Question for mathematicians/theoretical physicists

[Elizabeth I]

The Science Channel (I think) recently presented a program on the eleventh dimension - apparently a Harvard physicist started wondering why gravity was comparatively so much weaker than other forces (except, I guess, the weak force?) and figured out that "the math works perfectly" to explain why if you assume 10 more dimensions besides ours. It sounded as if she were theorizing that gravity "comes from" some far-away dimension and gets "filtered" through several dimensions before it gets to us, thus diluting its strength.

Additionally, I have tried (emphasis on tried) reading books on chaos theory, string theory, quantum theory, and you name it. All those people seem to base their conclusions on the fact that "the math works."

My question is: how do we know the "math" really shows anything at all? If I assume that if X is true, then Y must be true, then say "X is true, therefore Y," what have I actually proved except that I can construct an internally consistent logic system? If I understand correctly, many paranoid schizophrenics' world view is internally consistent - and follows if you accept their basic assumptions - but that doesn't make any of it true.

Similarly, fractal diagrams make really cool graphics, but how do I know that they are really the graphs of some strange equations? Anybody could draw anything and say it's a graph.

Any thoughts will be sincerely appreciated. And then maybe I can try to read those books again.

 

[sanguine]

My question is: how do we know the "math" really shows anything at all? If I assume that if X is true, then Y must be true, then say "X is true, therefore Y," what have I actually proved except that I can construct an internally consistent logic system? If I understand correctly, many paranoid schizophrenics' world view is internally consistent - and follows if you accept their basic assumptions - but that doesn't make any of it true.

In this case, the "math works" means that, based on what we know of physical reality, if we assume the hypothetical case (e.g. "10 extra dimensions"), then calculations based on that case yield results that match things seen in physical reality. Note that this doesn't show that the assertion is true -- but it does show that the assertion does not violate our evidence-based understanding of the world, which is important.

One of the standing complaints about some areas of discussion in theoretical physics is that they do not yield testable hypotheses. That is, you can make an assertion, show that the conclusions derived from that assertion don't violate known reality, but then can't come up with any test that would falsify that assertion.

So the scientist in question isn't saying, "The math shows reality is this way," but is instead saying, "If this were the case, it would all fit." There's a big gap between the two, and the latter is highly reasonably scientific thinking -- it's where hypotheses come from.

 

[shadron]

A quick note, Elizabeth - the weak force is only weak when compared to the strong force. The weak force is what makes radioactive decay work (the conversions of neutrons to proton/electron pairs, and vice-versa) , which is the basis of radio-thermal isotope generators. Comparing the four basic forces is a somewhat apples-oranges idea, but after you make some simplifying assumptions, then you have the strong force (holding nuclei together, and involved whenever we split or fuse nuclei), the weak force, then the electromagnetic force, and, running far behind, gravity. It takes the whole mass of the planet earh to give you just the weight you have, after all (and what an anthropocentric argument that is).

That fact that "the math works" is a necessary but not sufficient reason to believe that the principle the math describes also works. It has sometimes been found, however, that math which was assumed to have no practical application in the universe when it was derived later on became useful when a new physical concept needed a theoretical framework. Non-euclidean geometries were considered useless amusements before Einstein's general relaiviy grabbed them as a useful explanation for he geometries that mass imposes on it's environment. That sort of indicates that math can be used a a signpost of which way to go in developing a theory.

 

[sol invictus]

My question is: how do we know the "math" really shows anything at all?

Here's the deal: mathematics works really, really, really well for describing the world. It works so well it let physicists make models for things as disparate as electrons and galaxies, smoke and airplanes. Those models let them build computers, TVs, nuclear bombs, and space shuttles. All of that would be completely impossible if the world didn't obey strict logical rules (at least to a very good approximation).

So when someone finds a set of mathematical equations that appear to describe some aspect of the world, they take them seriously. There's no way to be certain they're exactly right - what usually happens with successful theories is that they are very good approximations, but aren't perfect (and may eventually be replaced with a more exact model). Unsuccessful theories, on the other hand, tend to fail either because they turn out to be internally inconsistent or simply because they are inconsistent with experimental data.

