The origin of zero

[Severian]

It is possible and reasonable to define 0/0=1. 1/0 isn't really a well defined concept, because it involves an infinity. We can regulated the infinity by saying that it is the limit of a normal number. So [math]\frac{1}{0} \equiv lim_{x \to 0} \frac{1}{x}[/math].

 

In that case, using our definition of 1/0, [math]\frac{0}{0} = \frac{1}{0} \frac{0}{1} \equiv \frac{\lim_{x \to 0} \frac{1}{x}}{\lim_{x \to 0} \frac{1}{x}}=1[/math]

 

The difficulty (or ambiguity) arises when we realise that we needn't have used the definition of 1/0 to say what 0/1 is. We could have just done [math]\frac{0}{0} = \frac{1}{0} \times 0 \equiv \lim_{x \to 0} \frac{1}{x} \times 0 =0[/math].

 

But in the opinion of a physicist 0/0=1 is the most sensible choice since we use the same symbol for numerator and denominator, implying that they are the same thing. In a physical situation (which let's face it is all maths is good for) we only really get close to zero, so a limiting case approach is fine.

 

Professor Homunculus does seem to be a bit of an asshole.

[matt grime]

It isn't a sensible choice, it is a ridiculous choice and you're making the same mistake a lot of people make in dealing with limits. Actually, more than usual. Firstly we only deal with limits that exist within the reals - saying the lim1/x, as x geoes to 0, is infinity means exactly that it does not exist and cannot be used as a number like you're doing. secondly, when you write, lim/lim you are treating something that isn't a real number as a real number. And you're not even letting the x differ in numerator and denominator and act independently as one should in these cases.

 

If you want to do calculus stay in the reals (or complex, or even p-adics), if you want to do artihmetic with transfinite numbers can I suggest you read up on it first?

 

 

Your ad hominem attack on Homunculus seems odd and misdirected and unmathematical, and as this is maths, and not real life (your ideas of what maths is good for seem very odd and uninformed, not to say antimathematical, so why post in a maths forum?), one wonders what the point of it is.

[matt grime]

It isn't a sensible choice, it is a ridiculous choice and you're making the same mistake a lot of people make in dealing with limits. Actually, more than usual. Firstly we only deal with limits that exist within the reals - saying the lim1/x, as x geoes to 0, is infinity means exactly that it does not exist and cannot be used as a number like you're doing. secondly, when you write, lim/lim you are treating something that isn't a real number as a real number. And you're not even letting the x differ in numerator and denominator and act independently as one should in these cases.

 

If you want to do calculus stay in the reals (or complex, or even p-adics), if you want to do artihmetic with transfinite numbers can I suggest you read up on it first?

 

 

Your ad hominem attack on Homunculus seems odd and misdirected and unmathematical, and as this is maths, and not real life (your ideas of what maths is good for seem very odd and uninformed, not to say antimathematical, so why post in a maths forum?), one wonders what the point of it is.

[Severian]

It isn't a sensible choice' date=' it is a ridiculous choice and you're making the same mistake a lot of people make in dealing with limits. Actually, more than usual. Firstly we only deal with limits that exist within the reals - saying the lim1/x, as x geoes to 0, is infinity means exactly that it does not exist and cannot be used as a number like you're doing.

[/quote']

 

Which 'it' are you refering to? Clearly 1/x exists if x is non-zero - I can choose any large value of x I like.

 

secondly, when you write, lim/lim you are treating something that isn't a real number as a real number. And you're not even letting the x differ in numerator and denominator and act independently as one should in these cases.

 

This is exactly the point I made in my later comment, that on need not take the limits in the same way (or even use limits for both). I never suggested that 0/0=1 was the only result, only that it could occur, and is reasonable, for some definitions. Did you read my post?

 

If you want to do calculus stay in the reals (or complex, or even p-adics), if you want to do artihmetic with transfinite numbers can I suggest you read up on it first?

 

I am a physicist, not a mathematician. While I am no expert on the details of the mathematical definitions, I am certainly an expert on how maths is applied in the real world (no pun intended). I personally don't have much time for things which one cannot apply in the 'real world'. Perhaps you could enlighten me as to what abstract maths which have no application is actually good for?

 

This is actually a good example of the difference between a (pure)mathematician and a physicist. In physics, infinities like this arise all the time in our theories and we have to deal with them. This means that we have to regulate them in some way and come up with an answer that we can apply. We do this in various ways (such as extending space-time to non integer dimensions) and use these 'mathematically undefined' theories to predict the outcome of experiments. The fact that this proceedure works amazingly well makes a mockery of your prejudices. We cannot afford to stick our heads in the sand and say that the physical laws of the universe are not defined....

