Looking for what others could tell about it on Quora, the closest relevant question I found there is "Why are top mathematicians not interested in philosophy of mathematics, also top physicists not interested in philosophy of physics etc? The people who are working in the philosophy of these fields are usually philosophers.
Replies there essentially repeat the following 2 points:
To illustrate that such observations were already done by other mathematicians, here is a quote from Talk:Foundations_of_mathematics
"Anyone, a mathematician especially, who appreciates the “unreasonable effectiveness of mathematics” and the “unreasonable ineffectiveness of philosophy" to scientific endeavors must recognize the dangers of letting "philosophy of math" ride roughshod over "foundations of math" and as a last line of defense, of letting "philosophy and foundations of math" ride roughshod over proper pure and applied maths.Now the situation is diverse, so there are also some genuine professional forms of "philosophy of mathematics"... done by mathematicians. A reference of genuine works (appearing among top google results on feb 2019) is the article Philosophy of Mathematics (2007) by Jeremy Avigad (link appearing in his "publications" list). He wrote a more recent overview of the philosophy of mathematics: Does philosophy still need mathematics and vice versa? (24 September, 2018) explaining the bright but also the degenerate side of the philosophy of mathematics in long and interesting ways. Some excerpts gathering the main ideas:
Just look at the talk page for "philosophy of math"! What a mess. Note that some of these people actually believe the destiny of science can be mastered thru verbose semantics, concepts, schema, arguments, etc. The last time I looked, the language of science was still written in mathematics. Fortunately, bullshit had not yet taken over in the math journals.
Specialists in foundations and/or philosophy of math often over-estimate the importance of their work to those in other specialties."
"The philosophy of mathematics reached its heyday in the middle of the 20th century, buoyed by the previous decades’ successes in mathematical logic.(...) For a subject traditionally concerned with determining the proper grounds for mathematical knowledge, modern logic offered such a neat account of mathematical proof that there was almost nothing left to do. Except, perhaps, one little thing: if mathematics amounts to deductive reasoning using the axioms and rules of set theory, then to ground the subject all we need to do is to figure out what sort of entities sets are, how we can know things about them, and why that particular kind of knowledge tells us anything useful about the world. Such questions about the nature of abstract objects have therefore been the central focus of the philosophy of mathematics from the middle of the 20th century to the present day. (...) philosophers of mathematics were chiefly concerned with the question as to whether numbers and other abstract objects really exist.In the rest of this page I will mainly focus on some cases of bullshit, and how they are not marginal. Namely, online encyclopedia articles appear to fall prey to the above mentioned unhealthy fixation ; even when the same ideas and authors are reviewed, Jeremy Avigad does a much better job at putting them in the right perspective.
This fixation was not healthy. It has almost nothing to do with everyday mathematical practice (...) there simply aren’t that many interesting things to say about abstract mathematical objects in and of themselves. (...) set-theoretic idealisation idealises too much. Mathematical thought is messy.(...) we have a lot to learn about how mathematics channels these wellsprings of creativity into rigorous scientific discourse. But that requires doing hard work and getting our hands dirty. And so the call of the sirens is pleasant and enticing: mathematics is set theory! Just tell us a really good story about abstract objects, and the secrets of the Universe will be unlocked. This siren song has held the philosophy of mathematics in thrall, leaving it to drift into the rocky shores."
