Learning Math as a Language: Comparing to Reading

[pzona]

I imagine everyone here is familiar with the idea of mathematics as a language. I have a few questions regarding the way people look at equations and math in general, using this analogy.

I'm reasonably skilled in math up to basic calculus, and so I kind of understand what "math as a language" is all about. For most simple expressions, if something can be factored out, I see it immediately, et cetera. I was wondering whether this starts to happen with more complicated equations as your mathematical skill increases. For example (and I know this isn't terribly complicated), I can easily figure out the equation for kinetic energy in terms of momentum (as opposed to KE=1/2(mv^2), but it doesn't just show up in my head instantly like some other equations do.

I guess my main question is, how would you compare learning mathematics to learning to read? After a while, do the terms just fall into place? Do you find yourself performing substitutions (like in the KE equation, but with more complicated expressions) without thinking? I'm just interested in how everyone else views the process of learning how to do math.

Btw, mods, feel free to move this if you feel like it should be in the philosophy section or one of the math sections. I wasn't sure.

 

[TheStatutoryApe]

By my own interpretation math would be a syntactic language. It is essentially the rules (or grammar if you will) that define it. If you are familiar with the rules then you should be able to understand it. The use of familiar symbols makes it easier but even if you read a "foreign" set of symbols you would easily be able to figure them out as long as they are using the same rules. Sufficient familiarity with mathematical concepts should also allow you to decipher and understand "foreign" symbol sets that use differing rules. The only problem that you will encounter is with symbols that possesses a semantic meaning. So with your equation "KE=1/2(mv^2)" anyone with sufficient knowledge will understand the math but not necessarily what the equation "means".

 

[chiro]

For me personally I see mathematics as the words in a story and the fields as individual chapters while the "understanding" is basically like the back cover telling the whole story.

This is what I've learned in study of mathematics for what its "about":

1) Representation.

If you can represent something, you can represent it with mathematics. Whether its a scientific model, even a model that doesn't correspond to anything real, it can be represented using mathematical symbols.

2) Decomposition

This is similar to representation but mathematics helps us "break-down" a system into its components. Not only that but through areas such as graph theory and number theory we have decomposition of a system into "vertices" and a number into "primes". Our ability to decompose something effectively into a system of parts in which the decomposition appeals to our senses in that the particular classification and decomposition system seems "natural" is important. Decomposition plays a huge role in all the sciences including biology and physics. In fact decomposition of a system can tell us important things about that system and the implications of the system and why it even exists before further analysis has been done.

3) Analysis

Currently the term "Analysis" in mathematical learning is simply a formalized extension and framework of calculus. But analysis is more than this. In that context we can see analysis as things measuring changes in a system in various ways and also talks about things mention above including decomposition (example Fourier series) as well as representation. It also allows us to study the relationship between variables in given frameworks and let's us not only "break-down" its behaviour, it (through the breaking down) allows us to capture important pieces of information in a generalized and specialized way of a system. For example if you are modelling the stress of a given material you might find at turning point that it represents a fracture in the material. This might depend on say temperature and internal stress (I'm not an engineer though so if I'm wrong once again I'm sorry).

In thinking about these I go back to the book analogy. Consider the following evolution of ones ability to do math. On the one hand you have a child just learning to count. Later they learn to generalize the art of counting to abritrary objects and not just apples or oranges. Later on they generalize this to high school algebra. Then we generalize a linear system to that of a matrix. You can then look at complete generalizations of algebra to that of things like Grassman algebras, exterior algebras and so on.

What I have mentioned above is known as generalization. Its a key activity that many mathematicians get involved in and by understanding higher and higher levels of abstraction, it can make the lower levels seem easier to understand and also make them able to "fit into place". But generalization needs reasons behind a generalization. Specifically when you generalize you typically take some specific properties of a system and go on to find out what that implies for anything based on these properties.

For example take geometry. First we start off with the 2D cartesian coordinate system
and later go to 3d. Then we introduce the concept of a manifold and generalize through the notion of curvature and how that applies to geometry. Generalizing in this sense is
extremely difficult because typically we can usually only think in terms of a few dimensions and we are usually so familiar with the cartesian coordinate system that the
other systems (especially with the introduction of a manifold) seem very hard especially if they are comprised of surfaces in 4 or higher dimensions.

