"Zero has an inverse" is true IFF zero has an inverse.

Problem solved.

Assuming you have the semantics of "inverse" to hand, which evidently we don't. In mathematics, it's stipulated. It can be stipulated in other contexts as well.

The trouble comes of what fills the role of stipulation in everyday usage of a natural language.

The trouble comes of what fills the role of stipulation in everyday usage of a natural language. — Srap Tasmaner

I don't know. There are jargons everywhere, in sports, in the law, in engineering, in medicine, etc. But I don't think there's much stipulation going on.

The original creator of C-language spent most of the introduction of his book complaining that C is not a language because "language" refers to the use of the tongue, and your tongue is useless in computer programming. He was already subject to a jargon that developed somewhat organically.

But I don't think there's much stipulation going on. — frank

Yeah that's what I meant.

I'd forgotten Dennis Ritchie talks about that, but computer scientists (not coders) spend a fair amount of time thinking about semantics. When Jim Backus and his team at IBM invented the first high-level programming language, they had to simultaneously figure out what such a thing would be, and also invented a formal way of specifying its grammar, the Backus-Naur Form still used today.

I'd forgotten Dennis Ritchie talks about that, but computer scientists (not coders) spend a fair amount of time thinking about semantics. When Jim Backus and his team at IBM invented the first high-level programming language, they had to simultaneously figure out what such a thing would be, and also invented a formal way of specifying its grammar, the Backus-Naur Form still used today. — Srap Tasmaner

I once made a program in machine language and burned it into proms. I'm guessing the higher level syntax would follow necessity to some extent? The purpose was to speed things up so that bigger, more elaborate programs could be written. There's stipulation in that, I guess, with necessity as a rudder.

What is math's rudder? What necessity would inspire us to talk in terms of +0?

What is math's rudder? What necessity would inspire us to talk in terms of +0? — frank

Yeah that's a funny thing. Math often allows degenerate cases to pass through for the sake of generality. If you need to say every integer has a sign (for whatever reason) then you'll need 0 to have a sign. Which one? That strikes me as a deep question, in the sense that your reason for giving it a sign is probably not powerful enough to dictate which sign; you'll need some other reason for saying which, and that reason is likely to be "deeper" if you see what I mean.

Something like this problem comes up with 0⁰. We have one rule that would assign it the value 0, and another that would assign it the value 1. Which one should win is an interesting question.

If you need to say every integer has a sign (for whatever reason) then you'll need 0 to have a sign. Which one? That strikes me as a deep question, in the sense that your reason for giving it a sign is probably not powerful enough to dictate which sign; you'll need some other reason for saying which, and that reason is likely to be "deeper" if you see what I mean. — Srap Tasmaner

Deeper into what? Cognitive imperatives?

Cognitive imperatives? — frank

No, I was thinking more fundamental mathematical principles, or how mathematics as a system works. Things like harmony, symmetry, orthogonality, duality, that kind of stuff. You might want signs just for convenience, but there will have to be a deeper coherence to how a signed number system works, and that's where you'd look to decide whether 0 is +0 or -0.

I have no idea what you're talking about (BS in Electrical Engineering, higher degrees in Mathematics).

How a reflection is constructed :

Given a point of reflection A, let B be any point in the infinite domain containing A (the domain can be a line, plane, or space). Draw line AB and let x be the distance from A to B. Find point B' on AB which is distance x from A but on the opposite side of A from B. Then we say B' is the image of B under a reflection in A.

If you admit that the definition of "opposite" includes "across from", then clearly B' is opposite B with respect to A.

Now consider point A itself. Following the construction given above, A' (the image of A) will be the same point as A (here, distance x=0). That is, the reflection of A in A is A. Thus A is opposite to itself with respect to A.

So, let the domain be the number line and replace A with 0. Clearly each negative number is the image of its corresponding positive value under a reflection in 0 (and vice versa). Now here's the kicker : 0 is a reflection of itself. I.e., 0 is opposite (across from) itself.

By the way, 0 is neither positive nor negative, so let's drop that nonsense now.

Actually $0^0$ is called indeterminate and has no value. Any rule you're trying to use to assign a value is not applicable.

So, let the domain be the number line and replace A with 0. Clearly each negative number is the image of its corresponding positive value under a reflection in 0 (and vice versa). Now here's the kicker : 0 is a reflection of itself. I.e., 0 is opposite (across from) itself. — Real Gone Cat

Do you know of any practical use for this information?

frank

No, I was thinking more fundamental mathematical principles, or how mathematics as a system works. Things like harmony, symmetry, orthogonality, duality, that kind of stuff. — Srap Tasmaner

Aren't those things features of how the human mind works?

I was addressing the idea that 0 cannot be across from itself. Now you want applications? I don't get you at all.

I was addressing the idea that 0 cannot be across from itself. Now you want applications? I don't get you at all. — Real Gone Cat

I'll take that as a "no"

Why do you need an application? That was not the question being addressed. Most of your posts address the nature of 0 and mine have as well. What are you aiming at?

What are you aiming at? — Real Gone Cat

Meaning is use.

Then I take it you don't recognize pure math as having meaning. I wonder what this implies for philosophy?

Then I take it you don't recognize pure math as having meaning — Real Gone Cat

If what you're saying is meaningful, there should be some use somewhere.

I take it you know of no practical use, but maybe there are non-practical "pure math" uses. If not, then what you're saying is mumbo jumbo.

Aah, so math only has meaning if applied. Then why have you spent an entire page exchanging banal musings about the nature of 0 with Srap and others?

Hmmm, let's see ...

Farmers, doctors, lawyers, scientists, mathematicians, philosophers, ... Which of these has the least practical use? Which could society lose without blinking?

