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Decades-Old Graph Problem Yields to Amateur Mathematician (quantamagazine.org)
492 points by digital55 10 months ago | hide | past | web | favorite | 173 comments

This guy has a background in computer science, but taught himself biology to spearhead the regenerative medicine movement. If I'm not mistaken, he published a book in biology with no formal credentials (16 years after receiving his BA in computer science), after which Cambridge honored him with a PhD in the subject.

He's also Thiel backed.

> honored him with a PhD in the subject

Just to precise if you were confused like me, Cambridge gave him an actual PhD in Biology based on the book - meaning the book satisfied all the prerequisites of a PhD at Cambridge and he had to do a viva - not just an honorary degree.

That's quite an impressive fellow.

The way a Cambridge PhD works, I believe – and I think this is the same across the UK – is that the only things you actually have to do are;

a) write a thesis

b) find examiners (internal and external) who agree to examine you; if you're a grad student this is your advisor's job. The examination is by thesis review followed by pass/pass-with-corrections/fail oral exam. It can take as long as it needs to (typically three to four hours), and is in private; it's not like public defenses elsewhere, which are for show, this is a real live-fire exam and I know people who have failed

c) pass

d) ... that's it; there is no part d)

In particular, there are no quals or other requirements; the degree is an old-school medieval masterwork deal. You submit your thesis and get examined on it. Typically you write your thesis through an apprenticeship to your advisor, but that's not actually required, it's just conventional; in Aubrey's case, the thesis was the book and the research proceeded by unusual means over a long period of time. Nevertheless, he got examined in the same way everyone else does, he passed like everyone else did, so he has his PhD.

(source: my own memory, I know Aubrey and have worked with him on a couple of non-science things fifteen years ago, and I got my PhD from Cambridge through the traditional route of being a grad student – thesis at https://aspace.repository.cam.ac.uk/handle/1810/218854?show=...)

If you are seeking the Ph.D. Degree under the normal regulations, you are required to have "kept three terms at least by residence" (i.e. have lived within three miles of Great St. Mary's Church for a year) [1]. There is also a minimum time of 9 terms (ie. 3 years) that must be spent as a Graudate Student [2].

The alternative 'special regulations' apply only to graduates of the Cambridge University, or graduates of another university who have been admitted "to some office in the University or to a Headship or a Fellowship of a College" [3].

> d) ... that's it; there is no part d)

Well, the most likely outcome of the viva is a pass with minor corrections, so there typically is a part d - "make the corrections requested by the examiners and re-submit".

[1]: https://www.admin.cam.ac.uk/univ/so/2017/chapter02-section9....

[2]: https://www.admin.cam.ac.uk/univ/so/2017/chapter07-section13...

[3]: https://www.admin.cam.ac.uk/univ/so/2017/chapter07-section13...

I'd forgotten about the Great St. Mary's deal. :-) (I spent my PhD firstly in Jesus College and then living out in a couple of places in Castle and Abbey, so I'm glad in retrospect that I didn't breach those rules...)

I think the spirit of what I wrote is broadly accurate. d) was definitely a rhetorical indulgence!

It was only slightly unusual. de Gray got an undergrad (computer science) degree from Cambridge and married a Cambridge biologist and then worked at a biology-computing lab while writing the book. With the professor and the lab (and the upper class boarding school and multimillionaire mother), the only thing missing was a formal admission to the PhD program.

saw this amusing story about how Prince Charles's bodyguard got a degree from Cambridge https://www.quora.com/Why-do-members-of-the-British-royalty-...

Worth noting that "for a standard undergraduate degree there are specific and strict academic requirements for admission to Oxford and Cambridge" isn't strictly true. A College can vary the admission criteria as it sees fit, and EE/unconditional offers can and have been given.


I think those low offers were typical in the days of the separate entrance exam. I took the (4th term) undergraduate entrance exam for Oxford. When I got the telegram "not required for interview at Oxford" I thought I had failed. But then got accepted with an EE offer (two minimum-pass E grades at A-level) without the need for an interview.

Nowadays the low offers are for those competing in the International Maths Olympiad and similar competitions, so they can spend less time on the irrelevant A Levels.

> viva

I wasn't familiar with this term; apparently it refers to the "oral defense" of a PhD thesis.

One of the toughest experiences of my work life was my viva. The examiner picks at every hole in your work and you need to have an answer. Even if the answer is acknowledging it and proposing future work. It's a very useful experience, and when combined with the rest of a PhD, teaches you to be humble and accept that criticism of work is not an attack on the author. I still have nightmares about mine!

