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> one view is that mathematics is a stiff and fixed set of rules and algorithms while the other view is that mathematics is flexible and our understanding of ma

> one view is that mathematics is a stiff and fixed set of rules and algorithms while the other view is that mathematics is flexible and our understanding of math comes from questioning of why mathematics functions so effectively I havent read the book but this dichotomy is not an intrinsic feature of how people pursued mathematics historically. The "stiffness" and excess focus on rigor and accuracy developed gradually over the 19th century because people were being loose canons - primarily around calculus.

leetcodesucks 3y ago
Its still not stiff enough. Mathematicians really ought to settle on a sound and coordinated syntax
- eru 3y ago
  Syntax is about the least of people's worries when they worry about 'mathematical rigor'. Math is made for people to read and write, and they have different 'domain specific languages' for different parts of math. Now what would be useful is a tool, perhaps something like a large language model, to automatically translate from the notation used in one area of math into another. That can't be a fully mechanical procedure (hence the need for something flexible like an LLM), because it's part and parcel of human mathematics to abuse notation here and there in the name of ergonomics. My personal pet peeve is defining a function like f(x) := x + 3, and then treating f(x) as the name of the function, instead of just f. But really, it's just a harmless abuse of notation when done by some humans to other humans.
  - rudy6912 3y ago
    I do not think mixing up 'f(x)' and 'f' is harmless, given mathematics is all about clarity. In any quality text, 'f(x)' denotes the value of 'f' at 'x', and 'f' denotes the function 'f' itself. Also, speaking of notation, I wonder why you used ':=' instead of '=' to define 'f'. There is no computation going on, right?
    - eru 3y ago
      Mixing up 'f' and 'f(x)' is mostly harmless in practice. The underlying principles are still clear enough. (And I say that as someone who would _really_ like to make that argument that people who mess this up are somehow unclear in their thinking. No, they are mostly fine. Getting 'f' vs 'f(x)' right mostly is really important for programmers who deal with higher order functions in general all the time. Most mathematicians don't fall into that category. You could say calculus deals with higher order functions, like the derivative. And that's a valid way to look at it. But most people get by just fine using special purpose notation for the derivative and not thinking about it as a function just like 'f'.) I used := to emphasis that I am defining 'f' here, not just writing down any old equation. (Eg like like in the example "Find all functions f such that f(x + 1) = x * f (x).") Though if you wanted to be pedantic about notation, I could have written that as with the x on the other side of the :=, like f := \x -> x + 3 (for Haskell inspired notation) or f := (x |-> x + 3) where |-> means the little arrow I draw by hand to denote a mapping when I'm writing math on a chalk board or piece of paper. I'm not sure why := would denote a computation? At most you might want to use it to denote an assignment in a mutable context?
      - xeonmc 3y ago
        > Getting 'f' vs 'f(x)' right mostly is really important for programmers who deal with higher order functions in general all the time. Most mathematicians don't fall into that category. Operators and functionals?
        eru 3y ago
        It's usually clear from context what you mean, even if you work with operators and functionals.
      - bheadmaster 3y ago
        > Getting 'f' vs 'f(x)' right mostly is important for programmers who deal with higher order functions in general all the time. Most mathematicians don't fall into that category Mathematicians deal with higher order functions all the time, e.g. in functional analysis.
  - atoav 3y ago
    While I agree with the point you make, I think that syntax has a huge impact. Maybe not on mathematicians themselves, but I am certain that mathematical syntax has done more to scare people away from the field than the actual mathematical problems themselves. For example the convention to use the greek alphabet for certain things. This is totally arbitrary and you could have also used emoticons instead (had they existed). But what this means is that the pupil, before tackling the meat of the mathematical problem has to accept that weird looking letter they have never seen for no real reason whatsoever. And I say that as someone who can fluently read the greek alphabet.
    - 4gotunameagain 3y ago
      It is not arbitrary, it is heritage. Apart from the fact that Greek was the de facto scientific language of the west (which is no longer the case), I think we can agree that the characters of a single alphabet are not enough for notation, especially given the fact that it is very useful to be able to discern different types of entities, e.g. constants from variables or vectors from matrices. If we changed symbols now, it would create an even bigger mess. Because the people that learn the new symbols, could not read any textbook published before 0 A.D. (Anno Discombobuli)
      - atoav 3y ago
        So you read my comment and thought: "That guy who can read the greek alphabet doesn't know there are historical roots for greek letters in mathematical notation". Sure, back then when everybody who learned trigonometry had a classical education with ancient greek picking greek letters when the latin alphabet wouldn't do was a rational decision. It just hasn't aged well.
