[JunkDNA]

The discipline of science (and its sibling mathematics) has been one of the great achievements of human civilization. It has been so astonishingly successful, we can’t even fathom there being any phenomena beyond its reach, given enough time and enough foundational understanding. Because I like to stretch my mind in uncomfortable directions, I often wonder: is there something else? Something that can lead us to Truth other than mathematics and science? Has human civilization gotten stuck in a local maximum where it appears as though we now have a means for understanding the entire universe, but there is another whole level we are as yet too primitive to reach? I worry that as a species we haven’t spent enough time trying to answer this question.

[yesenadam]

It sounds like you may mean philosophy. You are getting philosophical, being meta..(although I'm suspicious of capitalizing things like 'Truth') Philosophy asks "What is X?" e.g. what is science, how does it work? Tries to understand things. When the answers philosophy gets in an area become complete enough, that area is called science/maths/logic etc.. That happened with science itself, which used to be 'natural philosophy'. Also, if there was some new way discovered for increasing scientific-type knowledge, it would itself become a part of science.

[olooney]

It seems likely that such truths (which are neither mathematical nor scientific) exist, as can be seen from the effort that philosophers of science have put in studying the so-called "demarcation problem." If they could have defined science as "not math" and left it at that, they surely would have. They have some very good ideas in this direction (Popper's falsifiability) but the debate rages on. In particular, it's very hard to answer the question, are there things which are not science, but still true or at least useful?

https://en.wikipedia.org/wiki/Demarcation_problem

Some examples:

1. C. S. Pierce believed he had discovered a third form of thought, distinct from deduction (math and logic) and induction (empirical science). He called it "abduction."

https://en.wikipedia.org/wiki/Abductive_reasoning

2. Certain aspects of philosophy seem to be outside what we consider math or science, such as ethics. The "is-ought" problem is currently open; we basically have no way of giving force to normative statements (you ought/should do a thing, it is right/good/just to do a thing). Kant's categorical imperative is probably the closest approach, but it's hard to accept it uncritically (or seriously entertain Kant's resulting system of ethics, which for example imply you should not lie even to save a life.)

https://en.wikipedia.org/wiki/Is%E2%80%93ought_problem

3. Semantics (study of meaning) also may not be neither math (which can avoid all semantics and adopt a formalist POV) or science (which can also be described as a formal attempt to predict empirical observations, i.e. operationalism and instrumentalism.) For example, the logical positivists attempted to define a verifiability criterion of meaning, but this can easily be show to fail its own test, leaving the question open.