As a re--cognizing fruit to certain knowledge system, paradox in our science system is a kind of amazing treasures closely related to the philosophy of science. In a special way, it purposely discloses certain defects in some of our scientific cognitive ideas and behaviors as well as some fundamental defects in our present science system. The exactly same producing mechanism and surviving conditions decide those defects and their relating paradoxes coexist with the same fate（birth or death）in our science system. So,（1）the paradoxes are unavoidable if the relating defects are not solved, not only those paradoxes are not be able to disappear themselves but their family members will increase more and more keeping disclosing the same defects from different angles;（2）the paradoxes relating defects are unavoidable, these fundamental defects will mislead our working ideas, result in mistaken cognitive behaviors and fruits in our science;（3）it is impossible to solve those paradoxes produced by the innate fundamental defects of the relating science system in the theoretical frame of this very same science system itself------the newly born “Paradox of Harmonic Series” has been a typical example.
The serious defects disclosed by Zeno’s Paradoxes Family have been there keep challenging us for more than 2500 years, unavoidable and unsolvable in present classical science theory system! The new studied and cognized fruits against such a “huge comprehensive syndrome group” are not only to bring a revolution to the foundation of infinite related mathematics but also a revolution to philosophy of mathematics and philosophy of science------solving “mathematical ‘time—space’ absence defect” and dispelling the “2500-year-old huge black cloud of infinite related paradoxes over mathematics sky”.
The five typical defects disclosed by the members of Zeno’s Paradoxes Family have been there challenging us for more than 2500 years, unsolved and unavoidable. This huge black cloud of infinite related paradoxes over mathematics sky” can only be dispelled with the help of philosophy of mathematics.
Five of my published papers have been up loaded onto RG to answer such questions:
1，On the Quantitative Cognitions to “Infinite Things” (I)
https://www.researchgate.net/publication/295912318_On_the_Quantitative_Cognitions_to_Infinite_Things_I
2，On the Quantitative Cognitions to “Infinite Things” (II)
https://www.researchgate.net/publication/305537578_On_the_Quantitative_Cognitions_to_Infinite_Things_II
3，On the Quantitative Cognitions to “Infinite Things” (III)
https://www.researchgate.net/publication/313121403_On_the_Quantitative_Cognitions_to_Infinite_Things_III
4 On the Quantitative Cognitions to “Infinite Things” (IV)
https://www.researchgate.net/publication/319135528_On_the_Quantitative_Cognitions_to_Infinite_Things_IV
5 On the Quantitative Cognitions to “Infinite Things” (V)
https://www.researchgate.net/publication/323994921_On_the_Quantitative_Cognitions_to_Infinite_Things_V
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