A Criticism and Evaluation of the Modern Mathematical Platonism

Some mathematical philosophers believe that we can achieve a new and better version of mathematical Platonism, by eliminating defects of original Platonism. According to Brown's version of Platonism, that here we call it “Modern Platonism”, the nature of mathematics can be formulated in these seven theses: realism, abstraction, particularity, Intuitiveness, priority, fallibility, and extensibility.
This paper criticizes and evaluates the New Platonism, according to two major criteria: the social acceptability, and the methodological acceptability. The social acceptability of a theory, according to my definition, is the interest and attitude of the people to that theory; and it can be measured on the frequency or percentage of interested parties. But the methodological acceptance of a theory means to match it with criteria such as consistency, simplicity and explanatory power; and its value can be assessed based on its success in solving philosophical problems related to it. According to Brown, the new Platonism, is the best philosophical theory about the nature of mathematics, both sociological and methodological. As for sociological criteria, we can be sympathetic and agree with Brown. That is, it seems that the new version of Platonism is still acceptable. But it needs to prove its methodological acceptability. Because the access problem and the certainty problem are still not resolved.