This chapter more than many of the others connects mathematics with other disciplines of thought and perception in a direct way. Having a glimpse into how the perceptions of space and the world (both the physical world and the world that exists in our minds and souls) has been connected with mathematics over the years is very interesting. For me, mathematics seems like one of the only things that has absolutes / is an absolute that we have. Chemistry and biology are plagued with the inability to travel deeper and deeper and really feel how things work; physics is also stuck with both uncertainty and the inability to travel to the deepest level. The more intellectual / soulful based sciences (philosophy, theology, sociology) are so perspective dependent that uncertainty is always a factor: we can't know what some things mean, why some things are the way they are, what is the best possible way for something to be done, how we should go about living our lives, or even what our lives are. I'm speaking in broad terms but the uncertainty in the above subjects and thoughts is very clear. The point is that many parts of mathematics feel like they are so certain it can be no other way. The fact that two lines in Euclidean space can be situated so they never intersect; that a circle is both an infinite number of infinitely small lines stuck together and a single "line; that if you have four apples in a basket and sell two apples from the basket at 25 cents a piece, you'll be left with two apples in the basket (4 - 2 = 2); that the Mobius band is really simple but really complex at the same time; that two planes in three space can either intersect in a line, a plane, or not at all, but that two planes in four space can also intersect at a point . . . you get my point. These things feel so real and certain and absolute unlike any other aspect of the world or knowledge that I've ever experienced. Yes, I love my mother and father and sister, but what does love mean? I just have to accept that what I feel for them is love and go on with loving them because I can't learn any more than that. I can have strong feelings for other people, but what do these feelings mean? where do they come from? what is the ramifications of these feelings? how can I make these feelings a part of my world and a part of other peoples worlds? Yes I can rip a single through the gap, drain a jumper from the corner, or make the final out popping up to the third baseman, but then questions of perception arise, and questions of reality and life in general: it can always come back to the questions of what is life? But when I think about this question, I can find uncertainty in everything (at this moment) except math (many parts of it): MATH FEELS LIKE IT IS THE ONLY THING THAT TRANSCENDS ALL QUESTIONS OF WHAT LIFE REALLY IS. Even if life is something different that what we've perceived it to be, can anyone imagine that math, or at least the mathematical principles like a curve, a line, a point, a space, the concept of a Flatland a Spaceland and a Hyperspace land, would be different?
Then one might argue: "look how people's perceptions of math have changed as time has progressed. How, if mathematics can transcend to a level of absolutism, can we learn or discover or prove something that we didn't know before, or didn't believe before, or even couldn't have possibly imagined even if we were the wisest person ever to have lived or ever going to have lived?" My only argument is that, at this moment, some things exist in math that are seemingly unchangeable, despite new discoveries. Unlike the Greeks, I'm not saying that our world or universe is this or that. I am saying that when we "discovered" the fourth dimension (accepted dimensionallity and what not), the absolutism that a line is a line and that two lines can be parallel and never intersect did not change. The contexts of the discussions changed, but the absolutism of the subjects resist change. How many other aspects can you say this of? It is not easy to say that _________ is right or wrong and be certain of that forever; that _______ is the way of the universe; that _________ is the position of the electron at this moment; that ____ is how our eyes work; that _____ is how we respire; that _____ is music; that ________ . . .
It feels like math is the strongest idea: that it would necessitate the largest shift in how we think / how we are/ what we are to change it.
on another note, the figure eight Klein bottle is pretty cool. Very simple. Another interesting way to accentuate it's regularity is to project it, like in the chapter heading wine glass, but in a more symmetrical "figure eight" type way. Things get simple then as well. Have you ever seen Ken Lao's two page write-up he did in high school presenting the Klein bottle in this manner (obviously comment directed to Banchoff)? Very nice. I'll bring it to class.
Prof. Banchoff's response