[Article] > He believed in ghosts; he had a morbid dread of being poisoned by refrigerator gases; he refused to go out when certain distinguished mathematicians were in town, apparently out of concern that they might try to kill him. “Every chaos is a wrong appearance,” he insisted—the paranoiac’s first axiom.
vs
[Book] > Gödel believed in ghosts; he had a morbid dread of being poisoned by refrigerator gases; he refused to go out when certain distinguished mathematicians were in town, apparently because he feared that they would try to kill him. Gödel said, "Every chaos is a wrong appearance."
This is kind of a bummer too, because the article was a wonderful read, and it actually flows a lot better than the text in the book. However, it does appear that a lot of the article is a re-wording of what's in the book, just weaved together into a better flow.
EDIT: Based on the other replies, I may have things reversed. It may be that the book is ripping off the article.
According to Amazon (not always reliable) that book came out in September 2005. The New Yorker article is from February 2005. Suggests the sourcing may be the other way around.
I might be missing something as well, but it appears to me that the article was published about 7 months before Mathematical Apocrypha Redux was (Feb 2005, vs Oct 2005). No?
TIL:
- Einstein and Godel were good friends
- Einstein coined the term 'photon'
- Einstein "failed to earn a master's in physics"
- Godel was quite paranoid and weirder than I realized
- Godel refused to argue (maths) unless he had an airtight proof of his correctness
- Godel discovered a loophole in the constitution that would allow a dictator to rise (timely for 2016!)
- Godel is the one who showed Einstein that backwards time travel was permitted according to relativity
- Godel's precise cause of death was starvation brought on by paranoia of being poisoned
It seems like this ordinary world is not enough for the genius, so their brains make new worlds and new scenarios (like "you get poisoned if you eat other people's food.")
I think they're super creative, and not just in science. In fact, they can come up with super intelligent methods for killing people, including themselves. That's why they're always paranoid, cause they think other people can come up with the same things to kill them.
Then, as a sort of encore, he published a three-page note in September containing the most famous equation of all time: E = mc2.
I can't believe the New Yorker can't typeset superscripts. The equation might be the most famous of all, but sometimes it's also the most misinterpreted and mistyped.
> Our mental powers, it is argued, must outstrip those of any computer, since a computer is just a logical system running on hardware, and our minds can arrive at truths that are beyond the reach of a logical system.
I find this argument to be an unsatisfactory against "algorithmic consciousness". There have been automated proofs of Godel's theorem for a long while [0]. I don't mean this to be evidence on the contrary, but Penrose seems to ignore the fact that a computer can realize multiple axiomatic systems, and use them to make statements like Godel's Theorem(s). Godel's Theorem(s) are often taken out of context for philosophical purposes, for better or for worse, and it's important to remember that Godel's Theorem(s) relies on a meta-language (ZFC) to make statements about PA.
For a much more intuitive explanation as to why consciousness is not algorithmic, I recommend "The Neural Basis of Free Will" by Tse. His argument is that neurons and neuronal circuits (and more) harness randomness to provide inputs to "criterial detectors" which are satisfied when the right combinations of inputs (spatiotemporal patterns) arrive at the detector at the right time. This can't be algorithmic, because of the requisite noise in the inputs and because the brain realizes true parallel processing. As a further note, he posits that free will is realized in the resetting of the input weights, so "current" actions set up the criteria for future actions avoiding the issue of causa sui in free will.
Is the claim by Penrose not that humans can possibly intuit theorems which are true but unprovable (Gödel's first incompleteness theorem) e.g. the Riemann Hypothesis, and thus how could an algorithmic process ever achieve such intuition?
Perhaps this is a Turing test for consciousness. "I can't prove this, but I've been thinking about these theorems [inserts true but unprovable list of theorems] and I think they're true".
>Is the claim by Penrose not that humans can possibly intuit theorems which are true but unprovable (Gödel's first incompleteness theorem) e.g. the Riemann Hypothesis, and thus how could an algorithmic process ever achieve such intuition?
A "true but unprovable" Goedel statement is true in standard models of arithmetic but untrue in certain nonstandard models. The "incompleteness" is syntactic, not semantic. The real and complex numbers, AFAIK, only have one model, up to isomorphism.
And sometimes statements are difficult to prove because they're actually independent of the foundational system. Or because a counterexample exists somewhere.
