Often philosophers seek refuge from doubt in mathematics; Plato, Descartes, Hume, Russell and many other philosophers have done so. The result for humankind has been disastrous. Some have tried to formulate moral goodness with numbers. If a theory of goodness could have mathematical precision we could scientifically equate intention to behavior or use it to trace backwards to find substantive moral order. Chaotic, fragile and complex creatures like us are going to abuse this idea. Our moral valuations can't be formulated in ordered intellectual patterns; the attempt is poisonous to the creative mind. Someone or some group will use it to practice a form of constructivism and restrict human liberty to get a result of a certain human behavior on a mass scale. Personal morality can't be attained by coercive acts; it must come from individual choices done voluntarily. All forced behavior is conformity to law and not moral freedom to live a life of goodness. We practice love through action and if some organization forces you to be charitable, your obedience doesn't make it so. Philosophers of mathematics don't often believe in that kind of moral freedom.
Math isn't entitled to its status as having analytic certainty anyway nor as any longer being regarded as an exact science any more than physics. That's a dogma and those days have passed. All knowledge is tentative.* It originates from human beings and we are all fallible. Most philosophers of mathematics are Platonists who suggest that in this evolving ongoing constantly-changing event of life there is one permanency. They write philosophical works like Approaching Infinity to deal with infinity and while discussing the modern assumptions of empirical knowledge, do not even once mention Popper, the philosopher who solved the problem of induction as Hume himself defines it. Mathematics concludes a concept of absolute, eternal truth, because of proofs but these are a formula inside strict logic with sometimes no application to the multiverse; like poetry, beautiful, concise and of no relationship or application other than proof for the sake of the proofs. "All mathematical statements are tautologies, Their conclusions are logical consequences of thier hypotheses. The hypotheses already 'contain' the information in their conclusions." Mandelbrot was cautious in overstepping the bounds of mathematics as all humble mathematicians should be, well maybe he wasn't the most humble but you know what I mean. All people, each and everyone of us, are fallible, especially intellectuals, many who have caused endless grief in the history of humankind, i.e., see Weapons of Math Destruction and Intellectuals and Society.
Pythagoreans once believed the world was created by numbers. No math can be known a priori; all knowledge comes to us in some manner through the senses. History teaches us that only human application creates or discovers ideas. This is our bitter choice: every bit of knowledge about the world: sweat, blood and a dear, dear price. Math the same as science, logic and all instruments of knowing, is a tool of greater reasoning. Even these tools have taken whole centuries to hone.
It is a myth that mathematics holds a special place in knowledge. By asserting certainty, it has led to calamity. The analytic movement's Platonic revolt against reason has basically been accepted carte blanche on faith. It has given credence to the perspective that the "Description of Physics" - the micro-world - was a superior view than the sensual one, sidetracking all metaphysics. Anyone foolish enough to buy into this Russellian reckoning of epistemology paid an enormous price, and almost everyone did. Why it is so appealing to be falling through space instead of sitting in that chair (that's apparently not there), is the license it gives to the act of free falling for intellectuals. Why let a fact stop you when they are a figment of your “unreliable” senses, at best a probability.
This is essentially why I suspect you'll never meet a fiscal-conservative Analyst or Positivist nor perhaps even one that is a critical realist. None of them give any credit to Aristotle. They don't think reason rules over philosophy, but believe rather that mathematics and science do. They don't believe in individualism and they think empiricism is better than continental philosophy by fiat. They think metaphysics is a word which should be transferred from philosophy to theology. They think that philosophy is inferior to science, should never arbitrate its disputes with better scientific explanations and with falsifiability. I could go on in this vein for some time but in lieu of more examples, I will simply point out that any opinion which takes in substantial facts and aims at tentative modest knowledge is a metaphysical theory. Like all theory it waits patiently to be adjusted, corrected or refuted, no different for scientific or mathematical ones. Notice how professional mathematicians acted in Nazi Germany or Bolshevik Russia. They were employees of the state - glorified technicians - how can the community claim superiority to philosophy. The real philosophers were the dissidents. How disheartening that so many of today's Analysts and Positivists are socialists and believers in the handmaiden state.
Today’s street philosophers are going to protest against modern philosophy, but not ever from within academia. There, right this minute, academic philosophers use their powers to drive genuine students of philosophy away. To be interested in metaphysics is to in effect come from the continent and be as irrational as Kirkegaard. Perhaps by world-wide edict, philosophers of mathematics should be scolded once a month in public to remind them that scientism, historicism, and constructivism are sins against reason, and therefore, when they feel compelled to speak out on non-mathematical matters they must leave arithmetic aside, and talk in organic metaphysical concepts based on actual existent facts, things or events.
Note: “[A certain named theoretical physicist] . . . . is revealing his adoption of a common conception that exists among today's theoretical physicists. They believe that the symbols that appear in their mathematical equations represent true reality while our observations, which always look like localized particles, are just the way in which that reality manifests itself. In short, they are Platonists. It is important to note that twentieth-century greats Paul Dirac and Richard Feynman were not part of that school.” God and the Multiverse, V J Stenger.
"Proof and observation are merely means by which we check our explanations. Roger Penrose has drawn a further, radical and very Platonic lesson from Gödel's results. Like Plato, Penrose is fascinated by the ability of the human mind to grasp the abstract certainties of mathematics. Unlike Plato, Penrose does not believe in the supernatural, and takes it for granted that the brain is part of, and has access only to, the natural world." The Fabric of Reality.
Hence also – as the mathematician Kurt Gödel had discovered using a different approach to Hilbert’s challenge – almost all mathematical truths have no proofs. They are unprovable truths. The Beginning of Infinity,
"You might expect a book by a mathematician and a biologist to praise the precision of mathematics as an instrument for digging out surprising biological truths. On the contrary, we both warn you not to take mathematical models too seriously. Surprising consequences are fine, but consequences so surprising that they don't make any sense are almost certainly based on false assumptions. Don't be impressed by mathematics just because you can't understand it."
© 2022 - E. A. St. Amant