Dear Mozibur,
You presented a very interesting essay in the spirit of a deep Cartesian doubt, with historical analytics of the development of science and philosophy, important ideas and conclusions to overcome the crisis of understanding in the philosophical basis of fundamental science.
Morris Kline in "Mathematics: The Loss of Certainty" showed well the entire hundred-year-old epic to overcome the crisis in the foundations of mathematics. M. Kline quotes the words of Hermann Weil, said in 1946 (16 years after Godel's discoveries): "We are now less than ever confident in primary fundamentals of mathematics and logic. We are going through our own "crisis" just like all and everything goes through it in the modern world."
Doctor of Physical and Mathematical Sciences Alexander Zenkin (1936-2006) in SCIENTIFIC COUNTER-REVOLUTION IN MATHEMATICS noted:
"About thirty years ago, for the sake of" sports interest "I began to collect various" logics "used in modern logical-mathematical treatises. When their amount exceeded the second hundred, it has become clear: if the logic can be selected "on a taste" (or even can be constructed "on a need"), such notion as "science" becomes here simply inappropriate. Perhaps, the situation somewhat reminds the famous "Babylon" epic: the sounds - symbols of abstract speeches are almost the same, but the sense, if that is present, of everyone is peculiar. What was the end of the First Babylon is described in The Holy Bible ... " A.Zenkin concludes: "the truth should be drawn ... "
The problem with the justification of mathematics and logic ontological basification) is problem No. 1 for fundamental science and for cognition as a whole. One hundred years of trying to solve the problem did not lead to success. Some philosophers of mathematics consider this problem "eternal". But the whole saga of solving the problem of "foundations of mathematics" says that the methods and approaches were inadequate.
As Grigory Gutner notes in "Ontology of Mathematical Discourse», understanding is "grasping the structure". To «grasp» the primordial generating structure, common to Nature and thinking, dialectic ontologics is necessary. But the philosophy of science "sweeps the rug" of dialectics. There are also problems in philosophical ontology. Need breakthrough new ideas. To solve the problem of the foundations of knowledge, a dialectic-ontological basification of mathematics and logic is necessary. Mathematics is the "language of Nature", therefore, a deeper observation of Nature and its absolute forms of existence is necessary. At the same time, it's good to recall the philosophical testament of the philosopher and theologian Pavel Florensky: "We repeat: worldunderstanding is spaceunderstanding / Повторяем: миропонимание-пространствопонимание."
If you have time, look also at my dialectic-ontological ideas .
Yours faithfully,
Vladimir