A little math puzzle

[Gunboat Diplomat]

I know it's a mathematical problem, I was just making the observation that it would have a very real answer in reality due to the discontinuity of the Planck length (whether you could actually measure it is a different matter).

Why would you think that when we're talking about an infinite number of tries? Is an infinite number of tries more "realistic" to you than continuity?

 

[Master_Ghost_Knight]

Obviously no one denies that math is highly applicable to real world problems such as science. However, modern mathematics is not at all motivated by application. For instance, did you really think that imaginary numbers were invented to solve some possible scientific or engineering problem? No, they were conceived purely out of the curiosity of having a number system where even negative numbers have roots.

Actually we do use imaginary numbers to solve very real engineering problems, not that they were discovered for that. But my point is that you don't have to use imaginary numbers to describe the real world and nobody said that they were the best thing to model it, you could have picked another tool that could have potential done a batter job at it. You have raised the point that math is not aplicable to the world because a perfect circumference is continuous and obviously space isn't, but nobody said that you had to use the particular a continuous and infinite group to describe sapce, there are allot of descrete groups that do a better job at it. Just because you have picked the wrong tool it doesn't mean that all the contents of the toolbox is worthless. (being math very versatile, even if you had to acount recursively every aspect of it, you would still could get a mathematical model no matter how strange it may be)
That is why saying that math doesn't apply in the real world is pure nonsense.

Anyways, this is a mathematical problem, not a physics problem, and math is not bound by physical constraints, the physical description is just an hypotetical dysplay to transmit the nature of the problem.

 

[CkVega]

Why would you think that when we're talking about an infinite number of tries? Is an infinite number of tries more "realistic" to you than continuity?

I've already acknoledged that it's a math problem and not a physics problem, it was a semi tongue-in-cheek comment.

 

[Gunboat Diplomat]

I don't normally split up a paragraph but I'm doing so here 'cause there are so many disparate things said here that need addressing...

Actually we do use imaginary numbers to solve very real engineering problems, not that they were discovered for that.

What do you mean "actually?" Did you honestly think that I said we didn't use complex numbers in engineering? You need to go back and carefully reread what I wrote. Specifically, I wrote "modern mathematics is not at all motivated by application" and I stand by that. It's not. You even admitted yourself that they weren't discovered for that purpose and that was my point. How is it that you think you're rebutting me?

But my point is that you don't have to use imaginary numbers to describe the real world and nobody said that they were the best thing to model it, you could have picked another tool that could have potential done a batter job at it. You have raised the point that math is not aplicable to the world because a perfect circumference is continuous and obviously space isn't, but nobody said that you had to use the particular a continuous and infinite group to describe sapce, there are allot of descrete groups that do a better job at it.

That may be your point but it doesn't rebut my point.

It's clear from these two sentences that you think my point is that "math is not aplicable to the world" but what I'm saying is that modern mathematics is unconcerned about the world. The tools available in math are useful because they are perfectly logical deductions and are thus assuredly true (in some sense), so they can be of great help if you manage to "find the right tools for the job," as you like to put it. However, that these tools can help with real world problems is a coincidence. Just as with imaginary numbers or non-Euclidean geometry, no one created these things to help you build a bridge or an electrical circuit. These ideas were explored merely because they were interesting. Mathematicians obviously don't go out of their way to create useless ideas but I assure you that they are reality agnostic. They don't care!

Just because you have picked the wrong tool it doesn't mean that all the contents of the toolbox is worthless. (being math very versatile, even if you had to acount recursively every aspect of it, you would still could get a mathematical model no matter how strange it may be)
That is why saying that math doesn't apply in the real world is pure nonsense.

I never said that it "doesn't apply in the real world." My exact words were "modern mathematics has nothing to do with reality" and it doesn't! Mathematicians are not motivated by "the real world," they are motivated by whatever's interesting and logically consistent, the real world be damned!

I don't know to whom you're responding 'cause it doesn't appear to be me!

 

[Gunboat Diplomat]

I've already acknoledged that it's a math problem and not a physics problem, it was a semi tongue-in-cheek comment.

Well, that would be why I asked if you were joking. If this were actually your attitude, I'd expect a response more like "I was sort of joking," rather than "Why would it be a joke?"

 

[aeroeng314]

Mathematicians are not motivated by "the real world,"

If they were, they'd be engineers.

