By the turn of the 20th Century, “Philosophers” had realised that Aristotle’s too vague “Logic” was inadequate to the task of proving the theorems of Mathematics. Consequently, between 1910 and 1927, Bertrand Russell, the famous “Philosopher”, and his thesis supervisor, A H Whitehead, wrote their modestly-titled, three volume Principia Mathematica, asserting the primacy of “Logic” over Mathematics, the very idea which had been Aristotle’s fantasy.

The essence of Russell’s thinking, and of the Hilbert Program, which supports it, is that the language of Mathematics can be fully formalised in the sense that its claims/theorems are meaningless. “Logic”, as “Philosophers” conceive it, consists of rule-following processes/programs doable by a machine, without any reference to what the words and sentences stand for. This is the root of the Artificial “Intelligence” that is still promoted by Computer “Scientists” today. And it is the source of much of the evil that, in the early 21st century, has become evident to everyone.

That is the moral argument against “Philosophy”. Computer “Science” muddies the waters here, because some of its products are valuable for humanity.

The hubris of the many modern “Philosophers of Mathematics" who — still today! — promote Russell’s empty vision far exceeds the hubris of Aristotle. Nevertheless, the evidence against modern “Philosophy” is the same as the evidence against Aristotle. It is a fact about/of Reality, that a particular theorem of Euclidean Geometry has a correct proof if and only if that theorem, read as a prediction, does not fail empirical testing.

As to you grasping the meaning of a theorem of Euclidean geometry, as only you can do, your personal judgement must be decisive. That requires you, first, to repeatedly test the theorem as a prediction, or to accept the honest reports of others who have done that testing; and second to accept as correct/true a proof of the theorem which may be the work of others, but which you have somehow made your own.

The theorem of Pythagoras is a beautiful example of this equivalence, known already to Euclid. It is both an astonishing empirical fact, borne out by whoever does the work with ruler and compass to test it, and a provable fact, borne out as Proposition 47 of Euclid’s Elements of Geometry.

That brings us to a conclusion, the declaration of faith that an individual, like you/me, who has done the necessary work is free to make, and should make: the many sciences that constitute the Science that human beings have created/discovered are objectively true in the simple empirical sense that individuals who do the necessary work are compelled, as one, to accept those sciences.

The optional final page of this Essay, called its Conclusion, may be too difficult for some/many readers. On that page, going far beyond what was known to Euclid, I try briefly to show, with just one example, the remarkable power of the new way of thinking called Algebra that was invented/discovered in the 19th Century.