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Posted 27 May 2008
UNDERSTANDING MATH:
OTHER ASPECTS
Contents
Abstraction
An abstraction of a concept C is a concept C' with these properties:
¨ C’ includes all instances of C and
¨ C’ is constructed by taking as axioms certain assertions that are true of all instances of C.
¨ C may already have a mathematical definition. In that case the abstraction is typically a generalization of C. In other cases, C may be a familiar concept or property that has not been given a math definition. In that case, the math definition may allow instances of the abstract version of C that were not originally thought of as being part of C.
Examples
¨ A (binary) relation is a set of ordered pairs. That abstracts the everyday concept of relationship by focusing on the property that it holds between two objects.
¨ A continuous function is sometimes described as a function whose graph you can draw without lifting your chalk from the board. This is physical description, not a mathematical definition. In the nineteenth century, mathematicians came up with the definition we now use, which includes some surprising functions that you can’t draw without lifting the chalk. In fact you can’t draw them at all.
For other examples, see function, relation, equivalence relation, group and vector space.
Axiomatic method
The axiomatic method is a technique for studying math objects of some kind by formulating them as a type of math structure. You take some basic properties of the kind of structure you are interested in and set them down as axioms, then deduce other properties (that you may or may not have already known) as theorems. The point of doing this is to make your reasoning and all your assumptions completely explicit.
Nowadays research papers typically state and prove their theorems in terms of mathematical structures defined by axioms, although a particular paper may not mention the axioms but merely refer to other papers or texts where the axioms are given, except for some common structures such as the real numbers and sets, whose axioms are not only not referenced but the authors clearly don’t even think about them in terms of axioms.
The axiomatic method in practice
Typically when using the axiomatic method several things happen:
¨ You discover that there are other examples of this system that you hadn’t previously known about. This makes the axioms more broadly applicable.
¨ You discover that some properties your original examples had don’t hold for some of the new examples. Depending on your research goals, you may then add some of those properties to the axioms, so that the new examples are not examples any more.
¨ You may discover that some of your axioms follow from others, so that you can omit them from the system.
Example
Suppose you are studying the algebraic
properties of numbers. You know that
addition and multiplication are both associative operations
and that they are related by the distributive law: . Both addition and
multiplication have identity elements (0 and 1) and satisfy some other
properties as well (addition forms a commutative group
for example).
One way to approach this problem is to write down some of these laws as axioms on a set with two binary operations without assuming that the elements are numbers. You are abstracting some of the properties of numbers. This defines a type of math structure called a ring . You then prove theorems about such structures strictly by logical deduction from the axioms without calling on your familiarity with numbers.
When you do this you discover that there are systems such as rings of matrices that are not numbers but which form rings, and the theorems are true of them as well. You also discover that although multiplication of numbers is commutative, multiplication of matrices is not commutative, so you add commutativity of multiplication to your axioms and call the new type of structure a commutative ring.
Other examples
In abstractmath.org, these mathematical structures are defined by axioms:
Remarks
¨ Historically the first example of something like the axiomatic method is Euclid’s axiomatization of geometry. The axiomatic method began to take off in the late nineteenth century and now is a standard tool in math. For more about the axiomatic method see the Wikipedia article.
¨ Many articles on the web about the axiomatic method emphasize the representation of the axiom system as a formal logical theory (formal system). Mathematicians in practice create and use a particular axiom system as a tool for research and understanding, and state and prove theorems of the system in semi-formal narrative form. Typically it is not difficult to convert these narrated axioms into a formal system, in contrast to proofs, which are sometimes quite difficult to state as a formal object.
The ratchet effect
¨ Once
you acquire an insight, you may not be able to understand how someone else
can't understand it. It becomes obvious, or trivial to prove. That is the ratchet effect.
¨ Example
“How did you know that is never negative?”
“Because it is a sixth power, so
it is a square (of a cube of a number), and the square of a number is never negative.”
(Smiting forehead) “Of
course, why didn’t I see that!”
This observation stays in your
head. You can’t
forget it. Your neurons have been permanently changed and you can’t go back and
regain your former state of non-understanding!
This is an example of pattern recognition of ignoring the details and
seeing
as
. See also chunking. There is another example like this here.
Bad
Behavior
“How could you NOT see that is never negative?! It’s OBVIOUS
that it’s never negative. It’s a SIXTH
POWER YOU DUMMY!”
It is distressingly common that a
mathematician for whom a concept has become obvious because of the ratchet
effect will then tell someone else that the concept is obvious or trivial, leaving them to feel put down. I believe this behavior is a major
contribution to the dislike and fear of math that many people have.
When you are explaining abstractmath
DO NOT DUMP on those who are baffled
because they haven’t been RATCHETED UP yet.