Start here. You can work backwards from that post to earlier ones in the conversation, if you would like to do so.
What to make of all this?
First of all, philosophically speaking, Mises is correct: acts of valuation are not measuring anything. If I part with $40 for a steak dinner, I have not "measured" the value of the dollars or the dinner. I have made a judgment that I prefer the dinner to the $40, but a judgment is not a measurement. (I can of course, make judgments about measurements: "I think Bill is twice as tall as Joe." But that is not a measurement itself either.)
In fact, I think we can go further, and declare we have no particular reason to endorse Mises' claim that acts of choice place "all values on a single scale." Consider Socrates, sitting in his cell awaiting death, with the opportunity to escape before him. Mises' claim seems to imply that if Socrates had merely been offered enough olive oil and retsina, he would have been off for the boondocks. We can save Mises' claim by saying that Socrates valued obeying the laws of his polis more than an infinite amount of oil and wine, but that becomes an odd sort of "scale." No, we are better off here siding with Collingwood: ethical choices are of a different order than utilitarian choices: to judge something to be "right" is fundamentally different than judging it to be "useful."
But what about the argument of Bob's reader Chris, who writes:
"But if I prefer 1.51 ounces of Coke to 10 ounces of water and prefer 10 ounces of water to 1.49 ounces of Coke, is there any real problem in saying 10 ounces of water is worth 1.5 ounces of Coke (or whatever infinitesimal amount when I start to prefer one over the other)?"
Chris is correct here, if we interpret "real problem" in the right way: yes, as Mises notes, market exchanges do not measure value, and, in fact, there is no physical quantity present to be measured. But if we want to create an economic model in which we treat exchanges as if they measure some subtle substance called utils or "value," there is "no real problem." (And Chris is correct about the most likely way to generate such a model: capture changes in choice at the margin.) All models are false. But some of them are useful. The right question to ask about a model is not whether it is true, but whether it allows us to gain understanding of the phenomena we wish to explore by means of using it. And I suspect that models treating exchange as measurements of value can, indeed, help our understanding of economics. But the proof is always in the pudding.
By the way, this same analysis applies to Rothbard's rejection of the use of calculus in economics due to the fact that people choose discrete units of goods, and not continuously varying amounts of them. (Even when we seem to be choosing along the continuum, such as at a gas station, in fact we are merely choosing between very, very fine-grained discrete units: you can put 10.004 gallons of gas in your tank, or 10.005 gallons, but you can't purchase 10.00400000000000001 gallons.) So, philosophically speaking, Rothbard is correct.
But when it comes to modeling, so what? When I first came across Rothbard's argument (circa 2001), I was skeptical: it did not seem to me to be problematic to simply assume we could choose an amount of a good anywhere along the continuum. And then we would have a differentiable function.
I happened to be working for a mathematics PhD at the time: he was creating financial-market models, and I was programming them. I decided to ask him about this issue, but put my question into another domain, not knowing if he had a horse in the race concerning mathematical economics.
"Randy," I said, "let's say we are modeling the population of geese in a lake. Of course, there can only be an integral number of geese: but is it a problem to treat this number as though it were continuous, and then differentiate the resulting, continuous population-change function, so that we can get, say, an instantaneous rate of change for the population?"
He answered, "No, that is not a problem at all: absolutely standard to do that sort of thing."
Of course, one wants to remember that one is dealing with a model, and therefore, a useful fiction. It won't do to believe one's model is true, and head to the lake confident that one will find 47.348 geese swimming around in it. And neoclassical economists are sometimes guilty of this sin, and even worse, e.g., in the case of perfect competition, criticizing the real world for not living up to the unrealizable conditions characterizing the model. As Bob himself once memorably told me, "In in the model of perfect competition, it is as if the grocer wakes up in the morning and goes to his shop, only to find, to his surprise, that the supply and demand curves for milk themselves have changed the prices on all the cartons of milk on his shelves." (This is what it means for the grocer to be a price taker, and it is an assumption of the model that all market participants are price takers.)
That does not imply that we should not use the model of perfect competition, when it comes in handy: it means we should not believe it.