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We started the first day of our semester today by having a review of dimensional analysis. Viewing it afresh, I began wondering how it all “works”, i.e. what is the physics behind it all?

Nature sure doesn’t give a damn about how (or in which units) we choose to measure its properties. But anyway, we, according to our whatsoever knowledge, decided to “label” the observables in the universe with a finite set of “fundamental” units including meters, seconds, grams, etc.

And then the dimensional analysis applied to get time period ($T$) of a pendulum, assuming (reasonably) that it might only depend on gravity ($g$), length ($l$), mass ($m$) and amplitude ($\theta$), we neatly (and correctly) get that $$T=f(\theta)\sqrt{{l\over g}},$$

which I find humongously nontrivial.

How on earth can a “labelling system” devised by us put such severe limits on how a pendulum can oscillate and thus tell Nature how to behave? Just by noticing that adding grams to seconds is meaningless as adding apples to oranges?

(Note that the example of pendulum was just representative.)

The question which has been claimed mine to be the duplicate of is entirely different indeed, except the similar title. Let me show how.

That question asks why it is not justified to add quantities of differing dimensions. While I ask why dimensional analysis gives the correct physical form of the answer in terms of the observables.

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I think you believe the "labeling system" is arbitrary, frivolous, and of no consequence, so how can dimensional analysis predict anything about the period of a pedulum? Using Ben Crowell's language of "abstract dimension", the discovery of dimension is a fundamental empirical measurement made on the real world. Someone first observed that when you wait the world changes and quantified it's passage with the unit time. Someone first noticed an object can be moved over in 3 different directions and quantified the movement with the unit distance. No one has observed any movement that could be called a fifth dimension so we don't need it's unit when we do dimensional analysis. Identifying these transformations we can do to real world objects is a fundamental observation and provides the nontrivial units for dimensional analysis.

Further experimentation has led to the discovery that waiting, translations, rotations, and boosts form a particular mathematical group. This has allowed even more predictions about nature (eg: spin) than dimensional analysis.

Gary Godfrey
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