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/