Are Physicists Fudging their Math?

[Whack-a-Mole]

In another thread ( http://boards.straightdope.com/sdmb/showthread.php?threadid=46740 ) I managed a severe hijack so I thought I'd startup in a new thread. (BTW -- How do you do the link-in-a-word trick so I don't have to post the entire URL?)

Anyway, Arnold Winkelried wrote:

And from high school mathematics you may remember these four operations having indeterminate results:

infinity - infinity
0 * infinity
infinity / infinity
infinity / 0


In response I asked some questions about how some of those were supposed to work and got some good answers. I then, however, asked:
Does all this stuff on infinities (not being able to divide them, etc.) mean you can't simplify an equation where an infinity shows up? I thought scientists (physicists mostly) would occasionally work their equations so an infinity on one side cancelled out an inconvenient infinity on the other side. If you can't divide them and so on how can they do this (or are they fudging)?


Saltire chimed in with a good response:
Hopefully one of the real math geeks will fill in the details, but I can get things started. The renormalization of infinities is indeed a necessary mathematical technique in physics. Particle physics and cosmology often come up with infinite quantities in their equations, and can't really proceed along those lines unless they can cancel the infinities out, leaving only 'normal' numbers.

I don't really know the technique, but it involves enumerating an infinite set so that it can be matched with another. For instance, you can match each member of {even positive integers} with a member of {odd negative integers}. Thus, you can say that they sort of share an 'amount' of infiniteness. So, these two sets could be operated on.

However, some sets are more infinite than others. You can't assign each real number with a whole number, so the reals are 'more infinite' than the wholes. To cancel out an infinity of this order, you need to find something 'equally infinite' to balance it against.

So, there are strict rules about how you can do math with infinite sets. Just like other math, only a bit more arcane.

So...all that said...are physicist/cosmologists fudging their math? Saltire's explanation helps but it still sounds as if some hocus-pocus is being done. I mean, if you can't normalize an equation with an infinity on both sides (since infinity/infinity is an indeterminate number) how do physicists suppose they are getting meaningful answers?

FYI: I am sure that if and when scientists do this they all agree that it is acceptable so I'm not trying to second-guess them. I'm just trying to see what's actually going on.

[Strainger]

Well, L'Hopital's Rule (http://boards.straightdope.com/sdmb/showthread.php?threadid=2873) is one way to approach it.

[JeffB]

I'm not going to even try to answer the math question, but here's how to do the link thingy. You put the link inside the first bracket thus:

{url="www.link.com"}Display Text{/url}

but with [] instead of {}.

[Pasta]

From Jeff_42
So...all that said...are physicist/cosmologists fudging their math?

No. I think Saltire said it best:
So, there are strict rules about how you can do math with infinite sets. Just like other math, only a bit more arcane.

Algebra with infinities is bad. Math as a whole has no problem.

If I say A=(number of even numbers) and B=(number of odd numbers), then A/B=1 even though A and B are both infinite. Every occurrence of an infinity has an analogous "size" to it, but instead of always thinking about "how big is my infinity", you just set down the mathematically machinery and never worry about it again. This machinery includes limits, L'Hopital's Rule (for dealing with limits that are indeterminite at first glance), and more.

An example of a problem with infinities: Say you are doing some thermodynamics calculation about a volume of gas. Maybe you'd like to make the assumption that the walls of the container don't affect the system. This would be equivalent to thinking of a box of infinite size with an infinite number of particles (and thus no walls to worry about). But then you'll have infinities everywhere! What you do, then, is send both V (volume) and N (number of particles) to infinity in such a way that their ratio n=N/V (particle desity) stays finite and equal to your actual particle density.

Another way to think about this example is to imagine the entire universe is filled with this gas, and it has the same desity (n) throughout. Any physics that goes on in one location doesn't care how big the universe is or how many particles fill it. I mean, V and N could even be changing all the time (so long as n=V/N stays constant!) and I would never be able to tell, so my equations better not be able to tell either. That is, they should not depend on V alone or N alone -- only the ratio n can appear. Otherwise, we know we have an incorrect equation. (Recall the ideal gas law: PV=NRT ==> P=nRT. Pressure and temperature are related by the costant R and the desity n. Neither depends on N or V alone.)

The much more arcane example quoted in the OP (renomalization) has entire books written on it. A sketch: in quantum field theory, you write all your equations without every really establishing a normalization (or, scale) for all your quantities. When you try to calculate something useful, you get infinity, but you are still free to pick your normalization. You can rearrange stuff just right to hide these infinities in the normalizations. What it amounts to is that your original equation was actually full of quatities that had infinite and unphysical values, and not only does renormalization fix your answer, but it fixes those original values, too! Sounds fishy, but the details do work out.

