Question for mathematicians/theoretical physicists

[sol invictus]

That's certainly what the simplistic division of a volume by a length would imply. It's an area, measured in square metres.

Actually no. It's measured in square meters times meter/kilometer, or 10^{-3} m^2.

But in the real world, that is precisely what it is not.

So think of it as volume/length. What was the question again?

Here's a simple one for someone to explain.
In the expression "There exists an infinite number of real numbers between the integers one and two";
a)What is the exact meaning of the preposition "between" in this context?
b)Does the word "number" in the phrase "infinite number" mean the same thing as the word "number" in the phrase "real numbers"?
c)How many letters are there between a and b?
d)Is "infinite" a number or an adjective?

a) Either strictly greater than and strictly less than, or including 1 and 2, or one but not the other. The statement is true in all those cases.

b) Strictly speaking... no. Infinity is not a real number. But one can extend the reals to include infinity in various ways, and the statement would still be true. Or you could phrase it as "the number of real numbers between 1 and 2 is larger than any real number".

c) Zero, in the most obvious interpretation of "between". But as I said you could take the answer to be 0, 1, or 2 and it wouldn't affect a).

d) See b. It's not an adjective - for example one could rephrase it more precisely by saying what number (not a real number, a more general one) of real numbers there are in that interval. The answer is the cardinality of the reals, which is sometimes called c (usually written in some weird gothic font to make it look scary). c is not a real number - it's an infinite number.

 

[Elizabeth I]

I hate to say this, but that is true of all scientific theories and hypothesis, they are 'made up', they are adjusted to fit the facts.

In this case, as in much of higher level physics the question is 'what predictions does the theory make'?

We can not build a small planet to test plate tectonics nor a star to test hydrodynamics of fusing stars either.

In many cases you can observe those events and/or their results and compare with the math to see if it really does "work."

So far no one has observed a string, a brane, or another dimension (speaking strictly in the physics sense, of course.) Or am I wrong and have they?

SS, sympathy. I HATE dealing with that kind of problem. There was a reason geometry and I didn't get along...

 

[godless dave]

So far no one has observed a string, a brane, or another dimension (speaking strictly in the physics sense, of course.) Or am I wrong and have they?

No - which is why string theory has not been accepted.

 

[69dodge]

That's certainly what the simplistic division of a volume by a length would imply. It's an area, measured in square metres.

But in the real world, that is precisely what it is not.

I can't tell whether I really don't understand what physical problem you're trying to solve, or whether you're making a more philosophical point.

Can you give me an example, with specific numbers, where the answer to your problem, when given in units of m³/m, differs numerically from the cross-sectional area of the cylinder, when given in units of m²?

Or do you think that in every cylinder the two numbers just happen always to be the same, for no reason?

If there's an obvious reason why the numerical parts of the answers must necessarily be the same, it's not such a stretch to identify the two answers entirely, including the units. From this point of view, "cross-sectional area" and "volume per length" are just two different ways of referring to what is essentially a single property of a cylinder.

 

[69dodge]

Here's a simple one for someone to explain.
In the expression "There exists an infinite number of real numbers between the integers one and two";
a)What is the exact meaning of the preposition "between" in this context?

I can't quite tell what you're asking for. Do you have any doubt about which real numbers are between 1 and 2, and which ones aren't? Or is that knowledge still not enough for you to feel that you know the 'meaning' of "between"?

b)Does the word "number" in the phrase "infinite number" mean the same thing as the word "number" in the phrase "real numbers"?

That's a good question. I think there are important differences.

Cardinal numbers (including infinite ones like the number of reals), are sort of discrete. Each has a next one, with none others in between. They are for counting things. You could have five real numbers, or six, or infinitely many, or zero, but not five and a half.

Real numbers are for measuring continuously variable quantities. You could have a pound of flour, or two, but you could also have a pound and a half.

c)How many letters are there between a and b?

Am I supposed to say "none" here, but not "zero"?

"The number of letters between a and b is zero" means the same as "there aren't any letters between a and b."

d)Is "infinite" a number or an adjective?

An adjective, I guess?

That's not really a mathematical question, is it? Just a grammatical one. No?

I think I'm missing your real question here.

 

[shadron]

ETA Shadron- No argument , but consider this:-
I regularly have to calculate cylinder volume in cubic metres per kilometre. (I'm interested in fluid volumes in oil wells)
(My first order rule of thumb is "diameter (in inches) squared, divided by two, is very nearly the volume in cubic metres per kilometre.")