From Latin "viva voce", literally 'with a live voice' and roughly meaning 'aloud'.


At my university you also got a 'viva voce' as an undergraduate if your BSc is borderline between two grades. I received one and yes, I got the lower grade.

It's not strictly limited to just PhDs - at my university, I was required to take one for a year-end undergraduate project.

I'm not sure. Given that he was in a relationship with a professor there, he might have got an easy pass.

So then amateur mathematician is a bit misleading? It sounds like this guy is a marvelous person at whatever he does. How can he be an amateur ANYTHING but have a PhD?

Amateur comes from the Latin amar which means to love. An amateur is someone who does something for pleasure rather than money. Sometimes people use it to mean a novice or someone who is unskilled but in this case the original meaning applies.

I think amateur in this case refers to non-professional / research mathematician with a PhD working at a University or a Research Institute. You still need to have some serious math chops to get this kind of result, it just isn't reflected in your official title or degree.

"Independent" might be a better adjective for someone like that.

If you're not doing it for profit, you're an amateur. Independent would suggest he could still doing math professionally but on a contract or otherwise unaffiliated basis.

Hmm, that is a definition of amateur; just as "professional" can mean you make a living from it.

However amateur often means - in British English - simply "low skilled" as professional can mean "formally carried out with high competence".

IIRC this guy or his associates dug up soil from a medieval burial site near Cambridge (England) to try and find a bacterial enzyme capable of breaking down 7-ketocholesterol (and hence potentially to cure artherosclerosis). Maybe not Indiana Jones but certainly ingenius and intrepid!

I find that I’m much more open to teaching myself just about anything now. Years of learning every new thing under the sun in the software world will do that.

Doesn’t surprise me at all to hear this.

Gotta downplay his achievements to fit the narrative of the 'prodigy'.

What do you mean? How is it downplaying?

Downplaying in the sense that it's god given and not sweat equity.

Another such word: 'fringe'

> He's also Thiel backed.

Is having Thiel's association a positive thing still? Serious question.

Just don’t piss him off. Just got done reading “Conspiracy: Peter Thiel, Hulk Hogan, Gawker, and the Anatomy of Intrigue.” Starts a bit slow and pretentious, but once it gets going... I dislike Thiel immensely, but hole-lee-shit he pulled off one of the most complicated, ruthless, and complete revenge plots I’ve ever heard of. That he kept secret for NINE YEARS while planning and executing (quite literally) his scheme.

Even if you know the story well, the book is worth a read or listen.

Thiel's a brilliant person who's misguided in regards to some things - a few bad ideas shouldn't particularly mean a person should be shunned entirely.

I would assume so. I couldn't care less about his political affiliations

His political affiliations are in conflict with his desire for long life, given who has the nuclear football.

I don’t care, I’m an atheist with no kids who’d be bored out of my mind before I hit 300 years old, much less 1,000.

If you're afraid of the nuclear football, isn't "standing right beside the person holding it" the best place to be, if you hope to have any shred of control over whether it gets dropped?

If you think you can convince him not to pull the trigger. I’d rather be standing at ground zero. Dying of radiation poisoning is an experience I don’t need to see to believe.

My father in law was at the bikini islands for the test runs (died of cancer), my great uncle was on the ground in Japan shorty after they dropped Hiroshima (died of cancer), and great uncle’s kids has Japanese rifles he recovered from sites that glowed in the dark as late as the mid 70s.

How's that NK thing going?

From a bank balance perspective, yes!

(Probably anyway, I don't personally know anyone that is Thiel backed)

Oh, I thought the implication was the other way around, something to do with Thiel's political leanings. My bad.

Given Thiel's interest in longevity, yes

Oh great; an immortal Peter Thiel, just what the world needs...!

from a political perspective, no. if you care about actual technological accomplishments, as a technologist, peter thiel's endorsement should matter a lot.

Absolutely, especially depending on our audience. To your typical SJW crowd, probably not. To everyone else, yes.

Aubrey de Grey is chief science officer and co-founder at the SENS Research Foundation, which is the only charity that I personally support because I think it has small resources compared to the potential good it could do to increase healthy life and reduce human suffering.


