I have spent a long time arguing with someone over whether or not infinity times zero equals one. I would like to know if I have any hope on holding my position that it is not true, or if I am simply wasting my time.
His reasoning for the previous statement being true is as follows:
If 1/n=0 as n approaches infinity, he insists that 1/infinity=0 and thus 0 times infinity=1. (Also infinity/0=1)
I've tried to disprove this for my firm belief in the existence of the infinitesimal, and my disapproval of the number of properties which must be ignored for the statement to be true (which exemption from said properties may be justified by the concept that infinity is not a number (another statement he disagrees with)). I have tried countless analogies with no avail and his persistence began to make me doubt my own accuracy.
Please enlighten me on my err and misconception as they are in abundance.
QUOTE (thinker+Jun 6 2008, 09:14 PM)
I have spent a long time arguing with someone over whether or not infinity times zero equals one. I would like to know if I have any hope on holding my position that it is not true, or if I am simply wasting my time.
His reasoning for the previous statement being true is as follows:
If 1/n=0 as n approaches infinity, he insists that 1/infinity=0 and thus 0 times infinity=1. (Also infinity/0=1)
I've tried to disprove this for my firm belief in the existence of the infinitesimal, and my disapproval of the number of properties which must be ignored for the statement to be true (which exemption from said properties may be justified by the concept that infinity is not a number (another statement he disagrees with)). I have tried countless analogies with no avail and his persistence began to make me doubt my own accuracy.
Please enlighten me on my err and misconception as they are in abundance.
Ok, according to my old college calculus text, this is the definition of a limit at infinity:
"Let f be a funciton defined on some interval (a, infinity). Then
Lim x->infinity f(x) = L
means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large."
Definition is from Early Transcendentals 3rd edition page 90. James Stewart, McMaster University.
I believe there is a 4th or 5th edition by now, but I don't have one...
Here, infinity is clearly a "process" arbitrarily APPROACHING a number, and NOT the number itself.
This is relevant to the 0.9r arguments.
By the definition of a limit at infinity, 0.9r does NOT equal 1. It equals a number arbitrarily close to 1, but never EQUAL to 1.
The LIMIT is 1, but there is no real number that satisfies the limit.
His reasoning for the previous statement being true is as follows:
If 1/n=0 as n approaches infinity, he insists that 1/infinity=0 and thus 0 times infinity=1. (Also infinity/0=1)
I've tried to disprove this for my firm belief in the existence of the infinitesimal, and my disapproval of the number of properties which must be ignored for the statement to be true (which exemption from said properties may be justified by the concept that infinity is not a number (another statement he disagrees with)). I have tried countless analogies with no avail and his persistence began to make me doubt my own accuracy.
Please enlighten me on my err and misconception as they are in abundance.
Ok, according to my old college calculus text, this is the definition of a limit at infinity:
"Let f be a funciton defined on some interval (a, infinity). Then
Lim x->infinity f(x) = L
means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large."
Definition is from Early Transcendentals 3rd edition page 90. James Stewart, McMaster University.
I believe there is a 4th or 5th edition by now, but I don't have one...
Here, infinity is clearly a "process" arbitrarily APPROACHING a number, and NOT the number itself.
This is relevant to the 0.9r arguments.
By the definition of a limit at infinity, 0.9r does NOT equal 1. It equals a number arbitrarily close to 1, but never EQUAL to 1.
The LIMIT is 1, but there is no real number that satisfies the limit.
QUOTE (Quantum_Conundrum+Jun 7 2008, 05:39 AM)
Here, infinity is clearly a "process" arbitrarily APPROACHING a number, and NOT the number itself.
No, the 'limit' part can be considered a process. But once you've taken the limit, there's no process left. A number is a number.
And by the definition of limits, 0.9r=1. To claim otherwise is to demonstrate you don't understand your book on calculus.
infinity*0 is undefined. To equate it to a specific real number is inconsistent.
No, the 'limit' part can be considered a process. But once you've taken the limit, there's no process left. A number is a number.
And by the definition of limits, 0.9r=1. To claim otherwise is to demonstrate you don't understand your book on calculus.
infinity*0 is undefined. To equate it to a specific real number is inconsistent.
what about the dirac delta function - infinite height, infinitesimal width; total area = 1 by definition.
And that's to be found in just about any text on advanced physics or engineering maths.
And that's to be found in just about any text on advanced physics or engineering maths.
It's technically a measure and doesn't have a well defined meaning outside of an integral. It's the 'derivative' of the step function. Which isn't really differentiable, in the usual sense.
QUOTE (AlphaNumeric+Jun 7 2008, 04:05 AM)
No, the 'limit' part can be considered a process. But once you've taken the limit, there's no process left. A number is a number.
And by the definition of limits, 0.9r=1. To claim otherwise is to demonstrate you don't understand your book on calculus.
infinity*0 is undefined. To equate it to a specific real number is inconsistent.
Alphanumeric:
Tell us the value of infinity then.
Define infinity such that it is a real number, and not a process or a vague concept of something without a boundary. You can't actually do it.
It is not proper to treat a limit at infinity as a real number.
The LIMIT of 1/x as x approaches infinity is zero, but 1/x never actually equals zero. There is no such NUMBER as "infinity".
Divide 1 by ANY real number, and you ALWAYS get a result greater than zero. This result gets ARBITRARILY SMALL but will NEVER equal zero. It approaches zero, but will never equal zero.
Another example
the infinite limit f(x) = (x^2 - 1)/(x^2 + 1) = 1
The LIMIT equals 1, but I dare you to try to find a number to plug into x such that the function actually equals 1. It doesn't exist.
And by the definition of limits, 0.9r=1. To claim otherwise is to demonstrate you don't understand your book on calculus.
infinity*0 is undefined. To equate it to a specific real number is inconsistent.
Alphanumeric:
Tell us the value of infinity then.
