This is a summary of Russell’s logicism which was his attempt to show that mathematics could be boiled down to logic! That was the substance of his Principia Mathematica (1910-1913) which he wrote with Whitehead.
That first stark statement of mine may not be absolutely correct but it’s the best I can do. This post consists merely of excerpts, slightly paraphrased and edited from other sites (except where they are directly in quotation marks), that substantiate what sounds like an astonishing project by BR. In the end, it seems I haven’t got it wrong.
by an anti-philosopher and aspiring to be in plain English
BR wanted also to help show that mathematics was true!
I do not have a mind for the logic and mathematics that ‘Philosophy’ has always been, but I do for truth and profundity on matters of human life.
From here: Russell had hoped to show formally that mathematics was just logic ( I think that means that Russell had hoped that mathematics could be deduced from formal logic, i.e. logic written in symbols). But Godel in 1931 showed that there are some mathematical truths that cannot be so deduced. This is known as Gödel’s Incompleteness Theorem. W. V. O. Quine also criticized Russell’s thesis. However logicism is still alive throughout maths, philosophy, and beyond.
From here: ‘Their Mathematica tried to show that the mathematical notion of number rests on, or arises out of, the logical notion of class, that is, we come to understand what a number is through our grasp of what a class is.’ . (This begins to take us into what BR was doing: ‘Class’ means nouns,verbs, adjectives, adverbs, and perhaps other kinds of words.)
From here: He showed that all arithmetic could be deduced from a restricted set of logical axioms, presented in less technical terms in his Introduction to Mathematical Philosophy (1919).
‘…the goal of Principia Mathematica … is to find an undeniable reason for believing in the supposed truths of mathematics.’ (from here.) (I am amazed that some people didn’t consider maths ‘true’!)
‘The mathematical formulas of physics would be converted to symbolic logic.’ (from here). (Surely not only physics.)
From here: ‘The goal…is to defend the logicist thesis that mathematics can be reduced to logic. Russell believed that logical knowledge enjoys a privileged status in comparison with other types of knowledge about the world. If we could know that mathematics is derived purely from logic, we could be more certain that mathematics was true.’ (Again, I didn’t imagine that this latter was a worry to some people. This post also contains a lot of philosophy of mathematics. )
This same post also says: Logicism is a programme in the philosophy of mathematics, comprising 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.
Lower down it says: The aim of Principia was to show that all mathematical knowledge could be derived from purely logical principles. And also that Russell hoped to show formally that mathematics was just logic.
From here again, slightly edited: 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: 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’.
Or to go into it more fully, pretty much verbatim from here, though I am not up to understanding it: ‘In mathematical logic, [BR] 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 again, 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 first 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 is a paragraph 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….’ (I looked up Peano’s axioms but couldn’t understand them.)
The following is 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 are 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 electronic, 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. But that to my mind isn’t Wisdom. That’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.
Having a mind like BR’s, suited to philosophy, makes him unsuitable for human wisdom –the same throughout the whole history of philosophy.
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.)
BR’s logicism – an amazing thing to do but it’s not Wisdom a tour de force of pure braininess but it’s not wisdom
turning maths into logic.What on Earth has that got to do with Wisdom? Only a nerd or boffin would think it had.. Only someone with no human intelligence