Category Archives: Paradigms

Survey of Abduction, Deduction, Induction, Analogy, Inquiry • 1

This is a Survey of blog and wiki posts on three elementary forms of inference, as recognized by a logical tradition that extends from Aristotle through C.S. Peirce.  Particular attention is paid to the ways these inferential rudiments combine to form … Continue reading

Posted in Abduction, Aristotle, C.S. Peirce, Deduction, Dewey, Discovery, Doubt, Fixation of Belief, Functional Logic, Icon Index Symbol, Induction, Inference, Information, Inquiry, Invention, Logic, Logic of Science, Mathematics, Morphism, Paradigmata, Paradigms, Pattern Recognition, Peirce, Philosophy, Pragmatic Maxim, Pragmatism, Scientific Inquiry, Scientific Method, Semiotics, Sign Relations, Surveys, Syllogism, Triadic Relations, Zeroth Order Logic | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | 1 Comment

¿Shifting Paradigms? • 4

Re: Harvey Friedman • Foundational Crisis? 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

Posted in Foundations of Mathematics, Inquiry, Logic, Mathematics, Model Theory, Paradigms, Peirce, Programming, Proof Theory | Tagged , , , , , , , , | Leave a comment

¿Shifting Paradigms? • 3

Re: Harvey Friedman • Good Math 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 mind, if … Continue reading

Posted in Foundations of Mathematics, Inquiry, Logic, Mathematics, Model Theory, Paradigms, Peirce, Programming, Proof Theory | Tagged , , , , , , , , | Leave a comment

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, Combinatorics, Forest Primeval, Graph Theory, Group Theory, Integer Sequences, Lambda Calculus, Logic, Mathematics, Model Theory, Number Theory, Paradigms, Peirce, Programming, Proof Theory, 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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Objects, Models, Theories : 4

I need to stay with this problem a while … What are objects, models, theories, and how do they relate to one another? In contemplating this problem I always find it helpful to ruminate on the diagram shown above — … Continue reading

Posted in Adaptive Systems, Analogy, Aristotle, Artificial Intelligence, Biological Systems, Computational Complexity, Evolution, Gödel, Information, Information Theory, Inquiry, Inquiry Driven Systems, Learning Theory, Logic, Logic of Science, Mathematical Models, Mental Models, Model Theory, Natural Intelligence, Paradigms, Peirce, Pragmata, Semiotics, Sign Relations | Tagged , , , , , , , , , , , , , , , , , , , , , , , | 1 Comment