Define infinity such that it is a real number, and not a process or a vague concept of something without a boundary. You can't actually do it.
It is not proper to treat a limit at infinity as a real number.
The LIMIT of 1/x as x approaches infinity is zero, but 1/x never actually equals zero. There is no such NUMBER as "infinity".
Divide 1 by ANY real number, and you ALWAYS get a result greater than zero. This result gets ARBITRARILY SMALL but will NEVER equal zero. It approaches zero, but will never equal zero.
Another example
the infinite limit f(x) = (x^2 - 1)/(x^2 + 1) = 1
The LIMIT equals 1, but I dare you to try to find a number to plug into x such that the function actually equals 1. It doesn't exist.
QUOTE (AlphaNumeric+Jun 7 2008, 09:34 PM)
It's technically a measure and doesn't have a well defined meaning outside of an integral. It's the 'derivative' of the step function. Which isn't really differentiable, in the usual sense.
but still... what was it Dirac said about obtaining universal truth through literal mathematical interpretation?
but still... what was it Dirac said about obtaining universal truth through literal mathematical interpretation?
QUOTE (thinker+Jun 7 2008, 03:14 AM)
His reasoning for the previous statement being true is as follows:
If 1/n=0 as n approaches infinity, he insists that 1/infinity=0 and thus 0 times infinity=1. (Also infinity/0=1)
Ok, your friend is certainly wrong.
By using his logic:
If 2/n = 0 as n approaches infinity, he would insist that 2/infinity = 0 and thus 0 times infinity = 2.
This would imply that 0*infinity=1 AND 0*infinity=2, so 1=2. This is clearly wrong so there must be a problem with his logic. It turns out that 1/infinity (or 2/infinity, etc) is not valid as an operation on Real numbers, because infinity is not a Real number, and all the logic which follows is based on a false step.
Hope that helps.
If 1/n=0 as n approaches infinity, he insists that 1/infinity=0 and thus 0 times infinity=1. (Also infinity/0=1)
Ok, your friend is certainly wrong.
By using his logic:
If 2/n = 0 as n approaches infinity, he would insist that 2/infinity = 0 and thus 0 times infinity = 2.
This would imply that 0*infinity=1 AND 0*infinity=2, so 1=2. This is clearly wrong so there must be a problem with his logic. It turns out that 1/infinity (or 2/infinity, etc) is not valid as an operation on Real numbers, because infinity is not a Real number, and all the logic which follows is based on a false step.
Hope that helps.
Expanding upon AlphaNumeric's point:
Dirac was rather sloppy in talking of his delta as a function. It can be regarded as a "distribution" or as a "measure" but it is not a function. Unfortunately, the description of an infinitely tall, thin spike with an area of 1 has become the standard introduction to the Dirac delta. It is true that a VERY (not infinitely) tall, VERY (not infinitesimally) thin spike has properties that approach those of the Dirac delta, when such a spike function is integrated together with an ordinary function. However, when you try to take the limit of the spike function and get a "delta function" the limit simply does not exist in the realm of functions.
As a matter of fact, there is nothing infinite about the Dirac delta. If you think of it as a measure, it simply assigns a value of 1 to the point at the origin, and the value 0 to every other point. Therefore, when you integrate across a region containing the origin, the answer is 1, while if your region misses the origin, the answer is 0. Note that no infinities were used anywhere in this description. A related difficulty is that integration is often described as finding the area under a curve. This is a very unfortunate way of introducing the integral, because this image sticks in people's minds, and it is NOT TRUE. The area under a curve is merely ONE method of graphiclly displaying what an integral does. What an integral REALLY does is to ACCUMULATE VALUE. This value is called measure, and it can be either assigned to points or to sets (in which case the value is spread out into a density). When you integrate, you find the total value of the measure assigned to the region you integrated across.
Now in the special case that the value is spread out into a density, then you CAN, IF YOU WANT, represent that density graphically as the height of a function. The total value then shows up as the area under that graph. But both the graph and its area are just pictures illustrating what the integral is doing; they are not the integral itself. It is this insistance on thinking of the integral of a function as an area that forces people to try to set an area of 1 above a single point, which results in the unfortunate infinite spike picture. If one simply stops thinking of the integral as an area, the entire problem goes away, which shows the power of finding the right point of view.
About infinity:
Infinity as a number is not constructed by using limits. Like all finite numbers, it can be constructed from set theory. There are several constructions of infinite numbers: the original (and still my favorite) due to Georg Cantor circa 1890 in his book "Contributions to the Founding of a Theory of Transfinite Numbers;" the version of integer constuction due to von Neumann (don't know the year); the Abraham Robinson construction of non-standard analysis; and the J. H. Conway "surreal numbers."
Take for instance von Neumann's approach. He starts with nothing more than the empty set, and by repeated construction, defines the integers and also shows how to construct the operations of addition, subtraction, multiplication, and division entirely in terms of set operations, such as unions and intersections. One feature of his method is that allows him to construct infinity rather easily. The construction goes like this:
The empty set {} is named 0.
The set { {} } is named 1. (This is not the empty set; it is the set containing one element, and that element is the empty set. Think of box with an empty box inside it. Then the outer box isn't empty, is it? So another way to write this set is {0}.)
The set { {}, { {} } } = {0,1} is named 2.
The set (0,1,2} is named 3. From here on it's obvious what the pattern is: each new integer is DEFINED to be the set of all previous integers, starting with 0. Since 0 is the first integer defined, there are no previously defined integers, wo it contains nothing, i.e. 0 = {}, so the pattern holds all the way back to the beginning.
This allows one to define n+1 = {0,1,2,...,n}, so there is always a next integer. Now, what if you form the set of ALL integers, {0,1,2,...}. You CAN do this, because the set is described by a RULE OF MEMBERSHIP, and this rule is finite: x is in this set if and only if x is one of the integers constructed by the method above. For each particular x, you can always check directly whether it is in the set. For every integer, the answer will be yes, and for anything else it will be no. This means that this set is perfectly well defined and definite. It is the set of all non-negative integers. This set is called omega, which I'll print as "w."
