I wish I knew more about mathematics. I mean, I’m not feeling a need to sit around doing algebra, but I’d really like to understand at least conceptually what people do in higher mathematics. I think I understand that there’s something deeply mysterious about the fact there are mathematical systems that describe the physical world, but I don’t really even understand that.
There’s a book called “A Tour of the Calculus” that tries to explain calculus with a minimum of formulae, and I’ve been able to read the first half three times! Then I hit a wall and it’s beyond either my powers of concentration, or intelligence, or, very likely, both.
Many of the math courses will likely make a person’s head hurt if introductory calculus is beyond their ken. It is MIT after all, and many of the courses require Calc I and II, many times Calc I thru IV as prerequisites.
In my opinion, math, like physics is overrated as far as interestingness. The nitty-gritty of solving complex equations, constructing proofs is difficult work, and a person has to really have a talent and want to do it, and get their hands dirty so to speak. The “math gene” seems rare, and for that reason, for most it is not worth the effort.
I was good at math, at least up to Calculus I and II. However, like many students that test well at various levels, for much of it, I learned enough to do well on the tests, but don’t really absorb the material (starting on Trig and on up). Those that actually absorb the material and understand it, are like one-in-a-thousand at the advanced levels.
I consider myself slightly dysnumeric. I don’t know if that’s a real word, but it means my numerical abilities are sub-par in the same way a dyslexic isn’t quite up to snuff in reading. I can do it, and got through Algebra II and the basic college math requirements, but I perform calculations in a very clunky, most inelegant way. I sort of think of myself as a mechanical adding machine to the good mathematician’s electronic calculator brain.
I love how something like an apple can be described as an equation, but I utterly can’t comprehend it.
If it’s any consulation to you, I think many of those math geniuses and other genius don’t comprehend the things that we think are easy. Look at how many intelligent people can’t and won’t play a flute.
Simone Weil was a Catholic(ish) writer who tackled advanced mathematics in a way I have never understood. Her brother Andre was a famous mathematician.
I think calculus the single most beautiful, most useful, most mind-expanding subject in the entire universe. Single-variable calculus is not that complicated. There’s a lot of excess baggage that is often taught with it, apparently included to make it seem harder than it is.
Calculus is really just an efficient way to describe how things change. If you know the quantity at different times, you take the derivative. If you know how something is changing, you integrate to find out how much there is.
There’s plenty of other, weirder, higher mathematics. I wish it was taught differently in math departments. I had to take physics to learn to use any of it.
I have a friend who is studying physics and theology. Whenever I hear him speak about either of them, I am usually silent because I find myself simply amazed.
I always did well in math throughout K-12 schooling, but especially in high school I found it dull. I think that’s because I was never challenged – the poor teachers were trying so desparately to help the floundering that those few of us for whom it was easy weren’t pushed. I really enjoyed calculus in college, and ended up taking a lot more math than you’d expect when you hear I majored in philosophy. I didn’t develop a real love for math until my senior year as an undergraduate, when I took a non-Euclidean geometry class. Wow! Math IS philosophy!
After getting my B.A. in philosophy I went on to an M.S. in mathematics. (Yes, I skipped the bachelor’s degree in math.) I think most mathematicians would agree with me when I say that mathematics is all about truth and beauty. “Elegance” is highly valued in mathematics – we all love an elegant proof. There are elegant solutions at much lower levels, too: I coach a 6th grade MathCounts team, and a few of my 6th grade “mathletes” come up with some lovely solutions to problems. As for truth, well, mathematics allows one to explore deeply the truths that follow from a very simple set of assumptions, and even to change those assumptions and explore how the truths change. Finally, there is the astonishing USEFULNESS of mathematics – here it seems that we’ve made up all of these amazing mathematical structures, only to find that they can be used to describe our universe. (Some pure mathematicians shudder at application as somewhat dirty, but I think it’s pretty cool. A decade after the M.S. in math, I did an M.S. in civil engineering.)