 

Your ad hominem attack on Homunculus seems odd and misdirected and unmathematical, and as this is maths, and not real life (your ideas of what maths is good for seem very odd and uninformed, not to say antimathematical, so why post in a maths forum?), one wonders what the point of it is.

 

Lol. The word 'asshole' isn't mathematical enough for you? I normally don't post in this forum because, as I pointed out before, I think it is pointless. I only posted because I felt that Homunculus' attacks on the teacher were unjustified. (As are your attacks on me, but I suppose that just means that you are a little Homunculus...) He is defined to be an asshole by his attacks on the teacher, not by his mathematical opinions.

 

PS: These are Science fora. If you are not interested in science, why do you post here? In fact, if your definition of pure mathematics is that it should not be applicable to the real world, maybe we should remove all discussions of it from this site?

[Severian]

It isn't a sensible choice' date=' it is a ridiculous choice and you're making the same mistake a lot of people make in dealing with limits. Actually, more than usual. Firstly we only deal with limits that exist within the reals - saying the lim1/x, as x geoes to 0, is infinity means exactly that it does not exist and cannot be used as a number like you're doing.

[/quote']

 

Which 'it' are you refering to? Clearly 1/x exists if x is non-zero - I can choose any large value of x I like.

 

secondly, when you write, lim/lim you are treating something that isn't a real number as a real number. And you're not even letting the x differ in numerator and denominator and act independently as one should in these cases.

 

This is exactly the point I made in my later comment, that on need not take the limits in the same way (or even use limits for both). I never suggested that 0/0=1 was the only result, only that it could occur, and is reasonable, for some definitions. Did you read my post?

 

If you want to do calculus stay in the reals (or complex, or even p-adics), if you want to do artihmetic with transfinite numbers can I suggest you read up on it first?

 

I am a physicist, not a mathematician. While I am no expert on the details of the mathematical definitions, I am certainly an expert on how maths is applied in the real world (no pun intended). I personally don't have much time for things which one cannot apply in the 'real world'. Perhaps you could enlighten me as to what abstract maths which have no application is actually good for?

 

This is actually a good example of the difference between a (pure)mathematician and a physicist. In physics, infinities like this arise all the time in our theories and we have to deal with them. This means that we have to regulate them in some way and come up with an answer that we can apply. We do this in various ways (such as extending space-time to non integer dimensions) and use these 'mathematically undefined' theories to predict the outcome of experiments. The fact that this proceedure works amazingly well makes a mockery of your prejudices. We cannot afford to stick our heads in the sand and say that the physical laws of the universe are not defined....

 

Your ad hominem attack on Homunculus seems odd and misdirected and unmathematical, and as this is maths, and not real life (your ideas of what maths is good for seem very odd and uninformed, not to say antimathematical, so why post in a maths forum?), one wonders what the point of it is.

 

Lol. The word 'asshole' isn't mathematical enough for you? I normally don't post in this forum because, as I pointed out before, I think it is pointless. I only posted because I felt that Homunculus' attacks on the teacher were unjustified. (As are your attacks on me, but I suppose that just means that you are a little Homunculus...) He is defined to be an asshole by his attacks on the teacher, not by his mathematical opinions.

 

PS: These are Science fora. If you are not interested in science, why do you post here? In fact, if your definition of pure mathematics is that it should not be applicable to the real world, maybe we should remove all discussions of it from this site?

[matt grime]

Then why did you stoop to his level if you thought his attacks unjustified? Pointing out where someone's ideas are wrong is not the same as calling them an asshole, or, in the case where you refer to me, a "little asshole" I presume.

 

Surely as soon as mathematics gets a use it becomes, by its use, "unabstract"?

 

Modulo arithmetic is a good example of pure mathematics that has found a use - abstract ring theory becomes RSA.

 

How about the Riemann Zeta function? Abstract number theoretic object that appears to be of interest in describing energy levels in some dynamical system or other.

 

Symplectic geometry and quantum field theory?

 

Triangulated categories and string theory (though if you're of the experimentally verifiable school that's a little bit rubbish, isn't it?).

 

Lyapunov exponents and choatic systems?

 

Functional analysis and probability theory? Look at Rietz's representation theorem for something ugly and pure that probability people care about. Assuming prob/stats counts as real world.

