Mathematics and the Physical World

Galileo Galilei (1564-1642) formulated the relation between mathematics and the physical world by these eloquent and extensively meaningful words:

« Philosophy is written in that vast book which stands for ever before our eyes, I mean the universe: but it cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word».

Although the relation between mathematics and the physical world was known and recognized since the highest antiquity, no one before Galileo didn't express the intimacy and the importance of this relation, that bind the world of the concrete objects at the ideal world of the mathematics. He is a big pioneer for the establishment of the modern science.

Galileo had well seen the primordial role of the geometry whose figures, and especially the ellipses, will be used by Johannes Kepler (1571-1630), contemporary to Galileo, to be the orbits of the earth and planets when traveling about the sun, and for establishing the three fundamental laws of planetary motion.

He did well to put in relief the primordial role of geometry among the other sciences, because geometry is inseparable of the movement, and the real resolves in movement.

Besides, the affirmation of Galilee is a prophecy that is going to achieve itself in a vivid way in the period follow, and especially in the contemporary period, where, we are all of the witnesses. A special survey will be dedicated to Galilee to do a homage well deserved by this big genius.

In the present article we indicate the relations briefly between mathematics and the physical world. We will develop some topics illustrated by examples later.

Already, a long time before Galileo, the Babylonian used mathematics in their different activities, especially in astronomy, agriculture and architecture. They recorded their astronomical observations in tables which will be useful for the successor astronomers.

The Egyptian used mathematics for the measurement of lands and the construction of pyramids. Mathematics was present in the achievement of the Indian, Chinese, Maya, and other cultures.

Thales, the first Greek mathematician, astronomer and philosopher, measured the height of pyramid by measuring its shadow when the shadow of an object is equal to the length of the object. He surely applied the relation of proportionality in similar triangles. He also predicted an eclipse of the sun in 585 BC, probably by using recorded tables of astronomy done by Babylonians a long time ago.

The prophecy of Galilee will be achieved by Descartes (1598-1650) in 1637 when he published "La Géométrie".

This important discovery will bring a great contribution to mathematics. In the analytic geometry, algebra is used to find the points of intersection of lines by finding the roots of algebraic equations, while geometry by intersecting of the lines gives the roots of the algebraic equations.

Without the geometry of Descartes, it was almost impossible to discover the infinitesimal calculus by Newton (1642-1727) in 1687 and independently By Leibniz (1646-1716). The former, born the same year when Galileo died, will use his finding to establish the laws of gravitation and to confirm the saying of Galileo on the role of mathematics in the comprehension of the physical world. The infinitesimal calculation is a fruitful mathematical tool to discover the laws of the nature.

By applying the mathematics the mankind will dominate the nature and go to the moon, he moderates the bad effects in nature by constructing bridges, barriers etc …

For this reason mathematics, and especially geometry, must be a perfect science, holding the truth in the content of their first propositions. The Greek geometers understood, perfectly, this requirement. And the “Elements of Euclid” were considered during two thousand years like a sacred piece mathematical knowledge. Their axioms, self-evident and true, enable the geometer to derive the exact and necessary theorems describing the relations between the elements of the figures traced in the three dimensional real space.

The mathematicians have tried to
dismiss the geometry from giving eternal truths by the discovery of
Non-Euclidean geometry in the first half of 18^{th} century. The
incapacity of geometers to find a proof to the fifth postulate, or to find a
contradiction in the two Non-Euclidean geometries: hyperbolic and elliptic,
lead some mathematicians to propose the modern axiomatic theories, which all
expulse the meaning from the first propositions, and the principal victim were
the geometry, injured by the “Foundation of geometry” published by David Hilbert
(1862-1943) in 1899. Many readers will be scandalized by my affirmation, but
they should know that the fifth postulate is well proved in my book” The proofs
of the parallel theorem” who exposes more than one hundred proofs to settle the
problem raised by the fifth postulate. One proof is sufficient to show that the
Non-Euclidean geometries are wrong, and that the book of Hilbert cannot serve
as sound foundation for Euclidean geometry.

Conclusion

There is a mutual relation between mathematics and the physical world, and their interplay is benefic for both. Mathematics help to understand the universe whose facts play an important role in the development of mathematics.

The role of mathematics is not only confined to understand the universe, but it extends into many fields of the human activities: material activities, or mental activities.

Pythagoras (590-500) had well seen the intimate relation between mathematics and music.

Because mathematics is the language of the nature and of all human activities, either material or intellectual, it must be a true and exact science.

The readers are invited to visit the website dedicated to the mathematical truth:

**www.mathtruth-rachidmatta.com**

The next article will treat the question of the mathematical truth.

Rachid Matta MATTA

2008-06-09