[Lumpy]

Originally posted by Jeff_42
In another thread ( http://boards.straightdope.com/sdmb/showthread.php?threadid=46740 ) I managed a severe hijack so I thought I'd startup in a new thread. (BTW -- How do you do the link-in-a-word trick so I don't have to post the entire URL?)Here (http://boards.straightdope.com/sdmb/index.php?action=bbcode) is the Vb code page, which ought to be displayed more prominently in the FAQ list.

[Chronos]

Yes, it is fudging, and most physicists recognize it as such. However, the important point (for physicists, anyway... This sort of stuff gives mathematicians conniptions) is that it works. It's probably a sign that our understanding of field theory is incomplete, but in the meantime, it'll do.

[Chas.E]

Well of COURSE they're fudging the math. I once read a quote that I remember verbatim, by a famous mathematician discussing the relationship between Physics and Mathematics. He said "insofar as mathematics accurately reflects the real world, it ceases to be interesting."

[DrDoom]

Using infinity should not be associated with fudging. 1 divided by infinity or e^-infinity is a common occurance. The result of 1/infinity or e^-infinity is 0, e.g. 1-1/x as x approaches or "is" infinity equals 1 (critical tolerances being recognized). This does not mean that 0*infinity = 1 as some might suggest. The use of infinity in properly defined formulae is easily handle.

Only at singularities in ill defined problems do these undefined expressions show up. For example, we could ask what does 1-y/x equal, where y=x, as x becomes infinite? Regardless, 1 - x/x = 1 - 1 = 0. This is not conditional nor it is only valid except for x=0 or infinity. It is simply 1 - 1 = 0.

[erislover]

lets take a simple problem like
5 + x = 10
We are studying a problem which appears vaguely familiar to this like so:
5 + (infinity)x = 10 where x is an element of some set of numbers.

We got this second equation by a combination of theory and experimentation. In other words, the theory gave
5 + (infinity)x = y
but the measurment yielded
y = 10
Since we know what the answer needs to be, we fix the problem to suit it by fixing the infinity at some value. I do not feel this is fundging, so to speak, because I have only heard it in connection with the probabilitistic (is that an f'in word?!?!) sciences like quantum mechanics and thermodynamics. These have quantities that are not truly algebraic but probability functions in themselves. The overall tendency is most likely what the experiment is to yield, and so renormalization solves this.

Of course, I could be BSing that.

[Race Bannon]

I don't think it's proper to think of infinity as a real number. When physics problems require "infinite" numbers, I think it's common to apply the principles of real analysis and calculus, that can formally handle functions that have finite values even though values in their domains "go off to infinity". In this sense, there is no "fudging" of math.

When mathematicians talk about f(x) as "x goes to infinity" they are using shorthand speech for concepts about limits and unbounded values (look in a real analysis text, mine isn't handy). They do distinguish this phrase from things like "x goes to a, where a is a real number". The jargon is similar, but the concepts are distinct.

Now, there are things called transfinite numbers, which provide ways of talking about infinite sets, but these are not real numbers, and do not obey the algebra that we commonly expect from ordinary numbers. The "number" of natural numbers, for instance, is called aleph-0. The "number" of real numbers is greater than this (sometimes called c). If you can describe a set with an amount of members somewhere in between aleph-0 and c, you'll deserve some kind of mathematics award.

Do the physics examples you refer to actually use transfinite numbers, or just "really big" real numbers?

[Pasta]

If you can describe a set with an amount of members somewhere in between aleph-0 and c, you'll deserve some
kind of mathematics award.

How about the set of rational numbers (i.e., those numbers that can be written as a ratio of integers)? Seems like this set is larger than aleph-0 and smaller than c...

The more common physics infinities are true infinities because they are idealizations. Like with the gas example: I want to get a result that ignores the effects of the walls. I am okay in posing the problem that way, but any answer I get will have the footnote, "This answer ignores the effects of the walls." And, if I want to actually incorporate this assumption, I can just put the walls as far away as possible. Not 10 billion miles away, but infinity miles away. So in the math, the infinities are actually infinite. If they weren't I will have failed to incorporate my assumption fully. It does mean my answer will be an approximation, but that's okay. Most physics results are anyway. (You just have to keep track of when your approximation is valid and when it is not.)

I think there is a semantics issue at hand. Physicists are always doing things that are idealizations and approximations, but they aren't "fudging", as their answers come packaged with implicit (or explicit) disclaimers that they (the answers) only apply if the idealizations and approximations are valid! Circular, I know, but it becomes no longer a problem of solving the system but a problem of determining whether a real physical system meets the requirements for the solution to be applicable. Granted, in the extreme case of renormalization it seems we do lack some fundamental understanding of what's going on, but there is no fudging since logic still reigns supreme.