The Wikipedia article has one main quality : its list of "Recurrent themes", that is interesting questions, unfortunately not including relevant links to any answers. It also gives an interesting quote from Hilary Putnam :
"When philosophy discovers something wrong with science, sometimes science has to be changed (...) but more often it is philosophy that has to be changed. I do not think that the difficulties that philosophy finds with classical mathematics today are genuine difficulties; and I think that the philosophical interpretations of mathematics that we are being offered on every hand are wrong, and that "philosophical interpretation" is just what mathematics doesn't need."The main content appears to be its list of "contemporary schools of thought" defined by the different approaches to the nature of abstract objects. How miserable are these for contemporary times, if they are still there, which keep ruminating the relics of the history of mathematical research which led to current foundations. Something odd: one of these "schools", the "post rem structuralism" has one name (Paul Benacerraf, who indeed was a main actor of that pseudo-debate...), but only few google results : beyond Wikipedia and its multiple copies, only 4 more results could be found when writing this page (feb 2019), likely to add itself to this number. Then I'd be tempted to ask : do these "schools of thought" have any physical address, or at least statistical data on their relative popularity, otherwise would they be themselves mere abstract objects ? in which case why should I believe in their existence ? More precisely asking whether they would be indispensable to the understanding of our best scientific theories, well... I think they aren't.
So many philosophers still believe that the "foundational crisis" of
mathematics remains unresolved. Hahaha. My attitude is : the usual view
of something is not clear ? Well I have no time to philosophically
describe its state of unclarity because I'm too busy resolving this by
bringing clarity instead !
In first approximation following the working attitude of mathematicians, Platonism can be said to be the correct view, empirically supported by the famous indispensability argument (3.2 - I don't see what it may have to do with naturalism: that is just a particular case of empirical argument even if its object is some very general stuff which I further commented in another page). But for better accuracy some clarification, nuances and details would be needed. One aspect is structuralism, that is an important aspect of what mathematics is about : while individual mathematical objects are trivial and thus uninteresting, the point of mathematical theories is to describe systems which objects can form by means of structures. This combination of Platonism with structuralism is called Ante Rem Structuralism.
"In textbooks on set theory we also find a notion of structure (..) But this cannot be the notion of structure that structuralism in the philosophy of mathematics has in mind"Ah and why not ?
"For the set theoretic notion of structure presupposes the concept of set, which, according to structuralism, should itself be explained in structural terms."Well set theory is the natural reference theory encompassing all other theories, thus by which to define systems and their structures. Yet theories and their models with their lists of structures can be also considered in isolation. One of the roles (reasons) for using set theoretical interpretations is as a way to somehow more explicitly describe these structures of systems which the rest of theories are about. Now we need to start with something anyway to explain what we are talking about, don't we ? How can you start your explanations from nothing ?
"It appears that ante rem structuralism describes the notion of a structure in a somewhat circular manner."So indeed some crazy absolutist expectations of trying to start from nothing do not seem satisfiable anyway... still these expectations are what philosophers of mathematics put forward on why they dislike the way mathematicians understand structures, and they think that as philosophers they are much better placed than mathematicians to undertake, though not satisfactorily, to explain the exact concept of the structures that mathematics is really all about. Oh well...
The most interesting thing of the Stanford article is the whole section 3.3 Deflating Platonism which I invite you to fully read there (instead of copying it here). I agree with Tait as reported there : let mathematics speak for itself. To comment further : I see no sense asking what is the nature of mathematical objects. For me there is only one worthy answer which I gave there : that mathematical objects are objects whose only considered nature of is to be exact, unambiguous. There is no such thing as a lack of clarity of the status (nature or existence) of mathematical objects in need to be assessed with respect to any a priori more meaningful standard of "reality" : supposedly external references involved by the words "nature" and "real" in this context remain out of subject and a priori nonsensical, in lack of a meaning that may be compared to mathematical objects, but there is no point to look for such meanings, because as words without definition the question of finding a sense for these, if any, is a pure convention and matter of taste, not any worthy meaningful questions. Because mathematics is about clarity and exactness, and there is no sense trying to explain or justify what is clear and exact on the basis of what is fuzzy and unclear. Seriously, who is crazy enough to go look for what may be any real (not just metaphorical) physical standards of substances or ontology ??? Nobody could ever make any clear sense of such things. At best, the naive appearance of physical reality may be considered as an interesting metaphorical approach to discuss ontological views. Trying to refer to what may seriously be a true ontology of physical stuff would bring the discussion much too far away from the topic, into a long exploration of candidate views on the philosophy of physics.