 

[CkVega]

I've already acknoledged that it's a math problem and not a physics problem, it was a semi tongue-in-cheek comment.

Well, that would be why I asked if you were joking. If this were actually your attitude, I'd expect a response more like "I was sort of joking," rather than "Why would it be a joke?"

I guess sarchasm doesn't show very well across the interweb.

 

[Master_Ghost_Knight]

It's clear from these two sentences that you think my point is that "math is not aplicable to the world" but what I'm saying is that modern mathematics is unconcerned about the world. The tools available in math are useful because they are perfectly logical deductions and are thus assuredly true (in some sense), so they can be of great help if you manage to "find the right tools for the job," as you like to put it. However, that these tools can help with real world problems is a coincidence. Just as with imaginary numbers or non-Euclidean geometry, no one created these things to help you build a bridge or an electrical circuit. These ideas were explored merely because they were interesting. Mathematicians obviously don't go out of their way to create useless ideas but I assure you that they are reality agnostic. They don't care!

When you said that:

modern mathematics has nothing to do with reality

One is led to believe that you literally mean nothing.
It is true that math is based upon pure logic in order to establish non-other then absolut statments and that such statments veracity is independent of the happenings in the real world. But in the context your statment lead me to believe that you mean that "it is not aplicable" rather then "it is abstract".

Ps. Just as a note, by far the best contributers for the most usefull pieces of math are physicists rather then mathematicians (altough physicists are also mathematicians but i mean the group of mathematicians that are also physicists for phisics purpouses rather mathematicians for mathematics alone).

 

[Gunboat Diplomat]

When you said that:

modern mathematics has nothing to do with reality

One is led to believe that you literally mean nothing.
It is true that math is based upon pure logic in order to establish non-other then absolut statments and that such statments veracity is independent of the happenings in the real world. But in the context your statment lead me to believe that you mean that "it is not aplicable" rather then "it is abstract".

You're right, I see how you can interpret what I said as you did. I prefer to think of the utility of mathematics as a fortunate coincidence...

Ps. Just as a note, by far the best contributers for the most usefull pieces of math are physicists rather then mathematicians (altough physicists are also mathematicians but i mean the group of mathematicians that are also physicists for phisics purpouses rather mathematicians for mathematics alone).

Of course, this depends on what you consider "useful" and to what degree. I'm interested to know who you're thinking of when you say this...

Who was the last physicist to make up new maths? Lagrange (1736, 1813)? I disagree that physicists are mathematicians. To think that is to fail to understand the entire enterprise of modern mathematics. Obviously physicists use mathematics all the time but, really, who doesn't? Using mathematics doesn't a mathematician make any more than using science makes one a scientist...

 

[Master_Ghost_Knight]

Of course, this depends on what you consider "useful" and to what degree. I'm interested to know who you're thinking of when you say this...

Who was the last physicist to make up new maths? Lagrange (1736, 1813)? I disagree that physicists are mathematicians. To think that is to fail to understand the entire enterprise of modern mathematics. Obviously physicists use mathematics all the time but, really, who doesn't? Using mathematics doesn't a mathematician make any more than using science makes one a scientist...

Yeah, well, no.
Physicists are mathematicians, not because they very often use this tool called math, but because it is a fundamental requierment for any quatifiable endeavour (and if it is not quantifiable it is not good science) to know math and physicists need lots of it, and they need to know lots in order to be able to create their own mathematical tools on the spot so they are able to correctly quantifie and describe the expected phenomena. A physicists have a strong education in math, oriented to those fields of math that usualy tend to work but none the less it is more then enough math to make them mathematicians. Of course pure mathematicians can still serve the phycisist ass on a plate in terms of mathematical knowledge, but this knowledge is more wide spread and in fields that are interesting in mathematical terms.

 

[Dragan Glas]

Greetings,

I just saw this - but as others have already answered it, I can't win the virtual cake...

Oh alright then - I'll give my answer anyway...

Mathematically speaking:

The limit of 1/x as x approaches Infinity is 0.