Here's some info pulled right from https://www.againstmalaria.com/:

* Half a million people die each year and 400 million fall ill [with malaria]

* 70% of them are children under 5

* #1 killer of pregnant women

* Malaria is preventable

Malaria nets cost $2. Antimalarial drugs are dirt cheap in the developing world. If you want to impact human suffering and increase human health, this is a good place to start. Not with trying to extend the long and already pleasant lives of rich people in the West.

Yeah, someone always says this.

it’s an utterly worthless comment. I don’t want waste my time explaining why you’re wrong but I did blog a related response a year ago:


The short answer is that there are lots of people and lots of resources. It’s not going to hurt if some of those resources work on problems that you don’t think have a high priority.

I bet you can find plenty of people who aren’t working to solve any problems. Why don’t you berate them instead of someone who picked a problem that you think is less important.

> It’s not going to hurt if some of those resources work on problems that you don’t think have a high priority.

Probably not for most problems, but even ignoring the QALY calculations, pathogenic diseases (like malaria) are kind of unique amongst the landscape of global issues right now, in that you can actually permanently solve them with some level of attention, at which point they no longer require any donations but continue to provide benefits forevermore.

Polio is just... gone. We don't need to fight it any more. But that's only true because we (or rather, our governments) spent a whole lot of money over a very short period. If they had spread that money out over a longer period (equivalent to what would happen if fewer people were donating), polio would still be here, and not just because we "killed it slower"—it would still be thriving, because it was only being fought with half-measures. If someone only has enough money to pay for half a course of antibiotics, they may as well not take any.

Pathogens are the best justification we have for "focus firing" public resources toward a single goal. It really is a case of "we should be spending money on nothing else until we've solved this." (At least, if we're bothering to spend any money on the problem at all. 100% or 0%, but don't bother in between.)

Once we run out of "easily"-eradicable pathogens, though, the calculus of comparative advantage resumes, and it makes sense again to donate to a variety of things.

I think if you run the numbers, there aren’t many cases of malaria in the developed world. Treating the causes related to aging will save a lot more money.

There are major efforts to treat malaria, for example. Bill Gates is putting in billions.


There’s this great effort too:


I’ve even donated money for nets myself.

Part of the problem isn’t money but changing attitudes. A lot of nets, for example, end up being used for fishing.


Depends on your time scale. Solving malaria “early” might make a few decades’ difference in when we start seeing an explosion of developed, first-world African nations (as is theorized that introducing outhouses precipitated the productivity boom in the southern US, due to a generation born without parasite exposure.) And those African nations would be more sources of scientific talent to solve problems like aging.

On the other hand, aging is unique in that solving it (or even just pushing it back a bit) gives multiplicative effects to productivity against any problem that requires decades of expertise, because it lets researchers have more productive decades and/or spend more years learning to be productive and yet still spend just as many years being productive. Malaria, at this point, isn’t quite the type of problem this would help—but there certainly are many problems where more grabbing out more productive researcher man-years from the aether would be extremely helpful.

It’s rather interesting to think in these terms: trying to figure out which advance will have “unblocked the tech tree” the most 50 years down the line. I rarely see this kind of analysis being done, though, which I find disappointing; surely there are people far better equipped than I to do it.

I alway like to consider unlocking the tech tree the most. That’s why we want a lot of people working on a lot of problems. We can’t predict which discoveries will make the most difference.

Consider all the problems you need to solve to “cure death”.

The US GDP is $17 trillion. If people starting spending billions on a more youthful old age, we’d get more research. More venture capital would go to aging research.

That assumes your goal is to maximize utility with all humans as interchangeable. On the other hand, I'm more interested in my own life and longevity than people with malaria, as a result I'm more interested in this type of work.

It's also, in a sense, an unfair argument. You could say the same thing about any other startup. "Oh your goal is to create some new app/service/software? Why not instead focus your efforts on preventing malaria?" The fact that this startup is focused on longevity should not make it more deserving of your above criticism.

If your goal is to maximize your own benefit, then you're not really doing charity, so this comment is rather irrelevant.

No, it's not. Any interaction you have in the world can be measured against a counter-factual of interacting with the world in such a way to improve the plight of others.

Life pro-tip: your utility is correlated with the utility of a poor person in Africa. No really, it is. Climate change, war, disease, famine -- these things are directly connected to global poverty. And they are going to affect your life too, no matter where you live, no matter how insulated from them you feel in the developed world.

Separate from your argument, I find your lead-in of "life pro-tip" here to be offensive and condescending. It implies that you are an accomplished professional at living, while 'natalyarostova' is but an unthinking amateur. Did you intend it this way? Or was it meant to be humorous? Or does it serve another purpose that I'm missing?

