Inconsistency in textbooks

[Tom1992]

i liked the posts i read about how some people here learned calculus at a young age, so here i don't feel as isolated as from my 14 year old friends. i tried teaching myself calculus at age 11 with help from my dad, but had difficulty because i had to learn trig and factoring trinomials first. i managed that in a few months and then i finally learned calculus and vectors in R^3.

now 3 years later i have learned calculus in n variables, linear algebra, group theory, number theory, complex variables. now i am studying analysis and topology where everything is about proofs. i learn with help from my dad but he doesn't have time to teach me everything so i use his library of math textbooks.

i find it difficult to learn analysis and topology because different books give different definitions and notations, and definitions is very important to use properly in proofs. my dad said that often there will be a circle of definitions for the same thing because theorems prove that they are the same, and that's fine. but i get really annoyed when several definitions say the same thing but use different terms and symbols. this is very annoying for someone trying to teach himself from books. i heard that physicists have agreed on common units (called si units i think) so that there is no confusion, so why can't they do this in math?

 

[matt grime]

Historical reasons, perhaps. And probably because there are very few SI units compared with the terminology in mathematics.

 

[Crosson]

The SI units in physics cover numeric calculation, but not symbolic notation. Physics notation also fails to be standardized in many cases.

I would suggest that you study one book long enough to get used to its theorems and notations, so that when reading a different book you see how the previous books theorems 'fit in' with what you learned. Alternatively, keep doing what your doing and it will eventually all make sense.

I would strongly suggest studying Set Theory and Logic before doing Topology and Analysis. Start typing notes in $$\LaTeX$$ as soon as possible.

 

[Tom1992]

i know my set theory (de morgan's laws, etc...) and it was easy to learn because the symbols are pretty univeral there. i never said topology and analysis was hard, i said that it is hard to learn with the inconsistency in textbooks. i read most of topology from an old textbook but my dad said the book is too old and notations in the book was out of date, no paracompactness was there, no topological groups, no nagata-smirnov metrization theorem, etc... so i switch over to the more universal munkres textbook, but i was then overwhelmed by how different the definitions and notations were, and i had to start all over again as though i was taught incorrectly the first time.

 

[matt grime]

You were taught differently with a different emphasis. Something very similar might well happen if you tried that with absolutely any subject.

There are two very distinct things you're complaining about: nomenclature and notation. Pick up a textbook on physics from the 50s and see what that's like compared to a newer one. This isn't an attempt to justify it, but to say that a negative comparison with another subject is not really correct.

Actually make that three different things: nomenclature, notation, and content.

It would be good if you provided some examples, by the way so we understand what precisely it is that has annoyed you.

 

[mathwonk]

if you understand anyone book, the ideas will translate over.

in topology for instance the main point is understand when one point can approximated by points from another set.

the is the concept of closure. all books agree on the definition of closure of a set, but they use different words for concepts like limit point, accumulation point, neighborhood, and so on.

for instance, one book says a limit point of a set A is a point x such that every open set conrtaining x also meets A. Another book says that a limit point is a point x such that every open set containing also contains an infinite number of points of A, or soemthing else?

but anyway, the closure of the set A, is the union of A and all its limit points, either way.

Now if you define limit point the first way you do not have to say it is the union of A and its limit points, the closure is just the limit points.

so what, the idea is that the closure of a set is that set plus all points that can approximated arbitrarily well by points of that set.

so eventually you begiont to separate out the essential ideas from the variations in terminology.

for this it helps to read books that say what is going on, and what the purpose of the definitions is. then you realize there is more than one way to achieve your goal.

 

[chroot]

The truth is that the world of mathematics, physics, and engineering is nowhere near as large, conceptually, as it might appear from reading many different books. The truth is that there are only a relatively small number of fundamental concepts, but they're all dressed up differently in different contexts. The more you learn, the more you'll be able to recognize one concept as nothing more than a rehash of an old, familiar concept.

- Warren