5 ms·

I would argue that they need to by definition, and that if they don't then they are no longer the negative numbers

by joe__f 5y ago

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.
  - klodolph 5y ago
    > No, I'm not assuming any axioms. And yet, > I'm describing the process of observing properties of nature and then finding a formalised description of those properties. A "formalized description of those properties" is known as a set of axioms. That's just the name for it. If you "found something in nature" which you call the "negative numbers," what you've done is constructed something, labeled it "negative numbers", and derived axioms from its properties. It does not really matter if you come up with the construction first and derive the axioms afterwards, or if you come up with the axioms first and later derive a construction that satisfies those axioms. If you did not take upper division mathematics, you would likely not be familiar with this equivalence. Saying you "found something in nature" doesn't really relieve you of choice here, because there is more than one way to create a system of "integers" that contain negative integers, just like there are several different systems of natural numbers. > Aliens would also find negative numbers in this way, but they'd call them something different (presumably). Yes, that's exactly what I was saying when I said that this definition was "inevitable". It's so obvious and makes so much sense that there's just no reasonable chance that we'd come up with something else. But it's not inevitable in the sense that introducing additive inverses into your system of numbers forces you to conclude that, say, negative numbers satisfy the distributive property. > You're free to choose your own whatever other axioms for your 'negative numbers' that you want. If you think that the actual definition of "negative number" is somehow in dispute than you definitely misunderstood the argument.
    - joe__f 5y ago
      No, I haven't "constructed something, labelled it the natural numbers and derived axioms from its properties". For me, the negative numbers are an aspect of nature and I came across them by observation (well, other people did this and someone told me about them when I was a kid, but I can now observe them). Similarly to how I might observe trees and see that there's a property of greenness to their leaves, and then come up with a word to describe what I see. I think we're working on a different ontology of what mathematics is. Saying you "found something in nature" doesn't really relieve you of choice here, because there is more than one way to create a system of "integers" that contain negative integers, just like there are several different systems of natural numbers. I think you're talking about something like a non-standard model of the Peano axioms? My understanding is that this can only put in new 'numbers' c such that n < c for all natural numbers n. So, sure I've chosen a specific model of these axioms, but the model agrees with all other models for every finite number. Either way I don't really see why making a choice here detracts from my argument; numbers exist as a part of nature, and we can describe them with Peano axioms and a choice of model. I didn't assume any (specific mathematical) axioms in the first place, I assumed that reality exists and that anything that can be conceived of is a part of reality. I personally confirm for myself that this is true by observation, although you may not have done so for yourself. But it's not inevitable in the sense that introducing additive inverses into your system of numbers forces you to conclude that, say, negative numbers satisfy the distributive property. I don't really get what point you’re making here

I would argue that they need to by definition, and that if they don't then they are no longer the negative numbers

by 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

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.
- klodolph 5y ago
  > No, I'm not assuming any axioms. And yet, > I'm describing the process of observing properties of nature and then finding a formalised description of those properties. A "formalized description of those properties" is known as a set of axioms. That's just the name for it. If you "found something in nature" which you call the "negative numbers," what you've done is constructed something, labeled it "negative numbers", and derived axioms from its properties. It does not really matter if you come up with the construction first and derive the axioms afterwards, or if you come up with the axioms first and later derive a construction that satisfies those axioms. If you did not take upper division mathematics, you would likely not be familiar with this equivalence. Saying you "found something in nature" doesn't really relieve you of choice here, because there is more than one way to create a system of "integers" that contain negative integers, just like there are several different systems of natural numbers. > Aliens would also find negative numbers in this way, but they'd call them something different (presumably). Yes, that's exactly what I was saying when I said that this definition was "inevitable". It's so obvious and makes so much sense that there's just no reasonable chance that we'd come up with something else. But it's not inevitable in the sense that introducing additive inverses into your system of numbers forces you to conclude that, say, negative numbers satisfy the distributive property. > You're free to choose your own whatever other axioms for your 'negative numbers' that you want. If you think that the actual definition of "negative number" is somehow in dispute than you definitely misunderstood the argument.
  - joe__f 5y ago
    No, I haven't "constructed something, labelled it the natural numbers and derived axioms from its properties". For me, the negative numbers are an aspect of nature and I came across them by observation (well, other people did this and someone told me about them when I was a kid, but I can now observe them). Similarly to how I might observe trees and see that there's a property of greenness to their leaves, and then come up with a word to describe what I see. I think we're working on a different ontology of what mathematics is. Saying you "found something in nature" doesn't really relieve you of choice here, because there is more than one way to create a system of "integers" that contain negative integers, just like there are several different systems of natural numbers. I think you're talking about something like a non-standard model of the Peano axioms? My understanding is that this can only put in new 'numbers' c such that n < c for all natural numbers n. So, sure I've chosen a specific model of these axioms, but the model agrees with all other models for every finite number. Either way I don't really see why making a choice here detracts from my argument; numbers exist as a part of nature, and we can describe them with Peano axioms and a choice of model. I didn't assume any (specific mathematical) axioms in the first place, I assumed that reality exists and that anything that can be conceived of is a part of reality. I personally confirm for myself that this is true by observation, although you may not have done so for yourself. But it's not inevitable in the sense that introducing additive inverses into your system of numbers forces you to conclude that, say, negative numbers satisfy the distributive property. I don't really get what point you’re making here