Caplan’s writing is clear and readable.
I was confused by the terms “cardinal” and “ordinal” until I noted their mathematical origin. (I learned the distinction in set theory.) I think the truth lies in some murky area between the claims. I think there is a difference between the distances between different neighbors on the value scale. Quantifying these differences is indeed difficult. I guess I am firmly undecided on the issue! Perhaps this difference does not concern economics. I would like to be convinced that the interesting mathematical issues of the first few paragraphs bear on economic outcomes.
Just now I am concerned about the following considerations by an individual to whom either utility theory is applied. He encounters an option to acquire some apples and thus considers the utility thereof. He knows that he can eat only 20 apples before they go bad and this puts a sharp cutoff in his utility curve. Then he conceives a plan to trade some of the apples for some meat with someone who he expects to have a quantity of meat. Does his utility curve for apples thereby increase at the high end? This question may bear only on the definition of utility. I think that the two utility theories may make different assumptions about the answer to the above question. I have not heard this question explicitly addressed.
I find the discussion of indifference a bit silly. I agree that behaviorists ignore real phenomena whose consideration and modeling can produce better predictions as a mere practical matter. It now seems that we have even evolved mechanisms to model the thoughts of others. I don’t see the bearing of this on economic theory.
The note on continuity is interesting. Caplan’s observation about continuity and solving the supply-demand equations seems apropos. This is an issue that goes far beyond economics. The real numbers are a very peculiar concept, no matter how familiar. In the real world the subdivision of things comes to an end, but not in the world of real numbers. Infinite divisibility seems merely to be the simplest logical approximation by avoiding saying how things come to an end. It also seems to yield very practical results via sound but contorted logic. There are other approaches such as combinatorial topology for which the reals provide just a model. I don’t think the necessary math has been done to apply this to economics or other quantitative sciences.
Towards the end of “Welfare Economics” there is the phrase “While the justice of efficiency is far from evident”. This assumes that justice is prior to total utility or at least efficiency. This is not the usual economics perspective, I think. I like to think that much of ‘justice’ derives from utility, not the other was around.
All in all section 2 (Foundations of Microeconomics) while interesting, seems a bit disconnected from real economics. Perhaps I am merely an outsider.