This w has all the properties of an infinite integer. You can keep going beyond it, too: the set {0,1,2,...;w} contains all the integers up to and including w, so it is by definition w+1. By continuing in this way, you can build up all of Cantor's transfinite ordinal numbers using Von Neumann's construction. These are called ordinal because they depend on the ordering of the elements. If you now construct the equivalence class of all ordinals that can be mapped onto each other by 1 to 1 mappings, you get the transfinite cardinals, which act very much like the finite integers.
Once you have constructed these infinite numbers, you can then construct operations of addition, subtraction, multiplication, and division on them, just like the finite integers have. Of course, thinks turn out to work slightly differently with these new infinite numbers, but certain things remain true. For instance, the Schoeder-Bernstein theorem shows that if a<b and b<a, then a = b. Many known theorems show how these numbers work, and prove the logical consistency of the definitions.
To summarize, you certainly CAN construch infinite numbers and operations on them, several people have shown how to do so, the theory of them is well developed and known to be logically consistent.
Hope that helps!
--Stuart Anderson
Dirac was rather sloppy in talking of his delta as a function. It can be regarded as a "distribution" or as a "measure" but it is not a function. Unfortunately, the description of an infinitely tall, thin spike with an area of 1 has become the standard introduction to the Dirac delta. It is true that a VERY (not infinitely) tall, VERY (not infinitesimally) thin spike has properties that approach those of the Dirac delta, when such a spike function is integrated together with an ordinary function. However, when you try to take the limit of the spike function and get a "delta function" the limit simply does not exist in the realm of functions.
As a matter of fact, there is nothing infinite about the Dirac delta. If you think of it as a measure, it simply assigns a value of 1 to the point at the origin, and the value 0 to every other point. Therefore, when you integrate across a region containing the origin, the answer is 1, while if your region misses the origin, the answer is 0. Note that no infinities were used anywhere in this description. A related difficulty is that integration is often described as finding the area under a curve. This is a very unfortunate way of introducing the integral, because this image sticks in people's minds, and it is NOT TRUE. The area under a curve is merely ONE method of graphiclly displaying what an integral does. What an integral REALLY does is to ACCUMULATE VALUE. This value is called measure, and it can be either assigned to points or to sets (in which case the value is spread out into a density). When you integrate, you find the total value of the measure assigned to the region you integrated across.
Now in the special case that the value is spread out into a density, then you CAN, IF YOU WANT, represent that density graphically as the height of a function. The total value then shows up as the area under that graph. But both the graph and its area are just pictures illustrating what the integral is doing; they are not the integral itself. It is this insistance on thinking of the integral of a function as an area that forces people to try to set an area of 1 above a single point, which results in the unfortunate infinite spike picture. If one simply stops thinking of the integral as an area, the entire problem goes away, which shows the power of finding the right point of view.
About infinity:
Infinity as a number is not constructed by using limits. Like all finite numbers, it can be constructed from set theory. There are several constructions of infinite numbers: the original (and still my favorite) due to Georg Cantor circa 1890 in his book "Contributions to the Founding of a Theory of Transfinite Numbers;" the version of integer constuction due to von Neumann (don't know the year); the Abraham Robinson construction of non-standard analysis; and the J. H. Conway "surreal numbers."
Take for instance von Neumann's approach. He starts with nothing more than the empty set, and by repeated construction, defines the integers and also shows how to construct the operations of addition, subtraction, multiplication, and division entirely in terms of set operations, such as unions and intersections. One feature of his method is that allows him to construct infinity rather easily. The construction goes like this:
The empty set {} is named 0.
The set { {} } is named 1. (This is not the empty set; it is the set containing one element, and that element is the empty set. Think of box with an empty box inside it. Then the outer box isn't empty, is it? So another way to write this set is {0}.)
The set { {}, { {} } } = {0,1} is named 2.
The set (0,1,2} is named 3. From here on it's obvious what the pattern is: each new integer is DEFINED to be the set of all previous integers, starting with 0. Since 0 is the first integer defined, there are no previously defined integers, wo it contains nothing, i.e. 0 = {}, so the pattern holds all the way back to the beginning.
This allows one to define n+1 = {0,1,2,...,n}, so there is always a next integer. Now, what if you form the set of ALL integers, {0,1,2,...}. You CAN do this, because the set is described by a RULE OF MEMBERSHIP, and this rule is finite: x is in this set if and only if x is one of the integers constructed by the method above. For each particular x, you can always check directly whether it is in the set. For every integer, the answer will be yes, and for anything else it will be no. This means that this set is perfectly well defined and definite. It is the set of all non-negative integers. This set is called omega, which I'll print as "w."
This w has all the properties of an infinite integer. You can keep going beyond it, too: the set {0,1,2,...;w} contains all the integers up to and including w, so it is by definition w+1. By continuing in this way, you can build up all of Cantor's transfinite ordinal numbers using Von Neumann's construction. These are called ordinal because they depend on the ordering of the elements. If you now construct the equivalence class of all ordinals that can be mapped onto each other by 1 to 1 mappings, you get the transfinite cardinals, which act very much like the finite integers.
Once you have constructed these infinite numbers, you can then construct operations of addition, subtraction, multiplication, and division on them, just like the finite integers have. Of course, thinks turn out to work slightly differently with these new infinite numbers, but certain things remain true. For instance, the Schoeder-Bernstein theorem shows that if a<b and b<a, then a = b. Many known theorems show how these numbers work, and prove the logical consistency of the definitions.
To summarize, you certainly CAN construch infinite numbers and operations on them, several people have shown how to do so, the theory of them is well developed and known to be logically consistent.
Hope that helps!
