## Grundlagenkrise and metaphysics of mathematics

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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
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(ZF actually has infinitely many axioms)
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.

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.

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)
[...] 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.

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 (
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☐∀X(X is an ω-sequence satisfying PA's axioms → S holds in X)
) and b (
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◊∃X(X is an ω-sequence satisfying PA's axioms)
). 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).
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Yep, the Zalta from the SEP.

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.
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when mathematicians saw that Cantor's naïve set theory — an attempt to axiomatise mathematics — had contradictions,
Can you sketch simply and briefly what such contradictions might be? Or point to where they are laid out?
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Couldn’t a logicist also be a nominalist?

That is one very deep and dense OP. I'll acknowledge that I'm not a mathematician and that many of the concepts you're discussing are beyond me. I think, of the above, what makes sense to me is intuitionism.

I've become interested in the topic because of the argument that numbers are real but not material, which is an embarrasment to most typical forms of contemporary naturalism, which is dogmatically physicalist. For example:

What is Benacerraf'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?

It's not a problem for platonism, but a problem for naturalistic or empiricist accounts of knowledge. The question is, how can we know numbers, if they are purely intelligible in nature?

There's another article on this topic, The Indispensability Argument in the Philosophy of Mathematics which also leads with Benecareff. It starts:

In his seminal 1973 paper, “Mathematical Truth,” Paul Benacerraf presented a problem facing all accounts of mathematical truth and knowledge. Standard readings of mathematical claims entail the existence of mathematical objects. But, our best epistemic theories seem to deny that knowledge of mathematical objects is possible.

So, what are these 'best epistemic theories' and why do they seem to deny such knowledge? Why, it's because:

Mathematical objects are in many ways unlike ordinary physical objects such as trees and cars. We learn about ordinary objects, at least in part, by using our senses. It is not obvious that we learn about mathematical objects this way.

Doesn't that simply amount to problem for empiricism, i.e. the view that knowledge is attained mainly through sense-perception? So if mathematical knowledge is hard to square with empiricism, then so much the worse for it, I would have thought, considering how 'indispensable' it actually is for science itself.

It goes on:

(Rationalist) philosophers claim that we have a special, non-sensory capacity for understanding mathematical truths, a rational insight arising from pure thought. But, the rationalist’s claims appear incompatible with an understanding of human beings as physical creatures whose capacities for learning are exhausted by our physical bodies.

So instead of questioning why it is we can understand numbers, how about interrogating the claim that we are, in fact, 'physical creatures whose capacities for learning are exhausted by our physical bodies?' Or is that such an important principle in our 'best epistemic theories' that it has to be saved at all costs? That seems the point of the sophistry of the 'indispensability argument'.

So perhaps the 'crisis' is actually a manifestation of a problem at the foundations of naturalism itself, but it's kind of an 'emperor's new mind' type of scenario where nobody wants to admit it. Philosophical dualism and mathematical platonism have no such difficulties. But as it is, and as your cited SEP article on Platonism in Philosophy of Mathematics says,

Mathematical platonism has considerable philosophical significance. If the view is true, it will put great pressure on the physicalist idea that reality is exhausted by the physical. For platonism entails that reality extends far beyond the physical world and includes objects that aren’t part of the causal and spatiotemporal order studied by the physical sciences.[1] Mathematical platonism, if true, will also put great pressure on many naturalistic theories of knowledge. For there is little doubt that we possess mathematical knowledge. The truth of mathematical platonism would therefore establish that we have knowledge of abstract (and thus causally inefficacious) objects. This would be an important discovery, which many naturalistic theories of knowledge would struggle to accommodate.

Oh, and as for the question I quoted - from my limited understanding, Frege, who had quite a bit to say about that, believed in the reality of abstract objects, which nominalism explicitly does not. See Frege on Knowing the Third Realm, Tyler Burge (public domain.)
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[N]umbers are real but not material...
... except whenever they are instantiated.
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Goodness.

A wonderful topic, but I suspect that there is too much here for a single thread - it might have been better to choose one part of the issue at a time to deal with. The juggler will have. a hard time working out which ball is which.

Despite that, I'd throw another ball to the juggler: Wittgenstein’s Philosophy of Mathematics

On Wittgenstein’s view, we invent mathematical calculi and we expand mathematics by calculation and proof, and though we learn from a proof that a theorem can be derived from axioms by means of certain rules in a particular way, it is not the case that this proof-path pre-exists our construction of it.
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Good post, very informative.

