This thread will be centered around Ernst Snapper's article The Three Crises in Mathematics and around the following SEP articles: Nominalism in the Philosophy of Mathematics, Platonism in the Philosophy of Mathematics, and Platonism in Metaphysics. My personal highlights of Snapper's article are here. This thread connects two different topics and asks many question.

The below is a quick summary for reference, not to replace the reading but perhaps to preface it. This short slideshow "Foundations of Mathematics and Grundlagenkrise" by Vincent Steffan is in fact recommended before all, giving a historical run-down on the foundations of mathematics and the crisis. This webpage (archive) by Eric Schechter is also quite good. This all might seem like a lot, but it is introductory to the metaphysics of mathematics, a topic that is seldom touched in philosophy circles.

The Grundlagenkrise was the 20th century crisis on the foundation of mathematics that came to be when mathematicians saw that Cantor's naïve set theory — an attempt to give a foundation to mathematics — had contradictions, and thus could not be used as a foundation for mathematics. With that, three main schools came about to give a ground to mathematics that was free of contradictions.

What is logicism? Logicism is the idea that mathematics reduces to logic. Meaning, mathematics reduces to theorems that can be proven in first, or higher, order languages¹. If we are taking mathematics as the set of theorems that can be proven from ZF(C)'s axioms, for simplicity sake, the logicist's goal is to show that all nine axioms of ZF are logical propositions.
For the logicist, a logical proposition is a proposition that is true in virtue of its form (syntax), not of its content (semantics), such as P or not-P, where the content of P does not matter.
The problem is that the axiom of choice and the axiom of infinity are not logical propositions, so logicism seems to fail at about 20%.
Main figures are Frege, Whitehead, Ramsey and Russell.

What is intuitionism? Intuitionism is a collection of related claims about mathematics. Intuitionists believe that the beginning of mathematics are the natural numbers, because every human has an innate intuition (immediate awareness) of natural numbers. Indeed, for intuitionists, mathematics is purely a construct of the human mind. For them, mathematics cannot be reduced to logic.
Intuitionism refers to many theorems and laws (such as the LEM) as meaningless combinations of words because they are not made of constructs, which has led mathematicians to mostly reject intuitionism. For them, infinite sets are potentially infinite.
The intuitionists thought that the paradoxes in Cantor's set theory were not the result of incompetent mathematicians (as the logicists thought), but of faulty mathematics.
Main figures are Brouwer and Heyting.

What is formalism? The formalists wished to formalise the branches of mathematics. To formalise is different than to axiomatise. Euclid axiomatised geometry a long time ago. Formalising a theory means choosing a n-order language for that theory. A formalised theory has at least all those ∀, →, ∃, ↔, ∨, ∧, ¬, and also undefined terms. In Euclidean geometry, those undefined terms are line, point etc. In arithmetic, it is 0, +, ×. ZF is the easiest to formalise, as it has one undefined term only, membership (∈). Once the undefined terms are chosen, the theory is formalised, and all its axioms and theorems may be expressed.
Thus, formalists wished to formalise the branches of mathematics to verify they are free of contradictions. However, Gödel's incompleteness theorem arrived, mathematics can't prove that itself is free of contradictions, and the formalist program was recognised to be failed, but that is another topic.
The main figure is Hilbert.

One thing to keep in mind is that all these schools (theories) define their premises using philosophical (aka German/French/English/Hungarian), not mathematical language. Indeed, the "undefined terms" are given in natural language.

1 – A first order language is a formal system that expresses relationships between objects by introducing quantifiers and variables. Take the statement in first-order logic: ∀x(P(x) → Q(x)). First-order languages have ∀, →, ∃, ↔, ∨, ∧, ¬.

Now we want to go on to the ontology of mathematics, and connect it to the Grundlagenkrise. When we say things such as 2 plus 5 equals 7 or the zeroes of a quadratic equation are ±5, what is it that we are talking about? There are different schools of thought regarding that, especially: platonism, nominalism, immanent realism, and conceptualism.

