(See the first two paragraphs of my first post on BR, here.)
Russell’s attempt to show that mathematics is just glorified logic is called Logicism. It was a pretty convincing attempt. He published it in Principia Mathematica (1910-1913) which he wrote with Whitehead. This post is a summary of it, paraphrased from other sites that explain it for us.
Logicism is in the philosophy of mathematics and consists of one or more of these theses: that mathematics is an extension of logic, or some or all of mathematics is reducible to logic, or some or all of mathematics may be modelled in logic (from here).
The aim of Principia Mathematica was to show that all mathematical knowledge could be derived from purely logical principles; and also that Russell hoped to show formally (i.e. in symbols) that mathematics was just logic (from here again).
That is basically what Logicism is and all I can hold onto. What follows below within square brackets are individual statements that I just about understood and that may add substance and context to what he was doing. It all adds up to a tour de force of pure braininess like that of precocious schoolboys, but it just isn’t Wisdom. Just look here at Russell’s opinions on human matters including that of Israel/Palestine. That’s the kind of mind he had when applied to human issues.
[Frege had already begun reducing mathematics to logic, ‘continued it with Bertrand Russell, but lost interest….and Russell continued it with Alfred North Whitehead….inspiring some of the more mathematical logical positivists, such as Hans Hahn and Rudolf Carnap’, (from here).
BR also wanted to help show that mathematics was true, by showing that mathematics could be deduced from formal logic (from here). ‘Formal ‘means ‘written in symbols’, which logic had started being in the mid-19th century, from people like Boole, Frege and Gödel.
‘The mathematical formulas of physics would be converted to symbolic logic. (from here).
But it seems that Gödel subsequently showed against Logicism in his Incompleteness Theorem of 1931, that there are some mathematical truths that cannot be deduced from symbolic logic. W. V. O. Quine also criticized Russell’s thesis. Yet logic remains alive in maths, philosophy, and beyond.
From here: Russell and Whitehead ‘…tried to show that the mathematical notion of number… arises out of the logical notion of class…..’ ( ‘Class’ means a certain kind of word, such as nouns,verbs, adjectives or adverbs.)
From here: Russell also showed in less technical terms in his Introduction to Mathematical Philosophy (1919): that all arithmetic could be deduced from a restricted set of logical axioms. Axioms are statements regarded as self-evident. (So that relationship with logic seems different from mathematics being reducible to symbolic logic.)
‘…the goal of Principia Mathematica … is to find an undeniable reason for believing in the supposed truths of mathematics.’ (from here.)
Here is why showing that maths is just logic makes maths acceptable as truth: Russell and other philosophers believed that logical truths were special. First, they are true because of their form rather than their content. Second, we have knowledge of them a priori, i.e. without experience. (For instance, “penguins either do or do not live in Antarctica” is a logical truth: Regardless of whether we know anything about penguins, we are certain that this statement is true.) (From here, slightly edited. This post also contains a lot of philosophy of mathematics.)
Here is something, almost verbatim from here, that is beyond my understanding: ‘In mathematical logic, Bertrand Russell established Russell’s paradox, which exposed an inconsistency in naive set theory and led to modern axiomatic set theory. It crippled Gottlob Frege’s project of reducing mathematics to logic. But Russell defended logicism ( that mathematics is in some sense reducible to logic) and attempted this project himself, along with Whitehead, in the Principia Mathematica, a clean axiomatic system on which all of mathematics can be built, but which was never completed. Although it did not fall prey to the paradoxes in Frege’s approach, it was later proven by Kurt Gödel that — for exactly that reason — neither Principia Mathematica nor any other consistent logical system could prove all mathematical truths, and hence Russell’s project was necessarily incomplete.’
From here, with some editing: Logicians, beginning with Aristotle, have studied statements and arguments that have the quality of certainty and tried to discover what in their form makes them certain. The Principia is…an extension of this project from logic to mathematics. It aims to show that mathematical statements like “two plus two equals four” are true for the same reasons as our statement above about penguins.
‘Just as Newton’s Principia revolutionized physics, Russell and Whitehead’s treatise changed mathematics and philosophy’ (from here).
From here too:: Principia Mathematica brought forward mathematical logic as a philosophical discipline. It inspired metalogic which is the study of the properties of different logical systems. Most of the interesting results in logic in the twentieth century are in metalogic, and these have implications for epistemology and metaphysics (from here).
(From here.): One result of BR’s investigation of the logic of individual statements was that it led to proof that maths and logic are one (It would be useful to have an individual example, but I haven’t.)
From here: ‘…his work on the logical foundations of mathematics….. promised to establish formally the essential unity of logic and mathematics. …it is possible to begin with a restricted set of logical symbols and, using only simple inferential techniques, prove the truth of the Peano axioms for basic arithmetic….’ (I looked up Peano’s axioms but couldn’t understand them.) ‘…its ultimate success was significantly undermined by Gödel’s proof that some propositions necessarily remained undecidable…’
Paraphrased and edited from here, I think, for the next two paragraphs: Mathematical logic has had a great effect on the new analytic philosophy, which Russell helped found. Analytic philosophy is a philosophy by arguments, with assumptions and structure explicit and clear. This idea is directly parallel to the use of axioms and inference rules in formal systems [?] ….modern analytic philosophers try to justify each step of their arguments by some clear assumption or principle.
(That last paragraph is beyond me because I had thought Philosophy had always been by its very nature a rigorously logical enterprise, but apparently not!)
Both the technical apparatus of mathematical logic and its step-by-step reasoning have been used in such fields as computer science, psychology, and linguistics.
These excerpts from this site takes me into regions beyond myself: ‘…his work on the logical foundations of mathematics….. promised to establish formally the essential unity of logic and mathematics. …it is possible to begin with a restricted set of logical symbols and, using only simple inferential techniques, prove the truth of the Peano axioms for basic arithmetic….’
The following short paragraphs are a collection of BR’s thoughts as they appear on the Web, that I haven’t sufficiently understood:
He wrote about classes, types, tokens of words, which I think may be important in understanding mathematics as logic.
I think I have read that BR’s thoughts have been useful in creating Artificial Intelligence. This presumably includes the computer on my desk. I imagine bits of solid-state physics from Silicon Valley, populating a huge circuit and each going tick-tock, tick-tock, yes-no, yes-no, in a binary kind of way. It’s fiendishly clever and immensely useful. It’s the work of clever nerds and boffins. Accountants and actuaries do mathematical work far beyond me, but no-one says that’s Wisdom. Wisdom to me has to be a human wisdom, a human sensibility to human life, which to me logic and mathematics can’t give because they are in a different category.
This site also makes important comments that I don’t for the moment grasp: That BR didn’t think that either the ‘simplistic’ treatment of predicates in Aristotle or the ‘cryptometaphysical account of internal relations’ in Hegel, gave an adequate foundation for philosophy. Also that ‘the inductive reasoning of Bacon, Hume, and Mill offers grounds only for tentative empirical generalization’.
An ‘internal relation’ is an essential property of something that is also its relationship to something else. An ‘external relation’ is a relation to something else that isn’t an essential property of the original thing, (from here).]