Concerned about my mathematical maturity.

[lolgarithms]

I can't read abstract math textbooks/symbols quickly and i find myself skimming over them. I can't do any simple prooves.
I thought I was good at math, but then it struck...


With what textbooks/subjects can I get started?
Level of math: calculus 1,2, and 3 (non rigorous)

*Textbooks you recommend should be free
*It must emphasize prooves at some point, and various levels of prooves are a must

 

[dx]

One of the main reasons, in my experience, that people have a hard time with abstract mathematics is that they are not familiar with the concrete situations from which the abstract concepts arise. Abstract stuctures like vector spaces, hilbert spaces etc. are always introduced to discuss the common structures in various concrete situations. For example, if you can't think of atleast 5 concrete examples of a vector space, then you are not ready to learn about abstract vector spaces. If you can't think of atleast 5 concrete examples of a group, then you are not ready to discuss abstract groups.

 

[lurflurf]

^^
Can you give five (finite) examples of a vector space? Most linear algebra students can only give one, F^n. Examples are not sufficient for understanding. There are numerous mental hurdles. Assume a person has five examples Xi of an X she/he might then say
-Why should I care about any X but Xi?
-When I know I am dealing with Xi why should I assume X?
-Learning about X is hard why not just learn about Xi?
-I am not conviced there are any X except Xi.
-I think any X that is not Xi has something wrong with it.

 

[slider142]

I can't read abstract math textbooks/symbols quickly and i find myself skimming over them. I can't do any simple prooves.
I thought I was good at math, but then it struck...


With what textbooks/subjects can I get started?
Level of math: calculus 1,2, and 3 (non rigorous)

*Textbooks you recommend should be free
*It must emphasize prooves at some point, and various levels of prooves are a must

Theoretical mathematics texts cannot be "read quickly" by an undergraduate. Even when reading a math paper, you should have spare paper or a blackboard and make sure you understand the implications of every paragraph in the text. Unlike texts at calculus level and below, texts written at analysis level and above require the student to fill in theorems and factoids mentioned in the text themselves before they tackle the exercises. These theorems may not even be labelled "theorem", they will simply be statements the author makes without inline proof. If you try to read the text in a quick fashion like a calculus "recipe book", you will miss all of those little theorems, factoids, and motivations. Also unlike these books, typically the author is past the point of introducing the student to 5 examples of the same theorem, and will either not have an example or motivate only a single example, making these texts very dense. They can contain a great deal more in 5 pages than 5 chapters of most lower level calculus texts, as they expect the student to come up with more examples/applications on their own.
In short, the text is not meant to be simply read, it must be treated like an extended exercise in itself.

 

[dx]

Examples are not sufficient for understanding.

Where did I say examples are sufficient for understanding?

Assume a person has five examples Xi of an X she/he might then say
-Why should I care about any X but Xi?

Xi have a common structure, which is embodied in X. This structure can be discussed without reference to a particular example. The idea of a function can be discussed without talking about a particular function like cos or sin, and I'm sure you know that this is a very useful thing to do.

-When I know I am dealing with Xi why should I assume X?

If you can show (not assume) that Xi is a vector space, then various theorems and results from the abstract theory vector spaces become available for application when dealing with Xi.

-Learning about X is hard why not just learn about Xi?

Is it easier to learn about the derivatives of each function seperately, or to introduce the notion of derivative in context of general functions and then apply the results (product rule etc.) in each case?

-I am not conviced there are any X except Xi.

Then you better convince yourself that there are before you start studying X.

-I think any X that is not Xi has something wrong with it.

I have no idea what that means.

 

[dx]

Can you give five (finite) examples of a vector space?

Being a physics student, I only deal with real and complex vector spaces, and the theory of vector spaces over finite fields is not useful to me. That's another problem people face: worrying about irrelevant generality.

 

[lurflurf]

Where did I say examples are sufficient for understanding?
Yes that statement is too stong. In mathematics examples have value, but that value can be exagerated. There that is better. In addition care should be taken to chose diverse examples in order to prevent limiting ones imagination. In some areas it is nearly very difficult to grasp the many diverse elements a set can contain.


Xi have a common structure, which is embodied in X. This structure can be discussed without reference to a particular example. The idea of a function can be discussed without talking about a particular function like cos or sin, and I'm sure you know that this is a very useful thing to do.

A good point. Sometimes the trouble lies in thinking of x as potentially any element of X, sometimes the trouble lies in not thinking of some of the possible values.

If you can show (not assume) that Xi is a vector space, then various theorems and results from the abstract theory vector spaces become available for application when dealing with Xi.

That is true, but presumably the more general results must be proven before use. More general results are often more difficult and usually appear more difficult. For this to be worthwhile we would want to know that the more general result will be needed soon, or that it is available for low cost.