Lower-case platonism (not to confuse with Platonism of Plato) is basically the view that numbers are abstract objects — non-spatiotemporal, causally inert objects. The SEP explains shortly too:

Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. Just as electrons and planets exist independently of us, so do numbers and sets. And just as statements about electrons and planets are made true or false by the objects with which they are concerned and these objects’ perfectly objective properties, so are statements about numbers and sets. Mathematical truths are therefore discovered, not invented. — Platonism in Metaphysics – SEP

Nominalism is a bit tricky to define, but in broad lines it can be summarised into the two types of nominalism: the type that reformulates mathematical theories to avoid commitment to abstract objects, and the type that sets out to prove that no commitment to abstract objects is needed when mathematical theories is used. We can get a clearer picture of nominalism by overviewing the different nominalising programs for mathematics:
a. Hellman's modal structuralism puts that possible (not actual) structures are the object of study of mathematics and interprets mathematics in terms of modal logic, eliminating the need for abstract objects.

In order to articulate this point, the modal-structural interpretation is formulated in a second-order modal language based on S5. However, to prevent commitment to a set-theoretical characterization of the modal operators, Hellman takes these operators as primitive (1989, pp. 17, and 20–23). — SEP

b. Field's fictionalism is that mathematical statements are false, and mathematical statements are given with a fictional operator: "According to arithmetics, there are infinitely many prime numbers". Whereas without the operator, the statement would be false, as numbers don't exist (standard semantics).
c. Azzouni's deflationary nominalism does not require the reformulation of mathematical theories. The central point is that what exists are things that exist independent of our lingusitic and psychological practices; for Azzouni, mathematical objects do not exist independent of them. Yet, we can quantify over objects that do not exist, like we can quantify how many wands Harry Potter has.

I will also let SEP define immanent realism:

A more prominent strategy for taking number talk to be about the physical world is to take it to be about actual piles of physical objects, rather than properties of piles. Thus, for instance, one might maintain that to say that 2 + 3 = 5 is not really to say something about specific entities (numbers); rather, it is to say that whenever we push a pile of two objects together with a pile of three objects, we will wind up with a pile of five objects — or something along these lines. Thus, on this view, arithmetic is just a very general natural science. — Platonism in Metaphysics – SEP

Conceptualism (or psychologism) is the view that numbers exists within only the human mind, they are objects that depend on our psychology, generally to be identified with ideas. Frege gave strong blows against conceptualism in his book "Grundlagen der Arithmetik" (1884).

In the philosophy of mathematics, psychologistic views were popular in the late nineteenth century (the most notable proponent being the early Husserl (1891)) and even in the first part of the twentieth century with the advent of psychologistic intuitionism (Brouwer 1912 and 1948, and Heyting 1956). Finally, Noam Chomsky (1965) has endorsed a mentalistic view of sentences and other linguistic objects, and he has been followed here by others, most notably, Fodor (1975, 1987). — SEP

The SEP article also brings up Meinongianism.

What is Benecerraf's problem? Perhaps the main problem for mathematical platonism, or lower-case platonism in general, is, if numbers are causally inert objects, how could it be that we have any knowledge of them, given we don't interact with them at all?

Field claims that the correlation between mathematical facts and mathematicians’ beliefs is so remarkable as to demand an explanation. He further suggests that such an explanation is impossible if mathematical objects are abstract and acausal. — IEP

Given that platonism postulates the existence of mathematical objects, the question arises as to how we obtain knowledge about them. The epistemological problem of mathematics is the problem of explaining the possibility of mathematical knowledge, given that mathematical objects themselves do not seem to play any role in generating our mathematical beliefs (Field 1989). — SEP

One of the platonist's answer to the problem is full-blooded platonism (FBP). FBP is Balaguer's view that:

According to FBP, consistent mathematical theories are automatically about the class of objects of which they are true, and there is always such a class (where consistency is a primitive notion, and the notion of truth is a standard Tarskian one) — Justin Clarke-Doane, 2016

[...] thus, we might say that FBP is the view that all the mathematical objects which possibly could exist actually do exist, or perhaps that there exist mathematical objects of all kinds.7 (For rhetorical reasons, I will often use the first expression of FBP, in spite of its imprecision.) The advantage of FBP is that it eliminates the mystery of how human beings could attain knowledge of mathematical objects. For if FBP is correct, then all we have to do in order to attain such knowledge is conceptualize, or think about, or even "dream up", a mathematical object. Whatever we come up with, so long as it is consistent, we will have formed an accurate representation of some mathematical object, because, according to FBP, all possible mathematical objects exist. — Balaguer, 1995

2 – The lower-case view of platonism is NOT to be confused with upper-case Platonism, which is simply Plato's philosophy.