--Stuart Anderson
QUOTE (mr_homm+Jun 7 2008, 09:11 PM)
Expanding upon AlphaNumeric's point:
Dirac was rather sloppy in talking of his delta as a function. It can be regarded as a "distribution" or as a "measure" but it is not a function. Unfortunately, the description of an infinitely tall, thin spike with an area of 1 has become the standard introduction to the Dirac delta. It is true that a VERY (not infinitely) tall, VERY (not infinitesimally) thin spike has properties that approach those of the Dirac delta, when such a spike function is integrated together with an ordinary function. However, when you try to take the limit of the spike function and get a "delta function" the limit simply does not exist in the realm of functions.
Hi Stuart,
I'm not a mathematician, We were told that the Dirac delta is like the "unit function" in the space of functions. That is, when to take the Fourier transform of it you get 1. Surely the Dirac delta is a function, maybe it's not differentiable, but it maps a nuber to another number. In that sense it is a function right?
Dirac was rather sloppy in talking of his delta as a function. It can be regarded as a "distribution" or as a "measure" but it is not a function. Unfortunately, the description of an infinitely tall, thin spike with an area of 1 has become the standard introduction to the Dirac delta. It is true that a VERY (not infinitely) tall, VERY (not infinitesimally) thin spike has properties that approach those of the Dirac delta, when such a spike function is integrated together with an ordinary function. However, when you try to take the limit of the spike function and get a "delta function" the limit simply does not exist in the realm of functions.
Hi Stuart,
I'm not a mathematician, We were told that the Dirac delta is like the "unit function" in the space of functions. That is, when to take the Fourier transform of it you get 1. Surely the Dirac delta is a function, maybe it's not differentiable, but it maps a nuber to another number. In that sense it is a function right?
you could constroy several operaations using the idea of infinite,what is somethingpossible not of express "quantity".could use the idea of limit in term of infinity,thing that approaching as imaginary functions,and not imaginary numbers.that imaginary functions will obtain the complex spaces( or structures of groups,denominated by complex numbers,that are dual forms of express the real and imaginary numbers into the same space-could to think such union,as non-invariance in the tridimensional space,but the symmetry in 4-dimensions;this is the
operation of rotation in the spacetime,having the noncommutative property,as resulted) that are the connection of duality:comtinuity and discreteness,as form of thinking of the entity:spacetime.
operation of rotation in the spacetime,having the noncommutative property,as resulted) that are the connection of duality:comtinuity and discreteness,as form of thinking of the entity:spacetime.
QUOTE (prometheus+Jun 7 2008, 09:34 PM)
Hi Stuart,
I'm not a mathematician, We were told that the Dirac delta is like the "unit function" in the space of functions. That is, when to take the Fourier transform of it you get 1. Surely the Dirac delta is a function, maybe it's not differentiable, but it maps a nuber to another number. In that sense it is a function right?
To what number does it map zero?
You mention Fourier transforms: the mathematics that deals with such things is that of distributions (there's plenty on the web to google). These are intimately related to the notion of weak solutions to differential equations. These are things, in a loose sense, that only make sense when they're inside an integral, paired with a function of suitable niceness. It is rather likely, that without knowing it, you have spent a lot of time as a physicist dealing with weak solutions in the mathematics you use. (My proof would be: I bet at some point you've given a heuristic derivation of the validity of Greens' functions by differentiating under the integral sign. But you know the Greens' function isn't differentiable on the entire range of integration you're interested in! The notion of a weak solution to a PDE saves you). I recommend the book by Friedlander and Joshi if you'd like to learn about such things.
I'm not a mathematician, We were told that the Dirac delta is like the "unit function" in the space of functions. That is, when to take the Fourier transform of it you get 1. Surely the Dirac delta is a function, maybe it's not differentiable, but it maps a nuber to another number. In that sense it is a function right?
To what number does it map zero?
You mention Fourier transforms: the mathematics that deals with such things is that of distributions (there's plenty on the web to google). These are intimately related to the notion of weak solutions to differential equations. These are things, in a loose sense, that only make sense when they're inside an integral, paired with a function of suitable niceness. It is rather likely, that without knowing it, you have spent a lot of time as a physicist dealing with weak solutions in the mathematics you use. (My proof would be: I bet at some point you've given a heuristic derivation of the validity of Greens' functions by differentiating under the integral sign. But you know the Greens' function isn't differentiable on the entire range of integration you're interested in! The notion of a weak solution to a PDE saves you). I recommend the book by Friedlander and Joshi if you'd like to learn about such things.
QUOTE (thinker+Jun 6 2008, 10:14 PM)
His reasoning for the previous statement being true is as follows:
If 1/n=0 as n approaches infinity, he insists that 1/infinity=0 and thus 0 times infinity=1. (Also infinity/0=1)
Think about multiplication as successive addition.
u=n*0=0+0+0+...+0+0=0, with n terms.
As n increases, at what point does u go from nothing to something?
If 1/n=0 as n approaches infinity, he insists that 1/infinity=0 and thus 0 times infinity=1. (Also infinity/0=1)
Think about multiplication as successive addition.
u=n*0=0+0+0+...+0+0=0, with n terms.
As n increases, at what point does u go from nothing to something?
QUOTE (thinker+Jun 7 2008, 02:14 AM)
His reasoning for the previous statement being true is as follows:
If 1/n=0 as n approaches infinity, he insists that 1/infinity=0 and thus 0 times infinity=1. (Also infinity/0=1)
Here is a rigorous refutation of the above:
Case 1. 1/n * n =1.
Case 2. 1/n * n^2=n -> infinity
Case 3. 1/n * sqrt(n)=1/sqrt(n) ->0
This is why, in calculus, we know that 0 multiplied with infinity is indeterminate, it can take different values depending on the situation (see above). You can try using the counterexample above to convince your friend.
QUOTE (Euler+Jun 7 2008, 09:57 PM)
To what number does it map zero?