In my experience, people who proudly evince callous indifference to the suffering of others are usually one or more of a) privileged, b) sheltered, or c) quite young. In other words, naiveté. Rarely is it due to actual, willful malice. So yes, I meant it.

My answer was a rarely honest report on my actual interactions with the world. We live in a time where give well estimates something on the order of a few thousand dollars can save a life, through malaria donations etc. How much of my income do I actually donate to these people? About 5% I guess. How much would I pay for my own longevity? Way more than that.

Would you? If given the option between extending your life to the age of 200 through yearly payments of $10,000 -- or instead donating that money to save countless lives a year, which would you do? Maybe you would donate it, I don't know you, and I won't presume the answer. But is it a callous indifference to pick yourself instead? Maybe it is, and if so, I'm callously indifferent.

We could play with those numbers though, to see how you would value an additional year of your life vs. saving a year for one of the world's poor. Would you value them the same? It's only natural to value your life, and your family's life, higher than others. What's the number where it's not callous indifference? It's worth thinking about, although I don't think the answer to these questions is pure nihilism or anything, we should definitely work to improve the world, and certainly donate more than we do now.

But the point I'm arguing against is that this doesn't mean we also need to put on hold work to improve longevity of our lives.

(It's also worth noting that this entire argument is far more nuanced and second order than the simpler one I offered above, which is that holding this business to a different standard than, say, Facebook, doesn't make sense. Any time you choose to start a business or work, you can always instead choose to go work for a firm with a mission to help the world's poor.)

As for your previous claim that it's in my direct interest to support the world's poor, that's true, but it's also not weighing it against the appropriate benchmark. The appropriate benchmark being what else could I do with that money that would benefit me.

Here's what I don't understand. You are trying to convince someone that they are being "callously indifferent to the suffering of others" by telling them that "their utility is correlated with the utility of a poor person". Sounds like you both seem to have comparable goals, just that you believe that your "way" of accomplishing the goal is superior (i.e. help others to help yourself, rather than just helping yourself)

At risk of violating some HN etiquette about memes, "pro-tip" is a word of fuzzy and common meaning since it was memed back in the ancient era of memes ("Pro-Tip: To defeat the Cyberdemon, shoot it until it dies.")

This is how most people work. They value their own lives and their own happiness over that of others. They will gladly buy their luxuries and let dozens of people die as long as they don't have to see them suffer. We all know this.

Just don't fool yourself. The reason we value our own happiness over the lives of others is because we are weak. That's something we have to live with, but it doesn't make it good.

>You could say the same thing about any other startup.

And I do!

I don't know the answer to that. Am I weak? Is it wrong that I'm on a vacation right now, when I could have donated that money instead? Some EA say just donate 10% or so of your money and don't feel bad, which is a wonderful social convention, but it is just a convention.

If you say the same about other startups, then at least you're being consistent, and it's the inconsistency that irks me.

I don't know if I'm weak or if I should be comfortable with my selfishness. I donate, but not as much as I could. I'm not sure I'll ever know the answer. I find that wrestling with the question, at least, prevents me from going down the path of buying luxury goods and signalling, so I guess it's somewhat useful.

Serious question: where do you draw the line? At what point does life stop being a personal experience between life and death and start becoming a min-maxing of maximal utility on the planet?

While I totally understand your point, I think that the associated value judgment (you are weak if you value your own happiness over that of others) is misguided. Is strength to be solely associated with selfless altruism? What is the end-game of this ethic?

You could say the same thing about flu:

* Half a million die each year of influenza

* Most are very young or very old

* Flu is preventable ($1 face mask)

* Flu is easy to treat in the developing world

The way we think of flu in the west is roughly similar to how Africans think of malaria: a pain in the butt but not the end of the world. Imagine what we would do if rich Africans arrived and started handing out face masks during flu season. We'd probably do what Africans do with malaria nets: say thanks and throw them in a drawer.

>The way we think of flu in the west is roughly similar to how Africans think of malaria: a pain in the butt but not the end of the world. Imagine what we would do if rich Africans arrived and started handing out face masks during flu season. We'd probably do what Africans do with malaria nets: say thanks and throw them in a drawer.

