Bertrand Russell, The Analysis of Matter, Frege’s Theorem

… i have begun re-reading The Analysis of Matter… my reason for reading it in the first place, and attempting to read it again now, is that it contains what many consider to be an argument supportive of the idea that there is some basic quality known as consciousness that is a fundamental constituent of the universe… that all matter exhibits awareness, down to the smallest fractions of that matter… that is, it is a text that can be viewed as supportive of panpsychism…

… as i have mentioned, i started reading this before the pandemic and got close to the end during the pandemic, though, during the pandemic, i don’t think i had sufficient patience and concentration to take it in fully… it’s a philosophical treatment of the subject of matter, which, at the level of Bertrand Russell, is challenging to read… arguments and examples are developed mathematically throughout the book, which makes it a further challenge as i have very little knowledge of the mathematics involved…

… similar to what Robert Haas says he has done with translating Japanese haiku, which is learn just enough Japanese and Chinese to be able to translate, i will do my best to familiarize myself with mathematical (and other) concepts i am not familiar with, trying to drive down to some kind of knowledge of base concepts on which the work is constructed…

… towards that end, my first inquiry (rabbit hole) is into Frege’s Theorem and Foundations for Arithmetic, which i know nothing about… i immediately encounter the phrase “second-order predicate calculus” which i also need to look up… Wikipedia describes second-order logic as “_ an extension of first-order logic, which itself is an extension of propositional logic¹. Second-order logic is in turn extended by higher-order logic and type theory._²

… what is propositional logic?… more commonly(?) known as Propositional calculus, it is a branch of logic… logic is defined in wikipedia as: _ Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and argument forms using formal languages such as first order logic. Within formal logic, mathematical logic studies the mathematical characteristics of formal languages, while philosophical logic applies them to philosophical problems such as the nature of meaning, knowledge, and existence. Systems of formal logic are also applied in other fields including linguistics, cognitive science, and computer science._³

… the article on logic points back to Gottlob Frege as one of the progenitors of modern formal logic…

… so, back to Frege’s theorem…

… Frege developed a second-order predicate calculus which he used to define interesting mathematical concepts and to state and prove mathematically interesting propositions.… in doing so he included as an axiom Basic Law V from which he derived fundamental axioms and theorem of number theory… he believed it to be a logical proposition which, as it turned out, it was not… the resulting system was flawed as well, failing to be consistent as it was subject to Russell’s Paradox.⁴

… so, about Russell’s Paradox…

• _ Russell’s paradox is the most famous of the logical or set-theoretical paradoxes. Also known as the Russell-Zermelo paradox, the paradox arises within naïve set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself. Hence the paradox.
• Some sets, such as the set of all teacups, are not members of themselves. Other sets, such as the set of all non-teacups, are members of themselves. Call the set of all sets that are not members of themselves “R.” If R is a member of itself, then by definition it must not be a member of itself. Similarly, if R is not a member of itself, then by definition it must be a member of itself.⁵

… the good news for Frege’s legacy is that buried in the invalid propositions and arguments of his Grundgesetze, is all the essential steps of a valid proof (in second-order logic) of the fundamental propositions of arithmetic from a single consistent principle.⁶

… known as Hume’s Principle, it asserts that for any concepts F and G, the number of F-things is equal to the number (of) G-things if and only if there is a one-to-one correspondence between the F-things and the G-things⁷

… the derivation of fundamental propositions of arithmetic from Hume’s Principle do not depend on Basic Law V, allowing Hume’s Principle to be used as an axiom. At this point, his accomplishment can be appreciated: his work shows us how to prove, as theorems, the Dedekind/Peano axioms for number theory from Hume’s Principle in second-order logic. This achievement, which involves some remarkably subtle chains of definitions and logical reasoning, has become known as Frege’s Theorem.