Once so (not) specified the nature of mathematical objects, comes the question "Are mathematical objects real" ? Existence is a mathematical concept, and mathematics has its own standards of existence. It has some clear answers about it given by mathematical theorems. The most important theorems relevant to the most general existence questions about mathematical objects are
Now in that 3.3 of the Stanford article, is the note that "Of course not everyone
agrees with Tait on this point. Linsky and Zalta have developed a systematic way of
answering precisely the sort of external questions that Tait approaches with disdain
(Linsky & Zalta 1995)". A referenced work to which a whole further section is dedicated:
"3.5 Plenitudinous Platonism". More precisely the latter presents two versions of this
Plenitudinous Platonism, the other being "Balaguer’s version". Zalta is himself
the creator and principal editor of the Stanford Encyclopedia of Philosophy (yes, all that !!!!)
while Balaguer is himself the author of the Philosophy
of mathematics article at Encyclopædia Britannica, whose very short mention of the
issue indicates an absence of any difference between his and Zalta's version.
In the Stanford article which is a bit more detailed, "In Balaguer’s version, plenitudinous platonism postulates a multiplicity of mathematical universes, each corresponding to a consistent mathematical theory". Thank you but I see no need of postulation for this, since this is exactly what the Completeness theorem already says, once assumed the axiom of infinity. In more details I see another wrong aspect of his presentation of this so-called postulate : he does not anyhow elaborate on the condition for existence of a universe of a theory, which is that the theory needs to be consistent. Indeed by the incompleteness theorem, if a foundational theory is consistent then it is unable to prove its own consistency. This suggests that there has to be something unobvious in recognizing the consistency of a given foundational theory: unless formally approached by a stronger theory, it requires some kind of philosophy to get confidence in it.
If on the other hand, instead of focusing on the specific cases of "good foundational theories" with possible philosophical reasons for consistency, we want a method valid for the totality of consistent theories, to give proper meaning to the claim that any consistent theory automatically has a model, then as the only possibly suitable systematic method, we must admit having to first spend an eternity in fruitless search for contradictions of a given theory before concluding that it is consistent and thus letting our magic stick create a model of it (actually the proof of the completeness theorem involves an infinite series of such infinitely long checkings to create the model). Because, the consistency property of theories being not knowable in advance but somehow "random" in a sense well-studied by Gregory Chaitin, to dare giving systematic earlier existence to models of every theory that seems consistent just as we could not find contradictions there after a long search, would sooner or later end up also creating models for such theories which will turn out to be inconsistent, as contradictions would finally be discovered later. And that would put us in kind of a trouble.
To give a concrete example (if I don't mistake, otherwise we may need to modify this but the main idea still goes, as can be established by the method of Gödel's speed-up theorem) : if ZF is consistent then adding to it the axiom "There exists a contradiction of ZF with less than 1010100 characters" (rigorously encoded as a formula of arithmetic, making it look undecipherable to the "naked eye") the resulting theory can look pretty much consistent while it actually isn't, since a contradiction can be found by enumerating all strings of less than 1010100 characters and not finding any contradiction of ZF among them, but there cannot be any much shorter formal proof in ZF than this one of that result.
Now, regular mathematicians with their working mathematical Platonism basically don't have any problem packing up such infinite series of operations into their universe of discourse, but why would philosophers of mathematics like to work with that ? In his own words from the E.Britannica article, balaguer "argued that it is only by endorsing this view that Platonists can explain how humans could acquire knowledge of abstract objects". Good luck. He explains further in the second page:
"if full-blooded Platonism is true, then knowledge of abstract objects can be obtained without the aid of any information-transferring contact with such objects. In particular, knowledge of abstract objects could be obtained via the following two-step method (which corresponds to the actual methodology of mathematicians): first, stipulate which mathematical structures are to be theorized about by formulating some axioms that characterize the structures of interest; and second, deduce facts about these structures by proving theorems from the given axioms."Yes, like I did when I undertook writing (but did not finish) a proof concerning a particular variant of the Hadwiger–Nelson problem, which is that no almost-covering mapping of the plane with 5 colors and with the Baire property satisfying the condition of that problem can exist. The study went by first postulating that we had such a mapping with 5 colors, then progressively proving more and more theorems about its properties, until a contradiction would be reached. By the way, where was the wonderful postulate of existence of all possible abstract objects by this "full-blooded Platonism" so useful in this process ? the thread of the argument seems to have been lost here.
