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As somebody who's cursorily interested in what people would call "Foundation of Mathematics", I don't even think the "ultimate" part is justified. Why is 1 + 1

by hnfong 4y ago

As somebody who's cursorily interested in what people would call "Foundation of Mathematics", I don't even think the "ultimate" part is justified. Why is 1 + 1 = 2? This isn't really a mathematics problem (I know about axiomatic systems, they were invented relatively late in the timeline of mathematical history, invented as a fiction to avoid getting to the bottom of the messier things, i.e. the social context of how mathematics was actually invented and is used). Sidenote: I have written an article (not in English) about how numbers were a linear to log(N) time complexity reduction trick for recording information, and that 1+1=2 presumes a reductionist and somewhat capitalist view of the world... Anyway, to me, mathematics is a subject that invents abstract concepts and rewards those who are capable of understanding and believing in them. That kind of thinking is probably useful in a lot of areas that have business value. The make-believe part is, I would humbly suggest, quite an important feature too. It really fits nicely into our capitalist framework playing the money game where they'd want you to believe $1+$1=$2 is an axiom, and it's best not to reach out beyond the closed, axiomatic system and ask where those $ comes from or the messy social implications of the "math".

Koshkin 4y ago
> 1+1=2 presumes a reductionist and somewhat capitalist view of the world Hopefully, not all Chinese engineers think that... > mathematics is a subject that invents abstract concepts and rewards Not according to much of the history of math. Mathematics was borne of practical needs, and it has been extremely useful in science and engineering. (Sure, you can think of the number 2, say, as an abstract concept, but to most people it is a very concrete thing - as is the definite integral, etc.) Yes, a philosophical journey can be a journey down a rabbit hole, so one should be careful...
- hnfong 4y ago
  > Hopefully, not all Chinese engineers think that... :) The main point of divergence between "Western Capitalism" and "Socialism with Chinese characteristics" isn't whether it's ok to accumulate capital, but who controls it :) I think both agree that capital is a great idea for building strong nations. > Not according to much of the history of math. Mathematics was borne of practical needs, and it has been extremely useful in science and engineering. (Sure, you can think of the number 2, say, as an abstract concept, but to most people it is a very concrete thing - as is the definite integral, etc.) I agree, and... that's kind of my point. I intended to say modern mathematics is a subject that invents abstract concepts... Historically numbers were invented to solve a practical need, and then some hobbyists came and had too much fun with it, ending up with funny things like Peano and ZFC. I've seen too many people trained in mathematics that forgot the social-historical context came first, and jump straight to the newly invented axiomatic systems to explain things as basic as 1+1=2, which to me doesn't fit well with the "ultimate get-to-the-bottom-of-things" description. I mean, to cite a specific example, while the exercise is probably entertaining to some, anyone seriously believing 2+2=4 requires thousands of lines of "proof" is probably not very good at getting to the bottom of things in a general sense. (ref: http://us.metamath.org/mpegif/mmset.html#trivia http://us.metamath.org/mpegif/mmset.html#trivia )
edanm 4y ago
I'm... not sure I'm following most of what you're saying (in this and your other comment). > Why is 1 + 1 = 2? This isn't really a mathematics problem Umm, so first of all, depending on what you mean, "why is 1+1=2" is definitely a maths problem. It's exactly in the foundations of math, as you said. Or rather, it makes more sense to think of it as "what is our system for deriving true statements, that lets us do things like 1+1=2?". > (I know about axiomatic systems, they were invented relatively late in the timeline of mathematical history, invented as a fiction to avoid getting to the bottom of the messier things, i.e. the social context of how mathematics was actually invented and is used) That's not really why axiomatic systems were developed. It was part of trying to give better foundations to math, because mathematicians ran into a lot of actual problems, mostly beginning with calculus. They managed to reach paradoxes, weird conclusions, etc, and wanted to start being more rigorous in how they treated everything in math, which has led to lots of flourishing in maths, so it was probably a good idea. > [...] 1+1=2 presumes a reductionist and somewhat capitalist view of the world... This I don't understand at all. What does this have to do with capitalism? The rest of your comment is equally weird to me. Mathematicians like to invent concepts and play with them, sure, I'm fine with that description, but how does this have anything to do with the capitalist system? > [From your other comment] I mean, to cite a specific example, while the exercise is probably entertaining to some, anyone seriously believing 2+2=4 requires thousands of lines of "proof" is probably not very good at getting to the bottom of things in a general sense. (ref: http://us.metamath.org/mpegif/mmset.html#trivia http://us.metamath.org/mpegif/mmset.html#trivia ) I think you're confusing two things here. For the most part, most mathematicians don't need to bother with "foundations" questions, or reach a point where they're proving 1+1=2. It can be helpful to "peak under the hood" of various definitions to understand how they are defined rigorously, and more importantly, it can be fun. But most mathematicians use numbers as a black box, never needing the detailed definition of how numbers work. As for taking 2000 lines to prove this, that's probably only when you want to rigorously use a computer proof or something from first first principles. Given a set of axioms (e.g. Peano), I'm pretty sure it's easy to prove 1+1=2 in only a few steps. And again, this is mostly of interest to people who actually want to work on this kind of foundational thing, not the average mathematician I wouldn't think.

