This is a great article on the history of how we have thought about knowing itself in philosophical, logical, mathematical and scientific contexts/intersections.
This is also coincidentally a perfect explanation of how fascism can co-opt science (arguably science was the necessary host for fascism) but it can never co-exist with it since one way to describe the fundamental root of fascism is to set it in terms of a radical extremist ideology centered on violently denying the reality that everything cannot be perfectly categorized, segmented and controlled, a reality which science, philosophy, art and math has illuminated in undeniable, and beautifully strange ways over and over again.
also see https://www.quantamagazine.org/series/the-evolving-foundations-of-math/
Fascists cannot emotionally handle the universe being anything other than a complex yet elegant clock humming along perfectly according to its mechanisms and perfectly predictable so long as you understood them, and yet the pursuit of knowledge has thoroughly proven that the universe is not that way. From the perspective of science then, fascism is the most cynical religion and the only religion that is fundamentally incompatible with science as it desires to occupy the describable and destroy the idea of the indescribable completely rather than positioning itself across the divide of the indescribable from science as other religions have positioned themselves... which contrary to popular belief as this article establishes itself, is not a divide to which Science is necessarily hostile.
edit This article is also a superb explanation of why Science it is crucial to distinguish between Science itself vs. the residue of facts it leaves behind.
The day before his Königsberg address, at a small academic meeting in the same city, a 24-year-old Austrian named Kurt Gödel had quietly announced that no such system for arithmetic could exist. Whatever reasonable set of rules you fix, something true must lie beyond them and thus beyond our knowledge. We can’t know. We won’t know. Gödel’s results, known as the Incompleteness Theorems, are landmark achievements of 20th-century logic. Recent work in philosophy and mathematics has begun to illuminate the gap they opened, and with it something deeper about the foundations of knowledge itself. The story begins in the same city, but two centuries earlier.
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Does one unpack ‘add 7 and 5’ to find ‘12’? Kant thought not. Rather, one constructs 12 from 7 and 5 in the mind. More than a century later, a young Dutch mathematician called L E J Brouwer, following Kant’s thought, argued that, if the mathematician’s work consists in such mental constructions, then one cannot assume any statement is either true or false. That was controversial: to strip the mathematician of that assumption, Hilbert retorted, would be like ‘proscribing … to the boxer the use of his fists’.
Gottlob Frege, working in the late 19th century, found Kant’s account of mathematics intolerable for different reasons: arithmetic, he claimed, is not psychological. The numbers exist whether anyone counts them or not. He set about reducing arithmetic to pure logic, grounded in the mind-independent categories of True and False.
The project consumed two decades of his life and produced some of the most consequential advances in the philosophy of language and mathematics in history. It is widely regarded as having sparked the modern discipline of logic. And yet, as the second volume of his masterwork, the Grundgesetze der Arithmetik, sat at the printer in 1902, the foundations cracked. The British philosopher Bertrand Russell sent Frege a letter pointing out an inconsistency at the heart of the system. A melancholy note in his diary later summed up his defeat: ‘My efforts to become clear about what is meant by number have resulted in failure.’
Russell had been wrestling with the foundations of mathematics for some years before his letter. It was only after meeting the Italian mathematician Giuseppe Peano at the International Congress of Philosophy in 1900 that he acquired the tools to attempt the problem. In place of Frege’s edifice, Russell set out to build his own, together with his colleague Alfred North Whitehead. Principia Mathematica appeared in three massive volumes between 1910 and 1913: all of mathematics, grounded in pure logic.
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The Tractatus became the manifesto of a group of philosophers in Vienna including Moritz Schlick, Rudolf Carnap and Otto Neurath, who called their position logical positivism. A statement is meaningful only if there is some way of settling whether it is true. Analytic statements are settled by unfolding the definitions of the words involved: for example, ‘All bachelors are unmarried’ is true because a bachelor is, by definition, an unmarried man. Synthetic statements are claims about the world, and are settled by – in principle – going out and looking: ‘It rained in Vienna on 3 June 1929’ is false because one could have observed that the sky over the Ringstrasse stayed clear that day. Anything else is grammar without content.
The Vienna Circle, as they became known, claimed that mathematics is analytic and science is synthetic. The function of mathematics is to extract information from scientific claims: it carries what is already there without adding anything, the way plumbing moves water without supplying any. If the Vienna Circle were right, then Kant’s and Brouwer’s restrictions on scientific and mathematical knowledge were entirely unnecessary. If.
By 1929, after almost three decades of labours by Hilbert, Russell, Frege, Wittgenstein, the Vienna Circle and Brouwer, among others, the foundations of mathematics looked very nearly finished. One beam remained that would lock the whole frame together. In the plenary addresses for the International Congress of Mathematicians in Paris in 1900, Hilbert had set out 23 problems that would define the next century of the subject. The second problem was this: prove that arithmetic is consistent – that is, no chain of reasoning from the axioms, however long, would yield a contradiction. The beam would not go in.
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What he announced at the Königsberg meeting the day before Hilbert’s radio address, no one had expected. Taking Principia Mathematica as the articulation of mathematics within logic, Gödel showed that there were true mathematical statements that cannot be proved using the resources of Principia Mathematica alone. Among them is the statement ‘Principia Mathematica is consistent.’ The trick at the heart of the proof is to encode statements about the system within the system itself – a move with the same vertiginous quality as Russell’s letter to Frege. But where Russell’s letter had killed a system, Gödel’s theorem killed a programme.
His argument was so general that it applied to any reasonable system for arithmetic. Hilbert’s dream of an axiomatic system capturing all of mathematics – or even just arithmetic – was over.
