In response I asked some questions about how some of those were supposed to work and got some good answers. I then, however, asked:
Saltire chimed in with a good response:
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.
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.
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.
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.”
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.
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.
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?
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.
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.
A quote from David Griffiths, author of one of the standard textbooks on particle physics:
(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”.
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.
Owwwwwwwwwwwwwwwwwww!
full disclosure: I had 8 different algebra teachers in my freshman year alone.
damn Chicago public school system not paying for permanent teachers!
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:
