Some very diffrent conceptions of mathematics,believed by mathematicians,are presented and compared.Bourbakism is opposed to its critics from several points of view.Unconventional theoreies such as Non-standard Analysis,Constructive Analysis are analized in detail.At last also some non-standard philosophies of mathematics,believed by philosophers,are discussed.In conclusion there is a contrversy about mathematics and among mathematicians much more deep and important than it is usually believed.This controversy cannot be confided inside philosophy because it invests the whole mathematics.Epistemological criteria are unable to explain why some mathematical theories are generally accepted and other are considered heretical.

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