What is Mathematics 2

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Development of Mathematics

We have seen the development of various kinds of numbers and operations till now. Let us now take a bird&rsquo;s eye-view of the development.

It was discovered that the four operations were also related to one another. Hence expressions involving any operation could be rewritten in terms of other operations.

The place value concept helped in writing numbers of any magnitude using just the ten numerals! It also simplified the algorithms for the four operations so that even a child in primary school could do it.

The question of &ldquo;what is 5-5?&rdquo; led to the idea of zero as a number. Since it was not a number found in nature, 0 added to natural numbers the new set was called whole numbers.

The question of &ldquo;what is 3 – 5?&rdquo; led to the idea of negative numbers. When negative numbers were added to the set of whole numbers, the new set was called integers. The rules of operations with integers was worked out by application of the fundamental laws of arithmetic. This is the reason for the famous operation &ldquo;- &ldquo;X &ldquo;–&ldquo;= &ldquo;+&rdquo;!

Sharing things was a daily life activity in a family or society. It was clear that 8 fruits shared equally between 2 persons resulted in each getting 4 fruits. This was also written as 8 &divide; 2 = 4. Now a question was asked &ldquo;what is 8 &divide; 3?&rdquo;. This led to the idea of fractions. Operations on fractions were also worked out from logic and the application of the fundamental laws of arithmetic.

Geometers were using the concept of ratios to discover properties like similarity. Ratios were seen to be very similar to fractions but also different in many senses. From this emerged the idea of rational numbers. It was felt that anything in the Universe could be explained using rational numbers.

From the study of Pythagoras theorem, numbers were discovered which were not rational numbers! This revealed the existence of irrational numbers.

The study of geometry also revealed the existence of strange numbers like &pi; &amp; ϕ! which turned out to be irrational numbers.

As every student knows the rules of operation with fractions were not very intuitive. Hence there was a search for a better way to represent fractions. Decimal fractions were the answer. From decimal fractions came the idea of decimal number system. This system could represent all the numbers which had been discovered so far (and described above) on a common platform. These were called real numbers.

It was also discovered that all real numbers could be plotted on the number line which was now called the real number line. The number of points on this line was infinite!

Then a mathematician asked a question whether there is only one infinity or there are several infinities? If so then how do we compare infinities. Surprisingly the tool for doing this was found from the idea of &ldquo;one to one correspondence&rdquo; which is actually considered a pre-number skill!

What was surprising was that there were many kinds of infinities! Even on the number line, the infinity of irrational numbers was more than the infinity of rational numbers! And another surprise was that very few irrational numbers have been actually identified. The majority of them are totally unknown to us.

The study of integers had proved that &ldquo;- &ldquo;X &ldquo;–&ldquo;= &ldquo;+&rdquo;. This meant that the square of a positive or a negative integer was always positive.

Then a mathematician asked a question &ldquo;what is ?&rdquo;. This resulted in the invention of &ldquo;imaginary&rdquo; numbers which were numbers plotted on a line vertical to the real number line! This further resulted in the invention of complex numbers and a complex number plane.

Geometry studied the behaviour of figures drawn on a plane. One of the postulates of Euclid called the Fifth Postulate had been troubling mathematicians for over 2000 years. Then mathematicians asked the question &ldquo;what if the fifth postulate was wrong?&rdquo;. Surprisingly the answer was that it was not wrong but true only in one kind of geometry, the geometry of the plane.

When they pursued this question further, they came out with two different geometries which were different from plane geometry.

One of them called spherical geometry was found to be more suitable in explaining the geometry of large structures on the Earth which was a sphere. The other called hyperbolic geometry was found to be suitable in explaining the geometry of space and the Universe.

Another mathematician asked the question whether logic can be mathematized? The result was Boolean Algebra which is the mathematics of logical values (True &amp; False). The design of integrated circuits is based on principles of this algebra.

Some mathematicians tried to understand the root of mathematics. Since math is essentially logic, they asked if the entire field of mathematics can be logically derived from first principles. The answer, from Kurt Godel, was surprisingly, No. He proved that any field of mathematics has to be based on certain assumptions which themselves cannot be proved. They have to be just accepted. To that extent, math was subjective. Godel showed the difference between Truth &amp; Proof. Some truths cannot be proved. They have to be accepted.

Power of Mathematics 

Many of the ideas that have been discussed above were discovered much before any use or application was found for them. Imaginary numbers were discovered much before a discovery that they could be used in equations related to electric currents. One hundred years after the invention of Boolean Algebra, it was found to be useful in mathematizing electronic circuits and the flow of information inside computers.

This has led to a speculation that since mathematicians are looking for patterns, some of them also have a seventh sense in sensing future developments and inventing the mathematics necessary to work on these future developments.

Math – A discovery or invention

The answer to this question is not still settled. Mathematicians and philosophers are divided on this. The link given below is to a Ted Ex video on this issue. Leopold Kronecker, a German mathematicians is quoted to have said &quot;God made the integers, all else is the work of man&quot;

 https://youtu.be/X_xR5Kes4Rs 

Podcast on the structure of math

In this Mathematips podcast, Lucy Rycroft-Smith meets Dr Cathy Smith to delve into the world of mathematical knowledge

 https://www.tes.com/news/exploring-knowledge-structure-maths?utm_medium=40digest.7days3.20191218.home&amp;utm_source=email&amp;utm_content=&amp;utm_campaign=campaign 

< 33.5 What is Mathematics 1 | Topic Index | 33.7 Vedic mathematics >