In these examples I try to understand why the particular mathematician responsible (lets say for example Gauss with differential geometry) chose to do what he did. By doing that even if the guess is wrong at first you can begin to understand the mind of these people and understand it in a way that allows you to have the "bigger" picture.

For example look at someone like Rene Descartes. He basically invented analytic geometry as well as the idea of taking something x^r. Now think about when complex numbers came in the picture where x^2 = -1. There is a way to link the complex and the reals in harmony and that's through Eulers identity saying e^(i x theta) = cos(theta) + i x sin(theta). If you take the projection onto one axis (ie the sine) you get the
real component and on the other axis you get the complex component.

I see math in two camps. In one its about using the framework that currently exists and in the other its about reflecting on that framework or indirectly through say a problem, the framework is amended or a new one is created to help solve a particular problem.

In the first camp we learn tonnes of techniques and through that we eventually build a large repository of internal relationships so that as we do more problems we identify its structures and characteristics at a higher level. However there is a big downside to this:
if a problem requires a new infrastructure to solve it, then typically we become comfortable with the huge infrastructure we have and it becomes hard to think outside the framework that we have created for ourselves.

One the one hand I see the structures that we have created to be very useful and capable of solving quite a wide range of problems. But thankfully there are problems that have been unsolved for 100+ years that will require new analytical methods and frameworks.

In saying that sometimes it might be better for someone to discover something through themselves rather than being taught to "be inside the square" because for the majority of our lives the square is where we will be. But given the rate at which we learn and given the nature of the mathematical problems of today, it is in my opinion that we will need new frameworks that help target specific problem sets. I have no idea who will be responsible, but I know it will be someone on the kind of scale that Gauss was on.

Finally I just want to say that doing and thinking about math is what its all about. If you think too hard you become a philosopher and you do too hard you forget that you're a "fish swimming in the sea" as you are so immersed in that sea that you have no concept of what the "ocean" is.

 

[magpies]

From my experience it seems math is made harder then it needs to be. A lot of the time complexity is added to save on the amount that has to be written down. That is to say the maths could have what the gravitational constant is in it but instead they just put the letter and you have to go look up the gravitational constant. This basically makes it impossible for anyone who doesn't know where to look up the constant to do the math. It would for the most part be simple for a layperson if they didn't have to look up these terms. So in my opinion math is made complicated to keep laypersons from understanding it OR it could be just that nobody has taking the time to rework math so that it is easyer for laypersons. Either way frown town.

 

[jtbell]

Most math in physics is not about plugging numbers into formulas and punching buttons on a calculator. It just seems that way in poorly-taught introductory physics courses.

Symbolic equations give us a way of studying the general relationships among physical quantities without regard to their specific values. When I taught an intermediate electromagnetism course last year, I don't think we used our calculators even once!

 

[tyroman]

Quoting from OP;

"the idea of mathematics as a language"

To the extent that a mathematical expression can be depicted graphically, there is a significant parallel with language...

Think of a differential or integral and the corresponding graph showing the slope of a tangent or area under a curve. Now think of an illustrated novel. In both cases, the illustration reinforces or guides the understanding of the reader.

 

[pzona]

By my own interpretation math would be a syntactic language. It is essentially the rules (or grammar if you will) that define it. If you are familiar with the rules then you should be able to understand it. The use of familiar symbols makes it easier but even if you read a "foreign" set of symbols you would easily be able to figure them out as long as they are using the same rules. Sufficient familiarity with mathematical concepts should also allow you to decipher and understand "foreign" symbol sets that use differing rules. The only problem that you will encounter is with symbols that possesses a semantic meaning. So with your equation "KE=1/2(mv^2)" anyone with sufficient knowledge will understand the math but not necessarily what the equation "means".

It's interesting you should say this. I was reading something on Ramanujan a few months ago and it said that since he had so little formal training, most of his major discoveries were written in his own notation. When he sent them to someone in England (can't remember who he sent them to), it took them a while to figure out what he was trying to express, but it ended up being pretty important. I guess this is what fascinates me about math; it can be learned/derived by anyone who understands a few basic axioms, and it's the only language, as far as I know, that someone can (theoretically) teach themselves without help from another person.