3 ms·

Really? How do we know that?

by dave4420 4y ago

eimrine 4y ago
Why do we know that 1+1=2?
- ben_w 4y ago
  Wasn’t that “by definition” for a few millennia, then “that’s not good enough, let me invent set theory, oh no there’s an irreconcilable flaw and now I’m depressed” followed by nonchalant whistling by subsequent mathematicians?
- necovek 4y ago
  Pretty much because we define it as such. I.e. in traditional(*) axiomatic setup of the natural number system, "+1" is called the "successor" operation, and apart from "there being 1", we also assume that "there is a successor to 1" and then call that "2". Everything else (remarkably) flows from there: all the other numbers (whole numbers, rational, real, complex numbers), most of the Mathematical Analysis with the infinitesimals, differentiation and integrals and differential equations... After the summation is defined as an operation having a concrete set of properties, it's proven that "+1" is equivalent to "succ(1)". (*) A more recent approach is to use set cardinality instead of the "successor" approach, but regardless of the approach, even in maths, we simply take some things for granted. I find the successor approach much more approachable for the general public though, since it's so simple, and truly requires not too much (mathematical) intelligence.
  - eimrine 4y ago
    You gave a perfect answer, much better than I expected. But I mean, a long time ago, 1+1 meant, "Look, Harry, what that eccentric Pythagoras is talking about again." Some research can be difficult, like how to kill this mammoth, or how to kill any mammoth, or how to make a livestock farm so there is always fresh meat. But mathematical research is the most difficult, because only the most omnipotent knowledge is considered mathematics. That's why OP's point [1] is valid but maybe the definition of "mathematician" word is missing for making the question interesting. [1] https://news.ycombinator.com/item?id=32943723 https://news.ycombinator.com/item?id=32943723

Really? How do we know that?

eimrine 4y ago

Why do we know that 1+1=2?

ben_w 4y ago
Wasn’t that “by definition” for a few millennia, then “that’s not good enough, let me invent set theory, oh no there’s an irreconcilable flaw and now I’m depressed” followed by nonchalant whistling by subsequent mathematicians?
necovek 4y ago
Pretty much because we define it as such. I.e. in traditional(*) axiomatic setup of the natural number system, "+1" is called the "successor" operation, and apart from "there being 1", we also assume that "there is a successor to 1" and then call that "2". Everything else (remarkably) flows from there: all the other numbers (whole numbers, rational, real, complex numbers), most of the Mathematical Analysis with the infinitesimals, differentiation and integrals and differential equations... After the summation is defined as an operation having a concrete set of properties, it's proven that "+1" is equivalent to "succ(1)". (*) A more recent approach is to use set cardinality instead of the "successor" approach, but regardless of the approach, even in maths, we simply take some things for granted. I find the successor approach much more approachable for the general public though, since it's so simple, and truly requires not too much (mathematical) intelligence.
- eimrine 4y ago
  You gave a perfect answer, much better than I expected. But I mean, a long time ago, 1+1 meant, "Look, Harry, what that eccentric Pythagoras is talking about again." Some research can be difficult, like how to kill this mammoth, or how to kill any mammoth, or how to make a livestock farm so there is always fresh meat. But mathematical research is the most difficult, because only the most omnipotent knowledge is considered mathematics. That's why OP's point [1] is valid but maybe the definition of "mathematician" word is missing for making the question interesting. [1] https://news.ycombinator.com/item?id=32943723 https://news.ycombinator.com/item?id=32943723