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Universal Mapping Properties and Artificial Intelligence

 
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Last Post by posina 2 years ago
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posina
 posina
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Topic starter 22/08/2023 7:47 pm   [#202]

Can AI abstract the architecture of mathematics?

 

In the early 1960s there was a mathematical advance, an advance on par with Newtonian mechanism in physics and Darwinian evolution in biology. A mathematical theory, prior to F. William Lawvere’s Functorial Semantics of Algebraic Theories (Lawvere, 1963/2004/2013), was a list of statements, which together determined whether a given object is this or that. So, a theory of a universe of discourse, say, the category of graphs (consisting of dots and arrows), had no choice but to leave the given universe for one, with no readily discernible kinship with graphs, of arbitrary symbols, words, and sentences, i.e., language. Following Lawvere’s functorial semantics, a theory of a given category of objects is a [sub]category with their basic properties as objects and mutual determinations of properties as morphisms (Lawvere, 2003; see also Posina, Ghista, and Roy, 2017). Along with the functorial semantics of Lawvere, sketches of Bastiani and Ehresmann (1972), and Grothendieck’s descent (see Clementino and Picado, 2007/2008, p. 15) contributed to the monumental development of our mathematical understanding of mathematics, wherein the relationship between particulars, theory, models, presentations, and doctrine is spelled out in a spellbinding display of science: ever-proper alignment of reason with experience.


This topic was modified 3 years ago by posina

bardo


   
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Descent Functorial Semantics Model Sketch Theory
posina
 posina
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Topic starter 25/04/2024 7:04 am  

One thing I thought of adding, which you all may know, unlike the universal mapping property (UMP) definition of PRODUCT that can be used to calculate products, the UMP definition of SUM can only be used to test if, say, 2 is the sum of 1 and 1 (in the category of sets; Lawvere & Rosebrugh, Sets for Mathematics, pp. 127-128), which can be gamified such that the UMP definition of SUM can be abstracted, using reinforcement learning, from the scoring statistics, or so i thought many years ago.  Having gotten nothing other than "I don't get it", I am wondering if you could, your time permitting, let me know if the UMP definition of SUM is not evident in my golf game: Thing 1 & Thing 2 vs. The Cat.

Thanking you, Yours truly, posina


bardo


   
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