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Why don't we define “imaginary” numbers for every “impossibility”? (2012)

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Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)

#71

Post by JumpCrisscross »

ndsipa_pomu wrote: post #8

Well, we can define mathematical objects for every gap (impossibility), but most of them will turn out to be inconsistent with our existing mathematical objects, and thus not very useful or interesting. I'd consider that mathematics is the study of consistency and what can be discovered using the simplest possible starting points (axioms). The classic case would be if mathematicians wanted to assign a value to divisi…

> we can define mathematical objects for every gap (impossibility), but most of them will turn out to be inconsistent with our existing mathematical objects

Is the short answer it's not parsimonious or useful?

Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)

#72

Post by paulddraper »

ndsipa_pomu wrote: post #8

Well, we can define mathematical objects for every gap (impossibility), but most of them will turn out to be inconsistent with our existing mathematical objects, and thus not very useful or interesting. I'd consider that mathematics is the study of consistency and what can be discovered using the simplest possible starting points (axioms). The classic case would be if mathematicians wanted to assign a value to divisi…

You're right, though your example is weak.

Infinity and negative infinity can make a lot of sense. You can even allow imaginary infinities.

The caveats:

* You cannot multiply zero by infinity, or divide infinity by infinity

* You cannot add different infinities

Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)

#73

Post by syzarian »

topaz0 wrote: post #69
syzarian wrote: post #54

Earlier quoted context omitted.

The definition of division is multiplication by the multiplicative inverse. It may be the case that some elements don’t have such an inverse but the definition is analogous to that of subtraction. The analogy is not perfect because every element has an additive inverse while not every element had a multiplicative inverse.

What you're saying is that the analogy between subtraction and division is good as far as it goes. So why should "as far as it goes" end at zero not having an inverse, rather than division by zero producing something other than the multiplicative inverse of zero? The two choices end up having different structure, and so they end up being applicable to different things, but there is nothing wrong with either choice.

The word division means something in mathematics. There is general agreement in what that word ought to mean. You can define a binary operation in such a way that it doesn’t look like what we normally think of as division and label your operation division. In the same way you can define the symbol duck to refer to what most people call a chair. You won’t get anyone else agreeing with your new definition though.

Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)

#74

Post by thaumasiotes »

syzarian wrote: post #68
thaumasiotes wrote: post #67

Earlier quoted context omitted.

> It is not a stupid question. From their perspective mathematicians appear to do something and they wonder why it can’t be done in other situations. I mean, you've already gotten it wrong. This can be done in other situations. Where it isn't done, it isn't done because doing it is pointless, not because there's some bar to giving names to opaque labels.

How does your pedantry contribute meaningfully? If something doesn’t behave like 0 in a ring or other algebraic structure then using that label is confusing and simply not done. You are free to use any symbol you want but mathematics is a human endeavor and as such communication is important. Using the symbol 0 signifies something to those with mathematical training. Zero can’t have an multiplicative inverse because…

I'm having trouble following the argument from your premise "it is a stupid question to ask why I referred to a chair as a chair instead of a snkwoo" to your conclusion "it is not a stupid question to ask why, when we have no answer to a question, we don't just say that we do have one".

The answer (to both of those questions!) is, of course, that we could do that, but it wouldn't accomplish anything. Asking the question just means you have no idea what you're saying. Or in other words, it's a stupid question.

Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)

#75

Post by syzarian »

thaumasiotes wrote: post #74
syzarian wrote: post #68

Earlier quoted context omitted.

How does your pedantry contribute meaningfully? If something doesn’t behave like 0 in a ring or other algebraic structure then using that label is confusing and simply not done. You are free to use any symbol you want but mathematics is a human endeavor and as such communication is important. Using the symbol 0 signifies something to those with mathematical training. Zero can’t have an multiplicative inverse because…

I'm having trouble following the argument from your premise "it is a stupid question to ask why I referred to a chair as a chair instead of a snkwoo " to your conclusion "it is not a stupid question to ask why, when we have no answer to a question, we don't just say that we do have one". The answer (to both of those questions!) is, of course, that we could do that, but it wouldn't accomplish anything. Asking the ques…

I'm having trouble following…

I know. Please don’t become a teacher.

Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)

#76

Post by topaz0 »

syzarian wrote: post #73
topaz0 wrote: post #69

Earlier quoted context omitted.

What you're saying is that the analogy between subtraction and division is good as far as it goes. So why should "as far as it goes" end at zero not having an inverse, rather than division by zero producing something other than the multiplicative inverse of zero? The two choices end up having different structure, and so they end up being applicable to different things, but there is nothing wrong with either choice.

The word division means something in mathematics. There is general agreement in what that word ought to mean. You can define a binary operation in such a way that it doesn’t look like what we normally think of as division and label your operation division . In the same way you can define the symbol duck to refer to what most people call a chair. You won’t get anyone else agreeing with your new definition though.

We redefine multiplication for new contexts every day in math, I don't see why division should be any different. See also: https://en.wikipedia.org/wiki/Division_(mathematics)#Divisio...

Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)

#77

Post by scubbo »

thaumasiotes wrote: post #74
syzarian wrote: post #68

Earlier quoted context omitted.

How does your pedantry contribute meaningfully? If something doesn’t behave like 0 in a ring or other algebraic structure then using that label is confusing and simply not done. You are free to use any symbol you want but mathematics is a human endeavor and as such communication is important. Using the symbol 0 signifies something to those with mathematical training. Zero can’t have an multiplicative inverse because…

I'm having trouble following the argument from your premise "it is a stupid question to ask why I referred to a chair as a chair instead of a snkwoo " to your conclusion "it is not a stupid question to ask why, when we have no answer to a question, we don't just say that we do have one". The answer (to both of those questions!) is, of course, that we could do that, but it wouldn't accomplish anything. Asking the ques…

> Asking the question just means you have no idea what you're saying. Or in other words, it's a stupid question.

So, to be clear, you're saying that the only kind of question that isn't stupid is the one where the querent already has perfect knowledge of the discipline?

Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)

#78

Post by hgsgm »

comte7092 wrote: post #57
nextaccountic wrote: post #55

Earlier quoted context omitted.

> The classic case would be if mathematicians wanted to assign a value to division by zero. It turns out that if you do allow that to take a value, then it becomes possible to "prove" that any number is equal to any other number. Quite simply, it makes maths less interesting to allow that, but instead having division by zero be undefined appears far more useful/interesting. There are multiple extensions to the real n…

1 / 0 = +infinity Implies that 0 * +infinity = 1, so it does run into make of the same issues. There are instances that make it useful, but the extended real number line isn’t used heavily in practice.

What is the "issue"?

"/" means "* reciprocal of".

If "infinity" is defined as "reciprocal of 0", what is the problem?

Yes it is an exception to 0*n=0.

It won't work in every setting, but it works in some settings, like inversive geometry.

Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)

#79

Post by hgsgm »

2muchcoffeeman wrote: post #38
iamerroragent wrote: post #23

Earlier quoted context omitted.

Riemann Sphere: https://en.wikipedia.org/wiki/Riemann_sphere

This just goes to show that you really have to be careful when slinging out math facts. I've done some under grad maths and the only line on that page that I understand is "The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as 1 / 0 = ∞ 1/0=\infty well-behaved." It clearly does not satisfy a primitive understa…

1/0 is the limit of x/y as x approaches 1 and y approaches 0. It works fine if you choose to put a a point there called \infinity, with an appropriate notion of nearness.

Re: Why don't we define “imaginary” numbers for every “impossibility”? (2012)

#80

Post by hgsgm »

brookst wrote: post #49
momentoftop wrote: post #35

Earlier quoted context omitted.

Exactly. Functions in these logics are total, so if you want division to be a function (and you probably do), it has to assign something to division by 0. It would be acceptable to assign an unspecified object from the domain, for which you have no non-trivial theorems, and so all your real theorems must have a precondition about the denominator being non-zero. But if you specify a candidate like 0, you can get some…

I appreciate the explanation and I’m in no position to disagree, but ugh. Seems like it would work just as well to define x/0 as 6, or e, or -15. I’m sure that’s not the case. But as a long time tech person who’s always considered underflow/overflow to be a hack to get around limitations of hardware, it offends be a bit to find conditionals in abstract math. Undefined seems cleaner, like null, since it implicitly say…

Division is already defined conditionally in regular old elementary school field theory.
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