Can we understand the laws of physics without infinitesimals and infinities?

[geonat]

Has there been any attempts to formulate general laws of physics without involving infinitesimals and infinities? Would this be a better starting point? The continuous limit would of course be seen as an extremely useful approximation.

The general personal impression of today's theories is that they try to describe the interaction between discrete particles using continuous variables. This appears a bit absurd to me.

Or does interaction necessarily involve self-reference, and does self-reference necessarily introduce infinities?

 

[robphy]

Yes, there have been attempts.
You can google phrases like "discrete physics" (e.g. discrete mechanics, discrete electromagnetism, discrete quantum gravity, etc..) and "cellular automata physics", for example.

Conceptually, it might make a nice starting point... since counting is very natural [though possibly difficult and/or tedious]...and the use of the continuum for physics might just be a convenient approximation for mathematical analysis. But there's certainly a long road (physically and mathematically) ahead to make these ideas work... but it might be worth a try. (It's been argued that "continuum" calculus has had a relatively long history [read as "head start"], compared to "combinatorial" calculus... so our methods have been biased toward continuum calculus.)

 

[arildno]

It should be noted, however, that even if many could sympathize with a discrete approach, there will be serious mathematical problems occurring.

Just one example:
Difference equations (beyond the trivial ones) are nasty, and quite often, the analogous differential equation will behave in quite a different manner than the difference equation.

The powerfool tools of calculus are not directly applicable to the study of difference equation, and the analogous tools for discrete maths are on the whole, "weaker".

Not because the mathematicians dealing with discrete maths are dumber than the analysts, but because the topic is more thorny..

 

[robphy]

It should be noted, however, that even if many could sympathize with a discrete approach, there will be serious mathematical problems occurring.

This is certainly true... which amplifies my comment on things being "difficult" and that there is a "long road ahead to make these ideas work". Nevertheless, there are some researchers trying discrete approaches to deal with various infinities that show up in physical theories (e.g. general relativity and quantum field theory), possibly due to the implicit continuum assumptions.

For example,
http://www.math.ucr.edu/home/baez/penrose/
has links to transcriptions of Penrose's "Angular momentum: an approach to combinatorial space-time" and "On the nature of quantum geometry".
In addition, here's Loll's review of some discrete approaches to quantum gravity: http://relativity.livingreviews.org/Articles/lrr-1998-13/

 

[arildno]

This is certainly true... which amplifies my comment on things being "difficult" and that there is a "long road ahead to make these ideas work". Nevertheless, there are some researchers trying discrete approaches to deal with various infinities that show up in physical theories (e.g. general relativity and quantum field theory), possibly due to the implicit continuum assumptions.

For example,
http://www.math.ucr.edu/home/baez/penrose/
has links to transcriptions of Penrose's "Angular momentum: an approach to combinatorial space-time" and "On the nature of quantum geometry".
In addition, here's Loll's review of some discrete approaches to quantum gravity: http://relativity.livingreviews.org/Articles/lrr-1998-13/

Discrete thinking is, indeed, an exciting approach, and it might just be the thing to get rid of the "spurious" infinities cropping up in QM, even though today's techniques for handling many of these are adequate from the computational&predictive view.

 

[rbj]

but the fact that the space-time continuum hypothesis contains insuperable logical self-contradictions was cleared up already 2500 years ago when Zenon stated his aporias...

Blather.

didn't Zenon invent some kind of very bright pulse lamp? that's worthy of respect.

 

[arildno]

Did he?
I didn't think he moved about much.

 

[geonat]

But there's certainly a long road (physically and mathematically) ahead to make these ideas work... but it might be worth a try.

I see. Let's hope that the idea becomes more popular then. It is not discussed as a possible alternative in the literature that I've encountered up 'til now.

Thank you for the links in your follow-up; that serious researchers have suggested the approach for some time was exciting news to me.

One thing that naïvely comes to mind is that spherical symmetry has to be abandoned, since it involves an infinite number of directions. If this is true, it must have serious implications - mustn't it?

 

[Mentz114]

didn't Zenon invent some kind of very bright pulse lamp? that's worthy of respect.

That was his cousin Xenon I suspect.

 

[genneth]

Actually a much more interesting issue would be to do physics on a discrete spacetime. The issue of doing differential geometry on discrete simplices is actually a very active direction of research, with surprisingly direct application in things like computational physics.

 

[robphy]

Actually a much more interesting issue would be to do physics on a discrete spacetime. The issue of doing differential geometry on discrete simplices is actually a very active direction of research, with surprisingly direct application in things like computational physics.

It seems computational electromagnetism has moved into this direction...
for example,
http://www.lgep.supelec.fr/mocosem/perso/ab/bossavit.html
http://arxiv.org/abs/0707.4470
http://www.cs.uiuc.edu/class/fa05/cs598anh/

It might be interesting to pursue other field theories along these lines.
Of course, http://en.wikipedia.org/wiki/Regge_calculus was an early approach for studying gravity in this way.

For a truly discrete spacetime, there's the causal set approach
http://www.google.com/search?q="causal+sets"
which I am studying.