Mathematical Realism (PRO) vs. Mathematical Anti-Realism (CON)

Mathematical realism is the view that our mathematical theories are true descriptions of some real part of the world. However, Mathematical Anti-Realism is just a view that Mathematical Realism false. [1] Mathematical Realism can be divided into two categories: Mathematical Platonism - the view that, there exists abstract mathematical objects (non-spatiatemporal mathematical objects, and mathematical theories provide true descriptions of such objects); [1]Mathematical Anti-Platonism - the view that mathematical theories true descriptions of concrete objects (Psychologism and Physicalism). [1]I am Platonist, and when I use term "mathematical realism" I generally refer to Platonism. Mathematical Platonism can be defined as the conjunction of following three theses [2]:(a) There are mathematical objects (EXISTENCE)(b) Mathematical objects are abstract. (ABSTRACTNESS)(c) Mathematical objects are independent of intelligent agents and their language, thought, and practices. (INDEPENDENCE) The Frege's argument for Platonism asserts that it is only tenable view of mathematics [3]. Frege's argument can be written as following:(i) Most sentences accepted as mathematical theorems are true. (TRUTH)(ii) Let S be one of such sentence.(iii) The singular terms of the language of mathematics purport to refer to mathematical objects, and its first-order quantifiers purport to range over such objects. (CLASSICAL SEMANTICS)(iv) By Classical Semantics, the Truth of S requires that its singular terms succeed in referring to mathematical objects.(v) Hence there must be mathematical objects, as asserted by Existence.Classical Semantics enjoys strong prima facie plausibility. For the language of mathematics strongly appears to have the same semantic structure as ordinary non-mathematical language. For example, the statement "there exist prime numbers between 12 and 35" is the same as "there exist trees in forest".Another example, if statement "there exist solution for x-2=0" is false, then there is no solution. It is a simple statement and and it can be true only if the object to which term refers exists, the solution can exist only if 2 exists (x=2 is the solution). Therefore, 2 exists. If 2 exists, then it is abstract object (it is non-spatiatemporal and independent, x=2 is objectively true). We can generalize it for all arithmetic objects.When a random person says, there exists a prime number which is sum of cubes of two natural numbers, it is not a product of his imagination, or matter of his opinion, but rather it is just a simple fact (sum of cubes of 3 and 4 is 91). "There exists" is one of the most common phrases in mathematical discourse, and whether one is talking about a prime number with a certain property or the solution to a differential equation, a statement about existence is a statement of fact, not a matter arbitrary choice or opinion [4]. So numbers do have some sort of objective and independence existence. Quine's Indispensability Argument is also good argument for Platonism [5]:(i) We should believe the theory which best accounts for our sense experience. (science, empirical method)(ii) If we believe a theory, we must believe in its ontic commitments. (ontological commitment)(iii) The ontic commitments of any theory are the objects over which that theory first-order quantifies.(iv) The theory which best accounts for our sense experience first-order quantifies over mathematical objects.(v) Therefore, we should believe that mathematical objects exist.Basically, if you believe Newton's theory of gravitation to be true, and formula to calculate gravitational force to be correct, then you should accept the existence of mathematical objects.Putnam's Success Argument [5]:(i) Mathematics succeeds as the language of science.(ii) There must be a reason for the success of mathematics as the language of science.(iii) No positions other than realism in mathematics provide a reason.(iv) So, realism in mathematics must be correct.So, if we assume science is best arbiter of what exists, and we know that science depends on Mathematics. Therefore, if science says numbers exist, then numbers do exist. -------------------------------------------------------------------------------------------[1] - Philosophy of Mathematics ( http://books.google.az... ) - Mathematical Realism and Ant-Realism by Mark Balaguer (pages 35-101)[2] - Platonism in the Philosophy of Mathematics, Stanford Encyclopedia of Philosophy - http://plato.stanford.edu... [3] - The Fregean argument for Platonism, Encyclopaedia Britannica - http://www.britannica.com...[4] - Do Mathematical Entities Really Exist?, Lee Lady - http://www.math.hawaii.edu...[5] - The Indispensability Argument in the Philosophy of Mathematics, Independent Encyclopedia of Philosophy - http://www.iep.utm.edu...