Grundlagenkrise and metaphysics of mathematics

Couldn’t a logicist also be a nominalist?

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?

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.

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.

(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.

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.

[N]umbers are real but not material...

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.

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.)

A wonderful topic, but I suspect that there is too much here for a single thread

proof-path

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?

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.

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).

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.

(Rationalist) philosophers claim that we have a special, non-sensory capacity for understanding mathematical truths, a rational insight arising from pure thought

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

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.

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.

another layer of explanation that this theory would require.

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

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.

A wonderful topic, but I suspect that there is too much here for a single thread

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.

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

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

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?

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,"

Of course you can also trace the emergence of quantity to contradictions inherit in sheer, indeterminate being :grin:

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.

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?

It seems more reasonable to me than the inverse that mathematics was/is invented and that applications for it were/are discovered.

When he says proof-path, is he referring to the syntax which we use to prove theorems?

While its a little antiquated

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.

For example, certain principles or properties involving collections of sets may be more naturally expressed in SOL.

Second-order logic plays a significant role in model theory, which is the branch of mathematical logic that deals with the study of mathematical structures (such as sets) using formal languages.

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

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".

But then the claim "it is not the case that this proof-path pre-exists our construction of it", the syntax being the proof-path, and in our case being the FOL that we see in things such as ZFC, did we really construe relations such as ? and ?? If so, it would then bring up "how did we"?

I'm not following what you say here.

This would be impossible if the contradictory theory was erected on a logical foundation containing the Boolean principle Ex Contradictione Quodlibet ECQ, from a contradiction everything follows. So ECQ has to be abandoned, but fortunetely that proves possible, indeed mathematically straightforward. What remains is a rich field, of novel mathematical applications interesting in their own right, which sidestep the vexing questions of which foundational principles to adopt, by developing contradictions in areas of mathematics such as number theory or analysis which are far from foundations.

In a parallel with the above remarks on rehabilitating logicism, Meyer argued that these arithmetical theories provide the basis for a revived Hilbert Program. Hilbert’s program was the project of rigorously formalising mathematics and proving its consistency by simple finitary/inductive procedures. It was widely held to have been seriously damaged by Gödel’s Second Incompleteness Theorem, according to which the consistency of arithmetic was unprovable within arithmetic itself. But a consequence of Meyer’s construction was that within his arithmetic R# it was demonstrable by finitary means that whatever contradictions there might happen to be, they could not adversely affect any numerical calculations. Hence Hilbert’s goal of conclusively demonstrating that mathematics is trouble-free proves largely achievable as long as inconsistency-tolerant logics are used.

The arithmetical models used by Meyer and Mortensen later proved to allow inconsistent representation of the truth predicate. They also permit representation of structures beyond natural number arithmetic, such as rings and fields, including their order properties. Axiomatisations were also provided.

It should be emphasised again that these structures do not in any way challenge or repudiate existing mathematics, but rather extend our conception of what is mathematically possible. This, in turn, sharpens the issue of Mathematical Pluralism;

Various authors have different versions of mathematical pluralism, but it is something along the lines that incompatible mathematical theories can be equally true. The case for mathematical pluralism rests on the observation that there are different mathematical “universes” in which different, indeed incompatible, mathematical theorems or laws hold. Well-known examples are the incompatibility between classical mathematics and intuitionist mathematics, and the incompatibility between ZF-like universes of sets respectively with, and without, the Axiom of Choice. It seems absurd to say that ZF with Choice is true mathematics and ZF without Choice is false mathematics, if they are both legitimate examples of mathematically well-behaved theories.

Shapiro’s distinctive position has other ingredients: mathematics as the science of structure, and mathematical pluralism implying logical pluralism (on logical pluralism see also Beall and Restall 2006); but we do not take these up here.

The primacy of theories fits, too, with the natural observation that the epistemology of mathematics is deductive proof. It is only if one takes as a starting point the primacy of the mathematical object as the truth-maker of theories, that one has to worry about how their objects manage to co-exist.