 

[godless dave]

Elizabeth I, also keep in mind that when physicists talk about extra dimensions they are using the word "dimension" in the scientific sense, not the science-fiction sense. They aren't talking about anything "far away" or about alternate universes.

 

[Elizabeth I]

Elizabeth I, also keep in mind that when physicists talk about extra dimensions they are using the word "dimension" in the scientific sense, not the science-fiction sense. They aren't talking about anything "far away" or about alternate universes.

I know that (that's why I used "far away" and "dilutes" in quotations), but it's still something that they've made up and then twiddled factors until it agreed with itself.

That's what I don't understand. I could make up any damn thing and make up any number of equations to describe it, but unless and until it can somehow be tested in some other way than to see if the "equations work," it's still just made up. You can always make the equations work, as long as you don't have to test it real-world.

You can write the equations for the airflow over an airplane wing, then build a model to see if your equations are right.

Build me a model to test for the other ten dimensions, or to see if gravity really "travels" to our dimension from some other, or to figure out which one is the gravity "source."

 

[sol invictus]

That's what I don't understand. I could make up any damn thing and make up any number of equations to describe it, but unless and until it can somehow be tested in some other way than to see if the "equations work," it's still just made up. You can always make the equations work, as long as you don't have to test it real-world.

It's not like that. You see, we're not starting from scratch - we already know a tremendous amount about the world. So if you came up with your equations, to be interesting they would have to match everything we already know and add something new. That turns out to be incredibly difficult - only a very tiny fraction of the consistent mathematical theories you could write down match the data we already know. So when someone finds a new one, it's interesting.

It's also a bit of an art - good physicists (of which Lisa Randall, the person I think you're talking about, is certainly one) have a nose for what has a good chance of working and what doesn't. And there are some objective criteria too: any theory contains some number of parameters that need to be fixed by data before you can use it to make any predictions. The theories with the fewest parameters are preferred.

 

[godless dave]

Build me a model to test for the other ten dimensions, or to see if gravity really "travels" to our dimension from some other, or to figure out which one is the gravity "source."

"Our dimension" here isn't correct, and they aren't proposing that gravity "travels" here from any "source".

There are definitely four dimensions - three of space and one of time. We aren't "in" a dimension; we use dimensions to specify something's location in space and time. The other seven dimensions in the hypothesis in question are in our universe as well, but don't work quite the same way - if they exist at all.

 

[godless dave]

Similarly, fractal diagrams make really cool graphics, but how do I know that they are really the graphs of some strange equations? Anybody could draw anything and say it's a graph.

You could do some of the equations yourself. Or ask a mathematician friend to do them for you. I thought I had forgotten how to graph an equation, but when I took an economics class a few years ago I was able to dredge up the memory.

 

[Fredrik]

All those people seem to base their conclusions on the fact that "the math works."

My question is: how do we know the "math" really shows anything at all?

A mathematical model can't tell us anything about the real world on its own. In order to use a mathematical model in a theory of physics, you have to postulate some identifications between the model and the real world. E.g. you postulate that what a clock measures corresponds to a certain mathematical thing in the model. That collection of postulates is the theory. It's a set of statements that can be used to predict the results of experiments.

Physics is of course more than just theories. The experimental part of it is essential, but it's important to realize that the only thing that experiments can tell us is how accurately a theory predicts the results of experiments.

That's actually kind of annoying. A successful theory can be thought of as a logical implication "X implies Y", where X is the theory and Y is the agreement between thereotical predictions and experimental results. But "X implies Y" is not the same as "Y implies X", so we can't conclude that the statements contained in X are true (even approximately true) just because we have found that Y is (approximately) true.

Unfortunately, there's no other way to obtain knowledge about the real world. All the knowledge we have is in the form of "these predictions of this theory agrees with experiments to this degree of accuracy".

Similarly, fractal diagrams make really cool graphics, but how do I know that they are really the graphs of some strange equations? Anybody could draw anything and say it's a graph.

