Tag Archives: Foundations of Mathematics

Mathematical Demonstration & the Doctrine of Individuals • 2

Selection from C.S. Peirce, “Logic Of Relatives” (1870), CP 3.45–149 93.   In reference to the doctrine of individuals, two distinctions should be borne in mind.  The logical atom, or term not capable of logical division, must be one of which … Continue reading

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Mathematical Demonstration & the Doctrine of Individuals • 1

Selection from C.S. Peirce, “Logic Of Relatives” (1870), CP 3.45–149 92.   Demonstration of the sort called mathematical is founded on suppositions of particular cases.  The geometrician draws a figure;  the algebraist assumes a letter to signify a single quantity fulfilling … Continue reading

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C.S. Peirce • Syllabus • Selection 2

But round about the castle there began to grow a hedge of thorns, which every year became higher, and at last grew close up round the castle and all over it, so that there was nothing of it to be … Continue reading

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C.S. Peirce • Syllabus • Selection 1

Selection from C.S. Peirce, “A Syllabus of Certain Topics of Logic” (1903) An Outline Classification of the Sciences 180.   This classification, which aims to base itself on the principal affinities of the objects classified, is concerned not with all … Continue reading

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¿Shifting Paradigms? • 4

Re: Foundational Crisis? • Harvey Friedman 2014 Aug 22 Shock and surprise are relative to a prior state of belief.  The belief that mathematics reduces to logic, and that of a purely deductive sort from given axioms, seems to be … Continue reading

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¿Shifting Paradigms? • 3

Re: What Is Good Mathematics? • Harvey Friedman 2014 Aug 17 Speaking of mathematics in the context of “general intellectual activity” brings to mind Raymond Wilder’s take on “mathematics as a cultural system”. I would like to keep that in … Continue reading

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Forest Primeval → Riffs & Rotes

Re: Shifting Paradigms? • (1) • (2) • (3) • (4) • (5) • (6) Prompted by the discussion of Catalan numbers on the Foundations Of Math List, I dug up a few pieces of early correspondence and later discussions that … Continue reading

Posted in Algebra, Arithmetic, C.S. Peirce, Catalan Numbers, Combinatorics, Forest Primeval, Foundations of Mathematics, Gödel Numbers, Graph Theory, Group Theory, H.W. Gould, Integer Sequences, Logic, Martin Gardner, Mathematics, Neil Sloane, Number Theory, Paradigmata, Peirce, Planted Plane Trees, Programming, Riffs and Rotes | Tagged , , , , , , , , , , , , , , , , , , , , , | Leave a comment

¿Shifting Paradigms? • 2

Re: Timothy Chow • Shifting Paradigms? 2014 Jul 31 I can’t remember when I first started playing with Gödel codings of graph-theoretic structures, which arose in logical and computational settings, but I remember being egged on in that direction by … Continue reading

Posted in Algebra, Arithmetic, Combinatorics, Foundations of Mathematics, Graph Theory, Group Theory, Inquiry, Logic, Mathematics, Model Theory, Number Theory, Paradigms, Peirce, Programming, Proof Theory, Riffs and Rotes | Tagged , , , , , , , , , , , , , , , | Leave a comment

¿Shifting Paradigms? • 1

Re: Dana Scott • Shifting Paradigms? 2014 Jul 28 This is very interesting to me, but not all my posts make it to the list, so I will spend a few days reflecting on it and post a comment on … Continue reading

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Infinite Uses → Finite Means

The idea that a language is based on a system of rules determining the interpretation of its infinitely many sentences is by no means novel.  Well over a century ago, it was expressed with reasonable clarity by Wilhelm von Humboldt in … Continue reading

Posted in Automata, Chomsky, Descartes, Finite Means, Formal Grammars, Formal Languages, Foundations of Mathematics, Infinite Use, Innate Ideas, Linguistics, Pigeonhole Principle, Recursion, Syntax, Wilhelm von Humboldt | Tagged , , , , , , , , , , , , , | Leave a comment