"The Philosophy of Mathematics" translated from the "Cours de philosophie positive" [1851] of Auguste Comte in 1855 by W. M. Gillispie, Prof. for Civil Engineering & Prof. of Mathematics, published in New York.
Ordem e Progresso
After months of metaphysical studies, it is a breezy relief to deal with theories of positivism, which rejects metaphysic and theological explanations. Comte (1798 - 1857) was an influential French philosopher, mathematician and he was considered as the Founder of the doctrine of positivsm and the first modern philosopher of science. In the background of French Revolution, Comte sought to develop theories of social systems based on science (=positivism). Gillispie is underdrawing the importance of Comte's work and his want "(...) of a comprehensive map of the wide region of mathematical science". In his opinion Comte is capable to unite "clearness and depth, comprehensiveness and precision" and that those terms "have never been so remarkably united as in A. Comte's telescopic glance".
But not only Gillispie admires the ideas of Comte. Mill, writes about Comte in his "A System of Logic": "(...) by far the greatest yet produced on the philosophy of the science" and adds later on "this admirable work, one of the most admirable portions is that in which he may truly be said to have created the Philosophy of the higher mathematics." Again a praise for his commendable simplicity by Morell in his "Speculative Philosophy of Europe" [e-book here: https://books.google.sc/books?id=4uVRAAAAMAAJ&printsec=frontcover#v=onepage&q&f=false ]
"The classification given at the Sciences at large, and their regular order of development, is unquestionably a master-piece of scientific thinking, as simple as it comprehensive." Lewes titles Comte as "the Beacon of the ninetheenth century" in his work "Biographical History of Philosophy" and respectfully adds: "I unhesitangly record my conviction that this is the greatest work of our age."
Comte is not only examining the physical science of mathematics, but formulating theories in the light of positivsm, which are still forming the scientific system of validation (a philosophical current establishing that what is not proven by one's own experience cannot be considered valid, questioning any scientific statement that lacked solid evidence, but later in history criticized by Max Planck). Built and carried by similiar ideas, Comte established a theory of the three humane states of being (btw influencing Albert Camus' imagination of man and historicity):
I. Infancy of Humanity / The Theological State -> It was necessary to ask question, that would stay unanswered or would get an explanation through supernatural phenomena = presence of Gods. -> Social union by faith & no wars
II. Transitional Phase of Human Mind / The Metaphysical State -> Preparation of human mind for positive thinking III. Positive or Scientific State -> Breaking with previous two ideas for spiritually undivided focus on progress -> this concentration would eradicate all aggression, since everything and everyone is focussed on progress -> intellectual harmony = social harmony
"Love as a principle, order as a base, progress at last."
Comte had a big influence on many scientists of his time, but we should not underestimate, that this optimistic "belief" in science and unlimited progress [Note: What is progress? Which progress can be considered as worthy?] can reduce humanistic achievements.
