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Scientific approach and axiomatic approach

How to think like a scientist? Memorizing facts and laws is not sufficient. We should know how to deduce a rule from a large number of examples, not slavishly cling to repeating rules deduced by others.

Our school text books should help students think scientifically and creatively by questioning and criticizing established laws. A child should learn the thinking process of great scientists who discovered these laws.

That will give a deeper understanding of the laws as well as their limitations which will lead to new discoveries and scientific energy.

From “The Character of Physical Law” by Richard Feynman (theoretical physicist and famous teacher of physics)

“There are two kinds of ways of looking at mathematics, which for the purpose of this lecture I will call the Babylonian tradition and the Greek tradition.

relation-between-physics-and-mathematicsIn Babylonian schools in mathematics the student would learn something by doing a large number of examples until he caught on to the general rule. Also he would know a large amount of geometry, a lot of the properties of circles, the theorem of Pythagoras, formulae for the areas of cubes and triangles; in addition, some degree of argument was available to go from one thing to another. Tables of numerical quantities were available so that they could solve elaborate equations. Everything was prepared for calculating things out.

But Euclid discovered that there was a way in which all of the theorems of geometry could be ordered from a set of axioms that were particularly simple.

The Babylonian attitude – or what I call Babylonian mathematics – is that you know all of the various theorems and many of the connections in between, but you have never fully realized that it could all come up from a bunch of axioms.

The most modern mathematics concentrates on axioms and demonstrations within a very definite framework of conventions of what is acceptable and what is not acceptable as axioms. Modern geometry takes something like Euclid’s axioms, modified to be more perfect, and then shows the deduction of the system. For instance, it should not be expected that a theorem like Pythagoras’s (that the sum of the ares of squares put on two sides of right-angled triangle is equal to the area of the square on the hypotenuse) should be an axiom. On the other hand, from another point of view of geometry, that of Descartes, the Pythagorean theorem is an axiom.

So the first thing we have to accept is that even in mathematics you can start in different places. If all these various theorems are interconnected by reasoning there is no real way to say ‘These are the most fundamental axioms’, because if you were told something different instead you could also run the reasoning the other way. It is like a bridge with lots of members, and it is over-connected; if pieces have dropped out you can reconnect it another way.

The mathematical tradition of today is to start with some particular ideas which are chosen by some kind of convention to be axioms, and then to build up the structure from there.

What I have called the Babylonian idea is to say, ‘I happen to know this, and I happen to know that, and maybe I know that; and I work everything out from there. Tomorrow I may forget that this is true, but remember that something else is true, so I can reconstruct it all again. I am never quire sure of where I am supposed to begin or where I am supposed to end. I just remember enough all the time so that as the memory fades and some of the pieces fall out I can put the thing back together again every day’.

The method of always starting from the axioms is not very efficient in obtaining theorems. In working something out in geometry you are not efficient if each time you have to start back at the axioms. If you have to remember a few things in geometry you can always get somewhere else, but it is much more efficient to do it the other way. To decide which are the best axioms is not necessarily the most efficient way of getting around in the territory.

In physics, we need the Babylonian method, and not the Euclidean or Greek method”

– from the article The Relation of Mathematics to Physics, page 46,47

Dictionary definitions
theorem – a formula or statement that can be proved from other formulas or statements
axiom – a rule or principal that many accept as true

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