Oh, and as for the question I quoted - from my limited understanding, Frege, who had quite a bit to say about that, believed in the reality of abstract objects, which nominalism explicitly does not. See Frege on Knowing the Third Realm, Tyler Burge (public domain.)

I believe that what nominalism rejects is the idea that abstractions may be classed as "objects". So nominalism does not explicitly reject abstractions, it rejects "abstractions are objects".

If i remember correctly, the difference between nominalism and realism follows from a difference in interpretation of Aristotle's active intellect and passive intellect. Aristotle had determined that the intellect must consist of a passive aspect to receive forms from sense objects, and an active aspect to constrict conceptions beyond the limits of particular instances of sensation. Differences in how one "positions" these two aspects "within the world" result in the distinction between nominalism and realism.

Some thinkers wanted to maintain the intellect as pure separable Form, allowing that the intellect transcends the body in the immortality of the soul. That, I believe, is the realist position. This places the reality of the abstraction as non-spatial and within the intellect of the individual, but separable from the passive aspect which receives forms in sensation, and is attributed to the sense organs. Others assumed that the proper "active intellect" is already independent from the individual, as a feature of the collective 'people'. That, I believe, is the nominalist position. This makes the individual's intellect simply a passive intellect, receiving all it knows through sensation, passing that to the society where it becomes "knowledge" through justification.

You'll notice the nominalist representation is the more simple, at first glance, on the first level, dividing the passive intellect from the active through the two evident principles of 'property of individual', and 'property of the society'. It leaves a huge problem though, at a deeper level, as the reality of "the collective" the society, which must now be an entity with properties. The realist representation appears to be more complex because it exposes the problem of the duality of the intellect on the surface, as inherent within each individual intellect, instead of hiding it under the veil of an illusionary object "the collective".

The two render the "position" of the actual existence of the abstraction as fundamentally different. The realist places the abstraction as non-spatial, inside the fundamentally immaterial mind of the individual, while the nominalist places the actual existence of the abstraction as attributable to the society as a whole, and therefore independent from the individual mind. Therefore nominalism leaves the abstraction as a property of an object (the collective society) rather than as an object itself, while realism allows the abstraction to be a special type of object (non-spatial) which inheres within the minds of individuals.

Each has it's own problem. The nominalist has to validate the reality of "the collective", as an object, and the realist has to validate the reality of "the mind" as something within the individual, yet distinct from the physical body.
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It seems more reasonable to me than the inverse that mathematics was/is invented and that applications for it were/are discovered.
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A wonderful topic, but I suspect that there is too much here for a single thread

I had that in the back of my mind, which is why I considered calling it a "mega-thread" from where we could branch out other threads if there was enough interest in a particular subject. But I found that to be overly arrogant and dropped the idea.
The OP is introductory however, and focusing on only one part would leave context out in such a way to harm understanding. It is hard to talk about nominalism without talking about platonism. It had to be done in the whole package, but I hope that the thread will inspire more threads on the philosophy of mathematics here.

proof-path

When he says proof-path, is he referring to the syntax which we use to prove theorems? It seems Wittgenstein falls into nominalism then, which is that we build mathematics.
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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?

It's worth noting that Tegmark justifies his view with the claim that it fixes the Fine Tuning Problem in physics. I do not think this actually works. Basically, if every possible universe exists, then we seem to run into all sorts of undetermination problems and issues that are somewhat akin to the Boltzman Brain problem, although different.

Apparently, my objection is not novel and someone pointed it out to Tegmark, although they did it by only focusing on the problem as respects non-computable mathematical objects. This caused Tegmark to revise the hypothesis such that only computable objects exist. Aside from seeming ad hoc though, I do not think this actually solves the problem of how the MUH is not a good solution to the Fine Tuning Problem.

It is, nonetheless, an interesting idea. I think it's worth pointing out that Tegmark's 3rd (or maybe 4th?) level multiverses are themselves mathematical objects, singular. That is, universes are not composed of multiple discrete mathematical objects, which makes them a bit different than some other forms of platonism/ontic-structural realism.

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.

