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What is the ontological state of negative numbers?

Is it a human invention or a does it live with reality?

The Windmill King
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    If you have no money and you owe someone a dollar, then you have less than zero dollars. If you're standing below ground level, your altitude is less than $0$. – littleO Feb 09 '14 at 03:01
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    Define reality. – Emily Feb 09 '14 at 03:02
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    This question is probably more linguistics than mathematics -- specifically the analysis of what precise concept the questioner is trying to convey by the words "something" and "nothing", which most likely are not concepts of "nonzero integer" and "the integer zero" or the like. –  Feb 09 '14 at 03:03
  • Your question is unclear. That's the reason I downvoted it. I probably wouldn't have downvoted it for being duplicate because sometimes when somebody asks a duplicate, they have a real question. What is the specific problem you're wondering about? Did you mean something like "I've heard of negative numbers. Also a number always corresponds to an amount of something. From this, I can derive that there are no negative numbers. How is this possible?" I think it's fine for you to ask another question that's a fixed up version of this question to not invalidate its answers. I think it might also be – Timothy Jun 03 '19 at 00:10
  • fine for you to fix up this question if you're sure fixing it up the way you were going to won't invalidate any of the answers. – Timothy Jun 03 '19 at 00:12

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The modern view of mathematics is that of the axiomatic method. We don't care wether some concepts exist in some idealised plane of existence, we just define their properties in a matter that's unequivocal, and then proceed to derive further theorems based on certain rules of logical inference.

vonbrand
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Felipe Jacob
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  • Yes but axioms must relate to objects, concepts and phenomena that exist, should exist or are in some way rational. There should be concrete motivation behind the axioms. Or you could easily end up proving theorems on the axioms, and have developed something that applies to nothing i.e. you would be operating in a void.. – Ishfaaq Feb 09 '14 at 03:13
  • @Ishfaaq: What's the big deal? So what if we happen to develop an axiomatic system that applies to nothing in the world? There are worse things that can happen. – Zev Chonoles Feb 09 '14 at 03:34
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    @Ishfaaq Most of modern pure mathematics is done for intellectual pleasure and curiosity, and it does indeed operate in a void as far as applications are concerned. People develop new objects because they are curious about what they will imply by themselves, and not always to explain natural phenomena. – Felipe Jacob Feb 09 '14 at 03:36
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They can be seen as a pair of information; a magnitude (real, positive), together with a direction.