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What role does Hume's principle (HP) play in neo-logicism?

From what I understand Frege used Hume's principle in Freges theorem to create Peano axioms from HP by using second-order logic and that Freges theorem was the basis for neo-logicism.

But I don't understand what role HP play in the neo-logicism or in what way Freges theorem was the basis for neo-logicism more than that HP led to Freges theorem that led to neo-logicism?

Help would be much appreciated!

Thanks in advance!

ida
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  • Hume's principle (HP) was an assumption weaker than Frege's incoherent axiom V that was nonetheless sufficient to build arithmetic "from logic" his way, see [How does Frege's definition of number solve the Julius Caesar problem?](https://philosophy.stackexchange.com/a/41131/9148) Neologicists expand on his project to show how most truths of mathematics at large are analytic (independent of experience) because they are derivable from logic and HP, while arguing that HP itself is analytic, see [SEP, Neo-Fregeanism](https://plato.stanford.edu/entries/logicism/#NeoFre) for details. – Conifold Jan 15 '21 at 20:51
  • See [Frege's Theorem](https://plato.stanford.edu/entries/frege-theorem/): Frege's work can be recovered from the inconsistency following Basic Law V due to the fact that axioms of number theory can be proved from Hume’s Principle alone. "Heck (1993) showed that although Frege did use BLV to derive Hume’s principle, the subsequent derivations of the axioms of number theory from Hume’s Principle never made an essential appeal to Basic Law V. Since Hume’s Principle can be consistently added to second-order logic, we may conclude that Frege himself validly derived the basic laws of number theory." – Mauro ALLEGRANZA Jan 16 '21 at 10:08

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