"In this paper, we argue that our knowledge of abstract objects is consistent with naturalism. "Since basic math abilities are devoid of mystery and thus obviously compatible with any sensible view, the whole point is void then.
"Naturalism is the realist ontology that recognizes only those objects required by the explanations of the natural sciences."Nonsense. This claim is at the same time vacuous, i.e. tautological (just define the phrase "existing objects" as "whatever is required by natural sciences" = to explain facts observed in the world, and the claim becomes true whatever substantial metaphysical claims may be otherwise stated), and logically unrelated with more usual, substantial conceptions of naturalism (that the fundamental stuff of reality would be some kind of material or otherwise non-conscious stuff), so that it is totally crazy to so over-determine the meaning of "naturalism" by such a priori unrelated criteria. In other words, what would become of naturalism if experiments established the necessity of involving the influence of immaterial spirits as necessary ingredients to account for biological facts: would it proudly accept these as existing objects, or admit being falsified ????
" The problem is that traditional Platonists seem to rely on naive, often unstated, existence principles, such as that every predicate denotes a property (or picks out a class) or that a theoretical description of an abstract object is sufficient to identify it."Indeed there were some naive mistakes by some pioneers of mathematics, who did not understand how the mathematical reality was shaped. Now mathematicians with proper education in logic know that any theory may refer only at first sight (when taken literally) to a unique model, but in fact (looking at it from the outside of the particular theory but still mathematically), it is only a possible model in a range of other possibly existing models (if the theory is consistent). This does not contradict the reality of these models as abstract mathematical systems independent of the material world.
"The logical positivists articulated this worry by arguing that our knowledge is either empirical or logical in nature and that in neither case could we have genuine, synthetic (ampliative) knowledge of nonspatiotemporal abstracta."Hum... depends how we understand logical positivism and the kind of "nonspatiotemporal abstracta" to be considered. It is possible that some logical positivists wrote unclear or mistaken views beyond what I understand as the main and genuine rationality principles usually referred to as "logical positivism". I endorse a kind of light version of logical positivism where in the sentence "our knowledge is either empirical or logical in nature" I understand "logical" to mean "mathematical", as I cannot see significant or clear fundamental difference between logic and mathematics.
"Quine formulated a limited and nontraditional kind of Platonism by proposing that set theory and logic are continuous with scientific theories, and that the theoretical framework as a whole is subject to empirical confirmation."Quine is free to believe what he likes and I may think differently ; I don't like his NF system which I consider inappropriate as a foundation of mathematics, and hardly any specialist in set theory is interested in it either, so why should I care ?
" Most Platonists conceive of abstract objects on the model of physical objects. That is, they understand the objectivity and mind-independenceof abstract objects by analogy with the following three features of physical objects (...) We call those Platonists who conceive of and theorize about abstract objects on this model of physical objects Piecemeal Platonists. Historically, Piecemeal Platonism has been the dominant form of traditional Platonism, for traditional Platonists typically assume that their preferred abstract objects are “out there in a sparse way” waiting to be discovered and characterized by theories developed on a piecemeal basis."Without regarding physical objects as a "model" to conceive mathematical objects, but instead letting mathematics speak for itself, it just happens by coincidence that mathematical objects are rather well described in such ways. Then what ? This is perfectly coherent with the fact that the pioneers of mathematical Platonism and the foundations of mathematics did not immediately come up with the right, perfectly correct versions of all aspects of their respective topics, but corrections and improvements to these views had to be made later to better fit with the actual shape of mathematical realities that could be found after more detailed examination.
