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> We adopted them because we had already adopted the (very imaginary) rule that - * - = +. This is just an extension of ordinary addition and multiplication wi

> We adopted them because we had already adopted the (very imaginary) rule that - * - = +. This is just an extension of ordinary addition and multiplication with natural numbers. Starting with “2 x 3 = 6” it is inevitable that you’d come up with “(-2) x (-3) = 6”. It’s not some kind of imaginary or weird rule. If we extend our numbers to include negative numbers, and we want to preserve as much behavior as we can from what we observed multiplying positive numbers, then this is the only sensible way of doing things. When you start with something simple (like positive integers) and extend it, you keep some properties, gain some new properties, and lose some properties. For example, in the transition from rational to real numbers we gain the property that all Cauchy sequences converge. In the transition from real to complex we gain the property that the number of solutions to a nonzero polynomial is equal to the polynomial’s degree, but we lose the property that numbers are ordered. There are some very deep reasons why complex numbers are a natural choice for doing things in functional analysis. It’s definitely a sweet spot… surprisingly, there’s a concept called “holomorphic functions” which is a very tight constraint on functions, yet simultaneously a right field of study, and it’s the foundation of QM. If you move down the ladder to real numbers, the concept of holomorphic functions does not exist. If you move up the ladder to quaternions or octonions, you lose some critical properties like commutativity. > …we definitely didn't need them when we adopted them in the 16th century… They were necessary for solving polynomial equations… even finding real solutions to polynomials with real coefficients.

JadeNB 5y ago
> Starting with “2 x 3 = 6” it is inevitable that you’d come up with “(-2) x (-3) = 6”. To be clear (as you surely know but a reader might not), that's 'inevitable' in the sense of "a mathematical consequence", not 'inevitable' in the weaker human-events sense of "I can't imagine it winding up any other way". For one approach, 0 = (2 + (-2))3 = (2)(3) + (-2)(3) implies that (-2)(3) = -(2)(3); and then 0 = (-2)(3 + (-3)) = (-2)(3) = (-2)(-3) implies that (-2)(-3) = -(-2)(3) = (2)(3).
- deleted 5y ago
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- klodolph 5y ago
  The short version is “that’s begging the question”. That interpretation wasn’t what I intended. I intended the weaker “I can’t imagine it winding up any other way”. It’s not inevitable as a mathematical consequence. If we start with the counting numbers and formulate multiplication as repeated addition, 2x3=6. We can arrive at this conclusion by constructing a model of natural numbers, or from an axiomatic approach. If we decide that we want to introduce negative numbers, we are going to end up with a different model or a different set of axioms to accommodate them. The conclusion that (-2)x(-3)=6 is obvious only in the sense that our choice of axioms for negative numbers is obvious—it just seems like the right way to define negative numbers. For example, your proof relies on the distributive property of multiplication… something that we chose to preserve in our axioms. But we were not forced to preserve this axiom when defining negative numbers, hence it is begging the question. Just because an axiom is part of our formulation for natural numbers does not mean it is part of our formulation for positive and negative numbers. For more information on this topic, the “transfer principle” article on Wikipedia is really interesting. It dives into other extensions of numbers, such as the hyperreals. This principle is the missing ingredient here. https://en.wikipedia.org/wiki/Transfer_principle https://en.wikipedia.org/wiki/Transfer_principle
  - JadeNB 5y ago
    > The short version is “that’s begging the question”. > That interpretation wasn’t what I intended. I intended the weaker “I can’t imagine it winding up any other way”. It’s not inevitable as a mathematical consequence. I dare not argue with you what you meant, and apologise for assuming, but it very definitely is inevitable as a mathematical consequence. What I wrote was not a sketch or a hint, but a proof, from the ring axioms.
    - klodolph 5y ago
      > What I wrote was not a sketch or a hint, but a proof, from the ring axioms. Hence "begging the question"... you're assuming the ring axioms hold for the negative numbers. But they do not hold for the natural numbers in the first place... so where did the ring axioms come from, and why must they hold?
8note 5y ago
Inevitable? It could just as easily come out to -6, and have something else for -2*3 or 2*-3
- klodolph 5y ago
  No, it turns out that if you do that, it's hard to come up with definition of multiplication that people will accept. That's why (-2)x(-3)=6 is inevitable. The only "easy" thing is writing down an equation like (-2)x(-3)=6. That's not what we're doing, though. We're defining how multiplication works, and (-2)x(-3) is just an example.

JadeNB 5y ago

> Starting with “2 x 3 = 6” it is inevitable that you’d come up with “(-2) x (-3) = 6”. To be clear (as you surely know but a reader might not), that's 'inevitable' in the sense of "a mathematical consequence", not 'inevitable' in the weaker human-events sense of "I can't imagine it winding up any other way". For one approach, 0 = (2 + (-2))3 = (2)(3) + (-2)(3) implies that (-2)(3) = -(2)(3); and then 0 = (-2)(3 + (-3)) = (-2)(3) = (-2)(-3) implies that (-2)(-3) = -(-2)(3) = (2)(3).

deleted 5y ago
[deleted]
klodolph 5y ago
The short version is “that’s begging the question”. That interpretation wasn’t what I intended. I intended the weaker “I can’t imagine it winding up any other way”. It’s not inevitable as a mathematical consequence. If we start with the counting numbers and formulate multiplication as repeated addition, 2x3=6. We can arrive at this conclusion by constructing a model of natural numbers, or from an axiomatic approach. If we decide that we want to introduce negative numbers, we are going to end up with a different model or a different set of axioms to accommodate them. The conclusion that (-2)x(-3)=6 is obvious only in the sense that our choice of axioms for negative numbers is obvious—it just seems like the right way to define negative numbers. For example, your proof relies on the distributive property of multiplication… something that we chose to preserve in our axioms. But we were not forced to preserve this axiom when defining negative numbers, hence it is begging the question. Just because an axiom is part of our formulation for natural numbers does not mean it is part of our formulation for positive and negative numbers. For more information on this topic, the “transfer principle” article on Wikipedia is really interesting. It dives into other extensions of numbers, such as the hyperreals. This principle is the missing ingredient here. https://en.wikipedia.org/wiki/Transfer_principle https://en.wikipedia.org/wiki/Transfer_principle
- JadeNB 5y ago
  > The short version is “that’s begging the question”. > That interpretation wasn’t what I intended. I intended the weaker “I can’t imagine it winding up any other way”. It’s not inevitable as a mathematical consequence. I dare not argue with you what you meant, and apologise for assuming, but it very definitely is inevitable as a mathematical consequence. What I wrote was not a sketch or a hint, but a proof, from the ring axioms.
  - klodolph 5y ago
    > What I wrote was not a sketch or a hint, but a proof, from the ring axioms. Hence "begging the question"... you're assuming the ring axioms hold for the negative numbers. But they do not hold for the natural numbers in the first place... so where did the ring axioms come from, and why must they hold?

8note 5y ago

Inevitable? It could just as easily come out to -6, and have something else for -2*3 or 2*-3

klodolph 5y ago
No, it turns out that if you do that, it's hard to come up with definition of multiplication that people will accept. That's why (-2)x(-3)=6 is inevitable. The only "easy" thing is writing down an equation like (-2)x(-3)=6. That's not what we're doing, though. We're defining how multiplication works, and (-2)x(-3) is just an example.