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
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).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
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
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
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
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
Couldn’t a logicist also be a nominalist? — Lionino
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? — Lionino
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.
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.) — Wayfarer
A wonderful topic, but I suspect that there is too much here for a single thread — Banno
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 — Wayfarer
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. — Wayfarer
Philosophical dualism and mathematical platonism have no such difficulties. — Wayfarer
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. — Lionino
The problem is still how that faculty works to understand mathematical truths. It seems no one has given a satisfactory explanation. — Lionino
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. — Wayfarer
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. — Wayfarer
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 — Count Timothy von Icarus
This caused Tegmark to revise the hypothesis such that only computable objects exist. — Count Timothy von Icarus
RFBP—really full blooded platonism—can do the trick just as well, where RFBP differs from FBP by allowing entities from inconsistent mathematics. — https://academic.oup.com/philmat/article-abstract/7/3/322/1440511
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? — Count Timothy von Icarus
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," — Count Timothy von Icarus
Of course you can also trace the emergence of quantity to contradictions inherit in sheer, indeterminate being :grin: — Count Timothy von Icarus
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. — jgill
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? — Lionino
Same here.It seems more reasonable to me than the inverse that mathematics was/is invented and that applications for it were/are discovered. — 180 Proof
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.When he says proof-path, is he referring to the syntax which we use to prove theorems? — Lionino
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