I don't understand. Are you asking how you can know that mathematicians aren't lying to you? In the case of fractals you can probably download the source code to a program that generates one of those images, and make sure that you understand it before you run it. In other cases, like the theorem of Pythagoras, you can work through the proof yourself. In those cases where the proof is too difficult, you can either choose to trust the peer review process or you can choose not to. All the proofs are checked by other mathematicians. They are pretty good at finding flaws, but they're not infallible.

 

[Reality Check]

The Science Channel (I think) recently presented a program on the eleventh dimension - apparently a Harvard physicist started wondering why gravity was comparatively so much weaker than other forces (except, I guess, the weak force?) and figured out that "the math works perfectly" to explain why if you assume 10 more dimensions besides ours. It sounded as if she were theorizing that gravity "comes from" some far-away dimension and gets "filtered" through several dimensions before it gets to us, thus diluting its strength.

Additionally, I have tried (emphasis on tried) reading books on chaos theory, string theory, quantum theory, and you name it. All those people seem to base their conclusions on the fact that "the math works."

My question is: how do we know the "math" really shows anything at all? If I assume that if X is true, then Y must be true, then say "X is true, therefore Y," what have I actually proved except that I can construct an internally consistent logic system? If I understand correctly, many paranoid schizophrenics' world view is internally consistent - and follows if you accept their basic assumptions - but that doesn't make any of it true.

Similarly, fractal diagrams make really cool graphics, but how do I know that they are really the graphs of some strange equations? Anybody could draw anything and say it's a graph.

Any thoughts will be sincerely appreciated. And then maybe I can try to read those books again.

Hi Elizabeth I,
The Murray Gell-Mann: Beauty and truth in physics video has some relevance to this topic.

As an aside: Fractal diagrams are the consequence of some quite simple equations (nothing strange about them).
If you see a graph and it is labelled as a graph of an equation then you either trust the person who presented it to you or you do not.

 

[sol invictus]

I don't understand. Are you asking how you can know that mathematicians aren't lying to you?

I think what she's asking is why graphs of real equations are more relevant than graphs someone drew without any mathematics.

The answer is that math is symbolic logic, and the world seems to follow logical rules. So math describes the world - to arbitrarily good accuracy as far as we know.

 

[Dancing David]

I know that (that's why I used "far away" and "dilutes" in quotations), but it's still something that they've made up and then twiddled factors until it agreed with itself.

That's what I don't understand. I could make up any damn thing and make up any number of equations to describe it, but unless and until it can somehow be tested in some other way than to see if the "equations work," it's still just made up. You can always make the equations work, as long as you don't have to test it real-world.

You can write the equations for the airflow over an airplane wing, then build a model to see if your equations are right.

Build me a model to test for the other ten dimensions, or to see if gravity really "travels" to our dimension from some other, or to figure out which one is the gravity "source."

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.

 

[shadron]

Coincidentally, NOVA has a show on the history of fractals released just last month: http://www.pbs.org/wgbh/nova/fractals/program.html. A couple of things I learned from the show: a fractal line segment, fully expressed, is infinitely long, and an antenna folded into a fractal shape can emulate a much longer linear antenna; why we don't need a whip antenna any more in our cell phones.

You know, this is the third time in a week that I've had occasion to point to a NOVA show in threads on the forum. What is it about NOVA that they seem to anticipate our threads here? Now, there's a deep, parapsychological question!

Similarly, fractal diagrams make really cool graphics, but how do I know that they are really the graphs of some strange equations? Anybody could draw anything and say it's a graph.

As suggested above, you can grab the code (or if you are of the programmer persuasion, hammer out your own) which draws them. They are all computer generated from simple equations. The problem is not in the complexity of the equation or denseness of the equivalent code, but in the fact that the number of computations, simple as they are, increase with the square of the magnification used. If you dive into the mandelbrot set depiction and look to see what a given point looks like when magnified by 10, that requires 100 times more computer work, and it also requires the numbers that are computed have to have the ability to resolve at finer and finer detail. Commonly used computer numbers (called Floating Point numbers) have a finite accuracy, and will soon burn through that when diving into a Mandelbrot scene. The work-arounds consume even more time and memory to use. The beauiful animations you can find (like the ones in he NOVA program) require huge amounts of computer time to generate, even as simple as the code is. Here is a program for Casio calculators: http://www-xray.ast.cam.ac.uk/~jgraham/hypo/h10/mandelbrot.htm, and others can be had by just googling "mandelbrot set calculator".

