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When we differentiate the utility function with respect to some input $x_i$, we get a number that tells us how "fast" the utility function is changing at some point with respect to $x_i$. Doesn't that mean that when we compare marginal utilities, we are comparing with something that is based on the structure of the numeric output of the utility function, which shouldn't be the case since it's ordinal?

I know I'm missing something here intuitively, but I can't seem to figure out what it is.

To the pure ordinalist who believes that preferences are purely ordinal, the concept of marginal utility (MU) has no meaning. (And a fortiori, the concept of diminishing MU also has no meaning.)

However, the concept of the marginal rate of substitution (MRS) does have meaning.

In the course of our work, we may compute something that we call MU. But to the pure ordinalist, any such number found has no meaning in and of itself.

Example. Say an individual's preferences $succsim$ over two goods $A$ and $B$ can be represented by the utility function $U:(mathbbR^+_0)^2rightarrowmathbbR$ defined by $$U(A,B)=AB.$$

The intermediate microeconomics student may then carry out these computations:

$$MU_A=fracpartial Upartial A = B.$$

$$MU_B=fracpartial Upartial B = A.$$

$$MRS = fracMU_AMU_B=fracBA.$$

The above says that if, for example, my current bundle is $(A,B)=(200,1000)$, then $$MU_A=B=1000text and MU_B=A=200.$$ However, these two numbers have no meaning whatsoever.

The only number that has meaning is the MRS: To get another unit of $A$, I'm willing to give up (approximately) $$MRS=fracBA=frac1000200=5text units of B.$$

To the pure ordinalist, the above reasoning is completely legitimate, so long as one assigns meaning only to MRS. What's illegitimate is to assign any meaning to $MU_A=B$ or $MU_B=A$.

The pure ordinalist knows that if $hat U$ is a strictly increasing transformation of $U$, then $hat U$ is also a utility representation of $succsim$. So, for example, if $hat U:(mathbbR^+_0)^2rightarrowmathbbR$ is defined by $$hat U(A,B)=2AB,$$ then $hat U$ also represents $succsim$.

However, with $hat U$, our computations seem to differ slightly from before:

$$Mhat U_A=fracpartial hat Upartial A = 2B.$$

$$Mhat U_B=fracpartial hat Upartial B = 2A.$$

$$hatMRS = fracMhat U_AMhat U_B=frac2B2A=fracBA.$$

Any conclusions we arrive at with the new utility representation $hat U$ are the same as before.

If again my current bundle is $(A,B)=(200,1000)$, then $$Mhat U_A=2B=2000text and Mhat U_B=2A=400.$$ However and again, these two numbers have no meaning whatsoever.

The only number that has meaning is the MRS: To get another unit of $A$, I'm willing to give up (approximately) $$hatMRS=frac2B2A=frac2000400=5text units of B.$$

The quantity MU by itself has no meaning. Confusion arises only when one attaches meaning to MU and wonders how for example it is that $$Mhat U_A = 2MU_A,$$ and what the above equation means. (Answer: It means nothing.)

Out of convenience, the intermediate microeconomics student will often compute something called $MU_A$ and $MU_B$ and these can often be evaluated as actual numbers. But on their own, these numbers have no meaning (to the pure ordinalist). Only the ratio of the two quantities has any meaning: $$MRS=fracMU_AMU_B.$$

Some quotes. Hicks (1939):

We have now to undertake a purge, rejecting all concepts which are tainted by quantitative utility, and replacing them, so far as they need to be replaced, by concepts which have no such implication.

The first victim must evidently be marginal utility itself. If total utility is arbitrary, so is marginal utility. ...

The second victim (a more serious one this time) must be the principle of Diminishing Marginal Utility. If marginal utility has no exact sense, diminishing marginal utility can have no exact sense either.

Dittmer (2005, emphasis added):

Many introductory microeconomics textbook authors derive the law of demand from the assumption of diminishing marginal utility. Authors of intermediate and graduate textbooks derive demand from diminishing marginal rate of substitution and ordinal preferences. These approaches are not interchangeable; diminishing marginal utility for all goods is neither a necessary nor sufficient condition for diminishing marginal rate of substitution, and the assumption of diminishing marginal utility is inconsistent with the assumption of ordinal preferences.

What matters is not the utility function but the indifference sets. They produce an ordering of the outcomes. This ordering is a total order - reflexive, transitive, antisymmetric and total. If some conditions are satisfied we obtain indifference sets in the form of the the usual indifference curves. A sufficient condition for this is that the relationship is monotonic ($a > b Rightarrow a$ is preferred to $b$). Only after we have indifference curves do we get to speak about utility. A utility function is a mere representation of a particular partition of the outcomes into indifference curves. This is called utility representation of preferences.

Note that we are talking about marginal utility only when we are looking for the solution of the consumer choice problem. In this case we are not interested in the utility function in of itself. We only use the fact that a differentiable utility function with nice properties correctly represents the underlying indifference curves. This allows us to use the toolbox of mathematical optimization to find the optimal bundle.

A good example for the fact that the utility function is only a reformulation of the initial question, determined by the indifference curves, is the case of perfect comlpements - you have indifference curves, you have a set of utility representations for this indifference curves and you never talk about marginal utility in this case. As it is either zero or infinite, it is certainly useless to apply the utility representation to solve the consumer's problem here, but the consumer's problem still exists and has a well-defined solution.

Thus yes the utility function is cardinal, but the local ordering of preferences in a neighborhood of a bundle, which is what the marginal utility describes, stems from the indifference curves and is not caused by the cardinality of the utility function.