First, let me acknowledge Nik Weaver's objection. However, there is a notion of quantization in error correction coding. This notion might be compared with numerical methods that establish a policy by which some rational number is fixed to represent the real number of a given calculation. That is, an admissible value is accepted in relation to some actual value. So, Margaret Friedland's notion will be accepted for this response.

The initial objections to classical logic by Brouwer and others had to do with the definiteness of working with finite systems that differed from infinite systems. So, the sense of Margaret Friedland's notion would not seem to be the direct comparison to be made. And, in that same context, the original posted question seems badly construed.

Insightfully, however, Ms. Friedland thought the issue might lie with semantics.

The following remarks are from personal unpublished researches. As a philosophical matter, I reject logicism. The focus of my investigations had been the sign of equality and identity.

When Leibniz introduced the principle of identity of indiscernibles in "Discourse on Metaphysics", he did so by invoking geometric intuitions,

"What St. Thomas affirms on this point about angels or intelligences ('that here every individual is a lowest species') is true of all substances, provided one takes the specific difference in the way that geometers take it with regard to their figures."

Leibniz

My personal view on this is that numerical identity relies on geometric -- or, more precisely, topological -- notions. So, I interpret Leibniz' remarks along the line of Cantor's intersection theorem for non-empty, nested closed sets of vanishing diameter.

The semantic sense of Leibniz' remarks are to be found in another quote from one of his papers on logic (the name of which escapes me at this moment). Although somewhat deprecated in mathematical logic, the paradigm singular term in classical logic is the notion of a name. This is what Leibniz says about names,

"All existential propositions, though true, are not necessary, for they cannot be proved unless an infinity of propositions is used, i.e., unless an analysis is carried to infinity. That is, they can be proved only from the complete concept of an individual, which involves infinite existents. Thus, if I say, "Peter denies", understanding this of a certain time, then there is presupposed also the nature of that time, which also involves all that exists at that time. If I say "Peter denies" indefinitely, abstracting from time, then for this to be true -- whether he has denied, or is about to deny -- it must nevertheless be proved from the concept of Peter. But the concept of Peter is complete, and so involves infinite things; so one can never arrive at a perfect proof, but one always approaches it more and more, so that the difference is less than any given difference."

Leibniz

This, too, can be related directly to topological considerations in the guise of uniformities and uniform spaces.

Semantically, the identity relation is represented by the diagram or diagonal of the Cartesian product of a model domain. Under the received view, logical identity is not given by a metric interpretation. It is in the metrization lemma found in Kelley's "General Topology" wherein a system of relations containing the diagonal (say, a uniformity) and meeting certain other conditions generate a pseudometric. In other words, it is in the theory of uniformities where the original Leibnizian conception and modern semantical notions coincide.

It is for this reason that I am personally inclined to view the topological designations of open and closed sets (in particular, closed sets) in the context of the kind of "quantization" mentioned by Margaret Friedland.

For completeness with respect to the preceding remarks, let me observe that both Frege and Russell included descriptivist theories of naming in their logical analyses. Kleene reports that the eliminability of descriptions had been established in 1934 by Hilbert and Bernays. Robinson had been critical of Russellian description theory and discusses the use descriptions in relation to model diagonals in his paper "On constrained denotation".

Along similar lines, the logical interpretation of Leibniz' principle of identity of indiscernibles seems to have been established by the time of Kant. Kant criticizes Leibniz' application of that principle and asserts that identity associated with appearances is based on geometric notions. With respect to modern description theory, this notion of numerical identity in relation to geometry can be found in another critic of Russell -- P. F. Strawson discusses the matter in his book, "Individuals".

Let me reiterate that these are only personal views, and, that they are non-standard by every account.