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Leśniewski explicitly rejected set theory in favor of mereology on nominalist grounds. Could someone explain why classical set theory is usually regarded as Platonist (or “Platonizing”), and what makes mereology—especially Leśniewski’s version—ontologically non-Platonist? Why is set theory Platonizing?

I am particularly interested in how this difference relates to abstract entities such as sets, classes, and singletons versus individuals and part-whole relations.If I'm not wrong Goodman says that sets imposse a sort of "class-scheme" division upon things which arbitrarily or artifically "divides them" into sub-types or sub-sets. But why is this "platonizing"?

By platonic I do mean here accepting either the ontological existence of universals and/or abstract objects.

Maybe what does seem clear to me is that set theory can be less intuitive because it treats anything as forming a set/whole—for example, “let the set be a garbage bag, the number three, and my toothache.” However, I do not see why, in a formal sense, mereology could not also accept arbitrary compositions of the same kind, such as: “my toothache, the number three, and the garbage bag together compose a whole.”

Thanks in advance.

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    Could you please elaborate on your answer? @MauroALLEGRANZA Commented Dec 16, 2025 at 20:22

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Note: I use small-p "platonist" to clarify that platonist views were not the views of Plato. Thank Quine for the misnomer.

The two primary issues involve singleton sets and sets of sets. Non-platonists claim to have trouble grasping the concept of a set of one item that is distinct from the item itself. For example, what is {2} other than just 2? Sets of sets have a similar issue. How is {{2,3},{4,5}} different from the collection of 2, 3, 4, and 5? Both of these features of set theory require that sets be things distinct from a simple collection of their members even though there seems to be nothing to them besides their members.

Another issue that has caused worries is the empty set, and for much the same reason. If there's nothing in the set, what is the set? My sense is that this is less troubling to most non-platonists, though, because everyone needs some sort of token or place holder for the situation where nothing satisfies a condition.

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Mereology as a mathematical theory need not to be nominalistic:

while Leśniewski's and Leonard and Goodman's original formulations betray a nominalistic stand, reflecting a conception of mereology as an ontologically parsimonious alternative to set theory, there is no necessary link between the analysis of parthood relations and the philosophical position of nominalism [Actually, the original Goodman's Calculus of Individuals (1940) had variables for classes; a class-free, purely nominalistic version of the system appeared later in The Structure of Appearance (1951).

Stanisław Leśniewski (1886–1939) was a radical nominalist that rejected the philosophical concept of "general objects" (abstracts: classes, sets) in favor of common names.

He was the originator of an unorthodox system of the foundations of mathematics, based on three formal systems: Protothetic, a logic of propositions and their functions; Ontology: a logic of names, and functors of arbitrary order; and Mereology, a general theory of part and whole.

In 1916 (after the pubblication of W&R's Principia) he developed his own axiomatized system of foundations: his general theory of "manifolds" or "collective sets", later renamed "mereology" or theory of parts.

The key-popint of L's view on mereology is the rejection of the unintuitive distinction between an individual and the totality (the "collective class") of itself (the singleton).

For L, in order to describe totalities or collections of individuals having a specified property, like "social classes" that consist simply of their elements, "class" terminology is not necessary: we need only a language having names and operations that describe individuals and "concrete" collections composed of one or more individuals.

Any individual is the same individual as the totality of itself, and any ingredient (element) of an ingredient is itself an ingredient of that individual:

For example, only individual planets (Earth, ...) are planets. Whereas not only individual planets, but also arbitrary parts of planets (Australia, ...), arbitrary collections of planets or their parts, and even the totality of planets itself are ingredients elements of the totality (the "collective class") of planets, which consequently has no unique determinate numbers of ingredient elements.

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The first part of the question asks why set theory is generally considered as platonist. I don't think that it is "generally" seen as such; at least not anymore. I'll only address this false (or disputable) assumption.

Platonism is (roughly) the belief that mathematical objects or mathematical structures are real in the sense that they exist independently of the mathematical subject (humans). It is then also the belief that mathematical theorems are true, independent of whether or not they have been verified by us; their truth is not based on a mental construction, but is a discoverable fact of the world. Since these objects and these truths cannot, as such, ever be discovered in the empirical world, platonism then posits that there must be some kind of separate realm of (necessary, a priori) truth(s), in regards to which mathematical theorems are just as true (or even more so) as empirical statements are true of the physical world.

This is an informal philosophical, meta-mathematical belief that is independent of mathematics itself: it's not assumed in any mathematical theory, and it need not be assumed in mathematical practice, though it may have heuristic value if believed. It can also be used to informally justify certain logical principles, for instance LEM, while those who don't accept this belief (for instance intuitionists) may not find such informal justifications unconvincing.

Set theory is a mathematical theory that doesn't require either an affirmation or a negation of platonism. Even the classical form of set theory (ZFC) in which you have axioms like

There is an empty set.

There is an infinite set.

does not require platonism. It's for instance possible to take a formalist stance: we don't need to posit any "objects" in some mysterious, eternal realm, in order to work with those symbols. Or we can take a (related) fictionalist stance: As long as we don't run into inconsistencies, it's fine if we just treat infinite totalities as fictional objects, more or less similar to, say, fictional literary objects, like Hamlet.

If we look at the historical development of set theory (and number theory), then certain mathematical results can also be seen as (informally) undermining a strict platonism, or at least as making it less attractive. In particular the fact that Cantor's Continuum Hypothesis has been proven to be independent of ZFC. Results like that may make the view of a set-theoretic multiverse (Hamkins) seem more plausible, a view that is distinctly anti-platonistic (which btw doesn't make it nominalistic).

The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous diversity of set-theoretic possibilities, a phenomenon that challenges the universe view. In particular, I argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for.

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  • Most mathematical practice certainly does assume platonism. Mathematicians describe what they are doing as searching for certain objects or conditions and they describe things as existing or no existing. There are discussions about whether disputable axioms are true or not. Just because there is some hypothetical re-wording that removes this ontological language doesn't mean that the language doesn't mean what it clearly means. There is no justification for the claim that working mathematicians don't have in mind the plain meanings of their words. Commented Dec 17, 2025 at 15:48
  • "The plain meanings of their words" is obviously not platonism - The mere fact that there are radically different views about what this "plain meaning" would be, implies that it is not so plain. And working mathematicians/logicians can very well be realists about sets without being platonists. Example: Hamkins. Commented Dec 17, 2025 at 17:38
  • Apart from that, platonism itself can also be seen as a misguided extrapolation based on a pre-theoretic/everday intuition of "determinate object", or as a misguided solution to a question that strictly speaking makes no sense. Like asking if the rules of chess "exist" independently of players - the question itself as to the nature of math. objects seems illposed. Carnap and pragmatists have argued this. Commented Dec 17, 2025 at 17:45
  • Platonism is just the position that mathematical objects exist, so the plain meaning of talking about sets and numbers existing is platonism. Commented Dec 17, 2025 at 17:46
  • That's a simplification that does not capture what "platonism" in this context is.And perhaps it is clear to you what it even means to say "that mathematical objects exist", but to many this will not at all be clear. Which is exactly why there is philosophical debate about it... (including about whether such a statement can make any sense or is required to do math). Commented Dec 17, 2025 at 17:50

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