In synthetic approaches to mathematical subjects, it’s not necessarily meaningful to ask what a mathematical object “is”, or “what’s going on”. It’s not about things being less than rigorous—rather, all that matters is the axioms and rules of inference you get to use in that particular area. ISTM that extending “tendrils of knowledge” can be modeled as making such ‘synthetic’ inferences, whereas backfilling involves finding different models of the same theories, to make conceptual understanding more feasible.
In synthetic approaches to mathematical subjects, it’s not necessarily meaningful to ask what a mathematical object “is”, or “what’s going on”. It’s not about things being less than rigorous—rather, all that matters is the axioms and rules of inference you get to use in that particular area. ISTM that extending “tendrils of knowledge” can be modeled as making such ‘synthetic’ inferences, whereas backfilling involves finding different models of the same theories, to make conceptual understanding more feasible.