Invented or discovered is a debate in the philosophy of mathematics
If you have multiplication distributing over addition, then the product of (-1) and (-1) is 1 .
It isn't an arbitrary choice.
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.
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".
-1 * -1 = 1
-1 * x =
-1 * x + 0 =
-1 * x + -x + 1*x =
(-1 + 1)*x + -x =
-xThis 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.
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).
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.
> 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.
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?
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
The natural numbers don’t satisfy the ring axioms. Perhaps this is the flaw in your reasoning.
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.
They don't "need" to, but it is very useful that they do.
Rephrasing the argument only helps if I misunderstand your argument, and I don’t.
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.
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.
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
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.
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.