A friend of mine once discussed his experiences as a student of physics. He majored in physics as an undergraduate, is currently working at a national laboratory, and is on his way to graduate school - so I'd say he knows at least a little bit about things. So one day he remarked to me that he was starting to realize that the point of most physics classes is to keep you from figuring out just how much you don't know, how much you've yet to learn - because if you ever figured that out, you'd just give up then and there, knowing that it would be impossible to ever learn it all.
Once upon a time, I saw a student ask a famous professor of philosophy a question. The student began by discussing his experience reading Wittgenstein's Tractatus Logico-Philosophicus. He said that as he read, he felt all his concepts slipping away from him, that everything he'd thought he'd known and understood was just evaporating - it was tremendously disconcerting. Good, good, the professor said - that's exactly the experience you're supposed to have. No no! the student replied. It was a terrifying experience, and he's not sure that he can handle facing it again. Well, the professor said, philosophy isn't for everyone.
My own experiences of philosopy have been similar. When I first began, I had strong opinions about everything, and would argue them against all comers. But, as time passed, I started to realize that my arguments weren't always that good, that my conviction that I was right preceded the arguments I made. And so, as time passed, I felt knowledge slipping away from me. It's been a long journey, probably incomplete, towards realizing just how much I don't know and don't understand.
I got to talking philosophy with a friend of mine the other day. At the end of the discussion, they remarked that our conversation had brought up a lot of issues that she hadn't thought about since high school - and what was odd is that she realized that she hadn't gotten any further since then towards figuring them out. Realizing just how much there is to be known is an experience that, she said, she lusts after.
Sooo... what is the point of all this? Nothing other than to recount some stories about ignorance - about what it amounts to and the attitude we can take towards it.
* If you wanted, you could discuss the difference between some of these stories, possibly, through the lens of the distinction adumbrated in the following paper (which is excellent, by the way).
"Varieties of Skepticism"
by James Conant
in Wittgenstein and Skepticism
edited by Denis McManus
New York: Routledge
2004
1 comment:
Here's my experience:
When I first took Honor's Calculus, we went over mathematical fields. In math, fields are a special kind of set with two binary operations defined. For example, the real numbers are field with addition and multiplication as the two binary operations.
Fields are also required to have other special properties. For example, both of the operations in the field must be associative and communicative, and both must have multiplicative and additive identities (for the real numbers, these are 1 and 0).
Ok. Our teacher introduced these concepts, with the real numbers as an example, and all was going well. Ok... addition is associative - I learned that in elementary school... blah blah blah.
But the next thing that the prof showed us confused me. He said: now consider the following field:
1. The field has only two elements ('numbers', if you will) - 1, and 0.
2. The binary operations on this field, 'addition' and 'multiplication' are defined as follows:
1 + 1 = 0
1 + 0 = 1
0 + 1 = 1
0 + 0 = 0
1 * 1 = 1
1 * 0 = 0
0 * 1 = 0
0 * 0 = 0
I didn't get it, initially. On the one hand, this kind of looks like ordinary arithmetic - anything multiplied by zero equal zero, zero plus anything results in the same thing...but wait a minute... how can 1 + 1 = 0?
And why is it necessary to specify that 0 + 1 = 1, when you've already said that 1 + 0 = 1.
I didn't realize the obvious until I got home (at the time, home was Max Palevsky): He said: "With addition and multiplication DEFINED AS FOLLOWS:"
DEFINED.
I spent the rest of the day scrubbing bits of my brain from the walls. Multiplication and addition, as we know them, are JUST DEFINITIONS WHICH HAPPEN TO SATISFY CERTAIN CRITERIA. One can define OTHER kinds of 'multiplication' and 'addition' which also satisfy these criteria. There's no REASON why 1 + 0 has to equal 0 + 1, unless we define it as such. One plus one can equal zero, if we define it as such.
In fact, it's deceptive even to call these things 'addition' or 'multiplication' - we are better off calling them operation A and operation B.
And what's more there are also operations which DON'T satisfy the requirements of a field (that is, they aren't commutative, etc). And you can define new kinds of mathematical entities (new 'numbers', if you will, although it's again deceptive to even call them that) based off of these operations.
Alternatively, you can define MORE requirements which the operations have to follow, and get yet other kinds of mathematical objects. You can have more than two operations. Hell, you can have as many as you want, each defined in it's own way.
Basically, what I realized is this:
ALL of the math that I learned in school up to that point described ONE PARTICULAR KIND of mathematical object, defined ONE PARTICULAR WAY: The real numbers, with a euclidean metric. But this one set is only an incomprehensibly small piece of a much larger (in fact, infinite) mathematical universe. A universe I knew absolutely nothing about.
It's like being a sheriff in some backwater town, and all of a sudden discovering the NYPD. Or having seen only one book in your life and walking into a library.
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