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> 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 'inevit

by 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?
    - joe__f 5y ago
      So I take it that you're happy that the natural numbers exist. Staring from there, this is where the ring axioms came from; Different people sat down over the last millennium or so and realised that the natural numbers could only express quantity and not direction. They looked around them at the physical reality that they found themselves in, and noticed that sometimes it was useful to count in the opposite direction (ie. decreasing). Eventually they realised that they could do this by adding in some new numbers that were ordered below 0, and which functioned as additive inverses to the natural numbers. This process was constrained by the properties of reality; so the properties of the negative numbers are not arbitrary, they are fixed by properties of the world that different people noticed. Eventually (around the end of the 19th century maybe?) people tried to formalise the properties of these numbers in axiomatic schema. They noticed that certain of these axioms could also describe other extended algebraic objects which are a bit like the integers, and called these objects rings. So that's where the ring axioms came from; by observing reality and then generalising
      - klodolph 5y ago
        Please don't reply to one person in two places in the same thread. The natural numbers don’t satisfy the ring axioms. Perhaps this is the flaw in your reasoning.
- joe__f 5y ago
  Based on "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"*, I think you're arguing that the axioms for the negative numbers are arbitrary. This is not true, as the axioms from the negative numbers are constrained by reality the reality we live in. Negative numbers need to model different aspects of physical reality; for example they need to describe where you end up relative to your starting point when you walk 5m in one direction, and then walk 6m back in the exact opposite direction. So the axioms for the negative numbers are not arbitrary. I think they are probably unique given certain physical requirements, but I haven't thought or read about this.
  - klodolph 5y ago
    > Negative numbers need to model different aspects of physical reality; They don't "need" to, but it is very useful that they do.
    - joe__f 5y ago
      I would argue that they need to by definition, and that if they don't then they are no longer the negative numbers
      - klodolph 5y ago
        Again, that’s begging the question. You are assuming certain axioms, and the question is “why did we assume these axioms?” Using the word “definition” rather than “axiom” is just a way of rephrasing the argument. If your answer to “why are negative numbers defined to work this way” is “because that’s the way negative numbers are defined,” I’m gonna keep responding by saying that you’re begging the question. Rephrasing the argument only helps if I misunderstand your argument, and I don’t.
        joe__f 5y ago
        No, I'm not assuming any axioms. I'm describing the process of observing properties of nature and then finding a formalised description of those properties. The negative numbers are what we call the system of numbers which arose in this [https://news.ycombinator.com/item?id=30736335 https://news.ycombinator.com/item?id=30736335] way, and we then found an axiomatic description afterwards. Aliens would also find negative numbers in this way, but they'd call them something different (presumably). You're free to choose your own whatever other axioms for your 'negative numbers' that you want. If they don't describe the process of counting downwards or walking in opposite directions like I described, then I won't call them negative numbers.

by JadeNB 5y ago

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?
  - joe__f 5y ago
    So I take it that you're happy that the natural numbers exist. Staring from there, this is where the ring axioms came from; Different people sat down over the last millennium or so and realised that the natural numbers could only express quantity and not direction. They looked around them at the physical reality that they found themselves in, and noticed that sometimes it was useful to count in the opposite direction (ie. decreasing). Eventually they realised that they could do this by adding in some new numbers that were ordered below 0, and which functioned as additive inverses to the natural numbers. This process was constrained by the properties of reality; so the properties of the negative numbers are not arbitrary, they are fixed by properties of the world that different people noticed. Eventually (around the end of the 19th century maybe?) people tried to formalise the properties of these numbers in axiomatic schema. They noticed that certain of these axioms could also describe other extended algebraic objects which are a bit like the integers, and called these objects rings. So that's where the ring axioms came from; by observing reality and then generalising
    - klodolph 5y ago
      Please don't reply to one person in two places in the same thread. The natural numbers don’t satisfy the ring axioms. Perhaps this is the flaw in your reasoning.
joe__f 5y ago
Based on "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"*, I think you're arguing that the axioms for the negative numbers are arbitrary. This is not true, as the axioms from the negative numbers are constrained by reality the reality we live in. Negative numbers need to model different aspects of physical reality; for example they need to describe where you end up relative to your starting point when you walk 5m in one direction, and then walk 6m back in the exact opposite direction. So the axioms for the negative numbers are not arbitrary. I think they are probably unique given certain physical requirements, but I haven't thought or read about this.
- klodolph 5y ago
  > Negative numbers need to model different aspects of physical reality; They don't "need" to, but it is very useful that they do.
  - joe__f 5y ago
    I would argue that they need to by definition, and that if they don't then they are no longer the negative numbers
    - klodolph 5y ago
      Again, that’s begging the question. You are assuming certain axioms, and the question is “why did we assume these axioms?” Using the word “definition” rather than “axiom” is just a way of rephrasing the argument. If your answer to “why are negative numbers defined to work this way” is “because that’s the way negative numbers are defined,” I’m gonna keep responding by saying that you’re begging the question. Rephrasing the argument only helps if I misunderstand your argument, and I don’t.
      - joe__f 5y ago
        No, I'm not assuming any axioms. I'm describing the process of observing properties of nature and then finding a formalised description of those properties. The negative numbers are what we call the system of numbers which arose in this [https://news.ycombinator.com/item?id=30736335 https://news.ycombinator.com/item?id=30736335] way, and we then found an axiomatic description afterwards. Aliens would also find negative numbers in this way, but they'd call them something different (presumably). You're free to choose your own whatever other axioms for your 'negative numbers' that you want. If they don't describe the process of counting downwards or walking in opposite directions like I described, then I won't call them negative numbers.