Alain Colmerauer Analogy : Rational Terms / Rational Numbers (Re: Analogy as a Core of Intelligence (Human & Artificial))

Hi,
An example of human intelligence, is of course the
name "Rational Term" for cyclic terms set forth by
Alain Colmerauer. Since it plays with "Rational Numbers".
A subset of cyclic terms can indeed represent
rational numbers, and they give a nice counter
example to transitivity:
?- problem(X,Y,Z).
X = _S1-7-9-1, % where
     _S1 = _S1-6-8-0-6-2-8,
Y = _S2-1-6-1-5-4-6-1, % where
     _S2 = _S2-0-9-2,
Z = _S3-3-0, % where
     _S3 = _S3-8-1
The Fuzzer 2 from 2025 does just what I did in 2023,
expanding rational numbers into rational terms:
% fuzzy(-Term)
fuzzy(X) :-
    random_between(1,100,A),
    random_between(1,100,B),
    random_between(1,10,M),
    fuzzy_chunk(M,A,B,C,X,Y),
    random_between(1,10,L),
    fuzzy_chunk(L,C,B,_,Y,Z),
    Z = Y.
% fuzzy_chunk(+Integer,+Integer,+Integer,-Integer,+Term,-Term)
fuzzy_chunk(0, A, _, A, X, X) :- !.
fuzzy_chunk(N, A, B, C, Y-D, X) :-
    M is N-1,
    D is A // B,
    H is 10*(A - B*D),
    fuzzy_chunk(M, H, B, C, Y, X).
Bye
Mild Shock schrieb:

Hi,
 Rota often celebrated symbolic, analogical, and
conceptual understanding over brute calculation.
This philosophy has come full circle in modern AI:
 - Large Language Models (LLMs) like GPT-4 don't
   just store facts — they recognize patterns,
   make analogies, and generate new structures
   from old ones.
 - Rota’s work in combinatorics, symbolic logic, and
   operator theory is essentially pattern-based
   manipulation — exactly the kind of reasoning LLMs
   aim to emulate at scale.
 Rota had a clear aesthetic. He valued clean formalisms,
symbolic beauty, and well-defined structures. Rota wanted
mathematics to mean something — to be not just correct,
but intelligible and expressive.
 In contrast, modern AI (especially LLMs like GPT) thrives
on the messy, including: Noisy data , Inconsistency ,
Uncertainty, Contradiction. AI engineers today are mining
meaning from noise.
 What counts as “structure” is often just the best
pragmatic/effective description available at that moment.
 Bye
 Mild Shock schrieb:

Hi,
>
Spotting Trojan Horses is a nice example
of creativity that also needs ground truth.
Gian-Carlo Rota was phamous for this truth:
>
"The lack of understanding of the simplest
facts of mathematics among philosophers
is appalling."
>
You can extend it to GitHub acrobats,
paper mill balerinas and internet trolls.
But mathematics itself had a hard time,
>
allowing other objects than numbers:
>
- Blissard's symbolic method
   He was primarily an applied mathematician and
   school inspector. His symbolic method was a way
   to represent and manipulate sequences algebraically
   using formal symbols.
>
- Gian-Carlo Rota (in the 1970s)
   Gian-Carlo Rota (in the 1970s) gave Blissard’s
   symbolic method a rigorous algebraic foundation. Rota
   admired the symbolic reasoning of 19th-century mathematicians
   and often described it as having a “magical” or “mystical”
   elegance — again hinting at interpretive, almost poetic, qualities.
>
- Umbral calculus
   Modern formalization of this method, often involving
   linear operators and algebraic structures. "Umbral"
   means “shadow” — the power-like expressions are
   symbolic shadows of actual algebra.
>
Bye
>
>
Mild Shock schrieb:

Henri Poincaré believed that mathematical
and scientific creativity came from a deep,
unconscious intuition that could not be
>
captured by mechanical reasoning or formal
systems. He famously wrote about how insights
came not from plodding logic but from sudden
>
illuminations — leaps of creative synthesis.
>
But now we have generative AI — models like GPT — that:
>
- produce poetry, proofs, stories, and code,
>
- combine ideas in novel ways,
>
- and do so by processing patterns in massive
    datasets, without conscious understanding.
>
And that does seem to contradict Poincaré's belief
that true invention cannot come from automation.

>