[crimsonalucard]

Philosophy is the arbitrary categorization.

Usefulness has nothing to do logic or mathematics. Math and logic is statements, axioms and theorems. Usefulness is an arbitrary judgement that depends on context. It's "fuzzy" like you say.

Math and logic aren't fuzzy, ever. Categorizing things that are fuzzy and aren't makes sense.

[dogcomplex]

Even if math and logic were fundamentally without fuzziness, their applicability to the rest of the world is inherently fuzzy. Their invention has inescapable, inevitable repercussions which have ethical consequences - so yeah, those need to be considered by someone.

But I wouldn't even be that generous. The amount of inherently human decisions that go into the design, representation, learning, educating, and yes - categorization of math is far from a perfect un-fuzzy logical utopia. We inherently add our own human ideals of what we consider a beautiful (elegant, easily understandable) formula/proof/concept/whatever, and every time we do we affect the perception of subsequent discoveries. Any computer scientist who has developed a sufficiently complex system knows that it is is impossible to make something "purely rational" without making arbitrary categorizations, which another programmer might do completely differently to produce a similar result with far-reaching effects down the line. Likewise, Math as we know it is taught the way it is because of the specific way it formed historically and the way it's best understood by its human students. The fuzziness runs deep.

Maybe you want to escape to fundamental concepts with rigorous self-consistent definitions only, with no applicability to the real world. Just cold hard formulas - the "real math". 1+1=2. Well then I hate to break it to you, but unless you have mathematical definitions of elegance and complexity to replace the human ones, you have no choice but to treat every tautology the same. Even selecting which proofs and axioms we care about is inherently human. So 1=1 and 46332688=3+211+7774+... Oh, and forget base 10... Or for that matter, why we even choose to define particular operators... Or explaining the proofs of any of this in english without some self-bootstrapping rigorous definition of the proof language...

My point is just that to completely remove the human fuzziness you have to cut out a LOT. You have to sacrifice understandability, brevity, and conceptual usefulness, amongst many others. And even if you could make a perfect self-bootstrapping logic box (something famously proven to be impossible by Godel Esher Bach - no system can perfectly define itself, it can only be defined by its parent) then there's still the problem that whatever the discoveries of your self-contained system are, they'll still have an impact on the rest of the world - which we then judge the effects of through Ethics.

Sorry, but truly escaping the fuzzy swamp of human culture is pretty damn tough. Maybe it can be done, but I doubt it. All language is fuzzy, all understanding is fuzzy, representation is fuzzy, the choice to define one thing and not another is fuzzy, Mathematics and Computer Science as educational subjects are very fuzzy still (even as much as they believe they aren't). Fuzziness is the baseline, and the study of that fuzziness is Philosophy. Maybe there exists some pure kernel that is strictly without human fuzziness, but that remains to be seen, and even if it was - it would be very different from the subjects of Math and Computer Science as we understand them today.

[crimsonalucard]

Fuzziness is not baseline.

First off categorization and naming is a human endeavor. The logic that lies underneath is solid and independent of the nomenclature or any formal rigor needed to define the names. We understand math through using words and naming as tools, but the words themselves do not define the underlying structure itself which is as far as we know: Not Fuzzy.

Second our choice of what axioms and theorems to study don't make anything fuzzy. We are simply choosing a subset out of a set of all possible choices in which the subset remains not fuzzy. OUr arbitrary choices are human, but the logic within the framework of our choices still apply.

Godel Escher Bach is just a book. The incompleteness theorem you are seeking comes from Godel. Not only can a system not define itself, it cannot be fully provable and internally consistent at the same time. This does have a lot of interesting things to say about logic BUT it does not make mathematics or logic fuzzy and opinionated.

Fuzziness is not baseline. The baseline is unknown, but what we do know has been consistently observed to be not fuzzy.

Let's be clear, by fuzzy we mean things like ethics and religion. Things that are based on opinion and exclusive to the human experience. None of this type of fuzziness applies to science or math or computer science.

[dogcomplex]

I'm just reiterating now:

Sure, the underlying structure of math and computer science may be entirely logical. But to delve at that, we develop a language of terms and a culture of understanding that is inherently human and inherently fuzzy - which yes, connects to things like ethics and religion. The choices in how we study logic, which areas we care about, how we represent it and how we use it put our fuzzy cultural signature all over it - which is rife for study.

Now, can math and logic be defined without some arbitrary cultural signature? Maybe. But I've yet to see it, and as I've pointed out it would be quite hostile to human understanding - at least until we could define human understanding in logical terms. Also there's the no-self-defining Godel thing (sorry, just meant "made famous in Godel Escher Bach") which I'm pointing out might also imply that you can't really define logic without using our inherently fuzzy language and understanding as a parent.

The baseline I'm talking about is our human understanding (or ability to understand) - which is inherently fuzzy. Fuzzier still when we pass it on through education and it picks up all these unique cultural characteristics.

The baseline you're looking at is the foundational rules of how logic and mathematics work in a defined system, which we can glimpse at through our cultural lense.

Now, can your baseline exist independently of mine? Does it exist at all? I do not know. I certainly won't try to challenge that here - though I know it has been argued before. What I do know is the educational subjects of Math and CSC as they stand today are highly enmeshed with the fuzziness of human historical understanding, rife with choices that are based on human opinion, and pulled from particular moments in time and cultures. Ethics would study those choices and how they affect society. Religious studies might conclude the culture being created has its own similarities to worshippers of the past. We are unfortunately inescapably enmeshed in culture, and the choices we make inherently affect everything else and leave us open to alternative interpretations - we can't just claim we're free of Ethics because we (try to) study foundational logical rules without injecting human opinion. That's simply not true.

Though personally, I believe you're right and there is something unique and universal underlying logic - which probably reveals a fundamental property of the universe. But I think we have to admit at least that there's a strong observer effect where trying to look at it inherently imprints our fuzzy cultural understanding on it - which unfortunately is the only state we have ever really known as a species.

[crimsonalucard]

The observer affect is too abstract to be discussed. Simply because anything can be interpreted to be anything by an observer.

It only makes sense to discuss what the majority agrees to observe consistently and assume that, these observations are true independent of observation.