols you quite literally haven't done the math.
- bawolff 5y ago
  What a weird definition. Which fields of human endeavour cannot be rendered symbolically? If you are taking a very strict view of what it means to symbolize something (which i imagine you are if you think other fields cant be), then i wonder: do you consider geometric proofs math? Is euclid's elements with its prose proofs, math?
  - lisper 5y ago
    > Which fields of human endeavour cannot be rendered symbolically? All of them other than math. > do you consider geometric proofs math? Yes, because diagrams are symbols. They happen to be rendered in 2-D but they are symbols nonetheless. What makes something a symbol is that it is subject to a convention that allows it to fall into one of a finite number of equivalence classes, so that small changes in its physical details don't change its function within the context of the symbol-manipulation activity. Geometric diagrams have this property, so they qualify as symbols.
    - bawolff 5y ago
      I don't think i agree with that definition of symbol. I think that's closer to the definition of an abstraction, which i would consider to be different. Regardless, for the sake of argument lets accept it. Words also have this property. The word "Dog" refers to an equivalence class of a bunch of different physical manifestations which are considered equivalent in the context the word is used. Lets take a first aid textbook. This usually consists of a bunch of words describing situations, and a bunch of 2-D diagrams. This meets your definition of symbolic as far as i can tell. Is first-aid a sub-discipline of mathematics in your view?
      - lisper 5y ago
        > I don't think i agree with that definition of symbol. Really? How would you define it? > The word "Dog" refers to an equivalence class Yes, and if you wanted to, you could use dogs (actual physical dogs, not the word "dog") as symbols. It would be kind of a silly thing to do, but you could. (For the sake of completeness, because I'm not 100% sure this is what you were referring to, the word "dog" is unambiguously a symbol, one which denotes the equivalence calls of actual physical dogs. The equivalence class of the word dog includes things like "Dog" and "DOG" and "perro" and "hund".) > Is first-aid a sub-discipline of mathematics in your view? No, because the business of first aid is not the business of manipulating those symbols, it is the business of manipulating actual physical bodies. First-aid isn't math for the same reason that (say) building computers isn't math. This is not to say that math can't be a useful part of either of those activities, but they aren't equivalent.
- kmill 5y ago
  I see, so as a low-dimensional topologist I'm not a mathematician, thanks!* But more seriously, if you haven't read it already you might take a look at Thurston's "On Proof and Progress in Mathematics." He had a stunningly accurate intuition, and putting what was in his mind into a form that other experts could understand and consider to be a proof gave many mathematicians a job for quite a long time. So long as you had him around, you could query him for any amount of detail for why his claims were true, so in that sense he had real proofs. Symbolic proofs are sort of a lowest-common-denominator, serializing what's in one's mind to paper and reducing things to machine-checkable rules. It's also a rather modern idea that this is what math is. I think it's very much worth considering math to be the study of what can be made perfectly obvious, which is closer to what Thurston seemed to be doing vs formalism. * This is sort of a joke. At least I'm able to convince people it's math because in principle it can be written symbolically, and formalism is rather compelling as a foundational philosophy, if a bit nihilistic.
  - lisper 5y ago
    > as a low-dimensional topologist I'm not a mathematician Huh? How do you figure? > Symbolic proofs are sort of a lowest-common-denominator You are reading the word "symbol" too narrowly here. What matters here is not so much the symbols but the existence of rules for manipulating them. Letters are symbols, and so are words, and so proofs written in natural language can be math if those proofs respect a set of precisely definable rules. I don't know much about low-dimensional topology, but I'd be surprised if it didn't adhere to that definition. (And if it turns out not to then I would have no qualms saying that it isn't math.)

by kmill 5y ago

lisper 5y ago

The boundary where abstract noodling around with mental images becomes math lies precisely at the point where you can render those images symbolically. That is the thing that distinguishes math from all other forms of human mental endeavor.

