Comments on: Mathematical rigor, I

By: Kelli Garner

Kelli Garner — Sat, 03 Oct 2009 17:51:22 +0000

Great site, how do I subscribe?

By: Carlo Angiuli

Carlo Angiuli — Mon, 17 Aug 2009 11:44:00 +0000

Basically, I think his paragraph beginning "Of the three kinds of theory" is completely wrong and I may address its ideas in full at some later point. Same with his paragraph beginning "In place of the scientific method" where he outlines the "axiomatic method." In fact, the latter was a set of ideas I was going to deal with (a misunderstanding of "aesthetics," as I'm going to call it) once I tied up this whole rigor thing.

By: Matthew Gerken

Matthew Gerken — Mon, 17 Aug 2009 11:20:00 +0000

Yeah, I was a little doubtful about the idea of slapping "meta" in front of things and pretending like you've made a meaningful distinction that solves the problem. However, despite perhaps lacking knowledge of specific issues at stake, his negative critiques of the way science and math work seem pretty convincing to me. His positive solutions might be bunk, but if cared too much about that part I'd never listen to Nietszche either! I'd be interested to hear your thoughts about all the stuff that comes before Langan's "levels of proof" bits.

By: Carlo Angiuli

Carlo Angiuli — Mon, 17 Aug 2009 10:59:00 +0000

Lockhart's essay is one thing I was thinking of discussing in the future. I think he generally has the right idea but carries it a bit too far.As to that Langan essay, he seems to have basic misunderstandings of the shortcomings of science and mathematics. By its very nature, there IS no way to "resolve the self-inclusion paradoxes" in set theory; this is precisely what PM attempted to do, and which Godel proved to not work. It's not that nobody has been clever enough yet; the system, by its very nature, cannot be patched up.Likewise, issues he points to like "quantum nonlocality" are not things which need fixing. The EPR paradox defies logic but is well-characterized by quantum mechanics, through Bell's inequalities. In fact, he seems to be hinting at things like hidden variable theories, which have been systematically proven to be incorrect.So basically, I think Langan demonstrably doesn't know what he's talking about, and is just waving his arms around.

By: Matthew Gerken

Matthew Gerken — Mon, 17 Aug 2009 10:36:00 +0000

Unfortunately I'm a humanities major and have forgotten for too much of the math I knew in high school, but I'm nerdy enough to enjoy these things anyway.The question of rigor is a very interesting one to me because it seems related to a similar problem in philosophy- do we analyze arguments thoroughly and reduce all philosophy to linguistic analysis, or is there a better way?Issues I find interesting: how does the way we conceive of mathematics shape the way we think about wider metaphysical truths? http://www.frontporchrepublic.com/?p=4097 Is math education as awful as this man thinks?http://www.maa.org/devlin/LockhartsLament.pdfIs Chris Langan totally nuts or is he actually in the process of destroying the theoretical foundations of both math AND science? (Probably both, actually) http://www.ctmu.org/ (Click through to the "Theory of Theories" article- it's the only one I can really understand)

By: Carlo Angiuli

Carlo Angiuli — Mon, 17 Aug 2009 09:51:00 +0000

Tagged some people I've discussed this with recently.

By: Carlo Angiuli

Carlo Angiuli — Mon, 17 Aug 2009 09:19:00 +0000

Exactly — PM was an attempt to be so rigorous that nothing possibly could slip through the cracks. Of course, Godel showed that such efforts were fruitless, because of inherent limitations in any sufficiently powerful logical system.So while PM was absurd, it really intended to use rigor to, in some sense, "clarify the system" (by removing all ambiguity). There are much more useful and currently-mainstream examples of ways in which rigor actually clarifies the system greatly, and I'll be touching on some of those (and on PM) in the next post, I think.

By: Derek Cutting

Derek Cutting — Mon, 17 Aug 2009 02:34:00 +0000

Very interesting! I'm reminded of Whitehead and Russel's Principia Mathematica, in which it takes almost 400 pages to prove and define "1+1=2". I can't wait to read what you write next!

By: Anonymous

Anonymous — Wed, 31 Dec 1969 19:00:00 +0000