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Posted 8 April 2009
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.
Example
A continuous function
(from the set of real numbers to the set of real numbers) 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, mathe
maticians talked about continuous functions
but became aware that they needed a rigorous definition. One possibility was functions given by
formulas, but that didn’t work (some formulas give discontinuous functions and
they couldn’t think of formulas for some continuous functions.)
Cauchy produced the definition we now use (the epsilon-delta definition) which is a rigorous mathematical version of the no-lifting-chalk idea and which included the functions they thought of as continuous. To their surprise, some clever mathematicians produced examples of some weird functions that you can’t draw, for example the sine blur function. In other words, using the terminology in the discussion of abstraction above, C’ (epsilon-delta continuous functions) had functions in it that were not in C (no-chalk-lifting functions.) On the other hand, their definition now applied to functions between some other sets, for example the complex numbers, for which drawing the graph without lifting the chalk doesn’t even make sense.
Other examples
Abstractmath.org discusses some other structures that are abstractions of other concepts.
¨ A (binary) relation is a set of ordered pairs. That abstracts the everyday concept of relationship by focusing on the property that it holds or doesn’t hold between two given objects.
¨ A group is a mathematical structure which abstracts the idea of symmetry.
For other examples, see function, equivalence relation 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. For some common structures such as the real numbers and sets, the axioms are not only not referenced but the authors clearly don’t even think about them in terms of axioms: they use commonly-known properties (or real numbers or sets, for example) without reference.
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, and if x is any number, then
.
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. The properties listed in the preceding paragraph define 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 mathematicians did this, they discovered the following things (compare the list in the previous section):
¨ 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.
¨ Although multiplication of numbers is commutative, multiplication of matrices is not commutative. Now they had to decide whether to require commutative of multiplication as an axioms for rings or not. In this example, historically, mathematicians decided not to require multiplication to be commutative, so matrices remain an example of rings. They then defined a commutative ring to be a ring in which multiplication is commutative.
¨ You can prove from the other axioms
that for any element x of any ring, so you don’t need to include it as an axiom. (Try proving it: You use the distributive law.)
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Other examples
In abstractmath.org, these mathematical structures are defined by axioms:
¨ Groups
These pages contain explicit examples of some of the things that can happen when working with the axiomatic method.
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 are other examples like this here and 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. It is one thing for mathematicians to do this
to each other; that is a kind of normal competitiveness among
members of a tribe. I believe behaving
this way to non-mathematicians is a major contribution to the dislike and fear
of math that many people have.
When you are explaining abstract
math to a non-mathematician
DO NOT DUMP on those who are
baffled
because they haven’t been RATCHETED UP yet.