You're the one who seems to be insisting that the rules you've mentioned have no use even within the realm of math itself.

Interesting.

Aren't those things features of how the human mind works? — frank

No, I don't think so. Not exactly.

We have some basic intuitions about collecting and counting, about geometry, and so on, and we build mathematics out of those by making choices, our axioms, and then those axioms have logical consequences.

That means the consequences are implicit in what we made, but not in us as its makers; the properties of buildings are different from the properties of builders.

But really to get there I have to say that I don't think logic is just in our heads, anymore than the physics that underlies structural engineering is. And I don't.

I also don't think our intuitions about counting and geometry are just in our heads, but that doesn't matter for the point I'm making. I think.

Actually 0^0 is called indeterminate and has no value. Any rule you're trying to use to assign a value is not applicable. — Real Gone Cat

Okay. But isn't that just to say either there's no math that defines a value for it or that you're unfamiliar with math that does.

To just say, nope, is like saying negative numbers don't have square roots, or, for that matter, that 2 doesn't.

Math is a big place.

As for 0 carrying a sign, we could have such a system. 0 could be canonically positive and I don't think it would affect anything.

Also consider what happens if you're solving an equation and get all the way down to x = -x. You can see the answer, or you can take another step or two and get x = 0. Which means you can substitute back into the equations you had before, and in particular into x = -x, so that now says 0 = -0. You can make that go away, if you like, but even to do that you have to accept -0 as well-formed. And in fact, I could see having 0 = -0 be a theorem of a system that allowed 0 a sign, or even a definition of zero, that it is the only number for which this is true.

Math is a big place.

We have some basic intuitions about collecting and counting, about geometry, and so on, and we build mathematics out of those by making choices, our axioms, and then those axioms have logical consequences. — Srap Tasmaner

If you're talking about the axioms that protect set theory from paradoxes, you're right. There's nothing intuitive about those axioms. It's debatable whether math really needs set theory as a foundation, though. That's the danger of fiat. Once you're free of any rudder, anything goes.

A curious statement. All the years I've practiced math I can't recall using "opposite" in this way. But I suppose some do. — jgill

↪jgill

Major Edit : "Opposite" is perfectly fine when discussing positives and negatives — Real Gone Cat

This discussion revolves around the use of the word "opposite" in math. Apart from integers - the focus here - it arises in discussions of geometry, like a side opposite the hypotenuse, or in a complex vector field with an indifferent fixed point - on one side an attractor and on the other or opposite side a repellor. And so on.

Why not use a pair of these? $0+\text{ }0-\text{ }{{0}^{+}}\text{ }{{0}^{-}}$. They are commonly used in math. You could come up with the first being infinitesimals just to the right of zero, etc.
There are your "opposites" of zero.

Does quantum physics say anything about this?

How does he know other people exist? — Agent Smith

How you know they don't?

Come one man, we're both to old for that nonsense. Of course they exist.

If you're talking about the axioms that protect set theory from paradoxes — frank

No, not just those -- and they're not just to ward off paradox but are an attempt to capture our sense of how collections work, not the most naive sense, of course, but after we've been out on the road with it a while. But Euclid had axioms. Peano gave axioms for natural numbers. Axioms are everywhere in math.

It's known that classical logic is insufficient as a foundation for mathematics, but I remember reading somewhere that all you need to add is Hume's Principle, which is that two sets have the same cardinality if there's a bijection (a one-to-one correspondence) between them.

It's debatable whether math really needs set theory as a foundation, though. — frank

You'd prefer category theory? Nah, you just mean math doesn't need any foundation. And that's obviously true both as an historical and as a practical matter. But there is value in having the foundation. Sets are the lingua franca of mathematics, so they enable making connections, leveraging techniques developed in one context in another, and so on. I think they clarify the use of special axioms too, because there is the common set you can take for granted, and you get your little branch of math by specifying some *additional* constraints. Doing topology, for instance, feels a lot like doing set theory, but with just a handful of other properties in play, and out of that you can develop a rich set of properties and theorems for spaces.

You're the one who seems to be insisting that the rules you've mentioned have no use even within the realm of math itself. — frank

Citation please.

Don't know about quantum physics having multiple types of 0, but doubt it.

Why not use a pair of these? $0 +$ $0 -$ $0^+$ $0^-$. They are commonly used in math. You could come up with the first being infinitesimals just to the right of zero, etc.
There are your "opposites" of zero. — jgill

No good. $0^+$ and $0^-$ are used in limit notation to indicate one-sided limits but have nothing to do with opposites.

No good. 0+ and 0− are used in limit notation to indicate one-sided limits but have nothing to do with opposites. — Real Gone Cat

Picky picky. Of course they don't. But in the spirit of this discussion they could. There are enough ambiguities in math to satisfy MU. Take $f$ for example. What does it mean? A function? A constant force? Depends on the context. $f=f\left( x+1 \right)$ ? How about 4x/5y-1 ? If these guys want to play around with nonsense don't poop in their sandbox. :cool:

Okay. But isn't that just to say either there's no math that defines a value for it or that you're unfamiliar with math that does.

To just say, nope, is like saying negative numbers don't have square roots, or, for that matter, that 2 doesn't. — Srap Tasmaner

Are you arguing that some undiscovered branch of math looms out there waiting to give new meaning to well-established results? Possibly but very unlikely.

By the way, no one says negative numbers don't have square roots. What they DO say is that negative numbers don't have real square roots.

Actually, $0^0$ is an interesting case. Cauchy made the claim that it was indeterminate in 1821 and most calculus texts today echo that idea. Certainly the limit as $x\rightarrow0^-$ is undefined. But there are some contexts in which assuming $0^0=1$ helps to facilitate certain theorems.

Hee hee, very droll. Lost my sense of humor for a minute there.