What does it mean to explore the "ill-defined"?

We think it means something tantamount to “engaging with concepts or problems that lack clear margins or precise definitions.” Okay… But what does it mean to “engage” with them? Is it to think about them? To talk about them? To take them “seriously”? To “wrestle” with them?

Firstly, recognize this: the world is replete with intractable anomalies, so much so that ostensible inconsistencies are a banal norm. Just consider the glut of pop-sci references to “we don't know what 90% of the universe is made out of!” or the more sophisticated muon g-2 stories, or how many people reference Gödel's incompleteness theorem (albeit often without knowing its formal meaning, nor that there's a second incompleteness theorem, nor that there's a “completeness” theorem to boot).

Intellectuals like Neil deGrasse Tyson hold ill-defined and esoteric ideas in contempt. He's not the only one; I used to as well. It took several years for me to realize that this came from a place of orgulousness and a longing for the image that such a contemptuous stance toward religious affairs projects, rather than from a place of axiomatic reasoning, which I believed I was operating from. I'm deeply skeptical of people who haven't realized this about themselves as I know from experience how delusive it can be to think one is transacting a rational stance.

Generally speaking, we care far more about our position on a social echelon than almost anything else.

Philosophy tends to be sold as being split into the analytic and continental sort. The analytic are those of Russell, Moore, and the early-Wittgenstein. The continental are those of Kant, Nietzsche, and (arguably) the late-Wittgenstein.

The difference is not only in problem-solving but also in problem-defining. That is, how does one define their terms, if terms are defined at all?

One way to solve imprecise problems without making them precise is by “wrestling with it.” The willingness to “wrestle with” the ill-defined is one advantage that continental philosophers have over the analytical types, who see this as an unfortunate misstep that protracts “progress.” However, most problems in life aren’t of the mathematical or logically-stated-puzzle-kind, where each premise is made explicit or indubitable. Instead, they’re amorphous, turbid, and utility-laden, such as:

• Negotiations on what to pursue next in a startup

• Familial or friendship matters

• What so-called “tasty” meal to cook for dinner

• How to “maximize” one’s goals without even knowing what those goals specifically are

We all sip from the ill-defined’s brew. You notice this when you’re arguing with someone and you say, “wait, can you tell me what you mean by that term?” and then they irascibly fumble out, “well, look it up on Google.” Okay… but you’re using the term! It’s a tacit admission that they don’t know what they’re talking about, else they would have said so, as one-upping the interlocutor in this game of cerebral sparring is vital to most.

Technically, “well-structured” problems are those with well-defined inputs and outputs. The majority of the problems we encounter aren’t of this type. This isn’t even the case for most research problems, as defining the problem is a large part of solving it.

There’s an adage in math: “make your definitions complex and your proofs simple.” But what does this mean?

It means spending a significant amount of time thinking about the objects you want to manipulate and to do so simply, rather than manipulating what you already know in a convoluted manner. If I give you a bottle cap, a piece of paper, and a tooth, you can only create so much. Sure, you could create something that would be lauded by critics as a trenchant commentary on Duchamp, but other than avant-garde art that only has legitimacy because someone calls it “art,” there’s little that can be done. But if you’re given screws, a screwdriver, and some plywood, suddenly an entire world of construction opens up to you. This is one reason professor of mathematics at the University of Toronto (where I did my undergrad) Dror Bar-Natan told me that the whole “theorem / proof” emphasis of undergraduate and graduate studies is misleading.

Definitions are epochal. You learn this as a researcher more and more.

However, intellectuals will often repudiate some line of thinking about X as:

• “I don’t know what it means to X…”

• “I don’t know what the meaning of X is…”

• “X is ill-defined, so talking about it is useless…”

They’re fond of declaiming that they “don’t know what it means” when asked about a concept that potentially pokes holes in their theories or worldview. Not having an answer to such a fundamental question makes them appear foolish, and foolishness is the last thing an intellectual wants to be associated with publicly.

Questions such as “what exists?” or “what’s fundamental?” or “do we have free will?” may be met with a quizzical yet patronizing “I don’t know what such a question even means!” However, not knowing what something means is entirely different from saying, “I don’t have an articulation of that phenomenon that’s consistent with what I deem true (or at least, with what I state to deem true).” Do you dismiss the phenomenon of terror, hate, rancor, or despair simply because you can’t define it other than the indistinct and equivocal claim that it's “one-to-one” with neurological activity?

An intellectual would sooner reject their own existence by suggesting, “I don’t know what it means to be conscious,” than to admit that there’s something rational about the wrestling with the irrational. Technically, a purely “objective” view of science denies one’s own consciousness (if one takes the objective as metaphysically totalizing) since facts are meant to be subject-independent (while paradoxically having intersubjective agreement is part of the scientific method), which means your mind existing isn’t a fact (or all of your experience). They deny it not because of a commitment to truth or uncovering reality but because:

1. They have some commitment to eschew what seems supernatural or mystical, and they see accepting consciousness as accepting libertarian free will, which is a slippery slope.

2. They’re so ensconced in their powerful field that when they see what can’t be explained by it, they do away with the problem by saying it’s ill-defined (which it is, but the primary motivator is to retain magistracy in the field they’re comfortable with). This is what people refer to as the Streetlight Effect.

3. Perhaps there’s a Dostoyevskian angle that man would commit to the material world so much as to even deny themselves. It’s man that puts out his own light. Hell is a prison locked from the inside.

Edward Witten said that the beauty of mathematics is like the beauty of music; you can’t explain it, you can only experience it or feel it. No more so than you can ask someone to put music into words, while music is beautiful, could you state why some mathematics is beautiful. This is an appeal to intuition, as he and others have said they allow beauty to guide them—sometimes even in direct conflict with rigor and current experimental fact.

Polysemy is the whetstone of mind.