## ... NOT IN MODERN AGES

### CAN BAŞKENT

In his/her “difficult to digest” article, Y.Zheng discusses the possibility of revolution when it comes to mathematics.

S/he, first, discusses the proposition of Crowe: Revolutions never occur in mathematics. Zheng, at first stage, tries to make the notions of revolution and mathematics clear. For Crowe, revolutions entail that ‘some previously existing entity must be overthrown and irrevocably discarded’. On the other hand, Dunmore argues that, revolutions occur in mathematics; but only in meta-level. In my opinion, it is somehow the revival of old Platonic arguments, which consider the mathematical way of thinking as idealistic.

Consequently, Gillies claims that revolutions, in which the previously existing entity persists, but experiences a considerable loss of importance, occur in mathematics.

Mehrtens thinks that the elementary changes in mathematics –since the mathematics is inseparably- lead to changes in ‘substance’ or in ‘content’ of mathematics.

After examining those ideas, a brief interpretation should be made. In my opinion, non-science disciplines, such as philosophy, mathematics and history, contain a meta- level. They all have some improvable propositions, such as axioms, which the theory is based on. Immediately after the theory is constructed, strangely, the theory is trapped in these pre-assumed propositions.[1] As far as my comprehension allows me, this is the rule of consistency. I do not have a concrete idea if the rule of consistency is merely a pre-assumed proposition for the relevant theory or not; but an ultimate rule for a theory seems totally strange to me. How can a theory be forced to be consistent?[2] During all my elementary level mathematics education, we were taught any axiomatic system was supposed to be consistent to support the truthfulness of ultimate and divine characteristics of mathematics -keeping the Platonic approach on. But, at the higher educational level, we were supposed to learn the different systems of mathematics- namely different from each other, perhaps the conflicting systems may exist at the same time –or/and in the same space (??).[3] On the other hand, in philosophy classes the pupils were supposed to criticize the philosopher trying to find the inconsistencies of him/her. So, forming a similarity between philosophy and mathematics lead us to the point that:

i) Philosophy regards the inconsistencies as an expected part of a theory whereas existence of inconsistencies is the collapse of a theory in mathematics.

ii) In revolution aspect, inconsistencies or other gaps may lead a revolution or a break point in mathematics, but never in philosophy.

Philosophy and mathematics share almost the same meta-level paradigms. But, when it comes to revolution, they seem to differentiate. Prior one has no concept of revolution, but the latter one is assumed to have.

I put forward the claim that: Both mathematics and philosophy are equal in their meta-level, but not in object-level. Obviously, their problems and language are different in object-level. One deals with functions, algebraic structures etc. whereas the other deals with existence, knowledge, and cosmogony etc. It’s observable that, in the second half of the 20th century, advances in philosophy of mathematics and philosophy of science made us comprehend the ultimate (i.e. meta-level) similarities of philosophy and mathematics. But, it should not be considered as the revival of Platonic approach, although some philosophers are indeed Platonic. At this stage, I argue that mathematics and philosophy serve to a same meta-level truth. I will not go further claiming, especially logic of philosophy or philosophical logic (for instance in St. Augustine, P. Abelard, St. Anselm kind of medieval philosophers) can easily be reduced to a mathematical structure. For instance, it is largely known that 4th century philosopher St. Augustine realized the importance of mathematics i.e. numbers as a philosophical notion. Examples are numerous as Kant (theory of categories and a priori existence), Whitehead (Universal Algebra) and Russell (Principia Mathematica with Whitehead) excluding Hilbert -of course. In this manner, the idea of mathematical revolution may vanish. There is no possible revolution in mathematics, in an area such that the object of revolution may not exist. Conclusion is straightforward: Although without examining the relations between mathematics and meta-mathematics, possibility of revolution may arise in the meta-level i.e. only it is possible for the paradigm of mathematics. Possibility of Non-Euclidean geometry could only arise in that circumstances such that the change in meta-level had already occurred.

As Kline said in 1972, “All mathematicians until about 1800 were convinced that Euclidean geometry was the correct idealization of the properties of physical space and figures in the space”.

But, moreover Zheng claims that the non-Euclidean geometry has greatly changed the whole mathematics. But, I can identify it as the new paradigm of mathematics. It was totally different in ancient times and in Hellenistic period; consequently it is now totally different from those times. At the same time, Zheng used the word development for the mathematics. I cannot comprehend the development of mathematics. Do we need to consider the Lobachevsky more advance than the Eudoxus or Archimedes? Time interval in which they survived was different, so how to compare? Attributing a progressive property to mathematics is only valid for the pedagogical purpose never for the philosophical purposes. Even anthropology cannot be progressive. While we are trying to survive in metropolitan settlements, many indigenous people still live in their communal lands. Moreover, postmodern primitivist approaches do not call the cities as revolutionary structures. So, the “new and different” always does not mean the good in revolutionary aspect. In that account, we need to call the non-Euclidean geometry as “good” for it to be revolutionary. What were claimed making the non-Euclidean geometry is not its mathematical structure, but its avant-garde orientation. Non-Euclidean geometry is the revolution only if its avant-garde orientation is “good”. But, it is impossible to attribute it as good, since the “goodness” is not something related with the mathematical concept; but the rationalistic and idealistic philosophy. It was St. Augustine who claimed the circle is better than the triangle. It is in the sense that the circle is divine. I do not think, our discussion is related with the justification of Neo-Platonic Christianity.

As I pointed at a very early stage, the possibility of revolution in mathematics is the revival of Platonism in mathematics. [4]

[1] Since I am not well acquainted with the Gödel’s incompleteness theory, I shall skip this approach. But, I made a small investigation in the vein of Gödel’ian approach. However, at first glance, my own presented ideas may seem similar or plagiarized; but as I have noted beforehand, I am so insufficient on the philosophy of Gödel.

[2] At page 180, Zheng says: Generally speaking, a theory may taken to be determine a definite structure only when it is consistent. However it should be noted that, although the condition of being consistent is rationally theoritacally, it is neither necessary nor sufficient for pure mathematical research. (…) mathematicians in most cases do not pay attention to the problem ofconsistency at all. (…) consistency is not a sufficient condition for pure mathematics.

[3] What does exist mean is the question to be asked; although I am supposed to be the one to answer it. But, again, there is a risk of skipping to a irrelevant topic for our present discussion. However, I mean by exist that, an idea of a mind whether it is observable or not; experimentable or not; provable or not. Some forms of existence cannot be observed (due to Heisenberg, the sub-atom particles); some cannot be experimentable (some sub-atom particles of quantum theory); some cannot be proved (axioms?).

[4] At page 176, Zheng claims that, geometry used to be concerned with the study of the forms and relations of emprical space –only within the limitations of Euclidean geometry (…). At that point, author implicitly make us think that the Euclidean geometry was empiric i.e. non-Platonic. Excluding some astronomers and Archimedes, the geometry in antiquity was totally platonic and non-empiric. Trisecting an angle was not raised from the practical needs. So, the manipulation of the author can be observed trying to leading us to the claim of the drevolutianary nature of the non-Euclidean geometry.