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February 5[edit]

Godel's Proofs: Implications for Mathematics[edit]

I'm reading a very interesting book: "The Information: A History, A Theory, A Flood" by James Gleick for an anthropology seminar. It is an historical overview of the concept of information from the Greek philosophers to the Internet. I like it a lot. However, when the author discusses Godel he makes what I think are some fundamental errors. However, I'm not a mathematician and while I have a good working knowledge of logic and set theory Godel's proofs have always been difficult and I want to reach out to those of you who are mathematicians to make sure I'm not missing something before I bring this up in the seminar.

My understanding is the first proof proves that any system that can express basic (e.g., Peano) arithmetic will have some theorems that are valid but not provable and the second proof is that one can't prove a system to be consistent within that system, one needs a meta-system. However, what Gleick says seems significantly different.

When he is talking about a scientist who encountered Godel's proofs as a young man he says (bold is mine):

"This [Godel] was heady stuff for the boy, who followed the authors through their simplified but rigorous exposition of Gödel’s “astounding and melancholy” demonstration that formal mathematics can never be free of self-contradiction. The vast bulk of mathematics as practiced at this time cared not at all for Gödel’s proof. Startling though incompleteness surely was, it seemed incidental somehow—contributing nothing to the useful work of mathematicians, who went on making discoveries".

He's not stating what the person thinks because there are other parts of the book where he talks about Godel and talks about how he showed that math was "inconsistent". My understanding is that there is a major difference between saying a system is incomplete (a consequence of Godel) vs. saying it is inconsistent (not free of self contradiction). Once you have an inconsistency your system is worthless because you can prove anything. (I also don't agree with his assessment about Godel's impact on mathematics in general but that's more of a subjective issue). Then later in the book he says when talking about a scientist who came across Godel's proofs that the scientist wondered:

"Was Gödel incompleteness related to Heisenberg uncertainty?"

which made me shout as I was reading: NO! In Linear Algebra and Its Applications by Gilbert Strang on p. 250 at the end of a chapter introducing Eigenvectors Strang has a little aside about how "the uncertainty principle follows directly from the Schwartz Inequality". Nothing at all to do with Godel. Does what he said in either case make some sense that I'm missing? --MadScientistX11 (talk) 03:37, 5 February 2020 (UTC)[reply]

First, a spelling gripe: Either Gödel with the umlaut or Goedel with an e, but never Godel.

Second, you're quite right. Gödel did not show (and certainly did not believe) that mathematics is inconsistent. What he did show is that (subject to certain conditions), given a formal theory T, if T proves that T is consistent, then T is in fact inconsistent. There's a lot to unwrap there, as the kids say, so it's not too surprising that people elide some of it and just jump to the conclusion that everything is inconsistent, whatever that's even supposed to mean.

Third, I certainly don't see any connection with Heisenberg uncertainty. But the two results feel similar to a lot of people, being "limitative" results of a sort. --Trovatore (talk) 03:45, 5 February 2020 (UTC)[reply]

The bit you've "bolded" is, I think, a misunderstanding. Godel didn't show that mathematics was inconsistent. Rather, he showed that if you have a logical system that is powerful enough to do ordinary arithmetic whilst still being consistent, there are valid statements that can be expressed in the system that can't be proven within the system.

On the physics, I'm not able to comment with confidence. RomanSpa (talk) 15:29, 5 February 2020 (UTC)[reply]

Not only did Gödel show that there are then expressible valid yet unprovable statements, Gödel's second incompleteness theorem gives a concrete example of such a statement, namely (expressed informally) "I am consistent". This is expressed in the bolded bit. --Lambiam 04:57, 7 February 2020 (UTC)[reply]

I should also say that Godel's results in this area don't strike me as being particularly "limitative". It helps that I'm a neo-formalist. RomanSpa (talk) 15:35, 5 February 2020 (UTC)[reply]

RomanSpa your link doesn't really explain what a "neo" formalist is, so I'm not sure what part of your answer is hiding in the prefix. But actually it's formalists who were most devastated by the theorems. They really don't touch mathematical Platonism at all (and indeed Goedel was a Platonist, at least towards the end of his life). To the Platonist, the theorems just emphasize that truth cannot be fully captured by formal mathematical proof (they are "limitative" in that sense). The formalist, on the other hand, has a big job just explaining what the theorems mean; he/she keeps getting caught up in confusing levels of reference. By far the cleanest approach for the formalist is to think and speak of the theorems the way a Platonist would, then go back and add disclaimers at the appropriate spots. --Trovatore (talk) 22:13, 10 February 2020 (UTC)[reply]

