9 ms·

Invented or discovered is a debate in the philosophy of mathematics

injb 5y ago
Generally true, but harder to make this case for imaginary numbers, because we definitely didn't need them when we adopted them in the 16th century or whenever it was. We adopted them because we had already adopted the (very imaginary) rule that - * - = +. Had we decided to adopt a different rule, arithmetic would still have worked, and we'd never have needed imaginary numbers. That's why I find it hard to accept that QM needs them, even though I lack the capacity to really understand the QM argument.
- drdeca 5y ago
  uh, (-1) * (-1) = 1 holds in any ring. If you have multiplication distributing over addition, then the product of (-1) and (-1) is 1 . It isn't an arbitrary choice.
  - not2b 5y ago
    Here's a physical example for those who have trouble intuitively grasping why the product of two negative numbers is positive. Let's say we have a water tank. We can fill it by opening a faucet on the top, or drain it by unplugging the bottom. For simplicity let's say that if we fill it we add 1 liter per minute, and if we drain it we lose 1 liter per minute (or equivalently, add -1 liter per minute). Q1: if we open the top faucet for 2 minutes, how much water do we add? 1 liter/min times 2 minutes: 2 liters. Q2: if we open the bottom faucet for 2 minutes, how much water do we add? -1 liter/minute times 2 minutes, -2 liters. What about the volume two minutes ago? That would be -2 minutes, right? So suppose the drain was open, how much more water did we have -2 minutes from now, or 2 minutes ago? -2 times -1: 2 liters more.
  - Qem 5y ago
    It's arbitrary. It causes our math to be asymmetrical, dominated by negative numbers. It could be the reverse, positive-dominated, and there's even proposals to make it symmetric. See https://www.scirp.org/journal/paperinformation.aspx?paperid=111530 https://www.scirp.org/journal/paperinformation.aspx?paperid=...
    - drdeca 5y ago
      In what sense is it "dominated by negative numbers"? Half of the combinations of multiplying gives something positive and half gives something negative. That's balanced. If someone fails to count (+,-) and (-,+) separately, they're just counting wrong. Complaining that this is asymmetric is like complaining that even/odd is asymmetric, on the basis that "even plus even is even, and odd plus odd is even, but only if one is even and the other is odd, is the result odd. This is unbalanced in favor of producing even numbers".
  - nh23423fefe 5y ago
    -1 * -1 = 1 -1 * x = -1 * x + 0 = -1 * x + -x + 1*x = (-1 + 1)*x + -x = -x
- klodolph 5y ago
  > 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).
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      [deleted]
    - 5y ago
- JadeNB 5y ago
  > Had we decided to adopt a different rule, arithmetic would still have worked, and we'd never have needed imaginary numbers. As other sibling comments have said, what would your arithmetic look like that still worked, but with a different 'rule' (which is a rule just in the sense of a consequence of the axioms, not in the sense of an arbitrary choice)? If you wanted, for example, (3 + -2)(3 + -2) to equal (3)(3) + (3)(-2) + (-2)(3) + (-2)(-2), then you'd have a hard time getting that equality to hold.

injb 5y ago

Generally true, but harder to make this case for imaginary numbers, because we definitely didn't need them when we adopted them in the 16th century or whenever it was. We adopted them because we had already adopted the (very imaginary) rule that - * - = +. Had we decided to adopt a different rule, arithmetic would still have worked, and we'd never have needed imaginary numbers. That's why I find it hard to accept that QM needs them, even though I lack the capacity to really understand the QM argument.

drdeca 5y ago
uh, (-1) * (-1) = 1 holds in any ring. If you have multiplication distributing over addition, then the product of (-1) and (-1) is 1 . It isn't an arbitrary choice.
- not2b 5y ago
  Here's a physical example for those who have trouble intuitively grasping why the product of two negative numbers is positive. Let's say we have a water tank. We can fill it by opening a faucet on the top, or drain it by unplugging the bottom. For simplicity let's say that if we fill it we add 1 liter per minute, and if we drain it we lose 1 liter per minute (or equivalently, add -1 liter per minute). Q1: if we open the top faucet for 2 minutes, how much water do we add? 1 liter/min times 2 minutes: 2 liters. Q2: if we open the bottom faucet for 2 minutes, how much water do we add? -1 liter/minute times 2 minutes, -2 liters. What about the volume two minutes ago? That would be -2 minutes, right? So suppose the drain was open, how much more water did we have -2 minutes from now, or 2 minutes ago? -2 times -1: 2 liters more.
- Qem 5y ago
  It's arbitrary. It causes our math to be asymmetrical, dominated by negative numbers. It could be the reverse, positive-dominated, and there's even proposals to make it symmetric. See https://www.scirp.org/journal/paperinformation.aspx?paperid=111530 https://www.scirp.org/journal/paperinformation.aspx?paperid=...
  - drdeca 5y ago
    In what sense is it "dominated by negative numbers"? Half of the combinations of multiplying gives something positive and half gives something negative. That's balanced. If someone fails to count (+,-) and (-,+) separately, they're just counting wrong. Complaining that this is asymmetric is like complaining that even/odd is asymmetric, on the basis that "even plus even is even, and odd plus odd is even, but only if one is even and the other is odd, is the result odd. This is unbalanced in favor of producing even numbers".
- nh23423fefe 5y ago
  -1 * -1 = 1 -1 * x = -1 * x + 0 = -1 * x + -x + 1*x = (-1 + 1)*x + -x = -x
klodolph 5y ago
> 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
    [deleted]
  - 5y ago
JadeNB 5y ago
> Had we decided to adopt a different rule, arithmetic would still have worked, and we'd never have needed imaginary numbers. As other sibling comments have said, what would your arithmetic look like that still worked, but with a different 'rule' (which is a rule just in the sense of a consequence of the axioms, not in the sense of an arbitrary choice)? If you wanted, for example, (3 + -2)(3 + -2) to equal (3)(3) + (3)(-2) + (-2)(3) + (-2)(-2), then you'd have a hard time getting that equality to hold.