 

[shadron]

That's what I don't understand. I could make up any damn thing and make up any number of equations to describe it, but unless and until it can somehow be tested in some other way than to see if the "equations work," it's still just made up. You can always make the equations work, as long as you don't have to test it real-world.

Absolutely right. Newton developed a mechanics that seemed to successfully model mundane things here on Earth and the planetary orbits in space - that was the definition of "everything" at the time. Later, some close observations of Mercury's orbit showed a precession that Newton couldn't explain; Einstein figured out what was needed and added a patch onto Newton's mechanics that is only noticeable at very high velocities and/or very high gravitational fields. However, Einstein's equations were constrained - not only did they have to resolve into Newton's when the velocity and the gravity field was low, but they made predictions that then had to be observed and checked out before people accepted them wholeheartedly (for Einsten, many predictions have checked out; some predictions we do not yet have the means of checking, so they are held in abeyance for now, but compliance is not assumed; they are just unanswered questions as yet). The equations proferred also have to be internally consistent. If you take Einstein's equivalence equation E=mc^2 and substitute the units (or dimensions, a completely different meaning for that term) for the quantities, they have to resolve:

energy = mass * velocity ^ 2
length^2 * mass / time^2 = mass * (length / time)^2
check!

This is called dimensional analysis and all physically true equations have to comply - dimensional analysis is a necessary but not sufficient condition for truth. You cannot "say [just] anything"; as Sol pointed out, it must be consistent with what we know already, both in physics and in math, and it has to remain so as new evidence and new techniques are discovered.

 

[Soapy Sam]

Elizabeth, as another mathematical dunce, I sympathise totally.

The best mathematicians are not those who can "do the math", but those who are able to see how the math relates to real world phenomena. Such people are rare - and, sadly, it's even rarer to find one who can explain how he does what he does, in any language other than mathematics.

On the fractal question , you might like to try playing with Robert May's logistic equation . Even a very simple computer program can illustrate the reality of the period doubling effect leading to chaotic behaviour.
A link- http://hans.liss.pp.se/may.html

On string theory, I incline to agree with you- the mathematics can be made to fit anything and as we have no way to test it, the whole thing is hard to tell from fantasy. Some physicists don't see it as good science.

I'm simply unequipped to comment, except that a definitive equation, or a shred of experimentally verifiable / disprovable evidence would be nice.

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.

Incidentally- the link above also links to a movie of May's logistic equation plotted in a circular form. It's fascinating.
Do have a look.

 

[69dodge]

So dimensionally my answer is in square metres.
Which is mathematically correct, but makes no sense whatever.

It's the cross-sectional area of the cylinder.

 

[Paul C. Anagnostopoulos]

My question is: how do we know the "math" really shows anything at all? If I assume that if X is true, then Y must be true, then say "X is true, therefore Y," what have I actually proved except that I can construct an internally consistent logic system? If I understand correctly, many paranoid schizophrenics' world view is internally consistent - and follows if you accept their basic assumptions - but that doesn't make any of it true.

There is no truth except in mathematics. Science just comes up with models that are better or worse at predicting what most people observe to be the case.

As Dancing David said.

~~ Paul

 

[Soapy Sam]

It's the cross-sectional area of the cylinder.

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 have attempted to present your argument to at least thirty people (mostly engineers of one sort or another). While all agreed that m3 * m-1 =m2, none of them - not a single one- agreed that the answer is an area. (I agree with them. It's meaningless, in the context of the problem, to say it's an area.)

Now we may all be wrong from the mathematician's POV, but if we are all wrong over something so simple, I'd say our chances of agreeing with a mathematician about anything more complex , is slim to none.

This may be our failure to understand, or the mathematician's failure to explain.
Or both.

 

[Soapy Sam]

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?