 

We study it because it is interesting and who knows, maybe one day someone cleverer than us will put it all together and figure out something it is good for - Raul Bott started off as an electrical engineer and realized that he needed to understand differential geometry and ended up becoming an outstanding (pure) mathematician. Though I think we're starting to see a move away from such artificial and unhelpful labels.

 

My prejudice is that the issue of what 1/0 is or isn't, and why it isn't defined are that the question is almost always ill-posed. You may treat it in many ways, as physicists are wont to do (treating divergent series by approximating by the first few terms, for instance) that work out. That doesn't alter the fact that the question was probably ill-posed and knowing about the axiomatic structure of mathematics simply explains why it doesn't work within the real number system (ie a field), but multiplication by 0 is consistently defined within this axiomatics framework.

 

I apologize if it appears i misread your post - I almost certainly did. You did certainly say this though:

 

"But in the opinion of a physicist 0/0=1 is the most sensible choice"

 

and the question is about mathematics since that is where we refer to it as undefined.

[matt grime]

Then why did you stoop to his level if you thought his attacks unjustified? Pointing out where someone's ideas are wrong is not the same as calling them an asshole, or, in the case where you refer to me, a "little asshole" I presume.

 

Surely as soon as mathematics gets a use it becomes, by its use, "unabstract"?

 

Modulo arithmetic is a good example of pure mathematics that has found a use - abstract ring theory becomes RSA.

 

How about the Riemann Zeta function? Abstract number theoretic object that appears to be of interest in describing energy levels in some dynamical system or other.

 

Symplectic geometry and quantum field theory?

 

Triangulated categories and string theory (though if you're of the experimentally verifiable school that's a little bit rubbish, isn't it?).

 

Lyapunov exponents and choatic systems?

 

Functional analysis and probability theory? Look at Rietz's representation theorem for something ugly and pure that probability people care about. Assuming prob/stats counts as real world.

 

We study it because it is interesting and who knows, maybe one day someone cleverer than us will put it all together and figure out something it is good for - Raul Bott started off as an electrical engineer and realized that he needed to understand differential geometry and ended up becoming an outstanding (pure) mathematician. Though I think we're starting to see a move away from such artificial and unhelpful labels.

 

My prejudice is that the issue of what 1/0 is or isn't, and why it isn't defined are that the question is almost always ill-posed. You may treat it in many ways, as physicists are wont to do (treating divergent series by approximating by the first few terms, for instance) that work out. That doesn't alter the fact that the question was probably ill-posed and knowing about the axiomatic structure of mathematics simply explains why it doesn't work within the real number system (ie a field), but multiplication by 0 is consistently defined within this axiomatics framework.

 

I apologize if it appears i misread your post - I almost certainly did. You did certainly say this though:

 

"But in the opinion of a physicist 0/0=1 is the most sensible choice"

 

and the question is about mathematics since that is where we refer to it as undefined.

[Severian]

Then why did you stoop to his level if you thought his attacks unjustified? Pointing out where someone's ideas are wrong is not the same as calling them an asshole' date=' or, in the case where you refer to me, a "little asshole" I presume.

[/quote']

 

I didn't stoop to his level. I did not criticise him because of his mathematical opinions/knowledge. I critised him for his personal attack on the mathematical abilities of another. To attack a high-school teacher for this question is ridiculous. Homunculus presumably has a well paid cushy university job with (reasonably) well behaved students and has a lot more time to ponder such questions than some underpaid, overworked high school teacher stressed by a class of 50 screaming kids.

 

I happen to agree with his mathematics.

 

Surely as soon as mathematics gets a use it becomes, by its use, "unabstract"?

 

That is true, but it usually worked the other way around. The physics theories force the need for mathematics. Indeed, this is true for most of your examples:

 

Modulo arithmetic is a good example of pure mathematics that has found a use - abstract ring theory becomes RSA.

 

It was clear that the ability to factor large numbers was useful in the real world long before the RSA theorem came about. This was only 30(ish?) years ago.

 

How about the Riemann Zeta function? Abstract number theoretic object that appears to be of interest in describing energy levels in some dynamical system or other.

 

The RZ function is just an integral. The fact that it was calculated before it found use in physics is neither here nor there. If it had not been, the physicist doing the problem would have calculated it. There are plenty of examples of (much harder) integrals which were not known when physics encountered them.

 

Symplectic geometry and quantum field theory?

