Epistemologically is mathematics considered more of an invention, or a discovery?

[Mathochist]

@tim314

A structuralist who is a realist in both ontology and truth value would say:
There is a certain “correct” set of relationships among the natural numbers, for instance the relationships defined by the Peano axioms, and there is some object that fills the “zero” slot, some object that fills the “one” slot, etc., but these objects only exist in so much as they fill those positions. It is meaningless to ask anything about “two” beyond its role in the structure.

More or less. It should be emphasized that realism in ontology implies that the structure of the natural numbers “exists” in the same sense that a traditional Platonist would say that the number 2 does.

A stucturalist who is an anti-realist in ontology but a realist in truth value would say:
There is a “correct” set of relationships among the natural numbers, and if objects exist that fill the positions in this structure, then we can say that they have the relationships given by this structure. For instance, if the correct set of relationships is the Peano axioms, then we can say that if objects filling the positions of this structure exist then they would have to satisfy the conclusions of the theory of natural numbers.

A similar comment goes here: an antirealist in truth value would say that the structure doesn’t exist except insofar as it is exemplified by some instantiation.

A structuralist who is realist in ontology but anti-realist in truth value would say:
There isn’t any “correct” set of relationships, but there really are numbers “zero”, “one”, “two” etc., and if they happen to fill the slots in a structure defined by the Peano axioms, then the conclusions of the theory of natural numbers will apply to these objects.

A structuralist who is anti-realist in both ontology and truth value would say:
We can’t say that anything exists, but we can say that if there is a “zero”, a “one”, a “two” and if they exist in a structure where their relationships satisfy the Peano axioms, then the conclusions of the theory of natural numbers will be true for these objects.

Antirealism in truth value is best illustrated by set theory. Zermelo-Fraenkel set theory can be formally augmented by either asserting or denying the axiom of choice (ZFC: Zermelo-Fraenkel with Choice, ZF!C: Zermelo-Fraenkel without Choice). Both set theories are equally valid and neither one is any “more right” than the other. A realist in truth value would say that though both are logically consistent, the axiom of choice is “really” either true or false.

Even more radically, intuitionistic and orthodox mathematics are equally right. Orthodox mathematics is pretty much what most people work with, while intuitionism is the logical underpinning of the constructivist program. This is a clearer example because orthodox set theory leads to results which are false in intutionistic mathematics (e.g.: the Banach-Tarski paradox), while intuitionism gives results which are rejected by the orthodoxy (e.g.: all functions from R to itself are continuous). Both bodies results on R are seen by non-realists in truth value (there are subtler choices than antirealism) to be equally valid descriptions of the structure R, while a realist in truth value would say that there is only one “real” set of real numbers (only one structure which is the “proper” set of true statements).

In any case, no structuralist would say that the objects exist beyond the role they play in their structure – so questions like “does two contain zero” are meaningless – whereas a platonist would say that the objects are real things on the same level as any other object, and questions that go beyond their role in the structure are perfectly legitimate.

This is an (understandable from what I’ve said) oversimplification of traditional Platonism. Properly, Platonism fails to consistently account for what properties a given object of a theory “should” have. More to the point, if there is only one “real” number 2 and that number “is” (for purposes of argument) the set {{{}}}, then 2 has the property of containing the set {{}} and not containing the set {}. A Structuralist would say that to even talk about the natural number 2 having these properties is ill-founded, since it only exists as a slot in the natural number structure, which has no means of talking about set-theoretic containment.

Is this right? In particular, I’m a bit confused about the “realist in ontology but anti-realist in truth value” position. If in structuralism an object has no existence beyond the role it plays in the structure, then how can you have a real “two” but not a real structure with a “two”-position? Or are there no structuralists who are realist in ontology but anti-realist in truth value?

Realism in ontology but antirealism truth value would boil down to “structures exist, but are not necessarily unique”. This is a pretty scarce area because of a number of thorny issues that arise very quickly. More densely populated is what I’d call realism in truth value but obtainist in truth value. This would say that both ZFC and ZF!C set theory structures exist, but in the “real world” only one obtains, which puts it in a preferred position. Basically (assuming choice holds), theories of ZF!C are just as valid as those of ZFC in the abstract, but there are no models of ZF!C in the real world. Of course, maybe the constructivists are right and neither of these two specific structures obtain. Still, the point is that many structures exist (ontologically realist) but that only one describes the real world (non-realist in truth value).

Traditional Formalism comes closest to antirealism in ontology (the abstract idea of the natural numbers is just an abstract idea) but realism in truth value (every well-formed formula has a definite truth value, independant of the investigator).