However, if you mean "fudging" to be all the stuff I just said (approximating systems so you can get an answer at all, etc.), then, yeah, there's all kinds of fudging going on.

[Cabbage]

How about the set of rational numbers (i.e., those numbers that can be written as a ratio of integers)? Seems like this set is larger than aleph-0 and smaller than c...

Actually, that set has cardinality aleph-0, same as the integers.

The continuum hyothesis says that there's no cardinal number between aleph-0 and c. What kellymccauley was referring to is the fact that this is independent of the standard axioms of set theory. Back in the 30's, Kurt Godel proved that the continuum hypothesis is consistent with those axioms. In the 60's, Paul Cohen proved that the negation of the continuum hypothesis is also consistent with those axioms. Therefore, it can't be proved one way or the other. It's generally felt that the continuum hypothesis *should* be false (i.e., that there should be cardinals between aleph-0 and c), and there's been research into finding an additional, natural axiom that would decide it one way or the other, but as of now it's still inconclusive.

[Chronos]

A quote from David Griffiths, author of one of the standard textbooks on particle physics:
No one would deny that this procedure [renormalization] is artificial. Still, it can be argued that expression (6.59) merely reflects our ignorance of the high-energy (short distance) behavior of quantum field theory. Perhaps the Feynman propagators are not quite right in this régime, and M is simply a crude way of accounting for the unknown modification. (This would be the case, for example, if the "particles" have substructure that becomes relevant at extremely close range.)(Introduction to Elementary Particles, 1987 edition, footnote on page 209)
There's also a quote by Paul Dirac, who pioneered the method, where he says "When you get a number turning out to be infinite which ought to be finite, you should admit that there is something wrong with your equations, and not hope that you can get a good theory just by doctoring up that number." I'd say that "doctoring" counts as "fudging".

[Short]

Originally posted by Cabbage


Actually, (the rational numbers) set has cardinality aleph-0, same as the integers.



Hmm... I think the rationals have the same cardnality as the reals. The rationals, like the reals, are not countable (i.e. from any number, there is no "next" number. That is choose a next number and I will show you a number between the two). Splitting hairs, though.

Nifty math for fun and profit:
The cardinality of the set of numbers on the side of a square is the same as the cardinality of the set of pairs of numbers inside the square (as well as triplets for boxes and groups of 4+ terms for 4+ dimensional objects called hyperplanes).

Back on topic: who ever was first to say it: "physicists don't fudge math," was right. They do occasionally appoximate, however (substituting x for sin x when x is very small and the like), as do mathmaticians doing estimates. Generally this does not have to do with infinite sets though. In fact, I can't think of a single physics application for transfinite numbers. Limits going to infinity are used all the time and are very well defined (at least have been for about 150 years). Most calculus texts I've seen at least mention limits actually work, though some introductory analysis class is usually the first time they are applied.

Full disclosure: I am an undergrad math major.

[MikeG]

Owwwwwwwwwwwwwwwwwww!
full disclosure: I had 8 different algebra teachers in my freshman year alone.
damn Chicago public school system not paying for permanent teachers!

[Cabbage]

Hmm... I think the rationals have the same cardnality as the reals. The rationals, like the reals, are not countable (i.e. from any number, there is no "next" number. That is choose a next number and I will show you a number between the two). Splitting hairs, though.

The rationals are countable, but yeah, it is surprising that between any two rationals there's another rational, as you say, given one rational there's no "next" rational; the surprising thing being that they can do that and still be countable. Here's a standard proof of the countability of the rationals:

http://www.math.hmc.edu/funfacts/ffiles/30001.3-4.shtml

[Short]

Originally posted by Cabbage
The rationals are countable, but yeah, it is surprising that between any two rationals there's another rational, as you say, given one rational there's no "next" rational; the surprising thing being that they can do that and still be countable. Here's a standard proof of the countability of the rationals:

http://www.math.hmc.edu/funfacts/ffiles/30001.3-4.shtml

Suprising and a bit unsettling. I stand corrected.

[Short]

Originally posted by Cabbage
The rationals are countable, but yeah, it is surprising that between any two rationals there's another rational, as you say, given one rational there's no "next" rational; the surprising thing being that they can do that and still be countable. Here's a standard proof of the countability of the rationals:

http://www.math.hmc.edu/funfacts/ffiles/30001.3-4.shtml

Suprising and a bit unsettling. I stand corrected.

