Is it yet time to have an essay on “Hilbertism” vs. other concurrent forms of German nationalism? I thought that Gödel was the first to find some liberation and pleasure in the overthrow of the Hilbert program? And what about Einstein? Wasn’t this all happening around 1930, roughly the same year of the “Der Blaue Engel” debut? It is nice to find historical / factual quotations; however, history cannot always be so cut and dry, so explicit. That is, some stories are less amenable to writing down. But I look forward to hear what you come up with, It’s an interesting topic, yes. —Brad
On Jun 21, 2019, at 12:42 AM, Henry Baker <hbaker1@pipeline.com> wrote:
Hi Cris:
I'm not sure who your piece is intended for, but if it is intended for a non-mathematical audience, you might put in something for "lawmakers" and "policymakers" and "regulators" to chew on:
Goedel's Theorem (suitably interpreted and generalized) basically says that any sufficiently powerful language (presumably including English) will have lots of undecidability/noncomputability/computational-complexity problems, e.g.,
There may be no easy way (or no way at all) to determine whether an object is an element of a set, short of listing in sequence all of the (probably infinite) set of elements to see if yours pops up (by analogy with recursively enumerable, but not recursive).
This means that no matter how clever you think you are, you won't be able to "write a rule" that will cover every situation -- i.e., you may have to write down an infinite number of rules (some theories require an infinite number of axioms).
A currently hot example: why is it not possible to "write a rule" that distinguishes porn/hate-speech/fake-news/etc. from art/ethics-discussion/journalism/etc. ?
In short, if the Hilbert Program is doomed, then so is "the rule of law" (as opposed to "the rule of (wo)men" or more likely in the near future, "the rule of robots"). I'm no fan of Facebook, but asking Facebook to do the logically impossible isn't helping matters one whit.
The vast majority of the public understands "quantum" effects far better than "undecidability/noncomputability", but this is a very, very low bar, as the vast majority of the public doesn't understand quantum effects at all.
At 12:39 PM 6/20/2019, Cris Moore wrote:
Here's a history/philosophy question. We are often told that Goedel's incompleteness theorems were a defeat for Hilbert's program — that there was no method to fiind all the truths about mathematics, or even all the theorems. But in the modern age, we seem to view this as liberating — as a source of endless creativity, rather than a defeat — since it means mathematics will never be complete, and there will always be room for new proof techniques and new forms of reasoning. When did this shift first occur? Who first said "You mean no axiomatic system can prove everything, and no algorithm can tell us what it can prove? That's great!" Any help tracking down early quotes along these lines would be greatly appreciated, for a piece that a friend of mine and I are writing. Best! - Cris
p.s. I guess the same could be said about P vs. NP — who first said "some search problems arre [probably] really hard, including telling whether short proofs exist? that's great!"
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