Nominalism tries to be antirealist about mathematical ontology, and it may well eliminate “numbers” from science, but it ends up being very realistic. The basic objects are points and regions of spacetime, which end up becoming the place-fillers for a self-defined structure. Those who claim to have eliminated mathematics from physics, even in principle, have a very shallow concept of mathematics indeed. By translating everything into statements about “real” points and regions of spacetime and “real” relations between them, physics itself becomes a structure which, by its very definition, exists. Ultimately it’s realist (in its way) in ontology and truth value.

[Mathochist]

@Mathochist

Antirealism in truth value is best illustrated by set theory. Zermelo-Fraenkel set theory can be formally augmented by either asserting or denying the axiom of choice (ZFC: Zermelo-Fraenkel with Choice, ZF!C: Zermelo-Fraenkel without Choice). Both set theories are equally valid and neither one is any “more right” than the other. A realist in truth value would say that though both are logically consistent, the axiom of choice is “really” either true or false.

Actually, I just thought of a clearer way of saying this.

ZF is a structure. The statement of the axiom of choice is a well-formed proposition in this structure, and is independent of the axioms (neither the proposition nor its negation give a contradiction). An antirealist says that this proposition does not have a definite truth-value within the structure, while a realist says that it does: either the axiom of choice really does hold in ZF or it doesn’t, though the opposite case is not inconsistent.

[erislover]

@Mathochist

Ultimately, I think that it’s becoming clear (to me at least) that the main confusion is terminological (as it is so often). When I say “Platonism” I mean a very definite position. When you say “platonism”, you mean a class of positions which I call “realism in ontology” and which includes my “Platonism” as one of its “various sub-categories and nuances”.

I just mean any transcendental realism. I have to say that I appreciate these last quotes, I’ll chew on them, but I still don’t see it as an alternative to platonism/formalism (which are not really opposites).

Particularly troubling is:

@your cite

The Platonist view may be that one can state the essence of each number without referring to other numbers. The essence of 2 does not invoke 6 or any other number (except perhaps 0 and 1). If this notion of independance could be made out, we Structuralists would reject it. The essence of a natural number is its relations to other natural numbers.

Troubling… “The platonist platonist view may be that one can state the essence… of 2 [without invoking] 6 or any other number (except perhaps 0 and 1).” Troubling to follow that basic reading with the sentence he does.

[WordMan]

I have not read in-depth about the history of this investigation based on the links provided, but find the thread fascinating. My question is:

How is this discussion different from the “if a tree fell in the forest and no one was around, would it make a sound?” discussion?

My point - doesn’t the concept of math being discovered vs. invented rest on the existence of an observer? Just like whether the tree makes a sound depends, in part, on whether there are ears to receive the sound waves made by the fall of the tree, doesn’t the existence of math depend on an observer to observe math truths? So the potential for math truths exist without an observer, but become explicit when stated (whether the truths are correctly stated depends on observer accuracy, bias, etc…).

[Mathochist]

@WordMan

My point - doesn’t the concept of math being discovered vs. invented rest on the existence of an observer? Just like whether the tree makes a sound depends, in part, on whether there are ears to receive the sound waves made by the fall of the tree, doesn’t the existence of math depend on an observer to observe math truths? So the potential for math truths exist without an observer, but become explicit when stated (whether the truths are correctly stated depends on observer accuracy, bias, etc…).

You’ve really hit the nail on the head, though with the wrong end of the hammer. If what you said was correct, then math would definitely be invented. You’re a classic antirealist in ontology, in that these things “don’t actually exist” until someone thinks of them.

[Mathochist]

@erislover

Particularly troubling is:Troubling… “The platonist platonist view may be that one can state the essence… of 2 [without invoking] 6 or any other number (except perhaps 0 and 1).” Troubling to follow that basic reading with the sentence he does.

Well, even in Platonism the specific nature of the numbers isn’t really specified, though it’s assumed to exist. Really, the crux is that Platonists believe that 2 is something, while even realist-in-ontology Structuralists believe that 2 is nothing but a slot in a structure with no existance independant of this structure.

Further, something I hinted at but never really elaborated on earlier: the Platonist says that the number 2 in the natural number system is the same thing as the number 2 in the real number system, while the Structuralist says that the two are different slots of different structures. What can be said is that for any instantiation of the structure R, the objects occupying the nonnegative integral slots (“nonnegative” and “integer” defined in terms of the structure R) with successor s(n)=n+1 (addition defined by the structure R) will be an instantiation of the structure N. Deeper still, given an instantiation of N, one can construct (and, for intuitionists following along, I really mean “construct”) an instantiation of R in parallel with the standard buildup of R in any advanced calculus course.