But what I'm actually after is m^3 / m, so I move the decimal 3 places over.

A simple bit of geometry, well within the ability of most folk's mental arithmetic.
But wait a minute;- m3/m =m2 (In English "Cubic metres per metre equals square metres")
So dimensionally my answer is in square metres.
Which is mathematically correct, but makes no sense whatever.

You use a measurement which reflects the volume per length of well. While it doesn't really look like an area, it is directly proportional to the cross-sectional area of the well (and the proportionality constant has only to do with the specific units you use - it is the area when the units are all the same ones). Dimensionality is not violated, it is just that you are so used to the way you think about it that that truth has never been made apparentl to you. When you square the diameter, you computed the area:

d (in inches) ^2 / 2 ~= x (in m*3 / km) whatever x is...

[(r (in in) * 2) * (38.37 in/m)]^2 / 2 ~= x (in km^3/m) * 1000 m/km

[78.74 r] ^2 / 2 ~= x * 1000 (x and r in meters now)

6200 / 2000 * r^2 = x

3.1 r^2 = x but x is area of the x-section, so = pi r^2, so the proportionality between the true measure (to 4 places) and your estimate is

p = 3.1416 / 3.100 = 1.0134 or, about 1.34% off. That's a good rule of thumb. It is the result of the fact that the meters-to-inches conversion factor, squared and divided by 2000, is that close to pi. Pure coincidence, and handy for you.

Remember also, that a correct result from dimensional analysis is necessary but not sufficient to guarantee truth in the equation being analyzed.

 

[Dancing David]

In many cases you can observe those events and/or their results and compare with the math to see if it really does "work."

So far no one has observed a string, a brane, or another dimension (speaking strictly in the physics sense, of course.) Or am I wrong and have they?

SS, sympathy. I HATE dealing with that kind of problem. There was a reason geometry and I didn't get along...

What the Godless Dave said, string theory is self consistent but no predicted observations yet.

 

[sol invictus]

String theory is a good example of what I mentioned above - that it's almost impossible to find a consistent mathematical theory that accounts for all current data (which ST at least comes close to doing) AND adds something new (which it does in spades). When you do find something like that you take it very seriously - because it has an excellent chance of being right. But until you do an experiment that differentiates it from other theories you can't be sure.

By the way, testing string theory is easy - just drop something. The problem isn't testing it, it's differentiating it from general relativity.

 

[Soapy Sam]

I can't tell whether I really don't understand what physical problem you're trying to solve, or whether you're making a more philosophical point.

Can you give me an example, with specific numbers, where the answer to your problem, when given in units of m³/m, differs numerically from the cross-sectional area of the cylinder, when given in units of m²?

Or do you think that in every cylinder the two numbers just happen always to be the same, for no reason?

If there's an obvious reason why the numerical parts of the answers must necessarily be the same, it's not such a stretch to identify the two answers entirely, including the units. From this point of view, "cross-sectional area" and "volume per length" are just two different ways of referring to what is essentially a single property of a cylinder.

I suspect you are confused by the very simplicity of the question.
In traditional oilfield "API" units, the volume is calculated in barrels.
An 8.5" hole has an area of 72.25 sq ins and volume per thousand feet is about 70 barrels / 1000ft. To anyone in the business, that concept "barrels per thousand feet" is a important volume.
But of course, it's NOT a volume; it's a ratio of volume to length.

Using metric units brings this home more forcibly, as the dimensional unit is the same in all three dimensions. Cubic metres divided by metres gives square metres- which, as you and others have said, is clearly a measure of area.
Any mathematician seeing an answer in square metres emerge from an equation will be forced to come to the same conclusion, whether discussing volume per distance of a cylinder, or entropic information content of a black hole. Surfaces all the way. (What IS the radius of a black hole, anyway?)

Yet not one of the people I have pointed this out to would agree that any area is involved.

That's all I'm saying. :-that people who deal with real world phenomena and mathematicians who deal with logical systems may not agree on their interpretation of the data.

I can't quite tell what you're asking for. Do you have any doubt about which real numbers are between 1 and 2, and which ones aren't? Or is that knowledge still not enough for you to feel that you know the 'meaning' of "between"?

Honest answer- I'm unconvinced that there is any "between" between numbers, any more than there's a "between" between invisible unicorns.

That's a good question. I think there are important differences.

Cardinal numbers (including infinite ones like the number of reals), are sort of discrete. Each has a next one, with none others in between. They are for counting things. You could have five real numbers, or six, or infinitely many, or zero, but not five and a half.