Right, and if our minds produce such things, we might ask "why do they do so?" Trading off arguments made by enactivists, we can say that brains don't produce any consciousness in most environments. From development to death, there is a constant two-way flow of causes across any supposedly discrete barrier that constitutes a person. If we come to have mathematical intuitions and develop mathematical ideas, we do not do so in isolation, so how does this tie back to the world?

I can imagine all sorts of answers here, and many would not be anti-realist. IMO, it's really not that different from questions as to whether cats, trains, atoms, recessions, communism, etc. all really exist, if they are "mind-independent," etc.

My personal take would be that minds are somewhat unique in being able to use syntax to bring many properties of things together and make them phenomenologically "present" at once. However, this still represents a relation between things in the world, and is in no way a sort of "less real" relationship for involving phenomenal awareness. Indeed, I can see an argument for these relationships being "more real" in that the development of clear knowledge of "what a thing is," is itself the relation in which a thing "most essentially is what it is." In knowledge, many of a thing's properties come together at once, whereas normally any one thing is only manifesting some of its properties over any given interval.

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).

Naturalism seems generally to be defined loosely enough that I don't see an issue here. Consider Pinkhard's argument that Hegel is a naturalist for example. More broadly, it seems like there could be a naturalism that distinguishes between realms of being and becoming, existence/subsistance and actualization, actual and potential, etc., which might leave room for platonism.

Physicalism, if it's in the conventional package of "everything that exists is physical," superveniance, and causal closure seems more dicey unless a number of moves are made.

Gisin's application of intuitionism to physics adds an interesting wrinkle here, but I still think it might be consistent with a certain sort of naturalist platonism. If anything, it would seem to change physicalism more if it ever becomes mainstream.

Most physics theories are deterministic, with the notable exception of quantum mechanics which, however, comes plagued by the so-called measurement problem. This state of affairs might well be due to the inability of standard mathematics to "speak" of indeterminism, its inability to present us a worldview in which new information is created as time passes. In such a case, scientific determinism would only be an illusion due to the timeless mathematical language scientists use. To investigate this possibility it is necessary to develop an alternative mathematical language that is both powerful enough to allow scientists to compute predictions and compatible with indeterminism and the passage of time. We argue that intuitionistic mathematics provides such a language and we illustrate it in simple terms.

https://arxiv.org/abs/2011.02348
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(Rationalist) philosophers claim that we have a special, non-sensory capacity for understanding mathematical truths, a rational insight arising from pure thought

Let's examine it then. If our best epistemic theories are right (our physical bodies exhaust our knowledge), Benecerraf's problem stands. If they are not right (the physical plus the non-physical exhaust our knowledge), there is perhaps no such interaction problem.

Benecerraf's problem is that there is a chasm between the mathematical object and the human knower. Benecerraf's problem claims there is a chasm, and indeed there should be one if our best epistemic theories are right. But not only is there a chasm, but even if one bridges it, one must explain how that bridge works, instead of inserting an ad hoc substance that works exactly for that purpose (god of the gaps); the problem isn't the interaction itself but how interaction occurs, otherwise interactivist dualism would be a widely rejected view, but it is not.

But then let's say we have such non-physical faculty to attain knowledge. There is no guarantee that that non-physical faculty through which we gather knowledge is compatible in substance with the abstract objects we are supposedly interacting with¹ — in a way that it does not generate another interaction problem on top of it. Let's say it is compatible, and as the rationalists say "we have a special, non-sensory capacity for understanding mathematical truths". The problem is still how that faculty works to understand mathematical truths. It seems no one has given a satisfactory explanation. And it seems no one has yet given reasons (evidence) why one should believe that such faculty exists, as explanation for how we come to understand something can be summoned for a great variety of fictional objects, yielding a fictional explanation.

Frege, who had quite a bit to say about that, believed in the reality of abstract objects, which nominalism explicitly does not

True, there are logicists who are realists. The article howevers ties logicism to realism. I am wondering how that is the case, how is it that logicism requires realism? Or is it an inaccurate claim?

So perhaps the 'crisis' is actually a manifestation of a problem at the foundations of naturalism itself, but it's kind of an 'emperor's new mind' type of scenario where nobody wants to admit it.

Well, the crisis is that we are left without a mathematical foundation after the paradoxes of Cantor's set theory :smile:

Philosophical dualism and mathematical platonism have no such difficulties.

1 – As the SEP says on the same article:
The idea of an immaterial mind receiving information from an abstract object seems just as mysterious and confused as the idea of a physical brain receiving information from an abstract object.