"one must assert topic-neutral comprehension principles that yield a plenitude of abstract objects.... Some of these principles are distinguished by the fact that they assert that there are as many abstract objects of a certain sortas there could possibly be (without logical inconsistency)...We shall argue that a Principled Platonism and philosophy of mathematics based specifically on the comprehension principle for abstract individuals is consistent with naturalism. "I'm afraid he has no idea of the extreme discrepancy between what he claims to be doing and what he actually does. In mathematical standards, proving for example that the Axiom of Choice and the Continuum Hypothesis, or their negations, are consistent with the ZF axioms, is far from a small affair. We have to first specify the whole exact logical framework in which we work, then what we need to prove is that in this logical framework, all possible deductions from the given combination of hypothesis will never lead to contradiction (assuming that ZF alone was consistent). It requires a method to somehow anticipate and control the infinite set of all possible logical deductions from these hypothesis which can ever be made in the future (even if the actual method to ensure this does not proceed in these terms but through semantics: constructions of models...). It does not suffice to have a small look at the given statements, see them roughly agreeing on something, figure out a couple of possible deductions and not see a contradiction or disagreement simply popping up, to justify the conclusion that they are actually consistent.
Namely, while I am quite familiar with mathematical logic and of course not afraid of mathematical
notations, the formulas in his principles look like a big bunch of nonsense visibly impossible to
decipher as they are senseless anyway : just like it is possible and even sometimes appropriate to
express some very rigorous mathematics using ordinary language, it is also very possible to express
awful nonsense in mathematical symbols, a trickery I already noticed in some other case of pseudo-science.
I guess he actually does not give a shit about any consideration for mathematical rigor or
consistency in the formal expression of his principles, under the excuse that he is not working
inside mathematics but supposedly expressing some kind of extra-mathematical claim, through
which he thinks all mathematics could be founded on some extra-mathematical basis.
That reminds me the trick of Christian apologetic, that consists in providing bunches of
flawed, pseudo-rational arguments and at the same time claiming to be all about stuff beyond
the mere "flawed" human abilities of reason, as an excuse to avoid the burden of having
to defend the validity of their arguments in the face of rational criticism.
All I can do is skip unclear or possibly nonsensical stuff to search elsewhere for possibly substantial claims. And as far as it seems substantial, his 3rd principle seems totally inconsistent with some basic maths :
"3. If x and y are abstract individuals, then they are identical iff they encode the same properties.Let us take the example of 2 points in an Euclidean plane. Points are abstract individuals. In Euclidean geometry, every point has all the same geometric properties as every other point of the same plane (or of different planes, which we may need to distinguish for some reason). But if for this reason you decide to see all points as identical (a single point) then you collapse the plane to a single point and there is no Euclidean geometry anymore. I know, I know, it is the fault of all these awful school teachers who trained their pupils to lose any idea of the concept of point in a plane and replace it completely by that of oriented pair of coordinates... but why did some philosophers of mathematics never care to grow up from this shit since then ???
" The comprehension principle asserts the existence of a wide variety of abstract objects, some of which are complete with respect to the properties they encode, while others are incomplete in this respect. For example,one instance of comprehension asserts there exists an abstract object that encodes just the properties Clinton exemplifies. This object is complete because Clinton either exemplifies F or exemplifies the negation of F, for every property F. Another instance of comprehension asserts that there is an abstract object that encodes just the two properties: being blue and being round."Hum, in pure mathematics I know of no such property as "being blue". I guess one may safely say that Clinton is neither blue nor round, but can it really be said that Clinton really either exemplifies F or exemplifies the negation of F, for all other properties F as well, such as for example the property of being "complete" (just to pick an example of property named in that excerpt) ? and I once saw a report that different cultures (human languages) give different limits for giving names to colors, therefore raising some translation issues. As for "being round", well there is the property of "being a circle" in Euclidean geometry, but I doubt it may qualify any non-mathematical objects. As for "being round", well, where exactly is the limit supposed to be ? Being round is part of the definition of a dwarf planet. Ceres is considered round enough to be a dwarf planet, but Vesta isn't. I am not trying to start any controversy on these particular cases, but just to wonder how should we cope with these definitions if some intermediate case happened to be found.