Thanks to everyone for those great answers. On the name: yes I knew I needed an umlaut but didn't know how to make one. I agree with RomanSpa that the theorems aren't that limiting. That's another small gripe I have with the book: he describes the work of Gödel, Turing, and Church and sort of leaves it at that as if they discovered these problems which math never resolved but then people just went on using math anyway when in reality (at least based on my limited understanding) all their work ultimately led to ZFC set theory which is IMO one of the most amazing accomplishments in math or science, it provides an incredibly rigorous foundation to build the rest of standard mathematics on. It probably sounds like a terrible book but actually it's quite good, he just misunderstands some of the issues with math and logic something I think is unfortunately common in the humanities. I think in the Stanford Encyclopedia of Philosophy they have a whole section of either Gödel's or Turing's entry devoted to the ways people have misinterpreted the results. Anyway, thanks again for the excellent and quick responses and for giving me an "ö" that I can copy paste from now on when I need it ;-) --MadScientistX11 (talk) 16:49, 5 February 2020 (UTC)[reply]

@MadScientistX11: I think you're spot on in thinking that Gleick missed the boat with the comment about how people went on doing math after Goedel. What else would they do? I guess if Goedel had really proved what Gleick thought, then the comment would have some sense, but since he didn't, it doesn't. Gleick just flat misunderstood this; he's 100% wrong and 0% right. In mitigation, it is a confusing subject for the novice.
Now, as to what you wrote about ZFC, that part I don't really follow. ZFC was mostly if not entirely formalized well before the Goedel theorems. It's true that the theorems have been productive in further work in set theory (some of it by Goedel himself), for example leading to the notion of the hierarchy of consistency strength and by extension large cardinals, but that doesn't seem to be what you were getting at.
The only thing I can think of is that you might have been getting the Goedel theorems mixed up with Russell's paradox. The popular (but in my opinion incorrect) narrative is that the RP refuted Cantor's "naive" theory, and it was saved by formalization into ZFC. That isn't so. RP actually refuted an incorrect formalization of Cantor's ideas, and modern set theory is based on a picture (the so-called von Neumann hierarchy) which is arguably implicit in at least the later writings of Cantor himself.
You might also be interested in my reply of a couple days ago to RomanSpa. --Trovatore (talk) 19:51, 12 February 2020 (UTC)[reply]

It's interesting that the book you're reading is interesting, because I read Gleick's biography of Isaac Newton a couple of years ago and it struck me as quite unmemorable. The writing was pedestrian; I'm surprised I even finished it. -- Jack of Oz ^{[pleasantries]} 16:51, 5 February 2020 (UTC) [reply]

Gleick's biography of Feynman is pretty good. I remember his book about chaos theory also being good, but I may be conflating it with Mitchell Waldrop's book Complexity, which is good. 2601:648:8202:96B0:0:0:0:E118 (talk) 23:31, 5 February 2020 (UTC)[reply]

Just for future reference, for me the easiest way to get spell difficult names like Gödel, Erdős, Plücker, Ørsted etc. is to type your best approximation in to WP's search box, let the autocorrect/redirect system do its magic to find the article on the person, then just copy and paste. It even works for non-mathies like Erdoğan; the media constantly get either the spelling or pronunciation wrong. --RDBury (talk) 20:04, 5 February 2020 (UTC) [reply]

For German umlauts in general, though, if that's too much of a hassle, you can just use an e instead. The umlaut "diacritic" is not really a diacritic; it's a placeholder for a missing e. So Goedel works just fine if you can't find the umlaut-o quickly. Similarly Mueller if you can't find the ü to type Müller, Kaehler if you can't find the ä to type Kähler. --Trovatore (talk) 22:54, 5 February 2020 (UTC) [reply]

Gleick has written good stuff about physics, and mathematical logic is an area where physicists tend to get things wrong ;-). Our article Gödel's incompleteness theorems explains the theorems in more detail, and Torkel Franzén's book Gödel's Theorem: An Incomplete Guide to its Use and Abuse has a lot of examples of mistaken things that have been published about the topic. Here (by George Boolos) is an explanation of the Second Incompleteness Theorem in words of one syllable (shorter version). Fwiw there is no significant connection between Heisenberg uncertainty and mathematical incompleteness. 2601:648:8202:96B0:0:0:0:E118 (talk) 21:25, 5 February 2020 (UTC)[reply]

Gödel demonstrated that formal mathematics can never be both complete and free of self-contradiction. People usually omit one of the conjuncts and simplify this to "formal mathematics is necessarily incomplete". That is obviously an oversimplification. Gleick apparently made the other oversimplification. --Lambiam 04:57, 7 February 2020 (UTC)[reply]