        4gotunameagain 3y ago
        I'm just saying that changing the symbols will make things worse, not better. Virtually all the textbooks use those symbols. Do you have a viable and better alternative to suggest or are you just complaining? And I didn't assume that you know or don't know something, I just wrote it down for the sake of the argument. We are not having a private conversation, we are contributing to a public discussion.
- eigenket 3y ago
  Syntax and notation is an incredibly minor "problem". We have much more significant (and interesting) stuff to deal with. Judging maths on its syntax is like judging a poem or work of literature on its font. It really isn't a central thing.
ahhfgshando6698 3y ago
This is going to be a classic "computer programmer wants math to be like computers and doesn't get it" kind of take, but In my view the problem of mathematical rigour (or lack thereof) has only become worse over the 20th century and we certainly did not resolve any of the underlying issues in a foundational sense or a practical sense. In a practical sense, it's become much worse and we have many more layers now. In a foundational sense, we succeeded in giving up because we learned that we can in some sense pick and choose whatever is most convenient for our line of research. That's probably okay if we view mathematics in the way this book (I have not read it, going based on the description here) advocates, as a sort of toy for playing with arguments. And I'm certainly not saying Math should ever be viewed as an empirical discipline nor constrained by that kind of thinking. But I don't think I'm the only one that takes one look at things in the realm of say higher category theory and thinks it's mostly playing word and symbol manipulation games, and lacks any real mathematical content that could not be discovered at a lower and more understandable (and less likely to produce new research) level of abstraction. I guess I've sort of betrayed that I am pretty firmly a platonist in that respect so make of that what you will. Like I said, this is not an unusual opinion for a computer person to have and I'm sure it's fairly annoying to any pure mathematician at this point. But I think it's still fair if we want to understand what turns certain people off of pursuing mathematics further.
- bananaflag 3y ago
  This is a classical example of what happens when people give up on rigour: https://en.wikipedia.org/wiki/Italian_school_of_algebraic_geometry https://en.wikipedia.org/wiki/Italian_school_of_algebraic_ge... TL;DR: They started producing false results. Note that this was not about capital-F Foundations of Mathematics like (arguably) the foundational crisis of math that had its origins in the 19th century, but rather about lowercase-f foundations of a particular field, in this case algebraic geometry. Weil's foundations of AG in the eponymous 1946 book were horrible and messy but they solved the issue (even today there are a lot of celebrated results that can only be found as expressed in Weil's language) and later in the 1960s Grothendieck provided the elegant language of schemes in which people generally learn and research AG today, and which helped prove long-standing problems like the Weil conjectures and (to some degree) FLT. Category theory was, in this case, essential to proving theorems about "real mathematical content" like numbers and points.
reacweb 3y ago
IMHO, in order to be able to speak about these subjects, we must either be a mathematician, or have read logicomix (http://www.logicomix.com http://www.logicomix.com). It is the easiest book to read. Another usefull one is https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach
- throwoutway 3y ago
  IMHO, the book was too loose with the truth (even though it was purporting to write about mathematical truths). It's a work of fiction (beyond just filling the characters' words to each other), but that wasn't revealed until the end which left me sour. It pretending to be history right up until the end GED isn't really related to this subject although it contains some of the same themes and characters
- crabbone 3y ago
  This subject actually doesn't belong in mathematics: it's philosophy of mathematics. I.e. no matter how much you study mathematics you will not be able to answer (or even attempt to answer) these questions because no mathematical tools or disciplines are designed for that. I also believe that the emphasis on "rigor" here is misplaced. The argument isn't about whether mathematical rules are rigorous or not. The argument is about whether mathematics exists independent of mathematicians (and they discover it in a way how an astronomer peers into telescope and discovers new stars) vs mathematics being created by mathematicians' minds (similar to how an architect designs a building: there weren't one before, and now there's a concept of a new building with so many walls, floors, windows etc.) I believe that mathematics is art, not science. I.e. mathematicians create new rules, they don't discover them. The whole argument to support this point would be too long to write it in a single post, but the general idea is that mathematics is a system that can easily describe counterfactual worlds. We use it to also describe our physical world because, of course, it can do that. But then asking the question about the "surprising effectiveness" is moot: we deliberately made it to be as effective as possible, so how is it so surprising that it is?
  - AnimalMuppet 3y ago