Somehow a main idea emerges from all this pseudo-formalism : he starts from "properties" coming as never defined words of common language, but just supposedly qualifying (the heck knows how) some things that appear from the world, but themselves never defined either (as no care is given to analyze any ontology of physical or otherwise supposedly "real" stuff through some study of theoretical physics and choice of interpretation of quantum physics...). Then he defines "abstract objects" as any kind of pack of such properties whatsoever, without any distinction. In particular, under this conception, any pack of mutually inconsistent properties will form some "existing" abstract object just as well. This finally reveals what sense he is giving, in his mind, to the concept of mathematical object : he is conceiving these objects as nonsensical packs of nonsensical properties of supposedly real but never defined or analyzed external stuff. A nonsense over a nonsense over a nonsense. All the opposite, thus, of my above definition of mathematical stuff as clear, exact stuff. I understand: like all philosophers who are not mathematicians, there is no way he can ever admit the possibility for mathematical objects, which he is unfamiliar with (since he is not a mathematician) to be actually making any clearer, more meaningful or more consistent sense than all the more directly visible stuff from nature, which seems more real to him since that is what he is more familiar with.
"Consider a mathematician who at one time accepts the Continuum Hypothesis (CH), and then later rejects it, or consider two mathematicians who disagree about whether it is “true”. We claim that they are thinking about different objects—they just don’t realize it. "Please give names, or are you only talking about abstract hypothetical mathematicians from your imagination ? as I can hardly think of any concrete examples.
In some kind of concluding part of the article he writes
" But the most important ingredient of Platonized Naturalism is the argument that Principled Platonism is consistent with naturalism. Indeed, from a naturalist perspective, how can there be synthetic a priori truths like the comprehension principle? The answer is that there can be such truths if they are required to make sense of naturalistic theories, that is, if they are required for our very understanding of those naturalistic theories. To establish that our comprehension principle is required in just this way, we offer a very general argument that begins with two premises "again with this nonsensical definition of "naturalism" as some kind of epistemological tautology (basic empirical move). The first premise is some kind of triviality (or maybe not but it does not really matter).
"the second premise is simply that the comprehension principle and logic of encoding are an essential part of the logical framework required for the proper analysis of natural language and inference. In support of this premise, we claim that the comprehension principle and logic of en-coding are the central components of an intensional logic that offers the best explanation of ..."yes yes he is just here to boldly claim that his own ideas are the best explanation of everything, and that is his whole argument. Of course like everyone he is free to think that his own ideas are the best, and I understand that he may really believe so, as if he did not then he would have most probably adopted some other ideas that looked better to him instead.
Now more globally, what is are the main structure of that "work" ?
To first make a fuss about one crazy view named "Naturalized Platonism"
developed by some other dumb philosophers ; to point out that this view had some defects
(indeed). Then to present his own, different crazy view (Platonized Naturalism) and to feel
proud reporting that his new view does not display the same defects which the previous crazy
view had. So, just all about 2 crazy views accidentally picked up out of thousands of
possible crazy views (that he does not have himself the imagination to think about, since he
just happens to be so focused on these two), which other stupid philosophers might imagine as
well, each of which having a similar advantage of not having some precise defects which could
be pointed out in some other crazy view.
And what is the motivation of it all ?