[Bobort]

Originally posted by Chronos
A quote from David Griffiths, author of one of the standard textbooks on particle physics

Great book, that is. David's teaching my Elementary Particles class next semester :)

[Chronos]

But then, is there anything about Cantorian set theory that isn't a bit unsettling?

By the way, Short, welcome, and may your time with us be long, enlightening, and fun.

[The Ryan]

For instance, you can match each member of {even positive integers} with a member of {odd negative integers}. Thus, you can say that they sort of share an 'amount' of infiniteness. So, these two sets could be operated on.
No, no, no! Cardinality is not the size of a set. Take the set of even numbers and the set of integers. These two sets have the same cardinality. But that does not mean that there are just as many even numbers as integers. You can't divide aleph0 by aleph0; it's just not allowed.

[Cabbage]

I don't think it's really wrong to think of cardinality as the size of a set. Cardinality certainly corresponds to our idea of the size of a set when talking about finite sets, for example. Cardinality just extends that idea to infinite sets. And I don't think it's wrong to say that the set of integers is the same size as the set of even integers (that there are just as many of one as there are the other), since the two sets can be put into one-to-one correspondence with each other. I agree with you about dividing aleph-0 by aleph-0, however.

[Race Bannon]

The Ryan: No, no, no! Cardinality is not the size of a set.
What would you use to talk about the "size" of a set?
In my math education, there are two ways - one of which is cardinality. In some areas of algebra, it's convienient to talk about two sets which have the same cardinality, but (as in your eg. with integers and even integers) one set is a proper subset of the other. In this case, it's common to say that one is "larger" than the other, but it's understood that you're not talking about cardinality in this context.

[The Ryan]

Originally posted by Cabbage I don't think it's really wrong to think of cardinality as the size of a set.
Well then, clearly our opinions differ. I just don't think that it makes sense to say that if you take a set, remove a set of equal size from it, you are left with a set of exactly the same size. I'm willing to accept this with cardinality because it's an abstract concept that doesn't have follow our intuition, but size is a very concrete concept and it should follow intuition.

Cardinality just extends that idea to infinite sets.
Not really. The concept of size has no real extension to infinite sets because the definition of "infinite" is that it has no size.

And I don't think it's wrong to say that the set of integers is the same size as the set of even integers (that there are just as many of one as there are the other), since the two sets can be put into one-to-one correspondence with each other.
Intuitive ideas like 1-1 only work for finite sets. For instance, suppose I perform the following set of actions an infinite number of times (after first placing a ball marked "one" in the bowl):

1. Place a ball marked with the number one greater than the current ball in the bowl.
2. Remove the ball with the smaller number on it.
3. Return to step one.

The set of balls the remains in the bowl afterwards is clearly of size zero. Yet there is always a ball in the bowl! Intuitive ideas just don't work with inifinte sets.

kellymccauley
What would you use to talk about the "size" of a set?
If it's finite, cardinality is fine. If it's infinite, then there is no meaningful sense to "size", and I believe that it is deceptive to pretend that there is. The set of real numbers isn't "bigger" than the set of integers; it just has a larger cardinality.

[jcgmoi]

The Ryan said: Intuitive ideas like 1-1 only work for finite sets.

So you discount Cantor's arguments. Do you have anything to put in their place?

[DrDoom]

Re: infinity vs cardinal, ordinal, prime, aleph-0, etc.. Don't get it? How about using X= 24178.659! as a starting point for infinity, go up a notch to say X= 2315.39! x X, and see if the value of f(X) is converging. The try again with X= 4732.122! x X, to see if it is still converging, and so forth. Isn't there something like an Euler's theorem that if f(X) converges using an increasing X, then it will converge absolutely, otherwise it will diverge absolutely? Seems to have nothing to do with any "proper" numbers.

But then again, I might be diverging too .. :)

[Race Bannon]

because the definition of "infinite" is that it has no size.

Hm, I have never heard that one.

Intuitive ideas like 1-1 only work for finite sets.

Hm, you say the 1-1 relation between integers and even integers (for example) makes no sense? (or, what do you mean by "only work"?) I think it's pretty much a textbook case.

[Cabbage]

I just don't think that it makes sense to say that if you take a set, remove a set of equal size from it, you are left with a set of exactly the same size.

Actually, that's a standard definition of an infinite set. By definition, an infinite set is one which has a 1-1 correspondence with a proper subset of itself.

Intuitive ideas like 1-1 only work for finite sets.

But that's the definition of cardinality. Two sets have the same cardinality if and only if there's a 1-1 correspondence between the two sets, whether they're finite or infinite.

