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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

by 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.
      - bawolff 5y ago
        > Really? How would you define it? Well first off, i'm not sure why you're requiring finiteness or what that means in context. But more to the point, i would define it as something that represents (denotes) something else. The thing it represents may be concrete, it might be abstract (represent a group of all things that some property), it might not exist at all, etc. I suppose i'm actually a bit confused by your definition, i find it hard to parse. "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,": it sounds like what you're saying is that a symbol is defined as being a representative member of one or more equivalence classes, which seems obviously wrong, as symbols typically aren't members of the class of objects they represent. I was previously interpreting your statement as missing the word "denote". But denote is kind of the core of the definition of a symbol. If what you meant by your definition was that a symbol is something that denotes an equivalence class, you could probably drop the equivalence class bit, as all concepts can be thought of as equivalence classes with zero or more members, so it doesn't add anything to the definition. > No, because the business of first aid is not the business of manipulating those symbols, it is the business of manipulating actual physical bodies. Sure (although what about the activity of writing first-aid textbooks?), but that's a pretty dramatic shift in goal posts from "precisely at the point where you can render those images symbolically". Being able to "Render" something symbolically is very different from having the end goal of the activity being the manipulation of the symbols. I agree having symbol manipulation be the chief end, feels closer to a definition of math, but it still feels kind of lacking. Mathematicians don't just arbitrarily manipulate symbols, they choose what to do based on a variety of criteria, such as real world applications, beauty, connectedness to other areas, etc. The ultimate end for actual mathematicians isn't solely symbol manipulation. On the other side, poets manipulate symbols (words) following certain rules. They do so for a variety of reasons, but usually to evoke emotion or perhaps beauty in the minds of their readers. The final goal is a series of symbols manipulated just the right way. Are they mathematicians?
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 lisper 5y ago

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.
    - bawolff 5y ago
      > Really? How would you define it? Well first off, i'm not sure why you're requiring finiteness or what that means in context. But more to the point, i would define it as something that represents (denotes) something else. The thing it represents may be concrete, it might be abstract (represent a group of all things that some property), it might not exist at all, etc. I suppose i'm actually a bit confused by your definition, i find it hard to parse. "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,": it sounds like what you're saying is that a symbol is defined as being a representative member of one or more equivalence classes, which seems obviously wrong, as symbols typically aren't members of the class of objects they represent. I was previously interpreting your statement as missing the word "denote". But denote is kind of the core of the definition of a symbol. If what you meant by your definition was that a symbol is something that denotes an equivalence class, you could probably drop the equivalence class bit, as all concepts can be thought of as equivalence classes with zero or more members, so it doesn't add anything to the definition. > No, because the business of first aid is not the business of manipulating those symbols, it is the business of manipulating actual physical bodies. Sure (although what about the activity of writing first-aid textbooks?), but that's a pretty dramatic shift in goal posts from "precisely at the point where you can render those images symbolically". Being able to "Render" something symbolically is very different from having the end goal of the activity being the manipulation of the symbols. I agree having symbol manipulation be the chief end, feels closer to a definition of math, but it still feels kind of lacking. Mathematicians don't just arbitrarily manipulate symbols, they choose what to do based on a variety of criteria, such as real world applications, beauty, connectedness to other areas, etc. The ultimate end for actual mathematicians isn't solely symbol manipulation. On the other side, poets manipulate symbols (words) following certain rules. They do so for a variety of reasons, but usually to evoke emotion or perhaps beauty in the minds of their readers. The final goal is a series of symbols manipulated just the right way. Are they mathematicians?

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.)