Journey to the Edge of Reason: The life of Kurt Gödel
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(For definitions of these symbols and a fuller explanation of Gödel’s proof, see appendix.)
The particularly clever part of the coding scheme is that it is reversible. By using powers of prime factors to generate the integer representing a string of symbols in a complete formula such as “x = 0,” the resulting formula number, its “f-number” Gödel called it, can be “decoded” through a straightforward arithmetic process back to the statement x = 0.
Really nice book. It contains a very good description of arithmetization.
Which book? The one being reviewed? I can't read the whole article, so I might be missing some context.
EDIT: I didn't realize you were quoting an entire passage from Journey to the Edge of Reason. I thought it was from a part of TFA that was behind the paywall.
JOURNEY TO THE EDGE OF REASON
The life of Kurt Gödel
368pp. Oxford University Press. £20.
Stephen BudianksyBut it seems to me, we already have a perfectly good Gödel biography, "Logical Dreams" by John Dawson, who is himself a logician, catalogued Gödel's Nachlass which probably made him the world's foremost Gödel archivist, and was also one of the editors of Gödel's Collected Works (5 vols).
Obviously I haven't read the new book so I can't compare directly, but my Bayesian prior is that if you're going to read just one, go with Dawson's.
If you're more interested in understanding the incompleteness theorem and the great amount of nonsense and confusion surrounding it, try "Gödel's Theorem: An Incomplete Guide to Its Use and Abuse", by Torkel Franzén, reviewed here:
http://www.ams.org/notices/200703/rev-raatikainen.pdf
Dawson's biography of Gödel is reviewed here:
Was not aware of the previous biographies, thanks for sharing those. It is interesting that you’re saying the mental health story is not universally accepted.
My lay understanding is that Gödel's incompleteness put a dagger through Hilbert's vision of automating mathematics. And now we're automating mathematics.
I of course oversimplify, but what gives? You'd think all popular accounts of Gödel or proof assistants would name-check Hilbert (as this article does just before the paywall) then reconcile this apparent conflict.
Someone here understands this well. Can you explain?
If Gödel's incompleteness applies to the theorem you are trying to prove then it means that the theorem has been very carefully set up with reference to the axioms that you are using in order to be unprovable. It's not something you will stumble across otherwise.
Think of it like this: imagine knowing that maths will stop working if a certain very specific enormous number appears in your calculation. This would keep mathematicians up at night with worry but it would have no practical effect on anyone else because the enormous number simply never appears.
But if I understand it correctly, that's not a bad thing at all. It means that there might always be more out there to discover. Some dub Goedel's incompleteness a 'bug' of mathematics, I think it's actually a feature.
> Gödel proved that if a statement in first-order logic is well formed (that is to say, it follows the syntactic rules for the formal language correctly), then there is a formal proof of it.
That refers to Gödel's completeness theorem, which actually says something quite different (that a sentence is provable iff it is true in every models of the axioms). The way it is written, even if limited only to true sentences, it contradicts the incompleteness theorem that Gödel proved a few years later.
So I don't think the review reflects a really informed reading of the book. That's another Gödel biography by Rebecca Goldstein, which apparently does a good job describing Gödel's life and personality, but is full of terrible mathematical errors in the part that tries to describe Gödel's work. I haven't read that one but might someday, if the opportunity comes up. Gödel besides being a great mathematician unfortunately also suffered from serious mental illness, but I don't think we gain anything from sensationalism about that aspect of his history. I'd rather read a biography that gets the math right first.
This is not a nice way of saying that he was mentally sick. In his late life, his paranoia got so bad, that when his wife went to the hospital for a few days, he would refuse to eat the food prepared by anyone else, starving himself to death.
Suffering from a mental illness doesn't open up a gateway that lets you see into a mysterious universe of abstract concepts and explore things that others can't... to the extent that mental illness correlates highly with certain fields like math, computer science, it's likely due to the fact that people with good social skills were historically persuaded to enter social fields where they are more likely to deal with people on a day to day basis, such as sales, law, medicine, sports, etc... so that the bulk of people remaining who do not have great social skills are left studying subjects that historically could be studied in solitude.
Things are a lot different today than they were 100 years ago. Much of physics, math, computer science is now done in large teams and is a lot more social than it used to be. The people that will be looked up to in the 22nd century will almost certainly not have suffered from a mental illness or have their accomplishment in some sense be related to their illness. They will likely be much more extroverted, social and be recognized relative to a larger team/group of people as opposed to today where we often mythologize the mathematicians who were reclusive and had "mysterious" personalities.