leetcodesucks 3y ago

Its still not stiff enough. Mathematicians really ought to settle on a sound and coordinated syntax

eru 3y ago
Syntax is about the least of people's worries when they worry about 'mathematical rigor'. Math is made for people to read and write, and they have different 'domain specific languages' for different parts of math. Now what would be useful is a tool, perhaps something like a large language model, to automatically translate from the notation used in one area of math into another. That can't be a fully mechanical procedure (hence the need for something flexible like an LLM), because it's part and parcel of human mathematics to abuse notation here and there in the name of ergonomics. My personal pet peeve is defining a function like f(x) := x + 3, and then treating f(x) as the name of the function, instead of just f. But really, it's just a harmless abuse of notation when done by some humans to other humans.
- rudy6912 3y ago
  I do not think mixing up 'f(x)' and 'f' is harmless, given mathematics is all about clarity. In any quality text, 'f(x)' denotes the value of 'f' at 'x', and 'f' denotes the function 'f' itself. Also, speaking of notation, I wonder why you used ':=' instead of '=' to define 'f'. There is no computation going on, right?
  - eru 3y ago
    Mixing up 'f' and 'f(x)' is mostly harmless in practice. The underlying principles are still clear enough. (And I say that as someone who would _really_ like to make that argument that people who mess this up are somehow unclear in their thinking. No, they are mostly fine. Getting 'f' vs 'f(x)' right mostly is really important for programmers who deal with higher order functions in general all the time. Most mathematicians don't fall into that category. You could say calculus deals with higher order functions, like the derivative. And that's a valid way to look at it. But most people get by just fine using special purpose notation for the derivative and not thinking about it as a function just like 'f'.) I used := to emphasis that I am defining 'f' here, not just writing down any old equation. (Eg like like in the example "Find all functions f such that f(x + 1) = x * f (x).") Though if you wanted to be pedantic about notation, I could have written that as with the x on the other side of the :=, like f := \x -> x + 3 (for Haskell inspired notation) or f := (x |-> x + 3) where |-> means the little arrow I draw by hand to denote a mapping when I'm writing math on a chalk board or piece of paper. I'm not sure why := would denote a computation? At most you might want to use it to denote an assignment in a mutable context?
    - xeonmc 3y ago
      > Getting 'f' vs 'f(x)' right mostly is really important for programmers who deal with higher order functions in general all the time. Most mathematicians don't fall into that category. Operators and functionals?
      - eru 3y ago
        It's usually clear from context what you mean, even if you work with operators and functionals.
    - bheadmaster 3y ago
      > Getting 'f' vs 'f(x)' right mostly is important for programmers who deal with higher order functions in general all the time. Most mathematicians don't fall into that category Mathematicians deal with higher order functions all the time, e.g. in functional analysis.
- atoav 3y ago
  While I agree with the point you make, I think that syntax has a huge impact. Maybe not on mathematicians themselves, but I am certain that mathematical syntax has done more to scare people away from the field than the actual mathematical problems themselves. For example the convention to use the greek alphabet for certain things. This is totally arbitrary and you could have also used emoticons instead (had they existed). But what this means is that the pupil, before tackling the meat of the mathematical problem has to accept that weird looking letter they have never seen for no real reason whatsoever. And I say that as someone who can fluently read the greek alphabet.
  - 4gotunameagain 3y ago
    It is not arbitrary, it is heritage. Apart from the fact that Greek was the de facto scientific language of the west (which is no longer the case), I think we can agree that the characters of a single alphabet are not enough for notation, especially given the fact that it is very useful to be able to discern different types of entities, e.g. constants from variables or vectors from matrices. If we changed symbols now, it would create an even bigger mess. Because the people that learn the new symbols, could not read any textbook published before 0 A.D. (Anno Discombobuli)
    - atoav 3y ago
      So you read my comment and thought: "That guy who can read the greek alphabet doesn't know there are historical roots for greek letters in mathematical notation". Sure, back then when everybody who learned trigonometry had a classical education with ancient greek picking greek letters when the latin alphabet wouldn't do was a rational decision. It just hasn't aged well.
      - 4gotunameagain 3y ago
        I'm just saying that changing the symbols will make things worse, not better. Virtually all the textbooks use those symbols. Do you have a viable and better alternative to suggest or are you just complaining? And I didn't assume that you know or don't know something, I just wrote it down for the sake of the argument. We are not having a private conversation, we are contributing to a public discussion.
eigenket 3y ago
Syntax and notation is an incredibly minor "problem". We have much more significant (and interesting) stuff to deal with. Judging maths on its syntax is like judging a poem or work of literature on its font. It really isn't a central thing.