Kindest regards,

James

 

[Gunboat Diplomat]

Physicists are mathematicians, not because they very often use this tool called math, but because it is a fundamental requierment for any quatifiable endeavour (and if it is not quantifiable it is not good science) to know math and physicists need lots of it, and they need to know lots in order to be able to create their own mathematical tools on the spot so they are able to correctly quantifie and describe the expected phenomena. A physicists have a strong education in math, oriented to those fields of math that usualy tend to work but none the less it is more then enough math to make them mathematicians. Of course pure mathematicians can still serve the phycisist ass on a plate in terms of mathematical knowledge, but this knowledge is more wide spread and in fields that are interesting in mathematical terms.

I can't help but notice that you didn't answer any of the questions in my post. They weren't rhetorical, you know! I'm actually interested in knowing which physicists you were thinking of when you said:

Just as a note, by far the best contributers for the most usefull pieces of math are physicists rather then mathematicians (altough physicists are also mathematicians but i mean the group of mathematicians that are also physicists for phisics purpouses rather mathematicians for mathematics alone).

I like physics too, you know!

In response to your post, you pretty much simply said that physicists are mathematicians not because they use a lot of maths but because they use an awful lot of maths. Engineers use an awful lot of maths too but they are not mathematicians any more than they are scientists (they use an awful lot of science, too!)...

I can't remember the last time a physicist created some new maths, much less "on the spot." Can you give me an example? I know that physicists and scientists in general need to make up notation to describe whatever it is they're studying but that's not new maths. They discover new relationships and describe them using equations but that's not new maths either. Now that I think about it, I'm curious to know what you think mathematicians do!

 

[Ozymandyus]

I can't remember the last time a physicist created some new maths, much less "on the spot." Can you give me an example? I know that physicists and scientists in general need to make up notation to describe whatever it is they're studying but that's not new maths. They discover new relationships and describe them using equations but that's not new maths either. Now that I think about it, I'm curious to know what you think mathematicians do!

Well, the most often quoted example is of course Newton creating Calculus. Of course, Newton really was more of a mathematician than a physicist, so that hardly counts.

The line between theoretical physicist and mathematician is incredibly blurry, so its really hard to give credit to one or the other. Hard to really give credit to any particular area of study for new math concepts really - if you created a new area of mathematics I think its safe to call you a mathematician. But yes, many of these people also had other areas of study, and physics is probably the most common shared background.

 

[Master_Ghost_Knight]

I can't help but notice that you didn't answer any of the questions in my post. They weren't rhetorical, you know! I'm actually interested in knowing which physicists you were thinking of when you said:

Just as a note, by far the best contributers for the most usefull pieces of math are physicists rather then mathematicians (altough physicists are also mathematicians but i mean the group of mathematicians that are also physicists for phisics purpouses rather mathematicians for mathematics alone).

I like physics too, you know!

In response to your post, you pretty much simply said that physicists are mathematicians not because they use a lot of maths but because they use an awful lot of maths. Engineers use an awful lot of maths too but they are not mathematicians any more than they are scientists (they use an awful lot of science, too!)...

I can't remember the last time a physicist created some new maths, much less "on the spot." Can you give me an example? I know that physicists and scientists in general need to make up notation to describe whatever it is they're studying but that's not new maths. They discover new relationships and describe them using equations but that's not new maths either. Now that I think about it, I'm curious to know what you think mathematicians do!

A yes, I didn't taught in one in particular and I didn't taught that this was a relevant issue, there has been many contribution particularly in terms of diferential analysis, manyfolds there has been some contributions by physicist I believe but I think the vast majority has been from pure mathematics.
If this helps I consider a mathematician some one who is heavily trained in several fields of mathematics and is capable of indepedently derive mathematical relations and theorems. And the reason physicists are closely related to the mathematics that it is usefull is because they need the math that works on solving physical problems (i.e. that are usefull), there isn't a big mystery behind it.
This is in no way undermining the role that pure mathematicians have, has the vast majority of the theorems necessary to establish a good tool comes from pure mathematics.

 

[CrapsWithBears]

The probability of countably many events with prob 0, is still 0. Similarly, you can take any countable number of points you like, for example the set of all points on the circle with a rational angle. The probability of landing on any of those is 0. So, extending the original question, you could ask what the probability is of landing on one of these (countably) infinitely many points, given (countably) infinitely many tried. The answer is still 0.

Same problem as aeroeng314's proof.

Nah. The probability distribution is uniform over [0,1), or [0,2pi), or whatever interval you want. The probability of any finite or countably infinite event is 0. It really is that simple.

