&gt; &quot;proof&quot; in the day-to-day mathematical sense of a convincing argument<p>IIRC, that is the most that a so-called &quot;proof&quot; can ever be.  I remember reading Carl Friedrich von Weizsäcker&#x27;s &quot;Aufbau der Physik&quot;[1] in my younger years, and being somewhat flabbergasted when he starts by justifying classical logic, which he needs to do because later he talks about quantum logic, which is different.<p>Up to that point, it had never occurred to me to justify logic empirically, but now I can&#x27;t imagine naively accepting it.  That means that empirical evidence <i>always</i> trumps.  Everything.  If we find logical rules that we assume violated in the real world, we have to deal with it, not deny reality.  Just like we had to accept that other aspects of the world we held as self-evident were not just theoretically contingent, but actually turned out not to be true.<p>&gt; not in the formal sense for writing down an expression per line<p>&gt; such that each line can be logically deduced from some preceding<p>&gt; lines through the the use of axioms.<p>Hmm...that&#x27;s how we always did (simple) mathematical proofs:  line-by-line, with axioms or theorems justifying the transformation from line to line.  Last time I checked, so-called &quot;machine-proofs&quot; are definitely seen as different by mathematicians, yes, but they are <i>less</i> well-respected, not more so.  Has this changed?<p>[1] <a href="http:&#x2F;&#x2F;www.amazon.de&#x2F;Aufbau-Physik-Carl-Friedrich-Weizsäcker&#x2F;dp&#x2F;3423330848" rel="nofollow">http:&#x2F;&#x2F;www.amazon.de&#x2F;Aufbau-Physik-Carl-Friedrich-Weizsäcker...</a>