And this solution of an immaterial mind interacting with abstract objects then brings the issue of the integration of the physical (sense-perception) with the immaterial (reason), how we are able to mix mathematical knowledge with empirical knowledge when building scientific theories — which ultimately is just the mind-body interaction problem, which is beyond the scope of the thread, but it is another layer of explanation that this theory would require.
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Also quaint how I used a quote from this same https://plato.stanford.edu/entries/platonism/#5 article to prove a point in a mostly unrelated thread one day before I posted this thread https://thephilosophyforum.com/discussion/comment/890378 :razz: I had forgotten
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another layer of explanation that this theory would require.

I think the confusion revolves around the equivocation of the term 'object' in 'mathematical object'. Numbers are not actually objects in any sense but the allegorical. There is no object '2'. What there is, is the act of counting which is denoted by the symbol '2'.

Step back a little. You may recall an allegory that dates back to the 19th century, about how a two-dimensional surface of water would perceive a three-dimensional cone. Absent any idea of a third dimension, the surface could only ever perceive the cone in terms of concentric circles //and ellipses//. Then there'd be this big conceptual problem as to how all these circles are related. You could write an equation, presumably, that described the progression of circles, but you still wouldn't be seeing what they represent.

I think there's something analogous to that operating in this argument. The cone analogy is a metaphor for the limitations inherent in empiricist understanding (akin to pistis or doxa in Plato's terms), which cannot by itself grasp the sense in which mathematical knowledge is 'higher'. There is no higher! There's only the flatland of empirical reality mediated by physical senses. Just as the two-dimensional surface perceives only the cross-sections of a cone, without a conception of the three-dimensional object itself, empirical observation alone cannot fully comprehend the abstract realities that Platonists argue transcend physical particulars. This analogy underscores the empiricist critique of Platonism, suggesting that notions of 'higher' or 'abstract' truths, such as mathematical entities, are outside the domain of of empirical verification and as such cannot be explained in those terms.

And the diagram you provide illustrates the problem, as it's two-dimensional. I think that what happens in reality, is that rational inference (including counting) operates on a different level, but in concert with, sensory cognition (per Kant). Whereas the diagram seeks to treat them in the same way, that is, as objects, and then asks how they're related. It's a category problem, which ultimately originates in the 'flattening' of ontology that occurs with the transition to the modern world-view (hence the relevance of the 'flatland' argument.) Hence, it's a metaphysical problem, but as the proponents of empiricism are averse to metaphysics, they of course will not be able to acknowledge that.

The problem is still how that faculty works to understand mathematical truths. It seems no one has given a satisfactory explanation.

Mathematics is what explains. We don't need to explain arithmetical primitives, they are what provides the basis of explanation. Trying to explain mathematics by reducing those faculties to the sensory is the source of the problem.
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While its a little antiquated, I think a good starting point is the manner in which Frege considered it in The Foundations of Arithmetic. Philosophers from Mill, to Kant, to Frege were wondering two things:

“Are mathematical statements analytic or synthetic?”
and
“Are mathematical statements a priori or a posteriori?

Empiricists/realists like Mill thought we learned the meanings of mathematical statements by generalizing from experience. They are a posteriori. We see things in different groupings, and we generalize from there.

Both Kant and Frege disagreed with Mill, both thought that we learn the meaning of mathematical statements without any recourse to reality. Frege, to put it simply thinks it would be bizarre to suppose that my understanding of the number:

“6915027410”

is in any sense based on my experiences. To what experiences could I possibly point? This is true of mostly any and all numbers and combinations of numbers you might suppose outside the first few small numbers.

Kant thought mathematical statements were synthetic a priori and Frege thought that mathematics was analytic a priori. This is an important distinction, because it sets up the difference between psychologism vs logicism; the latter which, like empiricists hold mathematical statements to be mind independent facts versus psychologism which holds them to be mind dependent.

Kant thought, for example, that in the proposition:

“4+2=6”

Kant said that nowhere in the idea of “4”, “2”, or “2+4” is the concept “6” found. In order to show this, consider the following proposition:

“38102465 – 91042 = 38011423”

It’s obvious, that the random number “38011423” is in no sense containing the other individual numbers: “38102465”, “91042”

Then how do we know this statement is true? Kant's answer was that the necessary content was supplied by the mind. By the pure concepts supplied by understanding.

