Who wants to be a mathematician ?

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Who wants to be a mathematician ?

Postby Giacomo on March 15th, 2008, 2:59 pm 

I sound like a game show host.

Mathematicians tend to be one of two types, applied or theoretical.

Applied mathematicians try to solve specific real- world problems and may be physicists or engineers, working in areas ranging from defense to finance.

Theoretical mathematicians study math for its own sake and work almost exclusively in universities.

To do math all you need is 9 basic mental abilities:

1. Number sense
2. Numerical ability
3. Spatial-reasoning ability
4. A sense of cause and effect
5. The ability to construct and follow a causal chain of facts or events.
6. Algorithm ability
7. The ability to understand abstraction
8. Logical-reasoning ability
9. Relational-reasoning ability

If you view mathematics as a tool, then you could become a scientist, such as a physicist, chemist, a computer scientist, etc.

The ultimate difference between the mathematician and the physicist is the former is satisfied with a formula or even an existence theorem, the latter does not consider either an answer at all, until the work of numerical computation has been taken to a stage where comparison with observation is possible.

Proof is what makes mathematics different from other sciences. In physics, or biology, or economics, results come only after hard work with observations and experiments. In mathematics, there is proof instead.

For example, what is 1 + 2 + ... + 100

1 + 2 + 3 + 4 + … + 98 + 99 + 100

100 + 99 + 98 + 97 + … + 3 + 2 + 1

By adding each number in the top row to the one directly below it, we get 101. There are 100 pairs.
So the sum of all numbers is 101 * 100. But, this is twice the sum of the numbers 1 to 100. So

1 + 2 + 3 + 4 + … + 98 + 99 + 100 = 1/2 * 101 * 100.

A proof has three obvious advantages over simply getting the answer by putting the numbers into a calculator.

You understand the whole process completely, and hence get that sense of intellectual power which is one of the payoffs of studying mathematics.

As a result of your understanding, you have certainty that the result is correct. There is no prospect of a mistake in entering the numbers, or of the kind of measurement and observational errors that plague the other sciences.

Also because you understand, you can generalize: it is clear that the fact that the last number was 100 had nothing to do with it. It is just as easy to prove, with the same method, that

1 + 2 + 3 + 4 + … + n = n( n + 1)/2.

If anything, the problem is easier for arbitrary n than for 100, since there is no chance of being distracted by any particular facts about the number 100. So on proving an infinite number of facts can be easier than proving one.
Giacomo
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Location: New Jersey and Manhattan, NYC


Postby Giacomo on March 22nd, 2008, 11:54 am 

Mathematical Proof
=============

Poincare asked how it can be possible that many rational people do not understand mathematics.

A mathematical proof consists of a series of logical steps whose reasoning no sane person could deny.

How then can anyone fail to appreciate the whole ?

According to Poincare, it is not the individual steps that matter so much as the overall structure, or pattern, of the proof.

The question, then, is how such patterns can be created and perceived.
Giacomo
Resident Member
 
Posts: 4821
Joined: 20 Sep 2006
Location: New Jersey and Manhattan, NYC



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