You mention Fourier transforms: the mathematics that deals with such things is that of distributions (there's plenty on the web to google). These are intimately related to the notion of weak solutions to differential equations. These are things, in a loose sense, that only make sense when they're inside an integral, paired with a function of suitable niceness. It is rather likely, that without knowing it, you have spent a lot of time as a physicist dealing with weak solutions in the mathematics you use. (My proof would be: I bet at some point you've given a heuristic derivation of the validity of Greens' functions by differentiating under the integral sign. But you know the Greens' function isn't differentiable on the entire range of integration you're interested in! The notion of a weak solution to a PDE saves you). I recommend the book by Friedlander and Joshi if you'd like to learn about such things.
Cheers Euler.
That makes sense. I'll see if I can get hold of a copy of the reference you cite. A related question perhaps is how far would a mathematician say we can define a functional integral?
You mention Fourier transforms: the mathematics that deals with such things is that of distributions (there's plenty on the web to google). These are intimately related to the notion of weak solutions to differential equations. These are things, in a loose sense, that only make sense when they're inside an integral, paired with a function of suitable niceness. It is rather likely, that without knowing it, you have spent a lot of time as a physicist dealing with weak solutions in the mathematics you use. (My proof would be: I bet at some point you've given a heuristic derivation of the validity of Greens' functions by differentiating under the integral sign. But you know the Greens' function isn't differentiable on the entire range of integration you're interested in! The notion of a weak solution to a PDE saves you). I recommend the book by Friedlander and Joshi if you'd like to learn about such things.
Cheers Euler.
That makes sense. I'll see if I can get hold of a copy of the reference you cite. A related question perhaps is how far would a mathematician say we can define a functional integral?
mr_homm;
1. Isn't this going down the Russell path of a set containing itself?
2. The empty set has been defined as the set with no elements, so after all the iterations, where are the elements?
QUOTE
The set { {} } is named 1. (This is not the empty set; it is the set containing one element, and that element is the empty set. Think of box with an empty box inside it.
1. Isn't this going down the Russell path of a set containing itself?
2. The empty set has been defined as the set with no elements, so after all the iterations, where are the elements?
QUOTE (prometheus+Jun 8 2008, 12:07 PM)
A related question perhaps is how far would a mathematician say we can define a functional integral?
I'm not really sure I understand the question. Do you mean, how would a mathematician define an action, say? Or do you mean what would a mathematician make of the approach to QFT involving functional integration? Or something else entirely?
I'm not really sure I understand the question. Do you mean, how would a mathematician define an action, say? Or do you mean what would a mathematician make of the approach to QFT involving functional integration? Or something else entirely?
I mean the functional integral of quantum field theory. It's not a very nice object mathematically so I was just wondering how and in what circumstances it has a good definition.
QUOTE (prometheus+Jun 8 2008, 12:36 PM)
I mean the functional integral of quantum field theory. It's not a very nice object mathematically so I was just wondering how and in what circumstances it has a good definition.
It's not an area I know an awful lot about to be honest. I'm aware that it's far from complete. I've a feeling, from the QFT viewpoint, things are justified by adapting some results from Stochastic calculus - which is fair enough. In general though, the topic of "functional integration over an arbitrary function space" still has a loong way to go!
It's not an area I know an awful lot about to be honest. I'm aware that it's far from complete. I've a feeling, from the QFT viewpoint, things are justified by adapting some results from Stochastic calculus - which is fair enough. In general though, the topic of "functional integration over an arbitrary function space" still has a loong way to go!
QUOTE (Euler+Jun 8 2008, 12:44 PM)
It's not an area I know an awful lot about to be honest. I'm aware that it's far from complete. I've a feeling, from the QFT viewpoint, things are justified by adapting some results from Stochastic calculus - which is fair enough. In general though, the topic of "functional integration over an arbitrary function space" still has a loong way to go!
A joke my supervisor like to tell is if it's a Gaussian then it's easy. If it's anything else it's impossible. (physics humour is not the best it has to be said)
It's also pretty ad hoc. If you've got a scalar field p you get the expectation value of an operator O by doing
\int [dp] O Exp[i S] / \int [dp] Exp[i S]
Clearly, thats infinity/infinity which I really don't like. I guess it's not too different from renormalization where you take infinity from infinity to get something finite.
A joke my supervisor like to tell is if it's a Gaussian then it's easy. If it's anything else it's impossible. (physics humour is not the best it has to be said)
It's also pretty ad hoc. If you've got a scalar field p you get the expectation value of an operator O by doing
\int [dp] O Exp[i S] / \int [dp] Exp[i S]
Clearly, thats infinity/infinity which I really don't like. I guess it's not too different from renormalization where you take infinity from infinity to get something finite.
QUOTE (phyti+Jun 8 2008, 04:22 AM)
mr_homm;
1. Isn't this going down the Russell path of a set containing itself?
2. The empty set has been defined as the set with no elements, so after all the iterations, where are the elements?
1:
Actually, no. None of these sets contains itself. The set 0 contains nothing, so it certainly does not contain itself. The set 1 contains 0 but it does not contain 1. The set 2 contains 0 and 1 but not 2. And so on. Each set contains all the previously defined ones, but not itself. So in fact NONE of these sets contains itself. Even the infinite ones only contain all the previous sets, but not themselves.
2:
The sets constructed by this process ARE the elements. Nevertheless, no set contains itself, as mentioned above. It is true that you appear to start with nothing at all, and get all these numbers, but in reality this construction is driven by the axioms of set theory. So you aren't really starting from nothing, you are starting from the axioms of the theory. These axioms guarantee that you can put a box around ANYTHING, so long as you can specify exactly what you are boxing. So if you say, "I'll put a box around nothing at all," that is allowed by the axioms. Then you have a box, so you can say, "Now I'll put a box around my first box." and the rules allow this, too. And so on.
