CRITICS TO THE CLASSICAL APPROACH TO THE HISTORY OF MATHEMATICS
The definition comes with a memorial to Aristotle (Metaphysics (1090 a20)):
The Pythagoreans, because they saw many of the attributes of numbers belonging to sensible bodies, assumed existing things to be numbers–not seperately existing numbers, but that things are actually composed of numbers. Their reason was that numerical properties are inherint in a musical scale, in the heavens, and in many other things.
His worldwide reputation, as a geometer, was not enough to provide the deserved interest to Pythagoras’ philosophy. Mixed up with number mysticism, it was Pythagoreans who were probably the first introduced the non-materialistic view of existence and cosmogony. As stated in the article, the effects of Pythagoreanism has survived till the modern ages. To count some names, late Neoplatonists as Plutarch, some prescholastics as St. Augustine who endeavored to connect the truth with god by means of existence and knowledge, Thomas Aquinas who is one of the late scholars during the revival of Aristotle’s philosophy, made use of Pythagoreanism largely. Moreover author notes Russel and Frege.
On the other hand, while considering the Greek philosophy and mathematics, it is strongly needed to emphasize the speculative characteristics of that age when it comes to science. As a deep and great philosopher of the all ages, Plato seems to vanish when it is the science we talk about. As W.C.Dampier indicated in a funny manner “Plato was a great philosopher, but in the history of experimental science he must be counted a disaster.”
Plato argued that forms are real and do exist apart from the visible world. Thus, it is similar to the mathematical abstraction. Although two oranges exist in visible world the idea of being “two-ness” does not “exist”. Plato concludes, “Mathematics is real because it’s immaterial and eternal”.
As Pythagoreans’ philosopy lead, in some manner, to Plato’s forms theory, we may realize a visible and concrete relation between those and him. Many books on history of philosophy attribute both Pythagoras and Plato realists as Pythogoras claimed ‘number’ as the essence of existence in reality and Plato did it for the ‘forms’. In other words, both believe that “there is ‘something’ which exists”. In my opinion, after the relevant introduction I have noted till this point, this is the crucial point which needs to be investigated. As noted before, discussions on the essence of existence in ancient Greek culture were largely speculative, neither they were provable nor experimentable. Consequently, the same rule applies for Pythagoras and Plato.
Let me assume the philosophy be speculative, believing none would deny it. Here are some premises put forward with regard to ancient philosophy, again hoping none would deny it.
ASSUMPTION: Philosophy is speculative.
PREMISE 1: Ancient Greek philosophy is speculative. (Derived from the assumption)
PREMISE 2: Pythagoras and Plato are no scientists.
By using above premises, we may conclude that, what Pythagoras and Plato construct is no longer science, but rather philosophy which is speculative. So, what they did was just attributing ‘something’, which is real, as the essence of the existence; now, it is still not out of the borders we put with the premises. Hence, they both construct a realistic view of the universe making no use of science. On the other hand, they both mentioned mathematics as the primary tool of their philosophies. By Premise 1 and Preimse 2, we may manipulate the following conclusion: “Pythagoras and Plato, as non-scientists philosophers, made use of mathematics as a speculative tool.” I do not think one would reject the conclusion.
QUESTION: To what extend, it can remain as mathematics while being speculative?
Discussion is now forwarded to another point which is out of the objects of this article with many more questions regarding philosophy vs. mathematics and words vs. numbers. As an element of the language of the mind, which we need to consider first: number or word?
While, me myself, reproducing the speculations, how is it possible to find the answer of the question.
 Taken from the History of Science (Phil 182) course notes.
 Sir William Cecil Dampier, A History of Science, Cambridge:1966
 Title of the article regards word as speculations and number as mathematics.
 I see no irrelevance when replacing the word philosophy with mathematics in below brain-storming, since it is the antiquity we deal with.