The branch of philosophy that aims to study the foundations, assumptions and the philosophical assumptions of mathematics is called the philosophy of mathematics.
If one considers the historical evidences of thinkers contributing to the ideas that pertain to mathematics, the examples are aplenty. These include two basic categories of philosophers of mathematics: Western Philosophers and Eastern Philosophers.
Western Philosophers have some great names attributed to them such as Plato and Aristotle. Plato concentrated his studies on the mathematical objects, especially their ontological status. Aristotle, on the other hand, contributed to the field of logic of infinity.
It was the great mathematician Leibniz, who focused primarily on the relationship between logic and mathematics.
The study of philosophy of mathematics is made interesting due to the following aspects of mathematics:
o Mathematics is based upon countless number of abstract concepts.
o Wide application of mathematics: It governs many activities of our day-to-day life, besides its application in physics, chemistry and even biology!.
o Infinite: This notion is a peculiar one and has always aroused interest of many philosophers.
The relationship between mathematics and logic is one issue that has been a recurrent one in the philosophy of mathematics. In the 20th century, the philosophy of mathematics revolved around set theory, proof theory, formal logic and other such issues.
Around the break of the 20th century, there were several schools of thought that philosophers of mathematics held. At this time, three schools emerged, namely: intuitionism, logicism and formalism. In the beginning of the twentieth century, there was also an emergence of a fourth school of thought: predicativism. Any issue that would come up at that time, each school would aim to resolve that or claim the fact that mathematics is not as inevitable as opposed to those who believe mathematics to be “the most trusted knowledge”.
It is the thesis that mathematics can be reduced to logic, thereby making it a constituent of logic. According to the logistics, the foundation of mathematics lies in logic and hence all the statements in mathematics are nothing but logical truths.
Simply put, this thesis suggests that mathematics is nothing but logic in disguise.
This is attributed to the works of Brouwer. Intuitionism states that mathematics is an act of constructing. This involves mental constructions.
In this program of reforming the methodology of mathematics, it is believed that there exist no mathematical truths that have not been experienced.
This program is attributed to the works of David Hilbert. According to Hilbert, the natural numbers can be thought of as symbols, and not as mental constructions, as opposed to the theory of the Intuitionists. These symbols are basic entities. And as far as higher mathematics is concerned; its statements are the strings of symbols, which have not been interpreted as yet.
Ordinarily, predicativism would not be considered as one of the primitive schools. This program is attributed to the works of Russell.
Now let us focus our attention towards the other contemporary schools of thought that have emerged in recent times.
This program holds that mathematics is not invented by the humans, it is only discovered. For example, shapes like circles and triangles exist in the nature as real entities.
It is a form of realism. According to empiricism, mathematics can not be believed to be knowledge without experiencing (priory).
Mathematical facts can be discovered by empirical research. All the knowledge that is acquired is due to the observation that we make through our senses.
The followers of this program are of the belief that mathematical statements can be viewed as the consequences of a number of manipulation rules applied upon the strings of numbers. There is another version to formalism: deductivism.
There have been many cases of mathematicians been intrigued and drawn to this subject of mathematical philosophy because of the sheer sense of beauty that they perceive in it.
One can only reach to a fundamental philosophical question, which has begun to obtain the consideration that it is worthy of: what is mathematical understanding?