Dale, I admire you for trying with the calculus book, and I hope you get through it. And if you don’t, you can console yourself with this thought: Music is mathematics that we experience directly, without all the struggle of processing through the usual brain channels. When you’re hearing the beauty of music, you’re hearing the beauty of mathematics. And when you’re playing a whistle, you can say you’re doing applied mathematics.
I bought a book a few years ago with the intent of teaching myself calc.
The lack of fluidity between that subject and my brain caused me to switch to other reading material.
I’ve read a bunch of books on quantum theory, and I love it conceptually, but I just sort of skim past the multilayered formulae.
We’ve all heard that math is the language of science. I have no problem with that. What kills me is how someone like Einstein comes up with an idea like how time dilates with velocity or recent stuff about ‘quantum entanglement.’ I mean crazy stuff that doesn’t just pop into one’s head because it’s logical. So you go back and prove the concept with math, but how do you come up with the idea to begin with? I keep meaning to look into this.
Tony
Math has a lot in common with music, arts, playing chess or learning a language - it´s a skill and you´ll have to practice it every day! One can´t just read a book and then think one understands math. One has to practise it, much in the same way as practicing the violin. When one practices a skill it leads to questions, you can check up the answer in a textbook, or ask an expert, or try very hard to solve the problem yourself. But new questions will always arise!
The very best way to learn math, music, arts, chess or a language is to practice it! A famous writer (can´t remember who) was once asked which is the best way to learn how to write a book? Write a book! was his answer - write, write write! Instead of reading the same book three times I suggest you start with a problem and try solving it. If it doesn´t work read the part pertaining that particular problem in your texbook and then go back to your problem.
Well said, Cran. Mrs. Badger is currently occupied in a master’s program in math. Trying to teach herself non-Euclidean geometry, that sort of thing. She will get all excited at some discovery and want to explain it to me. I hardly ever understand her explanations, lacking as I do the required background knowledge. But she can love the beauty of it, and I can get brief glimpses of it through her. It’s moments like that when I realize how beautiful it is that we’re so different.
The situation reverses when I get all excited about the harmonic structure of a Beethoven sonata, say, and try to explain to her why it is so amazing.
The difference is, she’s much better at music than I am at math.
There is a play, Picasso at the Lapin Agile. I encourage you and others to see it. It is about an imaginary encounter between Pablo Picasso and Albert Einstein at a cafe, talking about the nature of genius. It is an entertaining play, and gives a glimpse into that world and where and how such ideas take root.
Often these kind of new ideas comes from some sort of problem (often a crises in it´s field of science) that can´t be explained within the existing paradigm. In Einstein´s case i belive it was the famous Michelson-Morley experiment, where they came to the conclusion that the speed of light is always the same. And at the time it was a complete mystery. (Imagine fiering a bullet from a train, different observers will measure different speeds depending on where they are standing.) He then set up a series of thought experiments - what would happend if I could travel on a photon (beam of light), etc? After having worked out the main lines of his theory he began, with some difficulty, to formulate the equations (but only then!) Trying to solve a problem or a mystery is often a good way to think new thoughts! A lot of science of art is just hard work! One common denominator for many great scientist and artists is that they are not afraid of breaking new ground in their field.
It wasn’t until the second semester of calculus that I started to appreciate
the usefulness of the stuff. Once I realized you could use an integral in polar
coordinates to derive the formula for the area of a circle, things were more
interesting…
Then I had the circuits classes that needed calculus (capacitors perform a
derivitive on voltage over time, and inductors perform an integral on voltage).
That really helped put 15 years of math education into perspective. Really,
I wish we could teach the uses of math at the same time as teaching the
math itself, but it’s hard to use math without a firm background in the subject.
I think that that was kind of the turning point for me too. At a certain point things just start to make sense and instead of memorizing formulas, you can derive them. Until then it just seemed painful (and the instructors were not all that great either) Unfortunately, that happened during my last quarter of calculus.
Math IS philosophy!
And if you don’t, you can console yourself with this thought: Music is mathematics that we experience directly, without all the struggle of processing through the usual brain channels. When you’re hearing the beauty of music, you’re hearing the beauty of mathematics. And when you’re playing a whistle, you can say you’re doing applied mathematics. 