"This problem is unresolved, therefore it's a Goedel Statement within our current foundational system" is extraordinarily unlikely. For one thing, that would imply that we could figure out the axiom we're missing and pass to the stronger system capable of resolving the conjecture straightaway, or that starting from some stronger foundation like homotopy type theory would resolve the conjecture right-away. Most unresolved conjectures are not unresolved for lack of proof-theoretic strength in our foundations.
Humans also intuit a whole lot of theorems (loosely speaking) which are false. There is nothing prohibiting an algorithmic process from generating statements which are variously (and, from the POV of the algorithm, indistinguishably) true but unprovable, true and provable, and false.
A test requires that you be able to produce an artifact which is witness to the results -- what could you produce as the result of the test to distinguish between a true, but unprovable theorem and a nearly true theorem with a single (very, very, VERY) large counter-example?
That sounds about as convincing as saying "If you find a watch in the woods, you will surely never think that it happened to be created through a natural process; a living organism is much more complicated, so it must have been intelligently designed."
This argument has a name. I've seen it presented before and it is rather good. A gross simplification: a computer could never "see" the truth of Godel's Incompleteness Theorems, but we humans can; therefore, we cannot be computers.
It's not a good argument at all. It's circular, or a tautology depending on how you define things. You're using different words, but what you start with is the same as what you end up with. You assert that humans are different from computers: computers can't "see" something that humans can. And you use that to "prove" that humans cannot be computers. Of course. If you start by asserting humans are computers plus some magic, then of course humans cannot be mere computers.
You could write a computer program to generate random propositions, and test those logically against known truths. After enough testing, the program could assert that propositions not disproven are true. Voila! You have a computer program that has "arrived" at truths outside of a logical system. Some of those truths will be wrong, and some will be right, much like humans and their "truths" that are not based on logical reasoning.
If you've never encountered a human claim to "see" the truth of something that turns out to be false, well, it happens... a lot.
It's quite a stretch to say that we find truths that are beyond every possible formal system, rather than those that we've simply managed to encode in computers.
> [Gödel] believed in ghosts; he had a morbid dread of being poisoned by refrigerator gases
I wonder how this relates to the refrigerator design Einstein worked on, "motivated by contemporary newspaper reports of a Berlin family who had been killed when a seal in their refrigerator failed and leaked toxic fumes into their home":
> Gödel believed that mathematical abstractions were every bit as real as tables and chairs, a view that philosophers had come to regard as laughably naïve.
It's not nearly as clear-cut. It's not the trendy theory du jour, of course, but it's an idea that perennially re-emerges in Physics circles. For a famous modern proponent (whose view is arguably even more radical), see Max Tegmark.
> When you are famous it is hard to work on small problems.
> The Institute for Advanced Study in Princeton, in my opinion, has ruined more good scientists than any institution has created, judged by what they did before they came and judged by what they did after. Not that they weren't good afterwards, but they were superb before they got there and were only good afterwards.
Pull up a chair and let me tell you a story about life before the internet, when people had far fewer reading choices and but time to savor the good ones... oh, wait, you don't want to hear a story.
tl;dr Gödel came up with an alternate model of the Universe but Einstein didn't like it. Gödel starved himself to death after Einstein died.
It's just a discussion of Gödel and Einstein written for the layperson. It has a little biographical detail and context of both, pretty reasonable brief explanations of their respective theories (incompleteness, photoelectric effect, brownian motion, special and general relativity) and then Gödel's contribution to General. A nice article
In typical New Yorker fashion it assumes you have never heard of platonism, formal logic or 20th century physics but assumes you remember the future conjugations of regular latin verbs.
=== In typical New Yorker fashion it assumes you have never heard of platonism, formal logic or 20th century physics but assumes you remember the future conjugations of regular latin verbs.===
So true and so well put - And I love the New Yorker. If I use this phrase in the future should I attribute it to you (although "according to gumby" might not fly so well with my other New Yorker aficionados).
I think the point of the article is summed up in one of the last paragraphs,
"If time travel is possible, he submitted, then time itself is impossible. A past that can be revisited has not really passed. And the fact that the actual universe is expanding, rather than rotating, is irrelevant. Time, like God, is either necessary or nothing; if it disappears in one possible universe, it is undermined in every possible universe, including our own."
Yeah, I get your point about the hivemind, but I think it's less damaging to see downvoting as being about the comment, not the person. iow, don't take it personally.