Frege argued that Kant misunderstood what the distinction between analytic and synthetic rested on. It was not based on the content of proposition, or the individual term's meanings, but rather, it related to one’s justification for accepting the truth of the statement.

Frege agreed with Kant, that the number “6” did not contain “4” or “2” or any number that can be added or subtracted to equal “6”, but we none the less know analytically that “6” must be the result of “4+2” because whether or not a statement was analytic or synthetic was based on whether or not it could be defined.

Mathematical statements are true analytically because they are provable, according to Frege. So, even if humans all died out, or never existed, the “Pythagorean theorem” will always be provable.
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And the diagram you provide illustrates the problem, as it's two-dimensional. I think that what happens in reality, is that rational inference (including counting) operates on a different level, but in concert with, sensory cognition (per Kant). Whereas the diagram seeks to treat them in the same way, that is, as objects, and then asks how they're related. It's a category problem, which ultimately originates in the 'flattening' of ontology that occurs with the transition to the modern world-view (hence the relevance of the 'flatland' argument.) Hence, it's a metaphysical problem, but as the proponents of empiricism are averse to metaphysics, they of course will not be able to acknowledge that.

It may be argued that the most fundamental use of numbers is in ordering, first second third, etc.. In this sense the number cannot be an object at all, but a place marker in a hierarchy or perhaps a temporal order. The hierarchy is a value structure, and it may be asked what it means to mark a position in such a structure. Even the question of what it means to mark a place in a temporal order is perplexing.
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Of course you can also trace the emergence of quantity to contradictions inherit in sheer, indeterminate being :grin:

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I recall an article about how geometry began in Egypt - obviously the construction of the Pyramids required advanced geometry but well before that it was used to allocate plots of farming land on the Nile delta. It will be recalled that this floods every year and the boundaries are erased, so every year the plots have to be allocated anew along the sides of the river-banks, which required sophisticated reckoning. The origins of arithmetic were likewise associated with the Babylonian-Sumerian culture - the ‘cradle of civilisation’ - for reckoning cattle numbers and storage of crops. You can see how arithmetic generally would be associated with ownership, building, cultivation of crops and conversely how it would not be of much relevance to hunter-gatherer cultures.
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A wonderful topic, but I suspect that there is too much here for a single thread

I agree. Logicism itself could be a huge thread itself on its own. I have 2 books by Dummett on Frege, and they are over 1000 pages. I have about 10 different books on Hilbert space only.
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I recall an article about how geometry began in Egypt - obviously the construction of the Pyramids required advanced geometry but well before that it was used to allocate plots of farming land on the Nile delta. It will be recalled that this floods every year and the boundaries are erased, so every year the plots have to be allocated anew along the sides of the river-banks, which required sophisticated reckoning.

Yes, I read similar information. I was researching the origins of the Pythagorean theorem, and "the right angle" and found that the general principle (probably the 3,4,5 triangle) was being used by the Egyptians to produce parallel lines for plots of land. In one of Plato's dialogues, in a sort of myth form, he talks about the movement of ideas from Egypt to what is now Italy. I believe it was part of Greece then.

The point I was making is that it may be the case that numbers first arose simply for the purpose of ordering, priority, before and after. Notice that when we learn to count, at a very young age, what we learn is an ordering of the symbols, count from one to ten for example. We learn to count (order the symbols) before learning about quantity. I believe this is why numbers are known as objects.

There is nothing else for the symbols to represent, because they do not represent a quantity of things. 1 is before 2, and 2 is before 3. To understand this meaning in terms of hierarchic priority, or even temporal priority would require very difficult conceptualization for a child learning how to count, so we simply say that "1" stands for a number, which is before the number represented by "2" etc., or "lower and higher". Employing these objects (numbers) allows us to avoid understanding what before and after, lower and higher, actually mean, leaving that to be determined by the context of application. The principal application became lower and higher quantity. So, by saying that the numeral represents an object (a number), and that the objects may be ordered, allows that "order" be left undefined, for unlimited applications.