Real numbers are for measuring continuously variable quantities. You could have a pound of flour, or two, but you could also have a pound and a half.

But not "minus one pound". The smallest real quantity is zero. But do numbers , once abstracted from quantities, mean anything at all?

Am I supposed to say "none" here, but not "zero"?

"The number of letters between a and b is zero" means the same as "there aren't any letters between a and b."

Where "between" implies "in alphabetical order". But is order a property of the letters, or of the alphabet? Is numerical order a property of the numbers or of the arithmetic?
a +a does not equal b. 1 + 1 supposedly is indistinguishable from 2.
This I find unlikely. It is not my experience of the world that any entity, doubled, is indistinguishable from something else.

An adjective, I guess?

That's not really a mathematical question, is it? Just a grammatical one. No?

I think I'm missing your real question here.

I'm a grammatical thinker. Perhaps Elizabeth is also. I think our difficulty understanding mathematical thinking is less a question of complexities than of a fundamental difference of point of view.

You use a measurement which reflects the volume per length of well. While it doesn't really look like an area, it is directly proportional to the cross-sectional area of the well (and the proportionality constant has only to do with the specific units you use - it is the area when the units are all the same ones). Dimensionality is not violated, it is just that you are so used to the way you think about it that that truth has never been made apparentl to you. When you square the diameter, you computed the area:

d (in inches) ^2 / 2 ~= x (in m*3 / km) whatever x is...

[(r (in in) * 2) * (38.37 in/m)]^2 / 2 ~= x (in km^3/m) * 1000 m/km

[78.74 r] ^2 / 2 ~= x * 1000 (x and r in meters now)

6200 / 2000 * r^2 = x

3.1 r^2 = x but x is area of the x-section, so = pi r^2, so the proportionality between the true measure (to 4 places) and your estimate is

p = 3.1416 / 3.100 = 1.0134 or, about 1.34% off. That's a good rule of thumb. It is the result of the fact that the meters-to-inches conversion factor, squared and divided by 2000, is that close to pi. Pure coincidence, and handy for you.

Remember also, that a correct result from dimensional analysis is necessary but not sufficient to guarantee truth in the equation being analyzed.

Oh I agree totally with your analysis. I choose the example because it's familiar to me and within my mental arithmetic ability. Yet I perceive no area and neither does anyone else I have asked- not one of whom is any more a mathematician than I. Yet to a mathematical thinker, the answer is self evidently exactly as you say.
It is this difference of perception that I feel illustrates the question Elizabeth asks in the OP. "How do we know the math shows what we think it does" To what extent is this a matter of conditioning?

 

[shadron]

Oh I agree totally with your analysis. I choose the example because it's familiar to me and within my mental arithmetic ability. Yet I perceive no area and neither does anyone else I have asked- not one of whom is any more a mathematician than I. Yet to a mathematical thinker, the answer is self evidently exactly as you say.
It is this difference of perception that I feel illustrates the question Elizabeth asks in the OP. "How do we know the math shows what we think it does" To what extent is this a matter of conditioning?

You are right about that. Especially when you are self-taught in a field, you tend to think in the same way as your mentor thought (be it human or some form of writing), and breaking the mold to see the commonality that underlies it and other systems that you already have mastered can be difficult. It is part of the experience (perhaps the main part?) which goes on from first experiences through to mastery.

'Way back, when I was learning Advanced Algebra, the people in the class often whined (myself included) that they just couldn't "see" the concepts. Our professor said that yes, for learning the field that being able to visualize what is happening is often very useful, BUT... but, visualization can become impossible when the concepts become abstruse, and so the requirement to be able to visualize something before it is understood is an intellectual weakness. One needs, in the metaphor of piloting, to "trust the instruments" without needing to look out the window in order to advance beyond a certain point in math, and since Einstein it has also become necessary in Physics.

 

[Soapy Sam]

I am faintly disturbed by your implication of familiarity with non-human writing, but perhaps that , as 69dodge would say, is a grammatical question?

 

[69dodge]

I suspect you are confused by the very simplicity of the question.
In traditional oilfield "API" units, the volume is calculated in barrels.
An 8.5" hole has an area of 72.25 sq ins and volume per thousand feet is about 70 barrels / 1000ft. To anyone in the business, that concept "barrels per thousand feet" is a important volume.
But of course, it's NOT a volume; it's a ratio of volume to length.