Ultimately, you have boxes of boxes of boxes.... The innermost boxes are empty, of course, but they are still there. I know that this seems rather strange, so look at it by analogy: suppose you owned a company that manufactured boxes. You would pack them into larger boxes to ship. One day, one of your customers complains that you are cheating him, because he paid you for a shipment, but when he opened every box he found, there was nothing inside, so you had shipped him nothing. That complaint would be unreasonable, because after all, you are a box company, and he paid for boxes and got boxes. So boxes of boxes of ... of boxes of nothing, is actually [I\something[/I].
Besides, these sets do have a structure governed by the laws of logic and the axioms of set theory, and this structure is rich enough to host the structure of the integers and their arithmetic. In mathematics, it pretty much doesn't matter what the substrate is that has the structure; the structure is the main object of study. This is not unlike computer programming, in which the actual hardware largely doesn't matter to the programmer. If I write a C code program that prints "Hello world!" I can compile it and run it on pretty much any machine in the world, and it will print "Hello world!" to the screen. The program itself is the same entity regardless of the underlying hardware, and its properties can be studied independently of the machine. In a similar way, mathematical structure can be studied independently of what things have that structure, as long as the structure itself is logically self-consistent.
While we're on the subject of programs, note that if I run the same program on several machines, that doesn't make it several different programs. It is several instances of the same program. Just because it is currently running on one machine, that is no reason why it cannot also be sitting on a disk somewhere else, too. This is in the nature of abstract objects: as many instances as you want can all be present simultaneously, without the abstract object being more than one thing. The relevance of this to the boxes in set theory, is that once you have put a box around something, you can still have another instance of that same something which is NOT inside that box. This is important, because without it you could not use von Neumann's construction. When you get to 2 = {1,0}, you will recall that 1 = {0}, so the 0 is already "inside 1", and someone might raise the objection that since it is already inside the box, how can it be outside of the box, too? Well, the same way a computer program can be on a disk and running on a computer at the same time. Abstract objects can do that.
Warning: the following is just for fun. It is not good mathematics, just wordplay.
You start with nothing at all.
To make each new number, you box up everything you have so far.
Since you put this box around everything you have, there is nothing outside the box.
But nothing is what you started with.
So you can start with nothing outside the box, and rebuild outside the box everything that was inside it.
So you have the same things both inside and outside the box.
Then you box that.
But this is just step 2 again, for the next higher number.
Continue forever.
You might want to look at this link, which describes the process all the way from basic set theory to the construction of the complex numbers in 14 pages.
--Stuart Anderson
1. Isn't this going down the Russell path of a set containing itself?
2. The empty set has been defined as the set with no elements, so after all the iterations, where are the elements?
1:
Actually, no. None of these sets contains itself. The set 0 contains nothing, so it certainly does not contain itself. The set 1 contains 0 but it does not contain 1. The set 2 contains 0 and 1 but not 2. And so on. Each set contains all the previously defined ones, but not itself. So in fact NONE of these sets contains itself. Even the infinite ones only contain all the previous sets, but not themselves.
2:
The sets constructed by this process ARE the elements. Nevertheless, no set contains itself, as mentioned above. It is true that you appear to start with nothing at all, and get all these numbers, but in reality this construction is driven by the axioms of set theory. So you aren't really starting from nothing, you are starting from the axioms of the theory. These axioms guarantee that you can put a box around ANYTHING, so long as you can specify exactly what you are boxing. So if you say, "I'll put a box around nothing at all," that is allowed by the axioms. Then you have a box, so you can say, "Now I'll put a box around my first box." and the rules allow this, too. And so on.
Ultimately, you have boxes of boxes of boxes.... The innermost boxes are empty, of course, but they are still there. I know that this seems rather strange, so look at it by analogy: suppose you owned a company that manufactured boxes. You would pack them into larger boxes to ship. One day, one of your customers complains that you are cheating him, because he paid you for a shipment, but when he opened every box he found, there was nothing inside, so you had shipped him nothing. That complaint would be unreasonable, because after all, you are a box company, and he paid for boxes and got boxes. So boxes of boxes of ... of boxes of nothing, is actually [I\something[/I].
Besides, these sets do have a structure governed by the laws of logic and the axioms of set theory, and this structure is rich enough to host the structure of the integers and their arithmetic. In mathematics, it pretty much doesn't matter what the substrate is that has the structure; the structure is the main object of study. This is not unlike computer programming, in which the actual hardware largely doesn't matter to the programmer. If I write a C code program that prints "Hello world!" I can compile it and run it on pretty much any machine in the world, and it will print "Hello world!" to the screen. The program itself is the same entity regardless of the underlying hardware, and its properties can be studied independently of the machine. In a similar way, mathematical structure can be studied independently of what things have that structure, as long as the structure itself is logically self-consistent.
While we're on the subject of programs, note that if I run the same program on several machines, that doesn't make it several different programs. It is several instances of the same program. Just because it is currently running on one machine, that is no reason why it cannot also be sitting on a disk somewhere else, too. This is in the nature of abstract objects: as many instances as you want can all be present simultaneously, without the abstract object being more than one thing. The relevance of this to the boxes in set theory, is that once you have put a box around something, you can still have another instance of that same something which is NOT inside that box. This is important, because without it you could not use von Neumann's construction. When you get to 2 = {1,0}, you will recall that 1 = {0}, so the 0 is already "inside 1", and someone might raise the objection that since it is already inside the box, how can it be outside of the box, too? Well, the same way a computer program can be on a disk and running on a computer at the same time. Abstract objects can do that.
Warning: the following is just for fun. It is not good mathematics, just wordplay.
You start with nothing at all.
To make each new number, you box up everything you have so far.
Since you put this box around everything you have, there is nothing outside the box.
But nothing is what you started with.
So you can start with nothing outside the box, and rebuild outside the box everything that was inside it.
So you have the same things both inside and outside the box.
Then you box that.
But this is just step 2 again, for the next higher number.
Continue forever.
