Help with definition and proportionality
Published by admin April 3rd, 2008 in ForceI wanted to ask your help with this mathematical question. Let me explain.
If I have an ugly expression and I don’t want to carry this ugly expression with its subscripts and superscripts and symbols from dead alphabets I may say “let me call this ugly thing L_Dirac”1 and from then on I would use L_Dirac whenever I need to mention that ugly expression.
When I am done moving around L_Dirac I would then eliminate L_Dirac and recover the original ugly expression. What is this process called in mathematics? (Definition? Substitution?)
To me L_Dirac is just a placeholder. There are not two different quantities: 1. L_Dirac and 2. the equation named L_Dirac. I defined the ugly expression to be L_Dirac so that I can move it around easily. In the comments Andrew says that every definition is a proportionality. To me L_Dirac is not a term in a proportionality, it is just a label. It is not a magnitude. It does not exist in nature or in the universe.
So where is my error?
Another example: If I define K=ma then I can write F=K. But this K is a placeholder for ma. There are not two different quantities here. There is a quantity (ma) and a label (K) for ma.
My rule for testing for labels is that if you can eliminate it then it is a placeholder.
K is eliminated, L_Dirac is eliminated, so they are placeholders, they are not magnitudes.
I would greatly appreciate your advice on this issue. Many thanks.
Note: I just read Alexandre Borovik’s article talking about something similar. I think the problem is resolved if proportionalities are written in the old fashioned way as the equality of ratios. So, like named numbers, names of equations, are not quantities. So L_Dirac/L_Dirac is as meaningful mathematically as, say, “Lagrangian/Lagrangian.” Let me know what you think.
- My apologies to Dirac Lagrangian fans. [↩]
This ranks as one of your best posts! Yes, that is exactly what is done.
Every now and then, one solves for something and the result is worthy in and of itself as taking a new name. For example, in special relativity, one computes the proper time s and that is something that all observers agree on. So it certainly deserves a new name; it is more fundamental than the things it was made from.
But for the examle, a Lagrangian is related to energy, and energy, unfortunately, is just as fictious as force, and for the same reasons. In fact, mass itself is only defined in terms of ratios, it’s not just force that is at issue. Energy of course is proportional to mass.
In short, everything that includes a “mass” unit will be, or should be, subject to your criticism. And I agree, that is why mass is such a mysterious thing (and also why I work on that problem).
Thanks, Carl. Do you know any formal discussion on this topic in math or physics sources? I don’t even know how to classify the subject.
Just coincidentally, when I was checking Newton’s Proposition III.41 in Cohen’s Guide I saw a footnote where Cohen explains that in Newton’s time “the adjective ‘occult’ was used to denote a line drawn in the construction of a figure, but not forming part of the finished figure, and also to denote a dotted line.”
Fictitious, imagined, hidden, occult, reified labels and names, confusing map with territory. . . . these things appear to be legal in physics.