Using metric units brings this home more forcibly, as the dimensional unit is the same in all three dimensions. Cubic metres divided by metres gives square metres- which, as you and others have said, is clearly a measure of area.
Any mathematician seeing an answer in square metres emerge from an equation will be forced to come to the same conclusion, whether discussing volume per distance of a cylinder, or [...].

Yet not one of the people I have pointed this out to would agree that any area is involved.

I still don't get it.

I'm sure everyone in the business knows that a bigger hole (i.e., one with a larger cross section) contains more barrels per thousand feet than a smaller one, and furthermore that the functional relationship is linear---twice the cross-sectional area means twice the number of barrels per thousand feet. So why would no one agree that any area is involved?

I'm not saying, the answer has units of area, so it must be some area or other. I'm just saying, the answer is in fact a particular area, namely, the cross-sectional area of the hole, so it's not surprising that it has units of area.

 

[Frinkiak7]

Elizabeth, one point I would like to make is that hypothesis always precedes experiment. If a hypothesis is not internally consistent, then there is no point in even attempting to experiment and measure. What the mathematicians are saying is, "This mathematical model appears to be consistent with phenomena we observe."

They have not reached a conclusion, as such, merely a jumping-off point for further exploration and experimentation. Unfortunately, technology has not yet advanced to a point where experiments to measure the accuracy of string theory, higher dimensions, and the strangeness that is quantum mechanics (among other things) can be performed.

For a concrete example, look at general relativity theory. GRT predicts an effect known as "frame dragging", where a rotating body in space will pull the fabric of spacetime around itself. This was not reliable experimentally observed until around 2004, but the idea was internally consistent, and ended up being just one more point in favor of GRT's usefulness.

 

[Elizabeth I]

By the way, testing string theory is easy - just drop something. The problem isn't testing it, it's differentiating it from general relativity.

I would like to hear more about that - any suggestions for books written in very short sentences with very small words?

And if it's indistinguishable from general relativity, how do we know it's not general relativity? And of what value is it to call it something else if it's really general relativity all the time? Isn't that like saying, "Yes, I know this looks like a light bulb and operates like a light bulb and can be used for the very same purposes as a light bulb, but I think I'll call it a begonia"?

I'm a grammatical thinker. Perhaps Elizabeth is also. I think our difficulty understanding mathematical thinking is less a question of complexities than of a fundamental difference of point of view.

You are probably right (and yet I did okay in my intro logic class...)

Way back, when I was learning Advanced Algebra, the people in the class often whined (myself included) that they just couldn't "see" the concepts.

That was my entire problem with calculus. (To begin with, I had a basic difficulty with f(x) - I kept thinking of it as "f times x," instead of "function of X.")

 

[MattusMaximus]

No - which is why string theory has not been accepted.

Well, it depends upon who you ask. I have the sense that there's a real split within the theoretical physics community on the question of ST's validity. There are plenty who claim it to be the real deal, while many exist on the other side who see ST as little more than hand-waving.

The only way to settle arguments like these between theoreticians is, in my humble opinion, to run an experiment. That usually has the desired effect

Of course, as has been mentioned by many here already, by its very nature ST would be exceedingly difficult to test with modern technology. Though I have heard some hold out hope that the Large Hadron Collider might be able to help out with this. Does anyone know whether it is definitive that the LHC could test aspects of ST?

 

[MattusMaximus]

By the way, testing string theory is easy - just drop something. The problem isn't testing it, it's differentiating it from general relativity.

Or quantum mechanics. ST is supposed to provide a long-sought unification of both GR and QM, if I'm not mistaken.

 

[MattusMaximus]

This thread reminds me of a joke...

Please pardon the derail, but I can't but help share this joke I heard long, long ago...

===================

An engineer, physicist, and mathematician are traveling together in a foreign land when they are arrested by the local authorities and thrown into separate cells. Isolated and alone, they are left in their dark and dingy cells with little more than an old rusty bed, a bucket, and (as a sick joke) an unopened can of beans for the week's food.

After a few days have passed, the guards decide to look in on their captives. They go to the engineer's cell first and open it. Peering inside, the guards see the engineer sitting against the wall, looking a bit shabby but not excessively hungry, as they observe the opened can of beans next to the engineer.

"How'd you open the can without any tools?" asked the curious guards.

"Simple," replied the engineer. "I just pried it open on the corner of this rusty old bed."

Impressed, the guards shut the door and went to the next cell, which held the physicist. Upon opening that cell, they saw the physicist sitting atop the musty bed seeming reasonably comfortable. Next to the bed on the ground was that cell's can of beans, also opened.