You might want to look at this link, which describes the process all the way from basic set theory to the construction of the complex numbers in 14 pages.
--Stuart Anderson
QUOTE (prometheus+Jun 8 2008, 12:54 PM)
A joke my supervisor like to tell is if it's a Gaussian then it's easy. If it's anything else it's impossible. (physics humour is not the best it has to be said)
I can imagine as much!
I can imagine as much!
QUOTE (prometheus+Jun 8 2008, 12:54 PM)
Clearly, thats infinity/infinity which I really don't like. I guess it's not too different from renormalization where you take infinity from infinity to get something finite.
Well I don't think the mathematics of renormalization was really sound until Alain Connes decided to look at QFT and did some work with Hopf algebras. And that was fairly hardcore stuff, from the mathematical viewpoint! We need the equivalent person in analysis and geometric measure theory to take up a similar interest and lay the subject to rest...
Well I don't think the mathematics of renormalization was really sound until Alain Connes decided to look at QFT and did some work with Hopf algebras. And that was fairly hardcore stuff, from the mathematical viewpoint! We need the equivalent person in analysis and geometric measure theory to take up a similar interest and lay the subject to rest...
@prometheus,
In regards to Euler's remarks, perhaps it is of interest to look at how the idea of weak solutions fits into the general schema of a mathematician's armamentarium.
The essential idea is extension. If a given problem has no solution, one asks immediately, "in what context?" Certain differential equations which were regarded as having "no solution" can be fruitfully regarded as having "no solution within functions." By adding the context specification, one is invited to see if there is a wider context in which a solution does exist. The usual (and very sensible) criterion is that the extended context should contain the original context or connect to it in some clear way.
The distributions generalize the differentiable functions, and so admit solutions to differential equations which are unsolvable within the context of functions alone.
Another case where Dirac's name is associated with this idea is in his solution of his Dirac equation. In order to solve this equation, he wanted to factor the differential operator which occurs within it. However, this operator did not factor by any of the usual methods, so he introduced matrix valued solutions into his equation (essentially the inception of spinor representations in physics, I believe), which allowed him to factor his operator.
The idea has a long history in mathematics, in reverse order:
Splitting fields allow one to factor polynomials which do not have roots in the rational numbers.
The complex numbers allow one to factor polynomials which do not have roots in the reals.
The real numbers allow construction of numbers which occur in geometry but cannot be expressed as rationals.
There are many other examples of this general approach as well.
Hope that was of some interest. The examples are pretty trivial; I just like to look at things in a historical context when possible.
--Stuart Anderson
In regards to Euler's remarks, perhaps it is of interest to look at how the idea of weak solutions fits into the general schema of a mathematician's armamentarium.
The essential idea is extension. If a given problem has no solution, one asks immediately, "in what context?" Certain differential equations which were regarded as having "no solution" can be fruitfully regarded as having "no solution within functions." By adding the context specification, one is invited to see if there is a wider context in which a solution does exist. The usual (and very sensible) criterion is that the extended context should contain the original context or connect to it in some clear way.
The distributions generalize the differentiable functions, and so admit solutions to differential equations which are unsolvable within the context of functions alone.
Another case where Dirac's name is associated with this idea is in his solution of his Dirac equation. In order to solve this equation, he wanted to factor the differential operator which occurs within it. However, this operator did not factor by any of the usual methods, so he introduced matrix valued solutions into his equation (essentially the inception of spinor representations in physics, I believe), which allowed him to factor his operator.
The idea has a long history in mathematics, in reverse order:
Splitting fields allow one to factor polynomials which do not have roots in the rational numbers.
The complex numbers allow one to factor polynomials which do not have roots in the reals.
The real numbers allow construction of numbers which occur in geometry but cannot be expressed as rationals.
There are many other examples of this general approach as well.
Hope that was of some interest. The examples are pretty trivial; I just like to look at things in a historical context when possible.
--Stuart Anderson
mr homm-what the relations between dirac 's delta function with the set theory of
banach-tarski?
banach-tarski?
QUOTE (mott.carl+Jun 15 2008, 12:05 PM)
mr homm-what the relations between dirac 's delta function with the set theory of
banach-tarski?
Stop asking irrelevent questions you idiot. Do you think your ability to ask idiotic questions makes you appear intelligent?
banach-tarski?
Stop asking irrelevent questions you idiot. Do you think your ability to ask idiotic questions makes you appear intelligent?
QUOTE (thinker+Jun 7 2008, 02:14 AM)
I have spent a long time arguing with someone over whether or not infinity times zero equals one. I would like to know if I have any hope on holding my position that it is not true, or if I am simply wasting my time.
His reasoning for the previous statement being true is as follows:
If 1/n=0 as n approaches infinity, he insists that 1/infinity=0 and thus 0 times infinity=1. (Also infinity/0=1)
I've tried to disprove this for my firm belief in the existence of the infinitesimal, and my disapproval of the number of properties which must be ignored for the statement to be true (which exemption from said properties may be justified by the concept that infinity is not a number (another statement he disagrees with)). I have tried countless analogies with no avail and his persistence began to make me doubt my own accuracy.
Please enlighten me on my err and misconception as they are in abundance.
It would be possible to construct a more precise definition of infinity that allowed infinity*0=1, but the identity for a multiplication by 0 would have to be altered as well and typically infinity isn't a sufficiently determinatively specified concept to rely upon all uses of the term to allow for this identity to be reliably used.
(Basically, people use the term infinity in many contexts that are not inherently required to be identical, but it could be quite useful to have a precise definition that allowed such operations to be performed in a deterministic manner)
The definition for "real" numbers specifies the multiplicative identity for 0 to result in 0 (always) and so appending an identity for infinity*0 to result in 1 would require an exception be made to this definition (the issue actually appears to be rooted in the loss of information when a product, such as x*0 is mapped to 0 instead of the x dimension being scaled linearly and this mapping is irreversible and not dependent upon x, so the result of 0 can't then be multiplied by any reciprocal term to later return x unless infinitesimal quantities are added, which then allow for the identity +0(positive infinitesimal)*infinity=1 to be constructed, though this is no longer identical to an additive 0 and an explicit differentiation between them would be necessary, though from the perspective of quantization of infinitesimal terms in Calculus for physical approximations, the final result of a computation could be "rounded" to substitute infinitesimal terms with 0).
