Resources for the mathematically inept?

[Cody Richeson]

As a hobby, I have spent years studying string theory, artificial intelligence, quantum mechanics, general physics, cosmology, astrophysics and other related subjects. I've read numerous books on the subjects mentioned and watched a number of lectures as well. The problem is that, in terms of mathematics, I have about a 3rd grade education, having never successfully passed rudimentary algebraic or geometrical courses. I have tried to read and understand some of the formulae associated with physical phenomena, but to no avail. I really don't understand why the material is not making sense, and was wondering if anyone knows of resources, books, videos, etc., that can help folks like myself understand the underlying math behind these fields of study.

 

[KiggenPig]

When I started learning math again, I went out and bought two ancient algebra books and worked through those. But Khan Academy is a good place to learn the basics.

Physics will not make much sense if you only have a 3rd grade math ability. To understand physics, you must know math. There is no way around that really.

 

[fresh_42]

If you find one, tell me. One of the basic problems is that physicists and mathematicians use their own language and notation. The main problem is to find a translation.

 

[Cody Richeson]

I think that's a big issue here, is that it's a language, and language by its nature is driven by abstraction. I find language to be very confusing. I can write and speak English well, but I do not understand the underlying structure of it at all. Consequently, I find the language of mathematics to be profoundly non-intuitive. It is far more abstract than written language tends to be, as the symbols can sometimes represent very complex, densely layered concepts. I want to be able to understand these concepts and work out the abstractions in my head. I worry about my ability to learn, because I can add and subtract and that's about it.

 

[fresh_42]

At least those languages are precise, spoken language is not. It all started with adding but in the last 400 years or so some concepts have been added. To start let's see what adding does. It leads to the natural numbers ℕ and multiplikation. Since we want to invert addition, i.e. substract, we come to the integers ℤ. The inversion of multiplikation is a division which leads us to the rational numbers ℚ. Now we found, that a diagonal of a square with length 1 can't be measured in ℚ, it's √2. So we have to complete ℚ with all limits of serieses of rational numbers: 1, 1.4, 1.41, 1.414, 1.4142, 1.41421 etc ... → √2. And of course π as well, the ratio between circumference and diameter of a circle and so on. So we got the reals ℝ. Almost done. We came from a half-group ℕ, to the additive group ℤ, which is also a ring, since we can multiply, to our smallest field ℚ, in which we can additionally divide. As the smallest field in which this can be done, it's a prime field. ℝ is its completion since a convergent serie has its limit in it. But there is still a problem. We cannot solve X*X + 1 = 0. So we introduced an imaginary unit i = √-1 to solve this equation and added it to ℝ receiving the complex numbers ℂ which turn out to be very useful in all computational processes. We therefore call ℂ the algebraic closure of ℝ, i.e. all algebraic equations can be solved in it. ℂ is a field extension of ℝ of degree 2, the power of X in X*X +1 = 0.
Now things become funny. Sir William Rowan Hamilton asked himself whether there is another field (+ - * /) containing ℂ. After years he found the Quaternions. But it came to a price. a*b is no longer equal to b*a. A quaternion can be written by four real numbers a + i*b + j*c + k*d where i,j,k fulfill certain equations. It is a 4-dimensional space over ℝ. Such spaces are vectorspaces which merely means arrows. Vectorspaces in which we can multiply are algebras. There are several forms of them: some with a 1, some without, some in which a*(b*c) = (a*b)*c is valid and some where it doesn't hold, e.g. Lie Algebras. Each of the described concepts has many generalizations, special cases, and unusual rules.
E.g. a light switch and this computer obey the rule 1+1=0, the clock 1+1+1+1+1+1+1+1+1+1+1+1=0. A total bunch of structures are out there.
Once you know what it's about it looses its secrets. I mean everbody uses a switch not thinking about the fact, that switching it twice is equal to not switching it. The problem is that meanwhile there are really many concepts, esp. added in the last century.

And to make it complicated, physicist write, e.g. their vectors different than mathematicians do.

 

[Cody Richeson]

I wasn't able to follow any of that.

 

[fresh_42]

I wasn't able to follow any of that.

In this case I have to apologize for my narrowed view and poor ability to explain things better. I just wanted to give an example how such a simple thing as addition become the source of more complicated structures.

 

[Cody Richeson]

In this case I have to apologize for my narrowed view and poor ability to explain things better. I just wanted to give an example how such a simple thing as addition become the source of more complicated structures.

It's okay. To be honest, I have trouble dividing and even adding/subtracting has to (generally) be done on paper if there's more than one digit in the number . I'd really like to get over this hurdle.

 

[fresh_42]

It's okay. To be honest, I have trouble dividing and even adding/subtracting has to (generally) be done on paper if there's more than one digit in the number . I'd really like to get over this hurdle.

But that is the nice part. In mathematics you don't have to do it. You just have to know whether it's allowed and which rules are upon it.
I knew some mathematicians who actually refused to compute something.

 

[symbolipoint]

It's okay. To be honest, I have trouble dividing and even adding/subtracting has to (generally) be done on paper if there's more than one digit in the number . I'd really like to get over this hurdle.

Basic Mathematics, Basic Arithmetic, and then "Algebra 1" or Beginning Algebra.

You are not the only person who needs to do calculations and computations on paper.

 

[Dewgale]

It's okay. To be honest, I have trouble dividing and even adding/subtracting has to (generally) be done on paper if there's more than one digit in the number . I'd really like to get over this hurdle.

Mental Math is a skill just like any other. There are several "for Dummies" books and the like on Mental Math that I recommend you either buy or take out from your local library. Work through it slowly and carefully, and practice every day. You'll find yourself improving, but give yourself time if it's not something you're naturally good at. Imagine if you were trying to learn to play basketball, and you started by trying to do slam dunks before you actually even knew how to dribble? You need to be really solid on the basics before you move on.

Believe in yourself. Nothing is impossible to learn :)

 

[Mark44]

I think that's a big issue here, is that it's a language, and language by its nature is driven by abstraction. I find language to be very confusing. I can write and speak English well, but I do not understand the underlying structure of it at all.

The structure of English is difficult to comprehend, primarily because of its roots in Germanic languages (from the Angles and the Saxons), and its French roots (from the Normans). These different roots make it very difficult to know how a word is pronounced; for example the "ge" in garage has a "zh" sound while the "ge" in Geiger has a "g" sound. The orthographic rules in French, Spanish, Italian and other Romance languages are pretty straightforward, but English, with its mix of Germanic words is a whole different matter.

Consequently, I find the language of mathematics to be profoundly non-intuitive. It is far more abstract than written language tends to be, as the symbols can sometimes represent very complex, densely layered concepts.

In a sense, mathematics is a lot simpler than natural language. Once you know the basic definitions and properties, things are relatively simple.

I want to be able to understand these concepts and work out the abstractions in my head. I worry about my ability to learn, because I can add and subtract and that's about it.

I concur with what symbolipoint said:

Basic Mathematics, Basic Arithmetic, and then "Algebra 1" or Beginning Algebra.

You are not the only person who needs to do calculations and computations on paper.

 

[symbolipoint]

Mark44 said:

In a sense, mathematics is a lot simpler than natural language. Once you know the basic definitions and properties, things are relatively simple.

Simple and Easy are two very different things. Right that natural languages are not so simple. They are usually easier than Mathematics because that is just how humans developed to be able to handle, naturally.