In David Deutsch's book The Beginning of Infinity, he argues, that the purpose of science is not to merely predict or to model the world, but to explain it. And he gives in my opinion convincing arguments, why a better explanation is needed to advance science. Therefore, he deduces that pure empiricism has failed. Because no matter how much data you have gathered, an explanation does not follow from it. The explanation is always an insight the scientist has to contribute. Then it is open to scrutiny by criticism or experiments.

I guess he would argue the same is true for mathematicians. Merely constructing proofs to arrive at theorem, is of no intrinsic value, if it doesn't spark an idea of a more elegant formalism that corresponds to a simpler and even more far reaching explanation.

I cannot recommend this book highly enough to anyone who is intellectually curious, not only about epistemology.

[1]: https://web.archive.org/web/20180318023156/https://www.viemu...

Learning to do long division by hand trains the mind. It prepares you to understand polynomial division. It teaches you the deep connection of division and subtraction. So, mastering it should be part of the curriculum. As soon as it is truly mastered, it should be allowed to use a calculator.

One could wrap a whole merkle-tree with an additional extension tree, that just adds the new hashes. That way both kinds hashes could be used to traverse all data. The new hashes could be used to check the consistency, the old hashes would still be there to use in UIs or old release documentation. The downside being that you introduce more nodes in the overall data-structure which will have to be supported basically forever. And if SHA-256 is to week a third layer would need to be introduced. But the point is, it could be done. Albeit it would loose some of the elegance of the data structures involved.