Chance hit. The German word for a right answer arrived at by luck.
It is not praise. Naming it marks the result as unrepeatable, and therefore as worth nothing beyond the single case.
The derivation
The derivation. In German physics and engineering teaching, the Herleitung is required — you show how the formula follows from what came before.
A formula that is merely known is not yet owned. The student is asked to bring it down from its origins, which is what the word says.
Critique of Pure Reason
Kant’s project in 1781 was not to reach conclusions about the world. It was to examine the instrument first — to establish what reason can and cannot validly do, before using it.
Critique, in his title, does not mean criticism. It means examining the apparatus.
The most influential work in German philosophy is an inspection of the method, carried out before the work begins.
David Hilbert
In lectures given in 1922 at Göttingen, Hilbert launched his Beweistheorie — proof theory. The aim was to take the proofs of mathematics and make them themselves the object of investigation, to establish that no contradiction could ever be derived within the system.
As he put it, we move to a higher level of contemplation, from which the axioms, formulae and proofs of the mathematical theory are themselves the objects of investigation.
Not: are these results correct. But: is this way of reaching results sound at all.
The conduct of a proof
Proof-leading. German separates the Beweis, the proof, from the Beweisführung, the conducting of it.
You can be right and have your Beweisführung attacked. The two are assessed separately, and the second is where the argument usually happens.
The duty to give reasons
Begründungspflicht. German courts and authorities are under a duty to give reasons. A judgment carries a Begründung — a grounding — as a mandatory part of the document, not a courtesy.
A decision without adequate reasons is defective as a decision, regardless of whether the outcome was right.
The reasoning is not an explanation of the ruling. It is part of it.
Able to be retraced
Literally: able to be travelled after. To be nachvollziehbar is to be such that another person can walk the same path and arrive at the same place.
English translates it as understandable or comprehensible, which loses the whole point. Those words describe a state in the listener. Nachvollziehbar describes a property of the reasoning — it can be retraced.
An argument that persuades but cannot be retraced has failed.
The calculation path
The calculation path. In German schools the Rechenweg is marked, not only the answer. A correct result with no working shown earns little or nothing, and a wrong result reached by sound method earns partial credit.
Every German adult sat through years of this. The habit of asking how you got there is taught before the age of ten.