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To be concrete and specific, let us say we are working in Peano Arithmetic. Is it possible to assign a partial order among theorems of Peano Arithmetic that agrees with our vague intuitive notion of profundity? So for instance, "0=0" would be ranked less than "SS0+SS0=SSSS0", and both would be ranked less than "For all x and y, (x+y)=(y+x)". What I am really asking for is if there is any text on this or a similar line of thought. I would love to read such a text.

  • is profundity the right word? – user6917 Sep 5 '14 at 14:36
  • Your example regards the lenght of the formulae, i.e.the number of symbols; I think this is not meaningful. Sometimes it cab be used the rank of a formula, i.e. the number of leading quantifiers. Very important is the Arithmetical hierarchy which "classifies certain sets based on the complexity of formulas that define them." – Mauro ALLEGRANZA Sep 5 '14 at 15:16
  • @MauroALLEGRANZA Oh, length is only incidental. There are long formulas that are not intuitively that profound. Such as "SSSSSSSSSSSSSS0+SS0=SSSSSSSSSSSSSSSS0". I would definitely call that less profound than a theorem like the commutativity of addition. – user107952 Sep 5 '14 at 16:12
  • What about taking the length of the shortest proof? – Tim kinsella Sep 5 '14 at 23:41
  • But I think all this "measure" are about some sort of "complexity"; they not involve somithing like "informativeness" or "difficulty" or ... which can be the possible elucidations of profundity. – Mauro ALLEGRANZA Sep 6 '14 at 17:10
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No, not particularly; for example one of the most profound developments in Number Theory comes from its geometrisation in Algebraic Geometry ie notions of curvature and so on that were also profound in the theory of GR; this would not come up in the your description.

The Peano Axioms are really part of logic & foundations and are not really part of Number Theory as such; though of course one should be careful to not overstate subject boundaries; it essentially denotes the introduction of a new question into mathematics or rather the possibility of answering it - which is how is logic and mathematics related.

One of the most profound theorems is the theorem by Pythagoras; it has many applications and developments; the famous theorem of Fermats, which is in a sense just a generalisation of the one by Pythagoras is, despite its fame, empty of any real content. Its significance is that it was impossible to solve with techniques then current so it 'spurred' on the development of alternative ones. In itself, the theorem is hardly worth the trouble of thinking about it.

Still, there are possibities that your question opens up that are intriguing. Once mathematics is formalisable in an intuitive manner by theorem-provers such as Agda and their cousains; and a sufficiently large fragment is formalised one can begin to systematically investigate using statistical techniques linkages and affinities between theorems which may express degrees of applicability, profundity or aesthetic charm.

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