My view is actually close to Mattheus's on what is actually true (preferences are ordinal, utility functions are made up, and interpersonal utility comparisons don't have a positive basis yet [that I know of - if a neuroscientist studying dopamine levels and human perception wants to challenge that I'm all ears]). But my views are quite far from Mattheus's on (1.) what it is mainstream economists think about these things, and (2.) how we should do microeconomics. With respect to those points, I can agree with Vienneau, stickman, and unlearningecon that a huge portion of Austrians have no clue what they are talking about.
So let's start at the beginning. On two separate points I think you really have to distinguish between two very different things:
1. You have to distinguish between "what mainstream economists think" and "what mainstream economists may or may not teach their undergraduate students"
2. You have to distinguish between "preferences" and "utility".
On point #1, I'm only going to concern myself with what mainstream economists think, because what they teach their students is a pedagogical issue and while I have my own thoughts on teaching undergrads, that doesn't really concern the discussion we're having.
Mainstream economists all agree that preferences are fundamentally ordinal relations. Indeed, that's how they're defined. It's a "preference relation" - it's not some preference scale. Everyone agrees that we can say "X is preferred to Y", or in some cases "X is indifferent to Y", but the sentence "X is twice as preferred as Y" doesn't make any sense because the preference relation is a binary relation. Now, we want to do something with that relation. So we usually assume certain things about preference relations. We say that agents are "rational". Now - a lot of critics of mainstream economics who don't know any better load a lot of junk onto that word "rational". But to say that someone has "rational preferences" is quite simple. It's just saying that (1.) the preference relations are complete, (2.) the preference relations are transitive. That's all.
The important question is - can rational agents settle on a maximal bundle given a set of constraints (typically a budget constraint) with these ordinal preferences.
Mainstream economists say "yes". I think different people learn a different answer to this question. I was taught the answer using Walker's Theorem (1977), because my professor thought it was particularly elegant. I understand there are other ways to show this, but learning Walker's Theorem was a bitch, so I've exerted exactly zero effort in learning any other method. This insight is the heart of mainstream micro. It's so important that we had to write the proof on our midterm and on our final. I think it was something like 1/8th of the points on our midterm and 1/5th of the points on our final. And since that was all of our grade except for 10% for homework, you can see how critical this point is. Ordinal preference relations can be maximized by a rational agent for any budget constraint. Please stop telling me mainstream economists don't think this.
OK, so this is settled. Now this is where Edgeworth's "grandest of generalizations" comes in.
We want to construct theories about the way the economy works motivated by action on the basis of these ordinal preference relations. But it's hard to do that with a collection of binary relations (X>Y, Y>Z, W>P, L>Q, etc.). So we ask ourselves: "can I think of a functional relation that represents these ordinal preference relations, which I can then analyze and use to theorize different things?". The key word here is "represents" - and this is also a point stressed by mainstream economists. What you want is a (cardinal) functional relation that will give you the same answer that you would get from a plausible set of ordinal preference relations. You want that because you can do calculus on a functional relation, but not on a set of binary relations.
Thus, we make up utility functions. We want utility functions that assign values to bundles that duplicate ordinal preference relations. If X is preferred to Y, we want U(X) > U(Y) for all X and Y under consideration. We also want to make sure we're talking about rational preference relations, so the function has to obey those rules as well. That's all that matters.
This gets to a point that stickman raised in some of the comment sections about monotonic transformations of utility functions. If U=x represents a preference relation, then U=ln(x) represents the same preference relation equally well. Indeed, there are lots of problems where you make that exact monotonic transformation because it's easier to solve U=ln(x) than U=x.
The point is, utility functions are meant to represent ordinal preference relations. They are not intended to give you information about the intensity of preferences. This is part of the reason that I'm somewhat skeptical of welfare analysis. It walks a dangerous line. Sometimes it's OK as a way of translating preferences into dollars. That doesn't cross the line of telling you how much more you enjoy something than another thing - it simply monetizes a preference relation. You don't know how much more you enjoy something unless you have a money-to-utils conversion, which of course is a conversion that doesn't exist (or at least that we don't have access to).
A lot of these discussions go wrong when people take what they think they were taught in undergraduate microeconomics and start assuming that's what mainstream economics is. Things are always simplified. The problems you spend a lot of time on as an undergraduate are very different from the problems you spend time on as a graduate student, and it's the latter that's somewhat closer to what economists actually spend time on. Undergraduates spend very little time on working with ordinal preference relations. Graduate students do. They work a lot with that because it's the underpinning of all the subsequent functional relations that we work with.
Critics need to stop and take a second to consider that maybe undergraduate economics doesn't spend much time articulating the step from ordinal preference relations to cardinal utility functions because that step - while necessary - is not important to get the fundamental insights of economics. If instructional time is scarce - and it is, because many don't continue on in economics - you disseminate more knowledge by expecting your students to trust you on the step between ordinal preference and cardinal utility. With scarce instructional time you are also probably better off finessing the difference between a utility function that is useful for theoretical applications and a preference relation that is what is actually motivating human action.
Anyway - trust me - this is not some incredible insight of the internet Austrians. We know this. We spend a considerable amount of time on it.
...and if you all are mean to me in the comment section my next post will be a proof of Walker's Theorem.