But the object, the number itself is actually redundant. The symbol may directly represent the quantity or order of the application without the requirement of the object as a medium. The object was only employed to facilitate understanding the usage. So the idea that there is an object (a number) between the symbol and the thing which the symbol represents in usage, is a sort of self-deception. This self-deception becomes very misleading when it becomes an axiom of set theory.
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if every possible universe exists, then we seem to run into all sorts of undetermination problems and issues that are somewhat akin to the Boltzman Brain problem, although different

That is true. Speaking of the Boltz, we had a leeenghty quarrel about this on the thread "Reasons to believe in the existence of the world", starting here, a few pages later the multiverse of probability is brought up and imo it is a very troublesome thought experiment.

This caused Tegmark to revise the hypothesis such that only computable objects exist.

This makes it appear that Tegmark's original theory was a sort of naturalised really-full-blooded platonism and later was changed to a simple naturalised full-blooded-platonism. If "computable object" here has some relation to consistency.

RFBP—really full blooded platonism—can do the trick just as well, where RFBP differs from FBP by allowing entities from inconsistent mathematics.

If we come to have mathematical intuitions and develop mathematical ideas, we do not do so in isolation, so how does this tie back to the world?

Good point, this feels like yet another troublesome question that arises from psychologism. A question whose answer might be a defeater for psychologism.

it's really not that different from questions as to whether cats, trains, atoms, recessions, communism, etc. all really exist, if they are "mind-independent,"

But then again, prima facie there is nothing necessary about the idea of cats, protons, or communism. It could be that numbers are innate ideas, being then "world-independent".
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Of course you can also trace the emergence of quantity to contradictions inherit in sheer, indeterminate being :grin:

Where is this from?
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The Stanford Encyclopedia article on Hegelian Dialectics, which is quite good too. I can see why people found similarities between Spencer Brown's "Laws of Form" and this. There is some interesting work on formalizing Hegel's Logic using category theory but it's sort of over my head. I would assume that if it can truly replicate the idea of can overcome many limits to formal systems in that contradiction is no longer an issue, but the very engine of changing the system itself.

There seems to be an interesting link between Hegel's Logic, Brown, information theory, and St. Denis/Eriugena I want to write about some day if I ever feel like I understand them well enough.
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I have about 10 different books on Hilbert space only.

Really? You must be a mathematician like I was. And one working in functional analysis. I have perhaps four books that speak of Hilbert spaces in certain chapters.
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Really? You must be a mathematician like I was. And one working in functional analysis. I have perhaps four books that speak of Hilbert spaces in certain chapters.

No I am not. :) I was trying to learn about it better. I kept getting more books on the topic until I find the best one. Ended up with about 10 of the books in the shelf. The last one I got was "Introduction to Hilbert Spaces with Applications" by Lokenath Debnath and Piotr Mikusinski 3rd Edition. It seems the best one so far. But the N. Young and Holland Jr. ones seem also good.
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Since you are an in-house mathematician, I will kindly ask you to take a look at question i. Of course a mathematician is not the same as a logicist, but it is as close as it gets.
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?
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sorry.just lost the internet for about a week.
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Plenty of time to think!
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First of all my complements on a very well constructed OP. Anyway, this is sort of a whimsical notion, but I'm seeing a sort of a poetic analogy between this and discussions of Quantum Physics. Just as a photon is either a wave or particle depending on how it is measured, it seems like these difference in math philosophy may all be neither wrong nor right - it depends on how the topic is approached.

Don't take this too literally :roll:
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OK, back unexpectedly quickly. What little I know of SOL is that it applies to set theory, and foundations wallows in that mire. I suspect if one wants to see how it translates into common mathematics one must start with 0={ }, and work up to sophisticated functions. I did this back in 1962 in my first grad math class. It was done on handout sheets, and we ended up defining the exponential function.

It was entertaining at the time, but not at all useful to me or 97% of the class. One guy went ecstatic and followed this dream. The rest of sighed relief and moved on.

Functions from sets might help.

Oh, here is an example of a Hilbert space: A Hilbert space
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It seems more reasonable to me than the inverse that mathematics was/is invented and that applications for it were/are discovered.
Same here.

When he says proof-path, is he referring to the syntax which we use to prove theorems?
Pretty much. So mathematical expressions are true only if there is a proof-path that shows it to be true. There are, one concludes, mathematical expressions that are neither true nor false. This is opposed to Platonism, in which mathematical expressions are either true or false regardless of our having a proof.

Arguably this approach is not subject to Benacerraf's problem because there need not be a thing to which each number refers.

Wittgenstein's own approach place restrictions on the creativity mathematicians.

But this topic requires a far more formal approach than is doable here.
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