MAY AN ALTERNATIVE EXIST?
CAN BAŞKENT
G.G. Joseph, in his article discusses the Eurocentric theory on the origins of Mathematics. He first gives an outline of the Eurocentric view, then passes to the Arabic and Babylonian civilizations. Frankly speaking, in this article, I did not find anything new, which we did not cover in our classes, or worth mentioning.
But, as a pedagogical aspect one should consider why the pupils are taught the Eurocentric theory on the origins of mathematics. In my opinion, there are two reasons worth mentioning.
1. The dominance of Marxist perspective on the history G.G. Joseph had already pointed it out: “… theories of modernization or evolution developed within some Marxist frameworks are characterized by a similar type of Euro centrism” (p.68).
2. Paradigm Scientific revolutions, in my opinion, cannot be identified within the dialectic approach. I have no intention to go deeper in this preposition. However, if we consider any scientific revolution as Pythagoras[1], Copernicus, Einstein, Gödel etc., we are not able to find any historical background behind it. Moreover, throughout the history of science, we can see the absence of scientific revolution although all the material conditions for a revolution were already present. In my opinion, this situation can be described as paradigm. Science requires something more than the historical conditions; this is what I call paradigm. Thus, Eurocentric (or be call it western) paradigm is the actual tendency of the science, as it was once the Arabic or Babylonian or Greek.
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Furthermore, in my opinion, being attached to the current Eurocentric mathematical system –which can be identified as axiomatic proof system (APS for short)- and the dominance of this system, do not let us to witness any further revolutions in mathematics. As many times criticized and somehow developed by the philosopher of science/mathematics; there is still no widely accepted alternative to this system.[2] But, apart from the discussion of being consistent or not; APS has a more crucial point in its body that needs to be investigated.
In the pedagogical sequence of mathematics, we have always been taught firstly the theorem and later on the proof of it as usual in APS. But, the language (in our context it is the way thinking in mathematics; namely APS); strongly and deeply affects our methodology. Moreover, we –as students- became more and more dependent to APS. In other words, we are taught that, although it does not seem so, the construction of mathematical knowledge follows APS. But, when I discuss it with many instructors in our university; I surprisingly found out that, this is not how it works for the mathematical discovery. I also amazingly found out that, it is not true when someone go deeper in the history of mathematical discovery. Happily my comprehension in some way coincides with Lakatos’. When Lakatos investigated the logic of mathematical discovery in his highly refuted article (PhD thesis); he comes up with the conclusion of the benefits and necessities of refutations since they are for the support of the theory. But, in the process of my mathematical education, I have never came across with such a methodology. I have always been forced to learn the theories with preceding lemmas and following corollaries. I always have intuitively the impression that, there occurred no gaps or refutations against that theory. So, being followed by the Pythagorean - Platonic tradition; theories are (accepted) taught as FACTS. It is clear that these results mystify the mathematics. One can consider the idea of mathematics among the overall communities: Most people think that mathematics deals with some transcendental things although everyone is acquainted with the first transcendental idea – number.
Everyone agrees with the idea that mathematics is an abstraction[3]. We, in that way, formed a bijection between the ideas and relations in our mind and with the symbols and notations we use. I call it the natural bijection of reasoning between two infinite sets. Within my primitive reasoning I always come up with the questions:
* Can we find any other infinite set to which we can apply the natural bijection of reasoning from our mind?
* Can we find any other form of bijection or homomorphism that can work for other forms of reasoning?
* By the method of proofs, we intuitively form another abstraction: Abstraction of the abstraction.
I have no further reasoning in this topic.
Throughout my article, I tried to show the possibility of existence for alternative of the actual mathematical methodology in a very naïve way. However, my ideas can easily be supported with G.G. Joseph’s ideas. He put forward that in early 20th century; it was though that there occurred no other system apart from Greek system in ancient times. It is exactly the same nowadays. Mathematicians keep attached to the actual APS. But, we remember the Indian, Maya, Egypt mathematics. They were largely different from Greek style. But, we still appreciate non-Greece mathematics although they are never axiomatic. Once, in a very narrow scale there existed a non-axiomatic system of mathematics. This is the very first conclusion I drawn from our classes. Now, I look forward to get my next conclusion from the history of science and philosophy of mathematics.
[1] The claim that, Pythagoras was a revolutionary mathematician worths discussing.
[2] This claim is obviously due to my little research in this field.
[3] This is never a definition, but merely a description.