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