This question already has an amazing amount of great answers. Being a physicist with very limited knowledge of mathematics, I certainly cannot expect to contribute something of equal value, however after reading through the answers I'm missing a certain aspect. The missing issue is: "What is a proof"? The "rigorization" process of Calculus with $\epsilon$/$\delta$ proofs was mentioned as a major progress. However I saw nothing doubting that the current foundations of mathematics might still be "improved", where it is of course an interesting question what that would mean. A while ago, when I was trying to find the answer to above question, I came across the following story:

Fields medalist Vladimir Voevodsky, when working on rather scary stuff that I do not begin to understand (motivic cohomology...), came across many cases where ground-breaking published papers with proofs contained errors, which would only be noticed years after. Sometimes this would render a lot of later work worthless. Errors were not noticed, although people were studying the papers in seminars. When it happened to him, it genuinely scared him and got him to start working on computer-assisted proof, as well as an axiomatization of mathematics that goes beyond ZFC, called "Univalent Foundations". It has several conceptual advantages but apparantly is very unknown (and not complete yet!). The aim is to produce a framework where computer verified proofs are a practical option (contraty to now, where such proofs are extremely cumbersome and impossible to use on a regular basis in publications).

Returning closer to the question at hand: A quote from his [motivation][1] (a lengthy article containing many references and statements relevant to this question) for the foundational work of his is: 
>A technical argument by a trusted author, which is hard to check and looks similar to arguments known to be correct, is hardly ever checked in detail.

Granted, the results I'm describing here are far removed from practical applications, such as would interest an engineer. However it isn't entirelly unreasonable that fields like theoretical physics would eventually come into contact with parts of mathematics which seem similar in terms of obscurity. The verdict is: Even in the 21st century, having a published result with a proof is not enough to be sure it's true, not even in the most renowned journals and by the most trusted authors. To achieve that, one would need to push the limits of rigour even further, thus acknowledging its importance (where of course it is up to debate whether and how we want to do this). 

*By the way, Voevodsky has a recorded talk of his where he considers the question "what if the current foundations of mathematics are inconsistent?", where he tries to imagine how one would work in a framework known to be inconsistent, rather than one where one hopes but can never prove that verything is fine*


  [1]: https://www.ias.edu/ideas/2014/voevodsky-origins