O.K., so many of you figured out that my claim, yesterday, that $1 = 1¢ is not actually true. And it was clear that you had to either (1) not square the dollar or cent symbol to begin with, in which case the “proof” wouldn’t be able to proceed, or (2) square them, but remember that the units become square dollars ($$) and square cents (¢¢) and the conversion factor is $$1 = 10,000¢¢.
But there developed a debate in the comments over whether the concept of square dollars (if you prefer the term “dollars-squared”, you can replace it throughout; same with similar terms) is even meaningful. I say clearly yes: as I explained yesterday, the variance of a dollar-valued function (which is the mean square of the deviation from the mean) is expressed in $$. If you don’t appreciate this (and appreciate the conversion factor of $$1 = 10,000¢¢), you won’t understand why your variance is 10,000 times bigger if you change all your dollar values into cent values. Similarly, you can have square grams, square degrees of temperature, square angular degrees — to say nothing of the usage of per-time-squared in acceleration, or per-anything-squared in derivatives of rates, or the gravitational constant G, which is 6.67384 × 10^-11 cubic meters per kilogram-second-squared. And these units “exist”, in the sense that anything in math exists if there’s a way of talking about it, regardless of whether it refers to some obvious thing you can look at in the real world. For instance, you can talk about whether one quantity expressed in terms of that unit is greater than some other quantity, or you can follow the rule that you can’t add or subtract quantities that aren’t expressed in the same dimensions. (Remember that dimensions are more general than units; length is a dimension, in which inches or meters or lis or versts are units.)
But at this point, I’m treading on more controversial ground: the question of whether mathematical objects actually exist. The view that mathematical objects actually exist is called mathematical Platonism, and one of the views that says the opposite is mathematical fictionalism. A good article from the Stanford Encyclopedia of Philosophy defending fictionalism (and describing and rebutting various other non-fictionalist alternatives to Platonism) is here; and you can click here to get an article from the same source about mathematical Platonism. These articles are fairly difficult, and slightly above my own fluency with these concepts; I don’t know whether I find fictionalism or Platonism more plausible myself. When I said that anything in math exists if there’s a way of talking about it, I’m mostly thinking of a fictionalist version of “existence”, but obviously if Platonism is true what I’m saying is consistent with that. Anyway, these are a good way to get an entry into the flavors of the argument.