Is it easier to learn about the derivatives of each function seperately, or to introduce the notion of derivative in context of general functions and then apply the results (product rule etc.) in each case?



Then you better convince yourself that there are before you start studying X.

or never study X

I have no idea what that means.

One may consider the new examples to be 'pathalogical' and as such unworthy of consideration.

These simple questions may not be all that make generality difficult to grasp, but the fact that they are often not adressed at all cannot be good.

 

[Werg22]

Being a physics student, I only deal with real and complex vector spaces, and the theory of vector spaces over finite fields is not useful to me. That's another problem people face: worrying about irrelevant generality.

Physics students have it tough: the vector space notation they use is terrible!

 

[lolgarithms]

Theoretical mathematics texts cannot be "read quickly" by an undergraduate.

i condemb your condescending tone.
you mean only grads can read those texts quickly? that's so unfair.

what is phisycist's vector space notation?

also can i get help on where to get started on more abstract math?

Is there nothing but prove, prove, prove?

 

[g_edgar]

what is phisycist's vector space notation?

Maybe he means "bra" and "ket" notation ... That is $\langle a \vert$ and $\vert b \rangle$ are a bra and a ket, put them together to get a "bracket" $\langle a \vert b \rangle$, which is what mathematicians call an inner product.

 

[lurflurf]

i condemb your condescending tone.
you mean only grads can read those texts quickly? that's so unfair.

also can i get help on where to get started on more abstract math?

Is there nothing but prove, prove, prove?

Of course you can read faster with practice.

There is more to math then proofs. There are also examples, definitions, disproofs, applications, computations, and much more. Proofs are important though.

The first thing to try would be to prove some obvious things you should know like
-sqrt(2) is irrational
-There is no largest prime
-exp(x) is not a polynomial

Next learn a subject in which simple proofs are use like
-Elementary Geometry
-Arithmetic (Number Theory)
-Linear Algebra

Though harder one could also use
-Calculus (though if you know some calculus it would be nice to do a few proofs)
-Topology
-Algebra

There also exist 'introduction to math' type books, I think the material they contain is better learned in concert with a particular subject as above, but it would not hurd to browse a few at your desired price of free.

Proofs and Concepts the fundamentals of abstract mathematics by Dave Witte Morris and Joy Morris

Sets, relations, functions by Ivo Düntsch

 

[daniel_i_l]

^^
Can you give five (finite) examples of a vector space? Most linear algebra students can only give one, F^n.

1) space of polynomials with a degree smaller than n
2) space of nxn Matricies
3) space of linear transformations from a space to itself (HomV,VV))
4) space of continuous functions from R to R
5) Finite fields: http://en.wikipedia.org/wiki/Finite_field

I'll let you come up with the appropriate fields and operations.

 

[Hurkyl]

Is there nothing but prove, prove, prove?

It might help you to realize that "proof" is a synonym for "doing calculations with logic".

 

[VeeEight]

i condemb your condescending tone.
you mean only grads can read those texts quickly? that's so unfair.

I'm not sure if that was what was actually meant. I think that most math books from Introductory Analysis and Abstract Algebra and up should be read in a detailed manner with some scrap paper for you to work out any details and examples you may find tricky. Skimming through books and attempting to even get the most basic understanding of the theorems and definitions is not suitable for most science students.

 

[snipez90]

I'm not sure if that was what was actually meant. I think that most math books from Introductory Analysis and Abstract Algebra and up should be read in a detailed manner with some scrap paper for you to work out any details and examples you may find tricky. Skimming through books and attempting to even get the most basic understanding of the theorems and definitions is not suitable for most science students.

This is the impression I get when someone mentions "mathematical maturity". But honestly, if you're not doing that, then you should either put in more effort or the material is too easy for you. I really doubt that "mathematical maturity" really has any more content than simply having the patience to work enough to understand a concept, which is true for other subjects beside math as well.

 

[Mark44]

Theoretical mathematics texts cannot be "read quickly" by an undergraduate.

i condemb your condescending tone.
you mean only grads can read those texts quickly? that's so unfair.

lolgarithms, What Slider142 said says nothing about graduates, and it is logically incorrect to infer from it some meaning about them. If I were to say that the New York City telephone book cannot be read quickly by an undergraduate, you would probably agree that this is a true statement. However, it would be logically erroneous to infer from this statement that graduates can somehow read this phone book quickly.

As was mentioned earlier in this thread, math texts typically are dense with notation that requires a lot of time to decode and understand. In addition, there are often steps that are only sketchily described, that usually require some work to verify. The extra needed for comprehension is something that is taken for granted by mathematicians, but probably not many/most math undergrads.


BTW, the word is "condemn."

 

[lolgarithms]

lolgarithms, What Slider142 said says nothing about graduates, and it is logically incorrect to infer from it some meaning about them. If I were to say that the New York City telephone book cannot be read quickly by an undergraduate, you would probably agree that this is a true statement. However, it would be logically erroneous to infer from this statement that graduates can somehow read this phone book quickly. BTW, the word is "condemn."

whatever.