Again, the guards asked how the captive opened the can of beans.

"Not too hard, actually," said the physicist, "I just did a few quick mental calculations, made some rough measurements on the can, and determined the optimum angle of impact with the wall which would open the can. Then I threw it against the wall - viola!"

Once again impressed with their captive, the guards closed the door to the physicist's cell and proceeded to the final cell, where the mathematician was housed.

When the guards opened the third door, they saw a much more grim scene. Laying on the floor, in obvious agony and seemingly near death from starvation, was the mathematician with the still-sealed can of beans on the floor nearby. As the guards shone a light upon the hapless captive, the mathematician's eyes feebly looked up at them.

Stunned and puzzled, the guards asked their third captive, "What's wrong with you? You are just as educated and intelligent as your fellows, so why didn't you open the can of beans and eat like they did?"

In a dry, raspy, and barely audible voice, the mathematician croaked...

"I... I d-don't under... stand. I defined... the... can... to be... open!"

Then the mathematician died.

 

[CaveDave]

... Isn't that like saying, "Yes, I know this looks like a light bulb and operates like a light bulb and can be used for the very same purposes as a light bulb, but I think I'll call it a begonia"?...

As an amusing example of renaming, check out Bell Labs Proves Dark Sucker

or Dark Sucker Theory

These even sound reasonable!

Cheers,

Dave

 

[Soapy Sam]

I still don't get it.

I'm sure everyone in the business knows that a bigger hole (i.e., one with a larger cross section) contains more barrels per thousand feet than a smaller one, and furthermore that the functional relationship is linear---twice the cross-sectional area means twice the number of barrels per thousand feet. So why would no one agree that any area is involved?

I'm not saying, the answer has units of area, so it must be some area or other. I'm just saying, the answer is in fact a particular area, namely, the cross-sectional area of the hole, so it's not surprising that it has units of area.

As usual, you make my point far more clearly than I ever could. It's precisely your inability to get it that I'm trying to impart. You are in the position of one who has seen the Dalmation in the pseudo random scatter of spots and shadows of a black & white photograph: You are now incapable of not seeing it!
You are absolutely right about the geometry of my example- as was Shadron. I'm not arguing with your analysis.
What I'm saying is that I can't appreciate that you are right (and nor can anyone I work with), except in a rather detached, theoretical way. It feels wrong.

I can't feel it as an area. I find it impossible to think of it that way.
You, 69dodge, either can, or the theoretical view is so clear in your mind as to make no functional difference. (I suspect an ability to mentally visualise spatial relationships is critical, something I lack completely. I can't visualise anything).

I know it's an area. But I don't believe it. It feels the same way some optical illusions do, where I know the lines are the same length, but my sensory system keeps insisting they are not. There's mind and there's gut.

And this example is something geometrically trivial, which I work with every day. Add another 8 dimensions and you may as well be talking in tongues;- indistinguishable from magic- or nonsense. I'm taking it on faith, which is not a happy position for a sceptic.
This I think is Elizabeth's point- either you grok this kind of thing, or you don't.

In the case of string theory, I find myself either forced to trust enthusiasts like Brian Greene who make a very nice living from it, or sceptics like Lee Smolin or Peter Woit who are "outside the charmed circle" and are doubtful aboutwhere it's going, scientifically and socially. I not only don't grasp the mathematical arguments. I can't.
It sometimes seems that those who can, simply don't believe that others cannot.

 

[Fredrik]

By the way, testing string theory is easy - just drop something. The problem isn't testing it, it's differentiating it from general relativity.

I would like to hear more about that - any suggestions for books written in very short sentences with very small words?

The only popular science book about string theory that I know about is The elegant universe by Brian Greene. They also made a TV documentary based on that. It's available here.

And if it's indistinguishable from general relativity, how do we know it's not general relativity? And of what value is it to call it something else if it's really general relativity all the time?

You have a point there. If two theories of physics make the same predictions, they are for all practical purposes the same theory, even if the underlying mathematical models are completely different. However, in this case, string theory makes lots of predictions that are different from what the current theories predict. Most of them aren't even about gravity. It's just very difficult to test the predictions. Also, string theory isn't a fully developed theory yet, so it's still not clear exactly what the predictions are.

Or quantum mechanics. ST is supposed to provide a long-sought unification of both GR and QM, if I'm not mistaken.

Nitpicking: It should be "quantum field theories" or "the standard model" (which is a quantum field theory), but not "quantum mechanics". String theory doesn't modify quantum mechanics in any way.