Is there a conflict between self-reference and ontology? (In relation to mathematics)

Question

I am a total layman when it comes to math, but I promise at least to clearly spell out my thought process.

1. Some like Elaine Landry say "mathematics is not metaphysics" https://youtu.be/JiLUiEmhAJc. If I understand her at all (I probably don't), Hilbert helped show the axiomatic method is about organization rather than fixing itself to any prior structures.

2. How can something be completed isolated from outside and have content unless it utilizes some form of self-reference or self-organization?

3. Math is almost completely isolated, or certain parts are completed isolated. The axiomatic method came from natural langauge and metalangauges, but is largely self-organizing now.

4. Ontology is a type or subfield of metaphysics. If math isn't metaphysics probably isn't ontology either.

Taken together, it seems like anything with or focused on self-reference/self-organizaiton is necessarily ontologically agnostic. Is there is such a conflict/exclusion, and is that the desired state of affairs? Is self-reference necessary to be ontologically agnostic?

Answer 1

it seems like anything with or focused on self-reference/self-organizaiton is necessarily ontologically agnostic

I can see how it may seem that way, but this is not necessarily so. Something, in general, can be self-referential without being only self-referential. We can consider some concrete examples as follows.

1. This statement is true, and there exists an integer x such that x > 1.
2. This statement is true, and Paris is in France.
3. This statement is true, and Paris exists.
4. I perceive myself to be hungry, and 1+1=2

The first part (clause) of each of these statements references the statement itself (exhibiting self-reference), while the second part provides external, new information that is either explicitly ontological (1 and 3), or implicitly (under “standard” interpretations, as in 2 and 4).

The first statement can be rigorously defined in formal mathematics.

We can also ask what does it mean for mathematics to be “isolated”, yet organic creatures can do mathematics? What does it mean for mathematics to be “isolated”, yet it is the language of the sciences for communicating about the empirical world? It doesn’t seem like mathematics is isolated. Even formally and rigorously, mathematics still requires logic.

Answer 2

Ontology can be 1) a discipline of knowledge, 2) the set of entities targeted by a branch of knowledge. The former is not really the problem here, but the latter, the ontology of mathematics.

A formal ontology, in a specific domain (case 2) is necessarily a formal system (see Wikipedia: formal system), where (domain) concepts and axioms are specified. Axioms essentially define relationships between concepts of the domain, but such concepts are never entities defined from scratch. So, they are never isolated.

Even if it is said that "a gwigwi is a wugwug" (a definition from scratch), it is assumed that both are objects, that both have boundaries, so, that they can be counted and organized in groups or systems. For a domain ontology, such definition already presupposes a subjective bias (e.g. arbitrary time and space scales, in addition to the human capability of bounding, of discriminating portions of nature one from the other, even if all nature is made of atoms interacting, where boundaries don't exist), and a dependency (preventing a possible conceptual isolation). So, both terms are not isolated concepts.

Domain concepts have strong dependencies on language, ergo, on subjective concepts. In addition, even if we consider that the object exists per se (e.g. according to Aristotle's approach of form and matter, circles would exist in nature as pure forms, and apples would be kind of evolved version of circles, expressed with matter), then in such case, perhaps, self-references would be possible: "for a gwigwi to be a wugwug, a wugwug must be a gwigwi". But no entities in mathematics are defined in such way. But if it's said that "a gwigwi is a flower that...", then the concept is useful; but the domain of knowledge becomes dependent on the domain of knowledge of flora.

Notice that in final terms, the dictionary is a circular set of references. Although it sounds funny, in order to understand words in the dictionary, recent life experience as a human is required. Ergo, not even the set of words in the dictionary is isolated. Words in a language depend on other languages, onomatopoeia, feelings, movement, a specific capability of memory, ability of inference, intuition of causality, etc. Kant: concepts without intuitions are empty.

Therefore, the ontology of mathematical objects can't be independent (isolated) at all. It is largely dependent on multiple human subjectivities. Moreover, it is interdependent on ontologies from other domains of knowledge, which might not necessarily be formal.

Answer 3

Mathematics without any meaningful content would be a diarrhoea of pure formal logic: φ ∧ ψ ⊢ φ ∨ ψ. As a matter of empirical fact, mathematical concepts are all profoundly meaningful. Mathematics should be regarded as the application of logical reasoning to whatever concepts we fancy applying logic to. Arithmetic and geometry demonstrate that these concepts are all inspired by our experience of the physical world. Even pure logic is self-evidently about the real world. A logical statement is supposed to be either true or false, meaning true or false of the real world, even if, in formal logic, the terms used do not actually refer to anything.

Logic does not bring any ontology into mathematics. Any mathematical ontology has to be introduced in the mathematical reasoning through the premises.