WhitneyLand 5y ago
So spending say 5 years, noodling around with abstract mental concepts and ending with symbols that turn out to be manipulatable and useful in a mathematical way means, you haven’t done math until year 6 starts? I’m trying to see the line as clearly drawn as your comment suggests. The exact boundary doesn’t seem as simple as, stops here / starts here.
- lisper 5y ago
  > So spending say 5 years, noodling around with abstract mental concepts and ending with symbols that turn out to be manipulatable and useful in a mathematical way means, you haven’t done math until year 6 starts? Yes, that's right. Until you write down the symbols you quite literally haven't done the math.
bawolff 5y ago
What a weird definition. Which fields of human endeavour cannot be rendered symbolically? If you are taking a very strict view of what it means to symbolize something (which i imagine you are if you think other fields cant be), then i wonder: do you consider geometric proofs math? Is euclid's elements with its prose proofs, math?
- lisper 5y ago
  > Which fields of human endeavour cannot be rendered symbolically? All of them other than math. > do you consider geometric proofs math? Yes, because diagrams are symbols. They happen to be rendered in 2-D but they are symbols nonetheless. What makes something a symbol is that it is subject to a convention that allows it to fall into one of a finite number of equivalence classes, so that small changes in its physical details don't change its function within the context of the symbol-manipulation activity. Geometric diagrams have this property, so they qualify as symbols.
  - bawolff 5y ago
    I don't think i agree with that definition of symbol. I think that's closer to the definition of an abstraction, which i would consider to be different. Regardless, for the sake of argument lets accept it. Words also have this property. The word "Dog" refers to an equivalence class of a bunch of different physical manifestations which are considered equivalent in the context the word is used. Lets take a first aid textbook. This usually consists of a bunch of words describing situations, and a bunch of 2-D diagrams. This meets your definition of symbolic as far as i can tell. Is first-aid a sub-discipline of mathematics in your view?
    - lisper 5y ago
      > I don't think i agree with that definition of symbol. Really? How would you define it? > The word "Dog" refers to an equivalence class Yes, and if you wanted to, you could use dogs (actual physical dogs, not the word "dog") as symbols. It would be kind of a silly thing to do, but you could. (For the sake of completeness, because I'm not 100% sure this is what you were referring to, the word "dog" is unambiguously a symbol, one which denotes the equivalence calls of actual physical dogs. The equivalence class of the word dog includes things like "Dog" and "DOG" and "perro" and "hund".) > Is first-aid a sub-discipline of mathematics in your view? No, because the business of first aid is not the business of manipulating those symbols, it is the business of manipulating actual physical bodies. First-aid isn't math for the same reason that (say) building computers isn't math. This is not to say that math can't be a useful part of either of those activities, but they aren't equivalent.
kmill 5y ago
I see, so as a low-dimensional topologist I'm not a mathematician, thanks!* But more seriously, if you haven't read it already you might take a look at Thurston's "On Proof and Progress in Mathematics." He had a stunningly accurate intuition, and putting what was in his mind into a form that other experts could understand and consider to be a proof gave many mathematicians a job for quite a long time. So long as you had him around, you could query him for any amount of detail for why his claims were true, so in that sense he had real proofs. Symbolic proofs are sort of a lowest-common-denominator, serializing what's in one's mind to paper and reducing things to machine-checkable rules. It's also a rather modern idea that this is what math is. I think it's very much worth considering math to be the study of what can be made perfectly obvious, which is closer to what Thurston seemed to be doing vs formalism. * This is sort of a joke. At least I'm able to convince people it's math because in principle it can be written symbolically, and formalism is rather compelling as a foundational philosophy, if a bit nihilistic.
- lisper 5y ago
  > as a low-dimensional topologist I'm not a mathematician Huh? How do you figure? > Symbolic proofs are sort of a lowest-common-denominator You are reading the word "symbol" too narrowly here. What matters here is not so much the symbols but the existence of rules for manipulating them. Letters are symbols, and so are words, and so proofs written in natural language can be math if those proofs respect a set of precisely definable rules. I don't know much about low-dimensional topology, but I'd be surprised if it didn't adhere to that definition. (And if it turns out not to then I would have no qualms saying that it isn't math.)