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On the other hand, Eleatics made the first paradoxes about the infinity in the universe. The paradox of Achilles and Tortoise is one of the most famous paradoxes, which states that if Achilles and a tortoise start running and the tortoise is from the beginning a distance in front of Achilles. So Achilles will never come up to the tortoise because the moment he will be at a place where the tortoise were, the same moment the tortoise will have covered a new distance and this distance between Achilles and tortoise is going to exist for ever… So do not be surprise with the idea that at the end of his life Zeno was crazy… Furthermore, Eleatics were the first ones whose cosmology included the word chaos meaning an area of absolute nothing like the space void.
With Ancient Greek people LOGIC was the beginning of all the science as Aristotle state the 4 rules of logic:
Principium identitatis, this axiom states that something is equal to itself, A=A, or the statement: 2+2=4.
Principium contradictionis, the logic is everywhere, because for example a human is human and not something else. So everything which is not no A is necessary A… (double negative)
Principium exclusi tertii, this axiom is based on Reductio ad absurdum method and states that A=B or A≠B.
Principium rationis sufficientis, everything has a reason if it exists, reason-result, for example the student did not write well, so they did not study.
And how all these ideas survived?
Arabs and Hindus saved a lot of the ancient passages with two ways:
1. They translated the ancient books in their language
2. They used them to create their own work.
One of their greatest contributions to Mathematics was: 0… And from zero we have the philosophy of nihilism (nihil=zero), which states that everything comes out of nothing …
Middle age in Europe
During the Middle age, every revolutionary or innovative idea was burned in the name of Christ. However the first idea that managed to survive was…
Thomas Hobbes (1588-1679) was one of the 1st empiricists. He supported that if we can’t imagine something, it doesn’t exist. This Theory was Plato’s Theory Reversed, because Plato stated that we live in an Idea World…
Moreover Quine and Putnam support the indispensability argument: Mathematics is indispensable to all empirical sciences, and if we want to believe in the reality of the phenomena described by the sciences, we ought also believe in the reality of those entities required for this description. That is, since physics needs to talk about electrons to say why light bulbs behave as they do, then electrons must exist. Since physics needs to talk about numbers in offering any of its explanations, then numbers must exist.
Formalism was one of the main theories created in 19th century. Its supporters believed that there is no problem without a solution, under the necessary circumstances such us to say the problem in a correct way. One of the main supporters, David Hilbert, said: in Mathematics there is no Ignorabimus which means that in Mathematics there is nothing we are going to avoid because we do not know it, this was the only statement accepted by everyone… Moreover Hilbert believed that we can create an algorithm which will give a solution for every problem. In Philosophical option it could be a way to find solutions for all the every-day problems… His ideas were laughed by Intuitionism, especially by Poincare who said that Hilbert’s ideas was like having a machine and you put on the one side pig and at the other side you get sausages…
Logicists believe that Mathematics are a part of our knowledge of logic, so every mathematical statement should be necessary a logical truth.