ahhfgshando6698 3y ago

This is going to be a classic "computer programmer wants math to be like computers and doesn't get it" kind of take, but In my view the problem of mathematical rigour (or lack thereof) has only become worse over the 20th century and we certainly did not resolve any of the underlying issues in a foundational sense or a practical sense. In a practical sense, it's become much worse and we have many more layers now. In a foundational sense, we succeeded in giving up because we learned that we can in some sense pick and choose whatever is most convenient for our line of research. That's probably okay if we view mathematics in the way this book (I have not read it, going based on the description here) advocates, as a sort of toy for playing with arguments. And I'm certainly not saying Math should ever be viewed as an empirical discipline nor constrained by that kind of thinking. But I don't think I'm the only one that takes one look at things in the realm of say higher category theory and thinks it's mostly playing word and symbol manipulation games, and lacks any real mathematical content that could not be discovered at a lower and more understandable (and less likely to produce new research) level of abstraction. I guess I've sort of betrayed that I am pretty firmly a platonist in that respect so make of that what you will. Like I said, this is not an unusual opinion for a computer person to have and I'm sure it's fairly annoying to any pure mathematician at this point. But I think it's still fair if we want to understand what turns certain people off of pursuing mathematics further.

bananaflag 3y ago
This is a classical example of what happens when people give up on rigour: https://en.wikipedia.org/wiki/Italian_school_of_algebraic_geometry https://en.wikipedia.org/wiki/Italian_school_of_algebraic_ge... TL;DR: They started producing false results. Note that this was not about capital-F Foundations of Mathematics like (arguably) the foundational crisis of math that had its origins in the 19th century, but rather about lowercase-f foundations of a particular field, in this case algebraic geometry. Weil's foundations of AG in the eponymous 1946 book were horrible and messy but they solved the issue (even today there are a lot of celebrated results that can only be found as expressed in Weil's language) and later in the 1960s Grothendieck provided the elegant language of schemes in which people generally learn and research AG today, and which helped prove long-standing problems like the Weil conjectures and (to some degree) FLT. Category theory was, in this case, essential to proving theorems about "real mathematical content" like numbers and points.

reacweb 3y ago

IMHO, in order to be able to speak about these subjects, we must either be a mathematician, or have read logicomix (http://www.logicomix.com http://www.logicomix.com). It is the easiest book to read. Another usefull one is https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach

throwoutway 3y ago
IMHO, the book was too loose with the truth (even though it was purporting to write about mathematical truths). It's a work of fiction (beyond just filling the characters' words to each other), but that wasn't revealed until the end which left me sour. It pretending to be history right up until the end GED isn't really related to this subject although it contains some of the same themes and characters
crabbone 3y ago
This subject actually doesn't belong in mathematics: it's philosophy of mathematics. I.e. no matter how much you study mathematics you will not be able to answer (or even attempt to answer) these questions because no mathematical tools or disciplines are designed for that. I also believe that the emphasis on "rigor" here is misplaced. The argument isn't about whether mathematical rules are rigorous or not. The argument is about whether mathematics exists independent of mathematicians (and they discover it in a way how an astronomer peers into telescope and discovers new stars) vs mathematics being created by mathematicians' minds (similar to how an architect designs a building: there weren't one before, and now there's a concept of a new building with so many walls, floors, windows etc.) I believe that mathematics is art, not science. I.e. mathematicians create new rules, they don't discover them. The whole argument to support this point would be too long to write it in a single post, but the general idea is that mathematics is a system that can easily describe counterfactual worlds. We use it to also describe our physical world because, of course, it can do that. But then asking the question about the "surprising effectiveness" is moot: we deliberately made it to be as effective as possible, so how is it so surprising that it is?
- AnimalMuppet 3y ago