Philsurvey2020 on the matter:
https://survey2020.philpeople.org/survey/results/4818
https://survey2020.philpeople.org/survey/results/5030

α: Things that we want to discuss:

a. Can some of these four views of the ontology of mathematics be reconciled with each other?

b. The article associates formalism with nominalism, logicism with realism, and intuitionism with conceptualism. The last one seems uncontroversial, but how true are the first two? Couldn’t a logicist also be a nominalist? Why does reduction of mathematics to logical propositions have to imply numbers as abstract objects?

c. Can a physicalist (or generally naturalists) be a platonist, or should they stick with nominalism or immanent realism? It seems they can't, because commitment to abstract objects seems to be a commitment to non-physical objects, but see for example naturalised platonism (3).

d. Conceptualism: really anti-realist? If we admit that the mind is part of reality, doesn’t research in mathematics equate with investigating our own minds? You might insist that it is still anti-realist because it’s not mind-independent, but the anti-realist label brings a connotation of fiction (not in the sense of fictionalist nominalism). In this case, the question is: does conceptualism really imply some sort of fiction (something we make up like stories, or perhaps useful stories like myths) or implies an investigation of our own minds as an object of study (cognitive science and psychology)? It seems to be the latter, given the fact that conceptualism turns mathematics into a branch of psychology.

e. Benecerraf’s problem: is it the same as the interaction problem of mind-body dualism, or the interaction problem of any not-monist ontology?

f. Full blooded platonism: nominalism with more commitments or nominalism without its problems? FBP defends that every possible mathematical structure exists, does it not seem like some forms of nominalism like fictionalism or structuralism, but with the extra ontological commitment to the existence of those structures?

g. Max Tegmark's Mathematical Universe (a type of mathematical monism) includes the view that every possible mathematical structure exists. Would the Mathematical Universe of Max Tegmark then be a naturalised FBP?

h. Azzouni's deflationary view puts that mathematical objects are creations of our languages and psychology. If so, how is his view different from conceptualism? After all, conceptualism says that mathematical objects exist but are dependent of our psychology, while Azzouni says they don't exist because they are dependent of our psychology — the only difference seems to be the semantics of 'existence'.

i. Modal structuralism puts mathematics in terms of a second-order logic, while logicism seeks to prove that mathematics can reduce to statements that can be proven in first-order logic. Why is the distinction between first-order and second-order so important that these two schools are distinct? Is nominalism trying to explain what mathematics talks about and logicism what mathematics is based on? If so, how does that connect with one using first-order logic and the other second-order? If not, then what is the distinction?
Edit: I have made a mistake with this question. It is not that modal structuralism reduces mathematics to second-order logic, but that modal structuralism expresses itself (neatly) in terms of a SOL, namedly the statements a () and b (). Modal structuralism in fact reduces mathematics to modal logic. So I change my question to:
ι. Modal structuralism avoids ontological commitments while also preserving the truth of mathematical statements by only assuming the possibility of the structures in question. Can we really say the statements are "true" if the structures are not assumed to be real but a mere possibility?

Relevant for FBP, really-full-blooded platonism: https://academic.oup.com/philmat/article-abstract/7/3/322/1440511

3. Naturalised platonism puts sets in the causal order, and says that philosophical claims about them are empirical and subject to reivision (Bernard Linsky & Edward Zalta, 1995).

β: Things that we don't want to discuss: whether there are self-reference paradoxes (especially the so called Russell paradox) and whether there are different infinities (Cantor diagonalisation), there are enough threads on that.

Educated suggestions on improvement of the OP are welcome, as I am not a specialist.