Let me give a very concrete analogy between analogies.
is to
as
is to
.And this analogy is called generalization.
und H.-P. Stricker berichtet, dass ChatGPT-4 auf Eingabe dieses Beispiels so reagiert:
The analogy you’ve presented is a complex one, involving multiple layers of relationships. Here’s how you might describe it:
1. The first part of the analogy,
,
is a simple proportional relationship. It represents a ratio where the second term is twice the first, and the fourth term is twice the third.2. The second part,
, generalizes the first part, showing the same proportional relationship but with arbitrary constantsand
, and a scaling factor
.
3. The third part,
, introduces a quadratic equation and its solution. The equationhas a solution.4. The fourth part,
, generalizes the third part, showing a general quadratic equation and its solutions using the quadratic formula.So, the analogy is moving from a specific example to a general form, first in the context of simple ratios, and then in the context of quadratic equations. You might call this an analogy of analogies, where the structure of the analogy itself is mirrored at two different levels of abstraction.
In more formal terms, this could be described as a meta-analogy that explores the relationship between specific and general forms in both arithmetic ratios and algebraic equations.
and 
, and a scaling factor 
.
Kurz: Eine künstliche Intelligenz ist bereits in der Lage, Analogien zwischen Analogien zu erkennen und ist damit (nach Banachs Definition) der ultimative Mathematiker.
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