μεταφορά and ἀναλογία

1. Is maths magic?
Is it right to draw inspiration from it, such as Jules Vuillemin wishing to transpose them to philosophy ? Or does mathematics / physics mimic a universal form of learning, whose language already offers the model, CF NL economics?
The modern mathematical point of view, that of structures (including functors and categories, CF 'on categories'), marks the triumph of algebra (CF algebraic geometry), i.e. rules / calculation algorithms. The constraints represented by these algorithms, the symmetries they encode, seem to represent a good compromise between tractability and demonstration power.
As already suggested in Symmetry Generalization I, II, in good science, tractability dominates the question of expressivity: tool conditions the explorable

2. the right point of view
As already discussed, CF 'against Vapnik' 13, learning is (art) to find the right point of view.
In a way, the good overview trivialises the field studied: the groups trivialize the resolution of the algebraic equations, the Game Theory trivialises most of the 'economic' problems, CF 'No equilibrium theorem'.
As if, by economic fact, calculation could not move us away from where we start, or that the point of arrival can hardly be more 'distant' than a 'rotation' of the point of departure. To be compared with the local / global exploration dilemma in optimization.
We find this trait throughout "récoltes et semailles" (RS), it is a well-known trademark of Grothendieck (RS p 669, Illusie https://lejournal.cnrs.fr/billets/grothendieck-and-dynamics-impressive, http : //www.cnrs.fr/insmi/IMG/pdf/Alexandre-Grothendieck.pdf)
In the Category paradigm, the good point of view is that of comparison and morphisms

3. Poincaré and the analogy
« Les faits mathématiques dignes d’être étudiés, ce sont ceux qui, par leur analogie avec d’autres faits, sont susceptibles de nous conduire à la connaissance d’une loi mathématique de la même façon que les faits expérimentaux nous conduisent à la connaissance physique. Ce sont ceux qui nous révèlent des parentés insoupçonnées entre d’autres faits, connus depuis longtemps, mais qu’on croyait à tort étrangers les uns aux autres ».
CF : « L’analogie algébrique au fondement de l’analysis situs », Herreman, in ‘L'analogie dans la démarche scientifique : Perspective historique’

4. The learning engine (learning to learn L2L) is therefore the comparison
Μεταφορά: trans-port
Ἀναλογία: (according to) ratio (ratio): proportion

5. 'Comparison' and 'Analogy' are fundamental aspects of knowledge acquisition, in
'Category: An abstract setting for analogy and comparison', Brown & Porter

6. ex 1: the concept of 'symmetry', CF SGII, must be understood first in the sense of group symmetry, ie of group morphism: a symmetry transports (rotates) a solid, or connects 2 positions of this solid , i.e. compares them

7. ex 2: homotopy / homology: from space to group
Here it is the comparison initiated by Poincaré / Betti between the topological spaces and the groups

8. ex 3: Galois theory: from (algebraic equations') fields to group
The Galois theory compares (in its Dedekindian version) (algebraic equations, more specifically) extension of fields and groups

9. The main point is perhaps less to be surprised to discover groups in topological spaces or algebraic extensions than to reaffirm Klein's point of view (and his Erlangen program) that one learns when one connects objects: when one studies its symmetries, or more generally its morphisms

10. enforcing comparison: this is the spirit of category theory