What topics are covered in the most rigorous undergrad curriculum?
besides calculus (including complex variables), ordinary and partial diffeqs, linear and abstract algebra, real and complex analysis, set theory/logic...

 

[Gear300]

whatever.

What topics are covered in the most rigorous undergrad curriculum?
besides calculus (including complex variables), ordinary and partial diffeqs, linear and abstract algebra, real and complex analysis, set theory/logic...

That is essentially what they cover. The analysis becomes more rigorous stepping into the graduate program (there are also other topics in the graduate program, such as topology).

 

[JasonRox]

One of the main reasons, in my experience, that people have a hard time with abstract mathematics is that they are not familiar with the concrete situations from which the abstract concepts arise. Abstract stuctures like vector spaces, hilbert spaces etc. are always introduced to discuss the common structures in various concrete situations. For example, if you can't think of atleast 5 concrete examples of a vector space, then you are not ready to learn about abstract vector spaces. If you can't think of atleast 5 concrete examples of a group, then you are not ready to discuss abstract groups.

Good response.

 

[Klockan3]

Maybe he means "bra" and "ket" notation ... That is $\langle a \vert$ and $\vert b \rangle$ are a bra and a ket, put them together to get a "bracket" $\langle a \vert b \rangle$, which is what mathematicians call an inner product.

That really isn't the idea. You have it like this:
Bra's are row vectors and kets are column vectors. This way you can easily and intuitively define both inner and outer products, and the vector algebra doing multiple vectors and matrixes gets a lot simpler.

Like $\vert a \rangle \langle a \vert$ for a as a unit vector becomes a matrix representation of a projection on a, since $\vert a \rangle \langle a \vert b \rangle$ will just pick out the a parts of b and give it out again as a's. IF you want projections on more advanced objects just add multiple of these together to form a matrix with higher rank.

 

[jasonRF]

One of the main reasons, in my experience, that people have a hard time with abstract mathematics is that they are not familiar with the concrete situations from which the abstract concepts arise. Abstract stuctures like vector spaces, hilbert spaces etc. are always introduced to discuss the common structures in various concrete situations. For example, if you can't think of atleast 5 concrete examples of a vector space, then you are not ready to learn about abstract vector spaces. If you can't think of atleast 5 concrete examples of a group, then you are not ready to discuss abstract groups.

This is a really good point that I think many professors overlook.

warning - long rant follows:

As an example, when I took linear algebra, we spent a few weeks on basic matrix stuff, then jumped into "abstract vector spaces" and the like. We covered lots of good stuff: inner product spaces, operators, matrix representations of operators, change of basis, eigen-stuff, etc. However, by the end of the semester I never understood why I should care, and had only seen contrived examples of vector spaces. We had no textbook so I had no other resource to get insight (typical arrogant prof: "there are no books that teach it right ..."). This is especially bad since this was the third semester of the required math sequence for engineers (the first two semesters were calculus, of course). I'm sure it was partially my fault, but I retained almost nothing I learned in that class, I think because I had nothing to "ground" it in. It wasn't until I took the subsequent differential equations class (and virtually every other class I took after that like quantum mechanics, circuits, signal processing, numerical methods, etc.) that I saw any good examples of vector spaces and saw what the abstraction does for you. As a result I re-learned the material on my own and finally started to gain a real understanding of what it was all about.

 

[lolgarithms]

Could I just have a list of hardcover or online textbooks that you recommend
on abstract algebra, linear and multilinear algebra, real analysis, complex analysis, topology, differential geometry? by varying levels of rigor/conceptual understanding required
doesn't have to be free, although it helps. so that i may do the 4 years worth of upper division math during my remaining high school years.

having some interesting examples helps in addition to proofs also.
I hope this will be the last time i am bothering you.

 

[Ben1220]

Mathematical maturity will come with time and practice. So long as you are in rigourous classes that you have the prerequisites for, and you do all the work, you should gradually become more comfortable with proofs and greater abstractions. Things start getting very abstract at the stage of linear algebra (generally taken in the summer after calc1 and calc2, 1st year with calc2 or 2nd year first semester) at my uni. This is when mathematical maturity really starts to develop. Then things get really hairy with real analysis 2nd year, followed by group theory in 2nd semester if you are up for it.

 

[lolgarithms]

Could I just have a list of hardcover or online textbooks that you recommend
on abstract algebra, linear and multilinear algebra, real analysis, complex analysis, topology, differential geometry? by varying levels of rigor/conceptual understanding required
doesn't have to be free, although it helps. so that i may do the 4 years worth of upper division math during my remaining high school years.

having some interesting examples helps in addition to proofs also.
I hope this will be the last time i am bothering you.

I just wanted a list of textbooks, not motivational material.