Dr. Givental,

Do you happen to have any additional advice to students seeking to study mathematics? Specifically about developing skills to understand proofs?

I picked up your texts as an intuitive impulse when I first attempted Calculus in Winter of 2007-2008 when I was studying mathematics up in Bellevue, WA. I somehow had this inherent realization (I think it was sometime when we introduced the idea of a tangent line) that a person really needed rigorous geometry if they were ever going to understand differential calculus. I've later come to learn, from my favorite professor of mathematics here in Portland, that my astute observations were correct. I also have come to realize that if I ever want to understand integral calculus, I must understand analysis, something I also realized a little beforehand. I'm coming to realize my learning style is incredibly deductive. I'm one of those guys who needs the broad explanation first, and an incredibly detailed explanation with justification at each step for why something is following a specific train of reasoning. I think it comes from my family (4 college professors, 7 engineers, several school teachers, a couple physicists, etc.), so I don't think I'd be wrong to assume that I'm the only one in my family with this problem.

I had a lot of problem with Spivak's *Calculus* when I attempted to study it because I didn't understand the mathematical mindset behind his writing. I had never studied any formal mathematics, except two terms with my favorite professor here in Portland who really got heads on with the subject, and forces us to think and write essays on the math, not just do problems.

As a learner, I need to see the big picture before I work my way down to the small stuff, and you can't make assumptions with me, I really hate that, about what I might or might not know about your reasoning. I've come to learn from a few colleagues in the philosophy department of my last college that what I want is rigorous proof. I want my professors to prove to me what they are doing deductively, or at least strongly in an inductive manner, void of fallacy and poor reasoning. I apparently ask a lot! In this thread of reasoning, I was curious to know if you had any suggestions for how a person could improve their reasoning?

I have been studying Harry J. Gensler's *Introduction to Logic: Second Edition* in an effort to get a well rounded study in Logic so I can begin to use proper reasoning when I try and tackle these problems in geometry. I was under the assumption, however, that when these subjects were taught, geometry was the means by which students learned how to perform proof, or has my more recent observation proved more correct (that logic was taught concurrently with geometry)?

An advisor of mine in physics, Dr. Allen, an interesting Englishman, informed me that geometry is probably the best way to learn deductive reasoning through application. The problem I have is that classically, I have observed, students learned logic along with geometry.

I'm not really sure if that a correct observation, but I know during the middle ages and in ancient times, an informal system of syllogisms were used. Later, in the 19th and 20th century, formal logic was developed. For a mind such as mine, I was eager to know your thoughts and suggestions on a prospective learning course.

I'm planning on entering Reed College in the next couple of years, and work through their physics, math and philosophy departments and hopefully head down to Berkeley for your Group in Logic and the Methodology of Science for my first doctoral work, before going somewhere else for my physics.

Would you suggest just going through Kiselev's texts and use the website for my numerous questions as an adequate preparation for Spivak, or would I be better to first study my formal logic, then work on your text while I am taking my courses at PCC in early undergraduate mathematics, and study Kiselev through the site and then hit Spivak and beyond?

I actually tried getting an independent study with one of my professors here to study Kiselev, but the department as a whole was against it. Apparently they like things how they are, and independent minds are a bit against such things. That is part of the reason my physics professor thinks I need a liberal arts college, where I will have some "freedom" and "save yourself from a [childish] hell."

I know this is probably off from the whole point here, but I found that your text was the starting point for this whole amazing mathematics journey of mine. One which started by formal inquiry, later resentment, followed by a stark rebuttal of the system in which I demanded more (namely rigorous mathematics and the means to learn it), then finally the peace that comes when you no longer care what the system wants or thinks about you and you just do what you think is right. I figured as someone who appears to be from a similar walk of life, you might have a few ideas that could help me.

I also wanted to thank you for all the efforts you put in here. I know you may get this a lot, but, as a student, I really had a life change after finding your books. They were the wake-up called, the moment when I realized "This…THIS IS MATHEMATICS!…I don't know what *that* is…that 'stuff' they've been trying to feed me, and *claim* is 'mathematics', but this *is* interesting. I love *this* stuff…I **want** to study *this*." I have a math professor here who is the same way, just constrained by the system. He calls it the math "educators" vs. "instructors" problem, where the instructors are there for themselves, while the educators are their to teach and help their students grow. It's a batty time to live, but I must thank you for giving me some hope. I actually wanted to attend Berkeley after finding your books. While it is Kiselev's masterwork, the afterword and introduction are yours. I found those to be the most inspiring part of the text. So, while my writing is poor and clumsy, I must give you the greatest thanks for the passion you put into teaching us poor, aimless geeks. Those students who *really* want to know more than what we are given, want to be shown something meaningful, but who are just given the shaft by the system that wants the "two column proof" vs. the "informal proof" or "formal proof" using real systems of formal symbolic logic, with rules of inference, syllogisms, etc.

If you can't think of any advice, then at least please take this message as my heartfelt appreciation for the amazing amount of effort you have put in to teaching. I was frustrated when there was no answer key, or manual to assist me with the text when I first began—I admit I hit my head against the wall, figuratively, for months just trying to give well reasoned answers to the first problems in the text. Problem one was a real doosey. But when the answer hit me, and Oh it hit me hard, after feeling like a complete idiot for how simple the answer was, it made me realize that it is within the simplest definition that the greatest implications can be found. It was in my misinterpreting of the question, specifically that to be bounded meant that a three dimensional object can have 1 side? How does this work? When I realized that to be bound by a plane could imply that the other sides are surfaces which are curved. Calculus helped some by making me realize the idea of the tangent line and the intersection of a tangent line. I felt like that was getting the fast ball my first time playing. It was a hard reality check, but a refreshing one. I remember problem 1, 7 and 8 and constantly use them as demonstrations to my friends and colleagues as an example of real mathematics. The students find it highly interesting, while the faculty seem impressed that I was able to figure such a complex set of problems out on my own. ^___^' I think they especially like the part where I can justify my answers and why I was having problems understanding it in the first place. What was wrong with my initial reasoning i.e. the part about my assuming that the object had to be flat or something. It couldn't exist. I had to think outside the box, consider some other facts. While the facts remain, I was right, that the object had to exist as a solid, and have three dimentions, it didn't have to have 3 planes as those sides. IT could have no planes bounding it, a sphere, one could be a cone or half-sphere. Two could be a Cylinder. I then came to find this interesting fact about spheres where you can cut them and there is a unique relationship about creating an infinite number of sides through slices of a sphere. By taking a half of a half of a half, we can increase the number of sides (parts bounded by the sphere). That was exhilarating to discover!

It took a long time to work on that section, but through that pain, I learned an entirely different kind of mathematics. I learned that a subject doesn't have to be long to be complex, nor interesting. It's those subtly simple things that you overlook that can have the greatest implications. Now that I realized you created this site as a means to actually help students like myself, as a group, with peers working together, and you putting in direct input, Wow…that just leaves me speechless. Thank you.

I hope that I can find out if I was correct in my analysis by posting my results here on the site. Again, my thanks!

Sincerely,

-Chris

PS. Please excuse my poor writing. It's about 2:30am and I stayed up wandering the site, stunned and overjoyed at all the new possibilities before me. ^_^' I can now study the text independently, without first having to burn down the Dean's office to get faculty teaching assistance, afterall! I'm so happy! -Sarcasm I assure you! ^_^'-