 

This depends on your definition of symplectic geometry. If you regard Hamilton's work as symplectic geometry, I must ask you if you regard Hamilton as a mathematicain or as a physicist? QFT is based on the Hamlton principle. Modern symplectic geometry postdates QFT.

 

Triangulated categories and string theory (though if you're of the experimentally verifiable school that's a little bit rubbish, isn't it?).

 

No - string theory is in principle testable, so I have no objection. But once again, the motivaton of string theory is physical.

 

We study it because it is interesting and who knows, maybe one day someone cleverer than us will put it all together and figure out something it is good for - Raul Bott started off as an electrical engineer and realized that he needed to understand differential geometry and ended up becoming an outstanding (pure) mathematician. Though I think we're starting to see a move away from such artificial and unhelpful labels.

 

I have no objection to you studying it. Who knows, you might do something which physicists find useful. What I do object to is the prejudice that anything which is not mathematically rigourous, in the pure mathematician sense, is wrong and not useful. You demonstrate this attitude yourself in your post:

 

My prejudice is that the issue of what 1/0 is or isn't, and why it isn't defined are that the question is almost always ill-posed. You may treat it in many ways, as physicists are wont to do (treating divergent series by approximating by the first few terms, for instance) that work out.

 

Ironically, this "treating divergent series by approximating by the first few terms" was the subject of my PhD. But the fact of the matter is that it works! My calculation is being used successfully by high energy physics experiments and correctly predicts the physics that they see. To suggest that this should not be done is the real prejudice. And my calculation involved the cancellation of lots of 'infinities'.

 

That doesn't alter the fact that the question was probably ill-posed and knowing about the axiomatic structure of mathematics simply explains why it doesn't work within the real number system (ie a field), but multiplication by 0 is consistently defined within this axiomatics framework.

 

You would presumably criticise Feyman's path integral formalism in this light? But again this has been tremendously successful in modern physics. Or how about the theory of Renormalization of QFTs? That won the Nobel prize a few years ago.

 

I apologize if it appears i misread your post - I almost certainly did. You did certainly say this though:

 

"But in the opinion of a physicist 0/0=1 is the most sensible choice"

 

and the question is about mathematics since that is where we refer to it as undefined.

 

I agree that it is undefined on its own. As I also pointed out, one has to make a clarification as to what one means by 0/0 before coming to a conclusion. I notice that you cut off the rest of the sentence, which is pointing out that if one defined the '0's on the left hand side in the same way then the answer is 1. In this light, your response seems a little disingenuous.

[Severian]

Then why did you stoop to his level if you thought his attacks unjustified? Pointing out where someone's ideas are wrong is not the same as calling them an asshole' date=' or, in the case where you refer to me, a "little asshole" I presume.

[/quote']

 

I didn't stoop to his level. I did not criticise him because of his mathematical opinions/knowledge. I critised him for his personal attack on the mathematical abilities of another. To attack a high-school teacher for this question is ridiculous. Homunculus presumably has a well paid cushy university job with (reasonably) well behaved students and has a lot more time to ponder such questions than some underpaid, overworked high school teacher stressed by a class of 50 screaming kids.

 

I happen to agree with his mathematics.

 

Surely as soon as mathematics gets a use it becomes, by its use, "unabstract"?

 

That is true, but it usually worked the other way around. The physics theories force the need for mathematics. Indeed, this is true for most of your examples:

 

Modulo arithmetic is a good example of pure mathematics that has found a use - abstract ring theory becomes RSA.

 

It was clear that the ability to factor large numbers was useful in the real world long before the RSA theorem came about. This was only 30(ish?) years ago.

 

How about the Riemann Zeta function? Abstract number theoretic object that appears to be of interest in describing energy levels in some dynamical system or other.

 

The RZ function is just an integral. The fact that it was calculated before it found use in physics is neither here nor there. If it had not been, the physicist doing the problem would have calculated it. There are plenty of examples of (much harder) integrals which were not known when physics encountered them.

 

Symplectic geometry and quantum field theory?

 

This depends on your definition of symplectic geometry. If you regard Hamilton's work as symplectic geometry, I must ask you if you regard Hamilton as a mathematicain or as a physicist? QFT is based on the Hamlton principle. Modern symplectic geometry postdates QFT.

 

Triangulated categories and string theory (though if you're of the experimentally verifiable school that's a little bit rubbish, isn't it?).

 

No - string theory is in principle testable, so I have no objection. But once again, the motivaton of string theory is physical.

