Cf: Higher Order Sign Relations • 3 https://inquiryintoinquiry.com/2019/12/23/higher-order-sign-relations-3/ Re: Ontolog Forum • Joseph Simpson https://groups.google.com/g/ontolog-forum/c/B9HpfImt3aQ/m/R7shee_3BAAJ Re: Relations, Types, Functions https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Rel_Typ_Fun All, JS refers to the following passage from my text on Inquiry Driven Systems. <QUOTE JA:> The subject matters of relations, types, and functions enjoy a form of recursive involvement with one another which makes it difficult to know where to get on and where to get off the circle of explanation. As I currently understand their relationship, it can be approached in the following order. • Relations have types. • Types are functions. • Functions are relations. In this setting, a “type” is a function from the places of a relation, that is, from the index set of its components, to a collection of sets known as the domains of the relation. </QUOTE> My 3-basket mantra recited above harks back to the mid 1980s when I took a course on “Applications of Lambda Calculus” from John Gray at Illinois. It was all about categories, combinators, and computation, focusing especially on cartesian closed categories, one of the hot topics of the day. We had a packet of readings from the classic sources and used J. Lambek and P.J. Scott's “Introduction to Higher Order Categorical Logic” as our main text. I followed that up with a supervised independent study where I explored various themes of my own. The directions I pursued and continue to explore all have to do with mutating category theory just far enough to encompass Peirce's 3-eyed vision in a more natural fashion. I'll make that more explicit when I next get a chance. Regards, Jon

Hi Jon, Is there any good contemporary reading of Peirce & James that you recommend? Their original works have been quite challenging for me. Best regards, Gary Sent from my iPhone

Cf: Higher Order Sign Relations • 6 http://inquiryintoinquiry.com/2022/08/14/higher-order-sign-relations-6/ Re: Cybernetics ( https://groups.google.com/g/cybcom/c/UwInmSnuv0c ) ::: Cliff Joslyn ( https://groups.google.com/g/cybcom/c/UwInmSnuv0c/m/w9ncBWKUAAAJ ) Cliff Joslyn recommended the following books. • Spivak, David I. (2014), Category Theory for the Sciences ( https://www.amazon.com/Category-Theory-Sciences-MIT-Press/dp/0262028131 ) • Fong, Brendan, and Spivak, David I. (2019), An Invitation to Applied Category Theory : Seven Sketches in Compositionality ( https://www.amazon.com/Invitation-Applied-Category-Theory-Compositionality/… ) Dear Cliff, All, The following Survey page gives a hint of the tack I've been taking with category theory since the early days but definitely moving into higher gear during my year at Illinois in the mid 1980s. John Gray taught a course joint between math and computer science on the Applications of Lambda Calculus and David Plaisted taught a course on Resolution-Unification Theorem Proving, both of which I took and followed up with independent studies. I spent a heady year making the circuit between math, computer science, and psychology departments and a lot of what I work on today goes back to issues raised in those days. • Survey of Precursors Of Category Theory ( https://inquiryintoinquiry.com/2020/09/20/survey-of-precursors-of-category-… ) I know that Survey from a couple years ago still looks a little sketchy but I'll be working to make it less so as time goes by, especially if I ever get around to unpacking my notes from the basement boxes. I have managed to sample contemporary approaches to categories at sundry sites around the web over the last couple of decades — John Baez, nCafe, nLab, Zulip Category Chat, Topos Group, etc. As great as all that is there's a reason why it bears but tangentially on the issues I've been pursuing. That has to do with the Peirce Factor and how far a given line of inquiry takes account of it. As luck would have it, one of the texts John Gray used for his course, Lambek and Scott's “Introduction to Higher Order Categorical Logic”, resonated strongly with themes I knew from Peirce and that led me to many adventures of ideas still in progress. The following set of excerpts I shared with the Standard Upper Ontology Group back in the day may suggest the character of that work. • Lambek, J. and Scott, P.J. (1986), Introduction To Higher Order Categorical Logic • Excerpts ( https://oeis.org/wiki/User:Jon_Awbrey/Mathematical_Notes#HOC ) • Discussion ( https://oeis.org/wiki/User:Jon_Awbrey/Mathematical_Notes#HOC_Discuss ) There's a lot more to say, but that's all I have time for today … Regards, Jon