by hnfong 4y ago

Koshkin 4y ago

> 1+1=2 presumes a reductionist and somewhat capitalist view of the world Hopefully, not all Chinese engineers think that... > mathematics is a subject that invents abstract concepts and rewards Not according to much of the history of math. Mathematics was borne of practical needs, and it has been extremely useful in science and engineering. (Sure, you can think of the number 2, say, as an abstract concept, but to most people it is a very concrete thing - as is the definite integral, etc.) Yes, a philosophical journey can be a journey down a rabbit hole, so one should be careful...

hnfong 4y ago
> Hopefully, not all Chinese engineers think that... :) The main point of divergence between "Western Capitalism" and "Socialism with Chinese characteristics" isn't whether it's ok to accumulate capital, but who controls it :) I think both agree that capital is a great idea for building strong nations. > Not according to much of the history of math. Mathematics was borne of practical needs, and it has been extremely useful in science and engineering. (Sure, you can think of the number 2, say, as an abstract concept, but to most people it is a very concrete thing - as is the definite integral, etc.) I agree, and... that's kind of my point. I intended to say modern mathematics is a subject that invents abstract concepts... Historically numbers were invented to solve a practical need, and then some hobbyists came and had too much fun with it, ending up with funny things like Peano and ZFC. I've seen too many people trained in mathematics that forgot the social-historical context came first, and jump straight to the newly invented axiomatic systems to explain things as basic as 1+1=2, which to me doesn't fit well with the "ultimate get-to-the-bottom-of-things" description. I mean, to cite a specific example, while the exercise is probably entertaining to some, anyone seriously believing 2+2=4 requires thousands of lines of "proof" is probably not very good at getting to the bottom of things in a general sense. (ref: http://us.metamath.org/mpegif/mmset.html#trivia http://us.metamath.org/mpegif/mmset.html#trivia )

edanm 4y ago

I'm... not sure I'm following most of what you're saying (in this and your other comment). > Why is 1 + 1 = 2? This isn't really a mathematics problem Umm, so first of all, depending on what you mean, "why is 1+1=2" is definitely a maths problem. It's exactly in the foundations of math, as you said. Or rather, it makes more sense to think of it as "what is our system for deriving true statements, that lets us do things like 1+1=2?". > (I know about axiomatic systems, they were invented relatively late in the timeline of mathematical history, invented as a fiction to avoid getting to the bottom of the messier things, i.e. the social context of how mathematics was actually invented and is used) That's not really why axiomatic systems were developed. It was part of trying to give better foundations to math, because mathematicians ran into a lot of actual problems, mostly beginning with calculus. They managed to reach paradoxes, weird conclusions, etc, and wanted to start being more rigorous in how they treated everything in math, which has led to lots of flourishing in maths, so it was probably a good idea. > [...] 1+1=2 presumes a reductionist and somewhat capitalist view of the world... This I don't understand at all. What does this have to do with capitalism? The rest of your comment is equally weird to me. Mathematicians like to invent concepts and play with them, sure, I'm fine with that description, but how does this have anything to do with the capitalist system? > [From your other comment] I mean, to cite a specific example, while the exercise is probably entertaining to some, anyone seriously believing 2+2=4 requires thousands of lines of "proof" is probably not very good at getting to the bottom of things in a general sense. (ref: http://us.metamath.org/mpegif/mmset.html#trivia http://us.metamath.org/mpegif/mmset.html#trivia ) I think you're confusing two things here. For the most part, most mathematicians don't need to bother with "foundations" questions, or reach a point where they're proving 1+1=2. It can be helpful to "peak under the hood" of various definitions to understand how they are defined rigorously, and more importantly, it can be fun. But most mathematicians use numbers as a black box, never needing the detailed definition of how numbers work. As for taking 2000 lines to prove this, that's probably only when you want to rigorously use a computer proof or something from first first principles. Given a set of axioms (e.g. Peano), I'm pretty sure it's easy to prove 1+1=2 in only a few steps. And again, this is mostly of interest to people who actually want to work on this kind of foundational thing, not the average mathematician I wouldn't think.