Mathematical proof is far more powerful and rigorous than the concept of proof we casually use in our everyday language, or even the concept of proof as understood by physicists or chemists. The difference between scientific and mathematical proof is both subtle and profound, and is crucial to understanding the work of every mathematician since Pythagoras.

The idea of a classic mathematical proof is to begin with a series of axioms, statements which can be assumed to be true or which are slef-evidently true. Then by arguing logically, step by step, it is possible to arrive at a conclusion. If the axioms are correct and the logic is flawless, then the conclusion will be undeniable. This conclusion is the theorem.

Mathematical theorems rely on this logical process and once proven are true until the end of time. Mathematical proofs are absolute. To appreciate the value os such proofs they should be compared with their poor relation, the scientific proof. In science a hypothesis is put forward to explain a physical phenomenon. If observations of the phenomenon compare well with the hypothesis, this becomes evidence in favor of it. Furthermore, the hypothesis should not merely describe a known phenomenon, but predict the results of other phenomena. Experiments may be performed to test the predictive power of the hypothesis, and if it continues to be successful then this is even more evidence to back the hypothesis. Eventually the amount of evidence may be overwhelming and the hypothesis becomes accepted as a scientific theory.

However, the scientific theory can never be proved to the same absolute level of a mathematical theorem: it is merely considered higly likely based on observation and perception, both of which are fallible and provide only approximations to the truth. Even the most widely accepted scientific 'proofs' always have a small element of doubt in them. Sometimes this doubt diminishes, although it never disappears completely, while on other occasions the proof is ultimately shown to be wrong. This weakness in scientific proof leads to scientific revolutions in which one theory is replaced with another theory, which may be merely a refinement of the original theory, or which may be a complete contradiction.

On the other hand mathematical proof is absolute and devoid of doubt. Science is operated according to the judicial system. A theory is assumed to be true if there is enough evidence to prove it 'beyond all reasonable doubt'. On the other hand mathematics does not rely on evidence from fallible experimentation, but it is built on infallible logic.

This is demonstrated by the problem of the 'mutilated chessboard'.

We have a chessboard with the two opposing corners removed, so that there are only 62 squares remaining. Now we take 31 dominoes shaped such that each domino covers exactly two squares. The question is : is it possible to arrange the 31 dominoes so that they cover all the 62 squares on the chess board?

There are two approaches to the problem:

(1) The scientific approach
The scientist would try to solve the problem by experimenting, and after trying out a few dozen arrangements would discover that they all fail. Eventually the scientist believes that there is enough evidence to say that the board cannot be covered. However, the scientist can never be sure that this is truly the case because there might be some arrangement which has not been tried which might do the trick. There are millions of different arrangements and it is only possible to explore a small fraction of them. The conclusion that the task is impossible is a theory based on experiment, but the scientist will have to live with the prospect that one day the theory may be overturned.

(2) The mathematical approach
The mathematician tries to answer the question by developing a logical arrangement which will derive a conclusion which is undoubtedly correct and which will remain uncahllenged forever. One such argument is the following:
* The corners which were removed from the chessboard were both white. Therefore there are now 32 black squares and only 30 white squares.

* Each domino covers two neighboring squares, and neighboring squares are always different in color, i.e. one black and one white.

* Therefore, no matter how they are arranged, the first 30 dominoes laid on the board must cover 30 white squares and 30 black squares.

* Consequently, this will always leave you with one domino and two black squares remaining.

* But remember all dominoes covert two neighboring squares, and neighboring squares are opposite in color. However, the two squares remaining are the same color and so they cannot both be covered by the one remaining domino. Therefore, covering the board is impossible.

This proof shows that every possible arrangement of dominoes will fail to cover the mutilated chessboard. A proven mathematical result has a deeper truth than any other truth becuase it is the result of step-by-step logic.Mathematics gives science a rigorous beginning and upon this infallible foundation scientists add inaccurate measurements and imperfect observations!