Dealing with the Formal Presentation of Mathematics
Date: 03/22/2008 at 10:52:45 From: Thomas Subject: Dealing with the overwhelming formalism When I was at school I found that I liked math very much. But I think (most of) today's mathematicians find it a great thing to hide their ideas behind all sorts of notation, definitions and the like. So what is it that "prohibits" writing in a paper things like "suppose we had this and this" or "what if we had ..."? I just can't imagine people doing this fiddling with symbols as "self-purpose"; this would in a way be like a writer writing just for sake of writing. So maybe you can give me some thoughts/ideas how to deal with this overwhelming formalism--or rather this lack of "developing ideas". I've already read some answers (for example "What is mathematics?") and found it a funny idea to treat it as sort of a game; but often I end up with the things that I discovered on my own that I enjoy most. Thanks.
Date: 04/05/2008 at 09:46:44 From: Doctor Vogler Subject: Re: Dealing with the overwhelming formalism Hi Thomas, Thanks for writing to Dr. Math. I don't think anyone is trying to hide their ideas. I've known some mathematicians who hide their ignorance behind fancy terminology and formalism, but these probably aren't the ones you're talking about, since you are probably referring to text books and college professors. I've also known many mathematicians who understand the math very well and are very smart, but they aren't good at teaching. Teaching and mathematics seem to be two skills that don't usually go hand-in-hand, perhaps because math is a left-brain skill and teaching is a right-brain skill. This might be more of what you're observing than deliberate obfuscation. I have seen a variety of papers and text books that use phrases like "Suppose we had this and this." Many good papers and most text books give examples of what they're talking about, and usually the examples are well-suited for showing what's going on. On the other hand, I've seen a few books that are really terrible places to learn a subject. (Hartshorne's "Algebraic Geometry" comes to mind.) More frequently, mathematicians use the formalism, the definitions, and the notation to simplify what they're dealing with rather than to obscure anything. The formalism allows you to learn about (and prove things about) many different related topics simultaneously. That allows hundreds of proofs to be combined into only one, which makes remembering them a whole lot easier. The definitions allow you to bring to mind a concept with just a word. For example, it's easy to say: The square root of a prime number is irrational. And you probably understood that completely! But if you didn't have those definitions, it would come out, "The number which multiplied by itself gives an integer bigger than 1 but having no integer divisors other than 1 and itself cannot be expressed as a ratio of two integers," which is far more confusing and obscure. And even that depends on other definitions, like "integer." The notation is really just more definitions, except it's for equations. Without notations, we'd be writing our formulas and equations using words and sentences like the ancient Greeks did, which was very cumbersome and hard to use. The notations allow us to write equations, which make algebra so much easier to do. When more advanced concepts are introduced (like limits, integrals, derivatives, sequences, infinite sums, and so on), we need notations to come along too or else give up our ability to write equations. All of these things become more natural with use, so just try doing some exercises and examples, and you'll become familiar with the notation and with the definitions and it will be easier to understand what's going on. Of course, you'll have more to get used to when you go on to a new subject. Of course, every book and paper assumes a certain amount of knowledge already. For example, your calculus book will assume that you know algebra, and it will be hard to understand if you don't. So you might have found yourself reading a book for which you have not yet learned the prerequisites. That's something one encounters a lot when learning independently, since it can be hard to know what knowledge the author is assuming before reading the book, and even hard afterward if you don't have the knowledge and therefore don't understand what the author is saying. When this happens, it can be useful to get a recommendation of books on the subject which are at your level from someone who knows the subject. If you have any questions about this or need more help, please write back and show me what you have been able to do, and I will try to offer further suggestions. - Doctor Vogler, The Math Forum http://mathforum.org/dr.math/
Date: 04/08/2008 at 00:10:33 From: Thomas Subject: Dealing with the overwhelming formalism Hi Dr. Vogler, Thank you for your answer. You wrote: "More frequently, mathematicians use the formalism, the definitions, and the notation to simplify what they're dealing with rather than to obscure anything." Well, then you're lucky :-). For example, I wanted to know more about group theory, so I bought one (Kurzweil and Stellmach), but their style of writing was so dry ... What makes the difference (at least to me) is the way how a concept is introduced. For example: a) I just write "A Permutation is cyclic if ..."; or b) I start out like this: Take a permutation p which is not the identical permutation. Then there's some k s.t. p(k!= k. Since p is injective, p(k), p(p(k)) etc. are different; but since I only have finitely many elements (numbers) I can map to, I finally "get back" to k. [This is just a sketch.] After this, the idea of a "cyclic permutation" would become more natural. The trouble is for me that concepts are presented more often like under a), and then a definition is "justified" by examples. An approach similar to b) would be more to my liking. Or: I remember a book of Richard Courant (probably written mainly for physicists?), but the way he explained the integral was easier to understand for me than just a purely "formal" treatment. So I think I could deal easier with books/articles which approach a subject not just stressing a "consistent theory" presentation--in case of binary quadratic forms just from the perspective of number theory, but taking ideas say from geometry. Best regards, Thomas
Date: 04/12/2008 at 10:30:25 From: Doctor Vogler Subject: Re: Dealing with the overwhelming formalism Hi Thomas, You wrote: >For example, I wanted to know more about group theory, so I bought >one (Kurzweil and Stellmach), but their style of writing was so >dry ... While I'm not familiar with this particular book, I can name several books in various subjects (such as "Abstract Algebra" by Dummit and Foote) that are very good books for learning the subject after you've had an introduction to it, but are terrible for a first exposure to the subject. It's entirely possible that your book is one of these. If it's the textbook for a class, then I would suggest talking to your teacher about things that you aren't understanding. If it's not, then I would suggest looking for a different book on the subject, such as one that has "introduction" or "first course" in the title. There are plenty of math books that do a very good job of introducing new ideas, but these are not, unfortunately, always the books teachers prefer to use, perhaps because the teachers *do* understand the subject and sometimes don't realize how hard the first introduction is to students. There is another thing you're fighting too, which is the Bourbaki method. Wikipedia: Nicolas Bourbaki http://en.wikipedia.org/wiki/Bourbaki In the early 1900s, a group of mathematicians in France exerted a great deal of influence in the mathematics community. They introduced some useful ideas, such as common notations for useful sets like the rational numbers and the empty set, but they also pushed hard for the type of dry teaching method that you are protesting. Some mathematicians even today still prefer the method, though we have gotten away from it for the most part. The method is often called the "definition-theorem-proof" style of book or teaching, which has no exposition (i.e. explanation) in between definitions, theorems, and proofs. Many books in the later half of the 1900s were written in this style, not just the Bourbaki books. And even as this style was fading away, it still held great sway, so that even books with explanation often had only a little bit of explanation. But even so, there were always authors who did not subscribe to this style and wrote books that were very easy to understand. Overall, work on getting a book that presents the material in the style you prefer. They're out there. - Doctor Vogler, The Math Forum http://mathforum.org/dr.math/
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