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Formal Proof Sketches (2003) [pdf]

30 点作者 larve超过 2 年前

3 条评论

pflanze超过 2 年前
(I&#x27;m a software developer, not a mathematician. I&#x27;m just starting to try to understand the proof world.)<p>I&#x27;m not sure about the &quot;disdain&quot; the authors have about &quot;program source code&quot;. The parallel between proofs and programs seems clear, a runnable program is only one for which all functions&#x2F;subroutines&#x2F;methods (in other words, abstractions) have been implemented.<p>We try to structure programs so that the abstractions are as sensible as possible, with good, preferably memorable names. A well-written program is one where the higher level is short and understandable without (always) having to dive into what the used abstractions do. It is an aim we seem to share with the writers of the paper. Sure, real world might be different, but the technology allows us to write well-readable programs.<p>Other attempts like UML&#x2F;flowcharts exist, but AFAIK attempting to connect these to runnable programs (i.e. make them <i>formal program sketches</i> in the sense of the proof sketches in the linked paper) has largely been a failure (correct me if I&#x27;m wrong). The idea to write the program and mark the parts that should show up in a flowchart (i.e. the inverse of what I&#x27;ve mostly seen talked about, program generation from UML), to get a &quot;human readable&quot; version of the program, might be a new one. Maybe if mathematicians succeed with that, it would be as applicable to software engineering?<p>In any case, am I correct in saying that the envisioned aim is the same? Does mathematics have peculiarities that make the situation different (other than, perhaps, mathematicians generally not being used to &quot;programming&quot;, or fully formalizing something)? Is this a &quot;teething&quot; stage in mathematics like UML was in software engineering and will largely be superseded by just &quot;structuring programs&#x2F;proofs well&quot;, which entails creating &quot;nice libraries&quot;?<p>PS. I have ignored that the paper is from 2003 (about the time of UML&#x27;s heyday, incidentally). What has happend to the idea of proof sketches in the mean time? Is this what &quot;tactics&quot; is about?
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mbrodersen超过 2 年前
&gt; formalizing mathematics currently seems far too much work to be worth the time of the working mathematician<p>A very general statement that isn’t true in all cases. I recommend taking a look at the LEAN mathlib project. It is a group of working mathematicians proving leading edge mathematics correct using a formal proof tool (LEAN).<p>The problem with leading edge mathematics seems to be that the proofs are too complex for anybody but a few specialists to verify the traditional way. And those specialists are already busy working on their own proofs. So having an automated assistant verifying the proofs seems to be necessary for the future of mathematics. I am not a working mathematician myself however that is the impression I am getting reading articles written by leading edge working mathematicians.<p>The same way that large complex software applications are too complex for humans to verify without the help of automated proof tools.
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fspeech超过 2 年前
We need to develope tools that can automatically fill small gaps (ones left as errors in the essay), because the point of having mechanized proofs is to not have gaps. Then practitioners can write lemmas and let the computers do the checking.
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