> Gödel began by devising a method that would transform statements about the process of proof into statements about the properties of numbers. Doing so would allow him to write metamathematical assertions about the construction of proofs as simple arithmetical formulas, which could thus be expressed within the syntax of a mathematical system itself. Turning metamathematics into simple mathematical propositions was the first of his self-referential feats of sleight of hand.
The scheme he devised made it possible to convert any string of symbols used to express a mathematical or logical formula in Principia Mathematica into a single, unique integer. Russell and Whitehead had used a deliberately limited set of a dozen “primitive” symbols in setting up their system, and they showed that with these symbols any proposition of logic or number theory could be expressed, albeit in an often extraordinarily cumbersome fashion. Gödel first assigned a numerical value to each symbol. He used several different coding systems in his different presentations of the Incompleteness Theorem; in the series of lectures he gave in Princeton in 1934 he used the scheme:25
(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.
> 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.
Budiansky is a good writer: his "Battle of Wits" about WW2 Enigma cryptanalysis is worth reading, even though it is mostly rehashed from other sources. I'll probably try to check out this Gödel biography, though the mental illness angle sounds pretty bogus.
But 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:
I just finished reading it last week. I feel he could have shortened the initial 30 pages of the Austrian geopolitical environment, but otherwise really enjoyed it. It was a smooth read.
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.
I think it is universally accepted that Gödel was a great mathematician who suffered from mental illness. Budiansky's book seems to espouse a romantic but bogus notion that Gödel was a great mathematician because he suffered from mental illness. It doesn't work that way.
I have not read Goldstein's book, but multiple reviews by mathematicians said that while she did a careful and sensitive job on the biographical parts of the book, she completely messed up the mathematical parts. It's not that a biography needs to focus on technical details (that's what textbooks are for), but the author should at least understand them before writing about their significance. See Torkel Franzén's book for more about that.
I'm a mathematician, but I don't really understand Gödel's incompleteness. I've recently been reading everything I can about proof assistants such as Coq and Lean, and Homotopy Type Theory and such.
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?
Gödel's incompleteness isn't a practical obstacle to proof assistants - it will never stop us formalizing proofs and searching for new theorems. It's irrelevant to their operation and will continue to be for all time.
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 as I also understand it, Goedel's incompleteness theorem also implies that for any given axiomatic system, if you have any finite number of theorems/proofs, you can never know if these are all theorems/proofs possible in that system. Hence it isn't possible to "finish" any axiomatic system, in the sense that no number of theorems/proofs will ever "complete" it. I might be wrong though.
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.
Thanks. The review is confused about a number of issues, e.g.:
> 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.
> Gödel besides being a great mathematician was unfortunately also a nutcase
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.
The article is behind a paywall so I didn't get to read the whole thing, but the little I did read suggests that Godel's accomplishment is intertwined with his mental illness. It is tempting to romanticize people's mental illness and make it out as a strength that allowed to them to accomplish great intellectual feats, but as far as all evidence goes, mental illness absolutely hinders one's mental capacity and works against it, rather than something that assists it.
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.
Nash is a counter to your assertion. Tesla as well. Einstein, too - he exhibited compulsive behaviors and may have had ocd. High levels of intelligence often accompany greater susceptibility to mental illnesses, resulting in lower quality of life and ability to care for themselves.
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.
Absolutely you can name incredibly talented people who suffered from a mental illness, but that does not counter my assertion. Instead it's an instance of Berkson's paradox [1] wherein you are unlikely to consider the group of people who failed high school, have a drug addiction, are homeless, or are otherwise suffering from their OCD. The only people with OCD who have a name worth mentioning will be among the most successful people in their field.
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.
I wouldn't just chalk it up to Berkson's paradox. A lot has been written on the connection between genius and madness and it seems to me that folk psychology gets it right on this point, at least in essence. It's easy to feel like we've solved that mystery by just denying that there's such a thing as a genius, that they're just incredibly talented people who were in the right place at the right time to produce that deep seismic shift in perspective that creative genius is associated with. This demystification might appeal to our competitive, status-seeking side as well, it's uwarranted.
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.
Yeah I'm reminded of Jason Padgett, who turned into a "math genius" after an accident involving head trauma[0]. Judging from that I rather think that us "normal" people have abilities that are hidden from us, filtered out by our normalcy.
i think there's some way in which the brain "crystallizes" as one grows up, and the modes of cognition which are the most used in daily life are deepened at the expense of others. i'd compare it to trails being worn into the dirt by foot traffic; the most trafficked paths will remain for years and years, while the unused paths slowly disappear, until eventually they show no sign of their existence at all.
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.
I know what you mean! My brother is very talented at drawing and he's been doing it his whole life. It's fascinating to see him start drawing something where I don't see what it's supposed to be until he's finished maybe 30% of it. Just seemingly random lines at first. It's fascinating to me as I can hardly picture even simple shapes in my mind. But I can picture music, because I've been doing that for most of my life, so I can somehow relate to what it's like.
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.
I think it is more complex than that. There are certain traits – for example, obsessiveness – which can manifest in different ways. If one can point one's obsessiveness in a socially useful direction, sometimes one can achieve great things with it. If instead it points in a deleterious direction, it can become rather disabling, and one may even end up with some sort of diagnosis as a result. Some people find it easier to control the direction their obsessiveness points in than others; to some extent, differences in that ability may be innate (genetic/etc); to some extent, it may be influenced by environment, upbringing, culture, social situation, opportunities, etc; to some extent, it is probably just good or bad luck, random chance. The difference between the person who achieves a lot and the person who is disabled by mental illness may be smaller than you think–they may both be people with unusually high obsessiveness (and various other traits besides), just representing two different outcomes that extreme can lead to. And of course, sometimes the same outcomes occur in the same person – at some times, a person may be able to direct their obsessiveness in the direction that enables them to achieve things; at other times in their life, they may lack that ability and be quite disabled by it.
> mental illness absolutely hinders one's mental capacity and works against it, rather than something that assists it.
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.
Maybe less useful, but in fiction, Ted Chiang's 'Division by Zero'[1] is a fascinating window into a human response to such a discovery as Gödel's, because it's not /just/ about math. Major breakthroughs in theory often have deep philosophical consequences as well, and the ramifications can readily break whole worldviews for those who actually and acutely understand what they've found.
I don't know whether Georg Cantor's investigations into higher infinities were related to his psychiatric difficulties, but I would surely have struggled with my sanity if I had been doing that work.
Having poor social skills != mental illness. Some people with poor social skills have no mental illness, and some mentally-ill people have excellent social skills.
The article is behind a paywall so I didn't get to read the whole thing, but the little I did read suggests that Godel's accomplishment is intertwined with his mental illness. It is tempting to romanticize people's mental illness and make it out as a strength that allowed to them to accomplish great intellectual feats, but as far as all evidence goes, mental illness absolutely hinders one's mental capacity and works against it, rather than something that assists it.
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.