It's a bit incorrect to state that 1/n=0 as n->infinity, because in this case n<infinity and we're working with an inequality. It's actually the limit of these reciprocal terms that's 0 (if we ignore irrational limits).
So we can simply rewrite this as:
For n<infinity, 1/n>0, in this case the limit of n is infinity and the limit of 1/n is 0, though notice that, for example, 1/(n^2) also approaches 0 as n approaches infinity.
In this case if we defined n=infinity and 1/n=0, then we could derive "infinity"*"zero" (in the context of a positive infinitesimal) to be equal to 1, but this would not extend as an equal to 0^2 (zero squared), though still an infinitesimal term approaching 0, because n*1/(n^2)=1/n, and in that case, though n and 1/(n^2) still can approach infinite and infinitesimal quantities respectively they aren't related in the same and manner the result (if we ignore the context of the references to infinity and a "scaling" by zero) we get approaches 0 instead of 1 (we could also have "infinity*0" approach infinity or negative infinity or be bounded by some finite constant as well, depending upon how each infinite and infinitesimal term was constructed).
So basically, no, without a more precise mathematical definition for infinity, such an identity isn't possible to construct, though the fault is not in the logic that 1/infinity should be equal to 0 or an infinitesimal, but in the definition of a multiplication by 0 for real numbers (and other numbering systems have been constructed to deal with such indeterminancies in much of the real number notations to allow for these constructions to be more precisely operated upon - See "Hyperreals" and "complex numbers" as some examples).
I posted a couple threads before regarding some (rather pervasive) areas of mathematics in which these issues can have a significant influence upon what results are typically derived by a computation (many results that are typically assumed to be entirely deterministically derived and provide no alternative solutions are not truly such but instead are just the typical results arrived at by following historical assumptions and conventions, but if you go back to basics and rederive the relationships, there are other solutions that can be made to solve the equations as well).
His reasoning for the previous statement being true is as follows:
If 1/n=0 as n approaches infinity, he insists that 1/infinity=0 and thus 0 times infinity=1. (Also infinity/0=1)
I've tried to disprove this for my firm belief in the existence of the infinitesimal, and my disapproval of the number of properties which must be ignored for the statement to be true (which exemption from said properties may be justified by the concept that infinity is not a number (another statement he disagrees with)). I have tried countless analogies with no avail and his persistence began to make me doubt my own accuracy.
Please enlighten me on my err and misconception as they are in abundance.
It would be possible to construct a more precise definition of infinity that allowed infinity*0=1, but the identity for a multiplication by 0 would have to be altered as well and typically infinity isn't a sufficiently determinatively specified concept to rely upon all uses of the term to allow for this identity to be reliably used.
(Basically, people use the term infinity in many contexts that are not inherently required to be identical, but it could be quite useful to have a precise definition that allowed such operations to be performed in a deterministic manner)
The definition for "real" numbers specifies the multiplicative identity for 0 to result in 0 (always) and so appending an identity for infinity*0 to result in 1 would require an exception be made to this definition (the issue actually appears to be rooted in the loss of information when a product, such as x*0 is mapped to 0 instead of the x dimension being scaled linearly and this mapping is irreversible and not dependent upon x, so the result of 0 can't then be multiplied by any reciprocal term to later return x unless infinitesimal quantities are added, which then allow for the identity +0(positive infinitesimal)*infinity=1 to be constructed, though this is no longer identical to an additive 0 and an explicit differentiation between them would be necessary, though from the perspective of quantization of infinitesimal terms in Calculus for physical approximations, the final result of a computation could be "rounded" to substitute infinitesimal terms with 0).
It's a bit incorrect to state that 1/n=0 as n->infinity, because in this case n<infinity and we're working with an inequality. It's actually the limit of these reciprocal terms that's 0 (if we ignore irrational limits).
So we can simply rewrite this as:
For n<infinity, 1/n>0, in this case the limit of n is infinity and the limit of 1/n is 0, though notice that, for example, 1/(n^2) also approaches 0 as n approaches infinity.
In this case if we defined n=infinity and 1/n=0, then we could derive "infinity"*"zero" (in the context of a positive infinitesimal) to be equal to 1, but this would not extend as an equal to 0^2 (zero squared), though still an infinitesimal term approaching 0, because n*1/(n^2)=1/n, and in that case, though n and 1/(n^2) still can approach infinite and infinitesimal quantities respectively they aren't related in the same and manner the result (if we ignore the context of the references to infinity and a "scaling" by zero) we get approaches 0 instead of 1 (we could also have "infinity*0" approach infinity or negative infinity or be bounded by some finite constant as well, depending upon how each infinite and infinitesimal term was constructed).
So basically, no, without a more precise mathematical definition for infinity, such an identity isn't possible to construct, though the fault is not in the logic that 1/infinity should be equal to 0 or an infinitesimal, but in the definition of a multiplication by 0 for real numbers (and other numbering systems have been constructed to deal with such indeterminancies in much of the real number notations to allow for these constructions to be more precisely operated upon - See "Hyperreals" and "complex numbers" as some examples).
I posted a couple threads before regarding some (rather pervasive) areas of mathematics in which these issues can have a significant influence upon what results are typically derived by a computation (many results that are typically assumed to be entirely deterministically derived and provide no alternative solutions are not truly such but instead are just the typical results arrived at by following historical assumptions and conventions, but if you go back to basics and rederive the relationships, there are other solutions that can be made to solve the equations as well).
StevenA is at it again!
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