# Discovery Science: Mathematics – Proofs

## New Mathematics

The foundation of mathematics is a logical proof based on pure thought, but at the same time, proven statements are generally accepted truths. A conflict exists in mathematics over whether it is a tool for science or an end in itself.

While there have been strong movements to reduce mathematical objects to their fundamental characteristics—to achieve the highest possible level of abstraction—modern life has made increasing demands on mathematics.

### Mathematics – Proofs

A mathematician is not satisfied until a statement is valid for all conceivable cases, not just hundreds, thousands, or millions of cases. A logical derivation that is true in all conceived circumstances is known as a proof.

According to astrophysicists, there are only about 1078 elementary particles in the universe. Why then do humans continue to try to prove a greater number of objects exist, or try to prove statements for infinitely more mathematical objects? The very nature of mathematics as a science is that it does not limit itself to finite entities.

For practical purposes many scientists and mathematicians are content to make approximations to obtain results that may be limited but work well for everyday life. However, human beings also have a natural compulsion to know everything, whether or not the knowledge gained has a practical use.

Using known, proven, or assumed statements, one can advance step-by-step to a logical derivation known as a proof. Natural numbers have an inductive character that allows statements about them to be proved. Because every natural number n has the successor n + 7, a proof can be constructed based on the principle of falling dominoes. All the dominoes fall if two conditions are satisfied: one domino must fall, and every falling domino must hit the next one.

When describing an infinitely large number of mathematical objects, a particular characteristic is specified, as in the theorem by the sixth century B.C. Greek philosopher Thales, which states that a triangle is always a right triangle if its base is a diameter of a circle and its vertex is a point on the circle.

Changing the logical structure is another way of proving a statement-instead of proving the statement true, one proves the opposite statement is false.

For example the theorem of the Greek mathematician Euclid, (born in Alexandria, Egypt, in 300 B.C.) that states there are infinitely many prime numbers can be proved by first hypothesizing the “greatest” prime number, and then proving that hypothesis false by finding another larger prime.

### THE WORLD OF PRIME NUMBERS

A prime number is a natural number that can be divided by only two particular numbers: itself and one.

Although the fact that there are infinitely many primes was demonstrated by Euclid over 2,000 years ago, unsolved problems remain and are the focus of many modern-day mathematicians.

### DIRECT PROOFS

Direct proofs are created using previously proven statements In combination with logical reasoning. For example, in this way the properties of angles can be used to prove that the sum of the angles of a triangle equals 180°.

As a demonstration a line parallel to one side of a triangle is drawn through the point formed by the other two sides. The three angles next to each other together form a straight angle of 180°.