Infinitesimals and differentials

[ehrenfest]

Homework Statement

I hate infinitesimals and differentials.

When I learned calculus, we used Liebniz notation df/dx only as a convenience for using the chain rule. In physics, apparently, people just play around with differentials and infinitesimals and expect to get the right answer. Physicists think that differentials are like regular numbers and you can just add them and multiply them and pretend they are meaningful outside of Liebniz notation. Even worse is the use of these "infinitesimal" quantities, which don't even have Liebniz notation to justify their use. Physicists simply made up these objects and go around pretending they are real mathematical objects that you can use in equations. THIS IS INSANE.

In thermodynamics and statistical mechanics, for example, many of the laws and relations are formulated in terms of differentials and infinitesimals. You can write down ridiculous things like
$$dU=\delta Q-\delta w$$
and
$$\mathrm{d}U = \delta Q - \delta W+\sum_i \mu_i\,\mathrm{d}N_i$$
and pretend they make sense.

In classical mechanics, you "define," the angular velocity vector as

$$\vec{\omega} = \frac{\delta \vec{\theta}}{\delta t}$$

That is probably the second most absurd equation I have ever seen. Apparently, no one who writes classical mechanics or statistical mechanics books understands the concept of something being "well-defined." When you define a derivative, you DO NOT JUST SAY "its just the ratio of the infinitesimal change of f to the infinitesimal change in x," you define it as a limit of a quotient AND YOU PROVE IT EXISTS. I have never seen any proof that $$\vec{\omega}$$ exists and therefore I think all of classical mechanics that uses [itex] \omega[/tex] is downright wrong and should be reformulated with mathematical objects that actually exist.

BTW, I am taking differential geometry next year. Is that where I learn these things?

EDIT: I am being somewhat sarcastic, but this has been constantly bothering me for a long time

Homework Equations

The Attempt at a Solution

 

[joelperr]

http://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis
http://en.wikipedia.org/wiki/Nonstandard_analysis

 

[ehrenfest]

http://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis
http://en.wikipedia.org/wiki/Nonstandard_analysis

It would be nice if the authors of my textbooks talked about that instead of just using the tools in those wikipedia articles and hiding the secret reason why their analysis works.

 

[dynamicsolo]

Physicists get away with this sort of "mathematical murder" because a lot of the functions we work with, at least in classical physics, are sufficiently continuous and differentiable to go around treating derivatives as if they were just ratios of differentials. For quite a lot of applications, it works quite well, even though we "know" this isn't really the "proper" mathematical definition.

But it is well to recall that, for around 150 years or so, this was the way most people used calculus. All of the rigorous mathematical machinery concerning limits, continuity, differentiability, and so forth, was invented in the 19th Century, when people increasingly ran into problems where the pitfalls and potholes in the simplistic approach led to wrong results (with Dedekind getting asked by a student what he meant by "continuity" and realizing he didn't really know, etc.).

As for textbooks... well, we know that until you get to a course where people actually worry about all these pesky details, most books are written to fit with the "rocket sled" tours of physics, mathematics, and applications...

 

[ehrenfest]

https://www.physicsforums.com/showthread.php?t=77178

From that thread, I get the sense that the authors of my textbooks are not hiding nonstandard analysis from us, but don't even know about it and therefore have no justification for the equations in their books.

PLEASE reassure me that SOMEONE has gone through classical mechanics and statistical physics and made sure that it is compatible with the axioms of nonstandard analysis. PLEASE tell me that the things my physics teachers writes on the blackboard are not ABSURD and MEANINGLESS. If not, I am canceling my physics major immediately and studying only pure math for the rest of my life.

 

[malawi_glenn]

the present physics don't deal with these issues as much as one did long ago. So don't overreact. What is the most important to you: The things that you examine, or the language with which you study them?

We also learned about differentials etc in our analysis and calculus classes, maybe you just have a lack in your mathematical background?

http://en.wikipedia.org/wiki/Differential_(infinitesimal)

 

[ehrenfest]

the present physics don't deal with these issues as much as one did long ago. So don't overreact. What is the most important to you: The things that you examine, or the language with which you study them?

I would not compare mathematics to a language here. Mathematics is a system of logic that is much more rigorous and exact than a language. How can you try to understand something as intricate as nature when you do not have a formal logical system set up?

We also learned about differentials etc in our analysis and calculus classes, maybe you just have a lack in your mathematical background?

http://en.wikipedia.org/wiki/Differential_(infinitesimal)

That could be true since I am only studying analysis now. But, as far as I know, there is NO math course at my school that studies infinitesimals that are not differentials.

 

[malawi_glenn]

I did not intend to compare math with german for example. math is the way we describe nature, the persons who foundated modern analysis where physicists.

By quoiting you: "Physicists simply made up these objects and go around pretending they are real mathematical objects that you can use in equations. "

And if you know of better definitions, then write your own book.

Have you read ALL books out there in classical mechanics and statistical mechanics? Don't think so..

For me all those things you wrote in your opening posts makes sense, since i have another background.

And in modern physics courses, the concepts of differentials and infinitesimals are seldom encountered. So don't worry.

 

[joelperr]

It has been proven that non-standard analysis is equivalent to the 'normal' formulation of real analysis (i.e. the epsilon-delta formalism). Thus, since the two formulations are in bijection, 'proving' that classical mechanics and statistical physics work using axiomatic non-standard analysis is unneeded; as is the usual case in mathematics, we've proved a more difficult theorem (equivalence of the two formulations of analysis) to solve a simpler one by corollary.

 

[ehrenfest]

I did not intend to compare math with german for example. math is the way we describe nature, the persons who foundated modern analysis where physicists.

The problem is that the authors of my textbooks DO think of mathematics as a language like german. They think that they can bend the rules of grammar and speak "colloquially" as long as what they are saying is reasonable and appeals to our intuition. I think that is bad because physics is too important to have the same sort of laxness and incoherence that you find in a language like German.

I am asking whether there is an axiomatic approach to classical mechanics that uses a logical system described at the wikipedia page below.

http://en.wikipedia.org/wiki/Formal_system

If not, then there should be and maybe I will write my own textbook someday.

 

[ehrenfest]

It has been proven that non-standard analysis is equivalent to the 'normal' formulation of real analysis (i.e. the epsilon-delta formalism). Thus, since the two formulations are in bijection, 'proving' that classical mechanics and statistical physics work using axiomatic non-standard analysis is unneeded; as is the usual case in mathematics, we've proved a more difficult theorem (equivalence of the two formulations of analysis) to solve a simpler one by corollary.

Excellent. I will keep my physics major. Where can I find the proof? What is a good introductory nonstandard analysis book to look at?

 

[genneth]

Excellent. I will keep my physics major. Where can I find the proof? What is a good introductory nonstandard analysis book to look at?

I would not bother. The methods go far beyond the necessary things for physics. You would end up chasing up a century's worth of meta-mathematical methods in model theory, logic and foundations.

For physics, remember that the ambiguity is rarely the mathematics --- the main problem is that of interpretation. Failure to build a suitable mathematical model that actually encodes the problem is the central problem. Mathematical subtleties rarely intrude.