We must understand the difficult situation and growing concern which many philosophers of mathematics are facing. While their work is supposed to be all about accounting for the nature, foundations and success of mathematics, it turns out to run into a worse and worse discrepancy with the actual works, practice and views of mathematicians themselves ; growing discrepancy that they are more and more desperate trying to resolve or even account for. This make them eager to dream about and attempt to propose any kind of resolution that looks good. Yet the solutions they can search for, propose and approve, cannot and will never be any real solution, that is solutions that can actually look good in the eyes of mathematicians. Because solutions that can be real or actually acceptable by mathematicians, would be both out of reach and none of their concern. Because their only concerns are to entertain their dreams (illusions) of resolution, and pass peer-review where peers are not mathematicians but peer philosophers, themselves similarly focused in their illusory goals and methods, with their minds deeply rooted in the same general misunderstandings on general ontology, the nature of mathematics and so many mistaken ways of thinking and questioning which philosophers kept entertaining in their long traditions.
And to desperate problems, desperate solutions. In the face of the triumphant working Platonism of mathematicians which they cannot account for nor even decipher (while they do not notice their failure to decipher it as they stay faithful to their traditional misunderstanding of it), the kind of "solution" thus proposed by the tenants of Plenitudinous Platonism may be described as follows : let us take the clothes of the super-nice guys ! We cannot understand the triumphant mainstream position ? Let us just offer a statement of extremism in its support (or at least in the support of what we think it is) ! We could not succeed to criticize the king ? Let us become more royalist than him ! So undertaking to behave as the mathematician's most faithful dogs eager to bark at any stranger without any sense of appropriateness, can anyway help us look brilliant in the eyes of our miserable peer wolves who were becoming desperate at how to resolve their conflict with humans.
"The thesis that set theory is most suitable for serving as the foundations of mathematics is by no means uncontroversial. Over the past decades, category theory has presented itself as a rival for this role. (...) Unlike in set theory, in category theory mathematical objects are only defined up to isomorphism"No they aren't. Category theory is just an approach in which the concept of isomorphism is given crucial importance. And while some category theorists may have considered category theory as a "rival" to set theory, such a view of these as "rivals" doesn't make any sense. Set theory and model theory are both steps in the cycle of mathematical foundations, each one being the natural foundation of the other. The concept of category is also very important somewhere in this cycle. It plays a crucial role, but is no rival. How could it be ? Some people don't like set theory because set theory is usually considered as not very friendly to the concept of isomorphism. But why ? Just because its traditional formalization, ZF, does not accept urelements nor functions as primitive objects, but only sets, and so builds all needed objects as sets, thus sets of sets and so on. However it just happens to be the traditional formalization of set theory, but not the only possible one. Why is it the traditional formalization, which everybody usually keeps following ? Just because it is supposed to be more elegant to have only one kind of objects, which are sets, and let all other objects be built out of this. It is usually supposed to be more elegant, except when it is not. So when it is no more, it can be time to consider a set theory with not only sets. There is no need to throw away the use of some set theory altogether, for the profit of some fanciful presentation of category theory as a starting point of mathematics, which as far as I could see, cannot look like any decent starting point of mathematics at all.
"A theorem may be provable in, say, classical logic, but fail in intuitionistic logic or paraconsistent logic. Because there is a substantial debate over what the ‘correct’ logic is, it follows that such theorems may not be entirely future-proof. Developments in philosophical logic, or indeed in meta-mathematics, may alter the landscape of what follows from what in a substantial way".No there is no such debate, and the issue of what follows from what cannot be altered. Classical logic (first-order logic) is the clear standard reference of what math basically means, in which all theories are normally understood (there is the other reference of second-order logic, where some things become more fuzzy, but this no way undermines the validity of theorems that were obtained from first-order logic). Theories in the ordinary sense must of course assume it to make the sense they are there for. You may of course choose to play nonsense by adopting another "logic" which denies the validity of proofs of classical logic, and consider "theories" in these other "logics" but even if you have the crazy idea of introducing there a "theory" by writing there "the same axioms" as a usual theory normally using classical logic, it won't make any more the sense of the initial theory, so that it is out of subject for what the initial theory was about.
Then the gap is huge between the security levels of maths and metaphysics for the following further ignored or under-estimated reasons