 

[Dragan Glas]

Greetings,

The probability of countably many events with prob 0, is still 0. Similarly, you can take any countable number of points you like, for example the set of all points on the circle with a rational angle. The probability of landing on any of those is 0. So, extending the original question, you could ask what the probability is of landing on one of these (countably) infinitely many points, given (countably) infinitely many tried. The answer is still 0.

With all due respect, I beg to differ.

Taking the above quote:

The probability of countably many events with prob 0, is still 0.

True.

Similarly, you can take any countable number of points you like, for example the set of all points on the circle with a rational angle. The probability of landing on any of those is 0.

False.

The probability "approaches 0".

So, extending the original question, you could ask what the probability is of landing on one of these (countably) infinitely many points, given (countably) infinitely many tried. The answer is still 0.

False for the above reason.

Kindest regards,

James

 

[CrapsWithBears]

Greetings,

The probability of countably many events with prob 0, is still 0. Similarly, you can take any countable number of points you like, for example the set of all points on the circle with a rational angle. The probability of landing on any of those is 0. So, extending the original question, you could ask what the probability is of landing on one of these (countably) infinitely many points, given (countably) infinitely many tried. The answer is still 0.

With all due respect, I beg to differ.

Taking the above quote:

The probability of countably many events with prob 0, is still 0.

True.

Similarly, you can take any countable number of points you like, for example the set of all points on the circle with a rational angle. The probability of landing on any of those is 0.

False.

The probability "approaches 0".

So, extending the original question, you could ask what the probability is of landing on one of these (countably) infinitely many points, given (countably) infinitely many tried. The answer is still 0.

It's not "approaching" 0, it IS 0. We're asking about the probability of a countably infinite set take from the unit interval with the uniform distribution. Take a look at "Uniform Distribution" at wikipedia. Should clear things up.

Cheers

 

[Master_Ghost_Knight]

Hey CrapsWithBears, nice to see you arround.

The answer is allways zero, it doesn't aproach anywhere because the number of point is set, inumerably infinite. If you are going to make the analysis by divinding the space as you go, you have cheated really bad because you tryed to trasform a inumerable infinite set into a numerable infinite one.

 

[CrapsWithBears]

Hey CrapsWithBears, nice to see you arround.

Thanks MGK, good to be seen

I looked over the wikipedia page on "Uniform distribution", and it really doesn't explain what I had hoped. So, I'm gonna get TECHNICAL!! In mathematics, the job of assigning numerical probabilities is done by what's called a "measure". A measure is a function defined on "measurable subsets" of a given set. In the context of this thread, the measure is the uniform measure on the set [0,1] (otherwise known as Lebesgue measure). Now it just so happens that EVERY countable subset of [0,1] is Lebesgue-measurable, and has Lebesgue measure 0. The probability is defined as the measure of the subset, which in this case is 0. So we need not consider limits, or "approaching" 0. The set under consideration is measurable, and has measure 0.

 

[Dragan Glas]

Greetings,

Hey CrapsWithBears, nice to see you arround.

Thanks MGK, good to be seen

I looked over the wikipedia page on "Uniform distribution", and it really doesn't explain what I had hoped. So, I'm gonna get TECHNICAL!! In mathematics, the job of assigning numerical probabilities is done by what's called a "measure". A measure is a function defined on "measurable subsets" of a given set. In the context of this thread, the measure is the uniform measure on the set [0,1] (otherwise known as Lebesgue measure). Now it just so happens that EVERY countable subset of [0,1] is Lebesgue-measurable, and has Lebesgue measure 0. The probability is defined as the measure of the subset, which in this case is 0. So we need not consider limits, or "approaching" 0. The set under consideration is measurable, and has measure 0.

Thank you for the explanation, CrapsWithBears (and thank you also, Master_Ghost_Knight) - you had me wondering if you were "taking the Michael out of me" when I read the Wiki articles (both "finite" and "continuous") and couldn't see how they related to what you were saying.

To be fair, I put my hand up and admit that I'm not a mathematician - my background is the computer industry (technical, rather than "sales" - just in case you're wondering!) - although I trust my grasp of mathematics is reasonably good!

So, are you saying that the answer to the topic's question is actually 0, rather than approaches 0, as I'd claimed in a earlier post!? Or was your answer directed to a possible discussion you were having with Pulsar (part of which I quoted but missed/hadn't read the full discussion) ?

Kindest regards,

James