I imagine the next generation of great thinkers will be more along the lines of a Feynman.
It's not a matter of romanticizing the crazy genius trope, but likely a consequence of genetics. Hundreds of genes are known to accompany higher intelligence, and many of them are correlated with mental illnesses as well.
It could very well be that mental illnesses can be the cause of varied changes in perspective and neural operation in ways that we view as increased intelligence.
Absent a rigorous theory of intelligence, it's like we're speculating about microbes when the best tools we have are magnifying glasses. I think within the next couple decades, we'll have the equivalent of SEM and germ theory of disease. We may be able to augment adult brains to operate at the levels of Einstein or Feynman without the accompanying neuroticism or illnesses.
the example i'd give is to think of a kid playing with hot wheels. the kid is not playing with the physical toy; the toy is an avatar from which the kid is building a vivid imaginative playground. the kid is playing within the world of their mind's eye. adults, for the most part, can't play with hot wheels and dive into that same mental landscape of imagination and association. an adult just sees a small die-cast car. rather than being an inherent part of growing up, i think this power of the "mind's eye" is simply a mode of cognition our society doesn't reinforce and select for; so, for most, it's lost as they grow.
however, there's really a lot of utility in the power of imagination, and so we also see people who either never lost the ability through repeated reinforcement, or went to great lengths to re-activate it - artists like Katsuya Terada or Kim Jung Gi. notable with these artists is the way they're able to draw photorealistic pieces of eye-watering detail, without construction or perspective layout; they simply start from any point and draw, somehow putting every line in its exact place. when asked about how they possess what is essentially an artistic superpower, they've both related that they are able to visualize the scene in a vivid, consistent, and complete way in their mind's eye, and they simply draw from the mental image. both have said that they drew for hours a day in this "rakugaki" (doodle) style as kids, and never stopped the habit. in doing so they were able to retain and reinforce the power of visualization, a mode of cognition that emerges naturally, but is not retained if not practiced.
The sad thing about this, and things like music, is that despite all the positive encouragement that "it's never too late to learn to play", it actually is (hard to impossible) if your goal is to achieve mastery resembling that of those who forged their paths at a young age.
And from this perspective it becomes so important to encourage and support children in creative endeavours and not just have them sit on a school bench memorizing things.
But that doesn't change the studies that show that unlike the myth first popularized by Pierre Janet and then popularized by Freud about the positive relationship between obsessiveness and intelligence, it turns out the correlation is negative [2]. Having OCD is overall detrimental to ones ability to perform intellectual tasks and statistically people with OCD are more likely to end up homeless or in poverty, the victim of abuse or otherwise taken advantage of.
Does this mean that a mental illness is a death sentence? No and I never claimed it was. My position is that historically if you did suffer from a mental illness, you were also more likely to be anti-social or introverted, and hence dissuaded from pursuing fields that involved working with other people. As such it's tempting to look back at many of the historical icons in said fields, see that many of them did suffer from a mental illness and conclude that said mental illness must have been a benefit to them. Absolutely not, it's just that people without a mental illness who were sociable likely did not study math or physics as they would have been encouraged both financially and socially to pursue other fields.
Going forward, as more fields that were traditionally studied by individuals become more team oriented, I hypothesize that we will no longer associate future leaders in math, physics, computer science, etc... with their mental illness. Of course we will still see some talented people who do suffer from mental illness, but the mystique that is often written about it, as if these people peered into a world only accessible to them because of their madness and saw things only they could possibly understand, will be considered farcical and instead we will appreciate that these people succeeded in spite of their mental illness rather than because of it.
There are those people who discover/invent a whole new field and those many many more people who are engaged in the vital and equally important collective effort of working out the consequences and filling in the gaps of those initial discoveries/inventions. I would expect the latter cohort (of very talented people), but not the former one, to be more psychologically and physically robust than the average.
Please show me this evidence.
What I know is that great things often require someone working obsessively hard. Which mental illness can lead to.
For example people with Asperger's can often work much harder, and have successfully lead teams. Examples include Alan Turing during the Enigma Project, and Elon Musk today. Bipolar is often nicknamed "the CEO disease" because so many leaders have it. Ted Turner and Winston Churchill are examples. In a UK survey, fully 40% of entrepreneurs suffered from dyslexia, with Richard Branson being the best known example.
No, mental illness does not make you magically smart. Yes, it is a disadvantage. But there is no reason to believe that having to work with a team will stop people with mental illness from rising to the pinnacle of intellectual performance.
This is too much of a generalization. I think that is mostly the case but there are obvious benefits to obsession and hypomania if that can be channeled, and sometimes it can.