Bourbaki and foundations
jodmos.horon at protonmail.ch
Thu May 12 21:22:02 EDT 2022
> So far I have not seen anyone give an explanation that makes sense of what is WRONG with set theory. [...] WHO is forcing anyone to use set theory in a way that requires any significant and unnecessary effort?
Aside from the point that categories has a more structuralist feel that points to a way to do mathematics that is more friendly to universal problems (a thing that set theory is not), I have a main big issue with set theory:
It's based on sets. Meaning it's based on elements. Meaning that, implicitly, all sets are equivalent to the boolean algebras they induce. Essentially, set theory has atoms. I believe it's not necessary and may even be flat out wrong.
Russell grew a set theoretic ideology by merging many ideas prior to him. I won't refer to Cantor who, in my view, ended up being biased towards topological questions rather than foundations of maths per se. But I'm referring to the school of the logic of relations of Peirce and Schröder that Russell seem to have semi-loathed or at a minimum heavily disliked.
Schröder following Peirce tried extensively to deal with "individuals", meaning roughly atoms as in boolean algebras, but more generally tried to dissect what such individuals were in general lattices and not merely boolean algebras.
Russell, on the other hand found, what I'd call a shortcut around the twists and turns of the school of Peirce and Schröder, a school that couldn't quite square these "individuals" with the calculus of binary relations they were developing. In essence, that shortcut was his so-called principle of abstraction:
Principle of abstraction, Russell's original version: any equivalence relation is the composition of a binary relation rho and its converse rho° such that when (z,x) and (z',x') are in the graph of rho, z~z' if and only if x=x'.
Principle of abstraction, contemporary version: one may quotient any set endowed with an equivalence relation into its quotient set.
Principle of abstraction, categorical version: in the category of binary relation endowed with the operation of conversion of binary relations, every self-converse (i.e. symmetric) idempotent binary relation splits.
These are all (essentially) equivalent. The second version earns the name of principle of abstraction proper as it guarantees that we may "abstract" an equivalence relation on couples of integers into a "set" of rational numbers. The first and third version are exactly the same, the first being phrased in a language akin to that of the logic of binary relations, the third is stating the same algebraic relation in the language of categories. So, in a sense, we are coming full circle with category theory.
This observation doesn't deal with the paradoxes at the root of the crisis of mathematics of the beginning of the 20th century. But it ties in with the ontology of naive set theory. And this principle of abstraction is what guarantees one may crush equivalence classes into a point in the quotient set, whose existence (of that set) is thus axiomatically posited by Russell's principle of abstraction.
I therefore believe it is a core statement of The Ideology of the Point.
And that notion of splitting of self-adjoint idempotents (some kind of Karoubi completion) is something that is studied by Peter Selinger in the context of dagger categories. Such as complex Hilbert spaces with bounded linear maps.
I'm not competent in that matter. Though I believe it is no coincide that the same notion of splitting of self-adjoint / self-converse idempotents was rediscovered by Selinger after Russell's early observation in The Principles of Mathematics.
There also are reasons why I believe set or equivalently boolean algebras are wrong. In quantum logic, distributivity fails whereas boolean algebras are always distributive. And, in fact, quantifying the extent to which distributivity fails is a way to compute the amount of qubits, from what I've read.
I'm under the impression that these are reasons to consider reworking naive set theory without the assumption that points or elements are fundamental.
There are other reasons why I believe items / points are not the right way to conceptualise a set / space, and they tie to interpretations of how one could think of Spec Z as a space. I'll say no more on that.
Another observation that makes me think category theory is in fact a way to go full circle back to the work of Peirce and Schröder on lattices is the observation that lattices and categories both are "partially defined semi-groups". It is curious how much one can do with a mere partially defined semi-group: it indeed is possible to unify the order theoretical properties of lattices and that of ordering of subobjects in a topos. And that is enough to reboot back onto Schröder's individuals in lattices, but now in category theory.
Bottom line: I believe that naive set theory, by focusing on items / elements / atoms (i.e. The Ideology of the Point) has blinded us to the idea that there could be another way to do maths that doesn't really require them.
It's not a simplification of mathematics, on the contrary. Doesn't solve the issues of paradoxes at the root of the early twentieth century crisis in foundations. But for reasons tied to quantum logic and arithmetics, I believe it's worthwile to rework set theory or at least naive set theory in an item-free or point-free context.
That's where I'm not satisfied with set theory: sets. And their items. I believe these items are expandable.
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