 

We study it because it is interesting and who knows, maybe one day someone cleverer than us will put it all together and figure out something it is good for - Raul Bott started off as an electrical engineer and realized that he needed to understand differential geometry and ended up becoming an outstanding (pure) mathematician. Though I think we're starting to see a move away from such artificial and unhelpful labels.

 

I have no objection to you studying it. Who knows, you might do something which physicists find useful. What I do object to is the prejudice that anything which is not mathematically rigourous, in the pure mathematician sense, is wrong and not useful. You demonstrate this attitude yourself in your post:

 

My prejudice is that the issue of what 1/0 is or isn't, and why it isn't defined are that the question is almost always ill-posed. You may treat it in many ways, as physicists are wont to do (treating divergent series by approximating by the first few terms, for instance) that work out.

 

Ironically, this "treating divergent series by approximating by the first few terms" was the subject of my PhD. But the fact of the matter is that it works! My calculation is being used successfully by high energy physics experiments and correctly predicts the physics that they see. To suggest that this should not be done is the real prejudice. And my calculation involved the cancellation of lots of 'infinities'.

 

That doesn't alter the fact that the question was probably ill-posed and knowing about the axiomatic structure of mathematics simply explains why it doesn't work within the real number system (ie a field), but multiplication by 0 is consistently defined within this axiomatics framework.

 

You would presumably criticise Feyman's path integral formalism in this light? But again this has been tremendously successful in modern physics. Or how about the theory of Renormalization of QFTs? That won the Nobel prize a few years ago.

 

I apologize if it appears i misread your post - I almost certainly did. You did certainly say this though:

 

"But in the opinion of a physicist 0/0=1 is the most sensible choice"

 

and the question is about mathematics since that is where we refer to it as undefined.

 

I agree that it is undefined on its own. As I also pointed out, one has to make a clarification as to what one means by 0/0 before coming to a conclusion. I notice that you cut off the rest of the sentence, which is pointing out that if one defined the '0's on the left hand side in the same way then the answer is 1. In this light, your response seems a little disingenuous.

[matt grime]

0 is defined in this case as the additive identity in a ring.

 

Your notion of regulating infinity is also an analytic one (removable singularity, pole of degree n, essential singularity, as you well know). We may certainly write 1/0 = [math]\infty[/math] if we are operating on the complex sphere, but we aren't.

 

Moreover, when we say that, say, sin(x)/x is 1 when x is 0 we are making several assumptions that merely boil down to the fact that as a function on R\{0} it has a continuous extension onto the boundary, we are not actually defining 0/0 in any absolute sense. We are not saying something about the arithmetic of the real numbers but about functions of a real variable.

 

I'm glad you feel that you can compellingly make such absolute claims as you have about the origin of maths/physics, though I disgree with some of your conclusions.

 

I don't think we should dismiss L-functions as mere integrals that casually. Nor did I claim the cases I gave were the first use of these "pure" things (RSA, though it is the manner in which we use groups and rings that is important, not counting things. I could also cite groups originally arising in a study of roots of polynomials that now has implications in electrical and chemical engineering), however it seems that nothing I can offer will satisfy your (unstated) requirements - whatever the origins of the motivation of a field of study the abstract study of other things can and is useful in it. Hamiltonian dynamics almost certainly predates the formal notion of a "closed non-degenerate two form on a manifold", but that doesn't stop that bit of "pure" mathematics being useful. I would accept some of these opinions if you could for instance categorically show that nothing produced in the abstract study of these objects has ever had any influence or bearing on the applications.

 

I could also add in about discrete mathematics and logic and their use in computer science.

The underlying influences in mathematics have shifted over the years, oringinally homological algebra arose from astronomy, and hence why there are syzygys (not that I think I've spelt it correctly). The second half of the 20th C saw a shift away to abstract study in its own right and now we're seeing the physicists, with string theory in particular, come back and give new impetus to maths. That is only a good thing. Pure mathematicians aren't as stuck on rigour as you seem to think we are, but once we've got a piece of maths sorted then we tend to get itchy about people misunderstanding it. (And, no I don't mean you. You evidently don't fall into that category.)

 

 

Actually I would hold that by treating infinite sums in such a formal way (ignoring convergence issues) is a good thing (pure mathematicians do it all the time). After all the model is just that, and anytime something like 1/0 appears it is not the "fault" of the underlying mathematical object, but in its application if we were trying to avoid an infinity.