The evolution of mathematics over the years can only be described as a fascination. The word itself, which is derived from the Greek word mathea has gone through all imaginable innovations and renovations to reach where it is today. The dynamic condition of mathematics has made it an interesting area of study for enthusiasts who may want to take a stake from discovering or innovating something new in the field. However, the earlier works of various personalities in the past have formed the foundation for future innovations. Some of the two celebrated personalities are Friedrich Ludwig Gottlob Frege (1848-1925) and Giuseppe Peano (1858-1932).
Frege, who was a German mathematician, is accredited for being one of the pioneers of modern logic and foundations on the field of mathematics. The philosopher is also through his works tagged as a “father of analytic philosophy”. One of his down falls was however the lack of recognition he failed to command for his exemplary works but the likes of Peano and Russell propelled his works and ideologies to logicians as well as philosophers.
Although Frege started had given his earlier works a mathematical bearing, he soon opted for a logic approach. His main concept in approaching his works in this manner was to show that “mathematics grew out of logic”. His invention of “quantified variables” assisted him a lot in coining out “axiomatic predicate logic” which Peano would soon advance in his latter works. Frege through his fundamental aspects tried to sink his points into the minds of his fellow philosophers that arithmetic was a branch of logic. His main reasoning was that arithmetic had no basis for intuition unlike geometry. This as he argued in his book Foundations of Arithmetic in 1879 was that arithmetic had no need for the so called “non-logical axioms”. Thereby, Frege attempted to derive all the laws of arithmetic with the use of “logical axioms” (Torretti, 1978).
Peano, who was an Italian mathematician, was born in the northeastern part of Italy near a village known as Spinetta. He enrolled at the University of Turin in 1876 where he graduated with a remarkable ‘High Honors’. Afterwards, he began his teaching career. Peano published his first paper in 1881. In the course of his career he managed to publish over 200 papers as well as books which were mainly based on the mathematics subject. He is mainly famed for formulating the “Five Peano Axioms” which formed the foundation for the definition of natural numbers in terms of set elements.
One of the main highlights for these axioms is the first axiom which states that “There is a natural number 0”. However, this axiom is sometimes paraphrased differently with the number 1 starting instead of 0. Another axiom states that if ‘a’ is a natural number, then it must have a successor which is usually denoted by “a+1”. Peano also goes further to say that 0 is not a successor of any natural number.
When Peano was formulating his five axioms, the term “mathematical logic” was still at its infancy. This proved as a major challenge since the system in which he formulated his ideologies was not popular. In 1889, he put his ideas into pen and paper and published a book Arithmetices principia, nova methodo exposita (The Principles of Arithmetic). The major concept of the book was to give his five axioms a definition of natural numbers which according to Peano come in sets (Torretti, 1978).
Peano incorporated his ideas and findings in an “artificial and canonical language’ in which he deemed fit to integrate all his ideas. Peano through his works has earned wide acclamation. This has mainly been attributed to his use of artificial language which has been accredited for the restriction in using unambiguous words or constructions. Peano was not a favorite especially among his peers in the sense that he always found mistakes. It is also said that during his teaching career, he never gave students any tests.