If it's finite, cardinality is fine. If it's infinite, then there is no meaningful sense to "size", and I believe that it is deceptive to pretend that there is. The set of real numbers isn't "bigger" than the set of integers; it just has a larger cardinality.

But the reason the idea of "cardinality" is useful is that it does describe the size of infinite sets. The idea of size is obvious when we're dealing with finite sets, it's when we're dealing with infinite sets that we need a rigorous concept such as "cardinality" to base our notion of "size" on. The whole breathrough, in my opinion, of Cantorian set theory is the conclusion that there are strictly larger and larger infinite sets. The real numbers are a bigger set than the integers, and that's the surprising thing about cardinality, and why it's useful.

[The Ryan]

Originally posted by jcgmoi
So you discount Cantor's arguments. Do you have anything to put in their place?
AFAIK, Cantor's arguments applied to cardinality not size. If they did apply to size, then nothing should take their place, because infinite sets don't have size.

kellymccauley
Hm, you say the 1-1 relation between integers and even integers (for example) makes no sense?
As far as it is used to establish that the two sets have the same size, it makes no sense. Yes, you can assign a relation between one and two, two and four, three and six, but you have to stop somewhere. It's not actually possible to assign a relation to every element of each set. You can define a relation over all of both sets, but it can't actually be implemented in its entirety.

Cabbage
Actually, that's a standard definition of an infinite set. By definition, an infinite set is one which has a 1-1 correspondence with a proper subset of itself.
Really? I certainly agree that if a set has a 1-1 correspondence to a proper subset of itself, it is infinite, but does the converse hold?

But that's the definition of cardinality. Two sets have the same cardinality if and only if there's a 1-1 correspondence between the two sets, whether they're finite or infinite.
Yes. That is that definition of cardinality. However, it is not the definition of size.

But the reason the idea of "cardinality" is useful is that it does describe the size of infinite sets.
I agree, if by "describe" you mean "give a symbolic account of".

The whole breathrough, in my opinion, of Cantorian set theory is the conclusion that there are strictly larger and larger infinite sets. The real numbers are a bigger set than the integers, and that's the surprising thing about cardinality, and why it's useful.
But simply making up a term and defining it to be something gives us no actual information. Just because cardinality is analogous to size does not mean it is size.

[RM Mentock]

Originally posted by The Ryan
But simply making up a term and defining it to be something gives us no actual information. Just because cardinality is analogous to size does not mean it is size.

Is there a mathematical definition of the term "size?" It appears that you are saying that there is one. Do you have a reference for that? What is the actual definition?

If we use, for instance, the American Heritage Dictionary definition, there is no problem with describing the cardinality of a set as its size. Its definition 3b is "Relative amount or number"

[The Ryan]

Originally posted by RM Mentock
Originally posted by The Ryan
But simply making up a term and defining it to be something gives us no actual information. Just because cardinality is analogous to size does not mean it is size.

Is there a mathematical definition of the term "size?" It appears that you are saying that there is one. Really? I was trying to express the fact that there isn't one. For instance, take set A to be the real interval [0,1]. Take set B to be the set of all numbers in set A which, when expressed in trinary, have no digits other than 0 or 2. Take set C to be the set of rational numbers in set A. The cardinality of A is the same as the cardinality of B, which is greater than the cardinality of C. But there's another property of sets called measure. The measure of C is the same as the measure of B, which is smaller than the measure of set A. So is B the same size as A or C? Neither. Cardinality and measure are analogous to size, but they aren't the same.

[Cabbage]

Well, of course it can depend on what "size" means, given what context you are considering "size" in.

Cardinality basically answers "How much does the set contain when the set is stripped of all properties other than that of being a set?" In the context of this thread, that's what I was taking "size" to refer to, and I believe that's perfectly acceptable, given that we all understand what aspect of "size" our attention is focused on. Essentially, "How many elements does it contain?"

Measure, on the other hand, is basically "How much space does this set take up?" A set, in and of itself, takes up no space; in fact, the question of "How much space does a set take up?" is meaningless when asked of a basic set. It requires more structure to be placed on the set, such as the idea of the length of a line segment. But if, in a given context, "size" is understood to mean "quantity of space filled", then certainly measure would be an appropriate description, not cardinality.

So it really comes down to, for example, comparing a line segment one inch long with a line segment two inches long. There is a one-to-one correspondence between the points of one with the points of the other; as basic sets they are the same size. If, of course, you then introduce the concept of length, then you can certainly say one is longer than the other.

But I still claim that cardinality, when referring to a set as a collection of elements, and nothing more, is essentially describing the size of that collection in "quantity" of elements, not in "space filling properties", which is what measure describes.