Mathematics leads in many directions and to many seemingly strange ideas. If the mathematics behind discoveries is consistent, the ideas are usually portals to new mathematical breakthroughs. Along the way, there have always been mathematicians who would not accept or even consider the possibilities and usefulness of new mathematical ideas that popped out of the expansion of mathematical concepts. Among these are:
- The Pythagoreans refused to consider and other irrationals as numbers.
- Early Europeans (even during the medieval era and the beginning of the Rennaisance era) generally dismissed the use of Hindu-Arabic numerals in favor of Roman numerals. They especially didn’t want to consider “zero” as a number — calling the notion of “nothingness as a number” as ludicrous. The only groups of people who use Hindu-Arabic numerals at the time in Europe were bankers and traders. They recognized the advatages of Hundu-Arabic numerals over Roman numerals. Imagine doing long multiplications and divisions using Roman numerals.
- It was Leonardo of Pisa, also known as Fibonacci, who popularized the use of Hindu-Arabic numerals during the 13th Century. In his book Liber Abaci (1202), he argued that the Hindu-Arabic numeral is more superior and flexible than the Roman numerals. Fibonacci was the first person who tried to convince the general public to switch to Hindu-Arabic numerals, and his efforts had paid off. But even then, many people still refused to switch to Hindu-Arabic numerals and several schools refused to include it in their curriculum.
- Most mathematicians of the 16th & 17th century refused to accept negative numbers, let alone imaginary, complex. quaternions, transcendental and transfinite numbers. During this era we have Nicholas Chuquet and Michael Stifel referring to negative numbers as absurd, and Girolamo Cardano giving negative numbers as solutions to equations even though he considered them as impossible answers. Blaise Pascal was reputed to have said “I have known those who could not understand that to take four from zero there remains zero.”, On the other hand. Albert Girard recognized complex numbers as formal solutions to equations which had no other solutions.
- In 1585, mathematician Simon Stevin published a pamphlet titled De Theinde (“the tenth”). The pamphlet explained the value and applications of decimal fractions like for extracting square roots. A number of ideas in the pamphlet was ahead of Stevin’s time. The metric system was introduced more than two centuries after the publication of the pamphlet, but there was already a discussion of decimalization of system of weights and measures in the pamphlet. Stevins stated that it was just a matter of time before the universal introduction of decimal weights, measures and coinage.
- From the late 1860s to the early 1900s, traditional mathematicians refused to accept what they called mathematical anomalies/monsters that surfaced from the works of such mathematicians as Cantor, Wierstrass, Peano, Koch and Sierpinski, which led to today’s evolution of fractal geometry.