That's incredibly misleading, ignorant, and even disrespectful. First of all, malaria is a scourge that's not even comparable with common flu (even thought the latter, through sheer virulence, rivals the former in number of anual deaths). Secondly, you are absolutely talking out of your butt when you say "Africans throw our nets in the drawer"; much to the contrary, these sort initiatives are constantly shown to be one of the most effective ways to reduce malaria infections, indeed one of the most effective ways to prevent suffering in general.

Malaria nets have been shown by many organizations as being one of the most effective use of funds possible. There are very few ways to more easily and effectively reduce pain and suffering in this world. Your comparison is ridiculous.


I would like to add, Givewell agrees with you (https://www.givewell.org).

For those who don't know, Givewell is a charity devoted to evaluating charities and writing recommendations for the most efficient and effective use of your donation money.

So my counter to that is that Givewell only recommends causes which have published evidence demonstrating their efficacy. It completely misses these more speculative “fat tail” causes, such as life extension or existential risk reduction.

Givewell's founders know this, which is why they spun off the Open Philanthropy Project: https://www.openphilanthropy.org/focus/global-catastrophic-r...

There are a lot of charities handing out mosquito nets in a driveby fashion. There are not many charities working closely enough with local communities to ensure that those mosquito nets will actually be used, and they won’t be turned into fishing nets instead. If you donate to malaria prevention, make sure it’s a truly hands-on charity.

Thank you for the information. I have no connection to the charity I linked (except I think I may have given them $50 once), and I don't know which charities, if any, are effective at preventing malaria. Maybe they should be giving out fishing nets as well. It's telling that for the communities receiving these nets, they see starvation as a more immediate threat.

Anyway, my point is just that if you're looking to make optimal use of money to save or improve human lives, there are many things that should be higher priorities than stopping people dying "of old age" in the West.

Clean water is an exponentially larger problem than any single disease. From Flint, MI to Malawi, no one is going to listen to any aid worker about anything else if they’re dying due to lack of clean drinking water.


Unfortunately anti-malaria nets backfired in a major way - people started using them for fishing, wrecking local ecosystems.


Relevant Quote:

> “There are two ways to make the world a better place. You can decrease the suck, and you can increase the awesome… And I do not want to live in a world where we only focus on suck and never think about awesome.” - Hank Green

You are advocating for a crude, and disturbing, form of utilitarianism here. Ultimately, I’m a moral relativist, so I even though I can’t say another point of view is “more correct” when it comes to morality, I think engaging some effort and discernment in choosing a moral system is “good.”

That said, connecting this utilitarian reasoning to the guidelines of what research to pursue, never mind the moral aspect would lead to an impoverished field of science.

On the everyday perspective, if one is suffering from ill health, but not from poverty and malaria, the idea that concern for your suffering should be set aside until the numeric quantification of the suffering of other populations is brought in line with your own, lacks compassion.

Value systems that lack compassion defeat their own purpose.

I have to admit, I love how that site incorporates HTML5 without shoving a bunch of animations in my face. Great design!

Graph Theory, unlike number theory, is a strangely shallow subject. By this I mean that a lot of the results don’t require a large number of previous results (there’s definitely some).

Compare this to number theory, where every interesting extant problem appears to require ten years study.

(I realise people may infer a value judgement from shallow/deep, but none is intended.)

>> Compare this to number theory, where every interesting extant problem appears to require ten years study.

I don't often come here to comment but as someone in progress on an original research masters in number theory I can say this is utter bullshit. I assume your 'interesting' qualification (somehow) excludes obvious candidates like Landau's problems [0]. Some examples. I was taught about the ABC conjecture as an undergrad. You can easily teach the Brun sieve [1] method of working out that the sum of the reciprocal of the twin primes converges. Novel solutions to Diophantine problems are sometimes accessible to undergrads. Richard K. Guy wrote a whole book on unsolved problems in NT, some of which have been solved using undergraduate number theory and someone's upper bound you can just use (as easy as apt-get installing this_dope_bound). You can start reading papers without a PhD, never mind ten years of study. I think it's possible to get an utterly unrepresentative sample of either field by only sticking to "elementary" results. There are some extraordinarily subtle results in graph theory! Conversely, you can find NT problems amenable to elementary techniques [2].

[0] https://en.wikipedia.org/wiki/Landau%27s_problems

[1] https://en.wikipedia.org/wiki/Brun%27s_theorem

[2] https://en.wikipedia.org/wiki/Quadratic_reciprocity

I’ll defer to your superior experience here. Only done fairly introductory number theory and graph theory stuff. That was just the impression I got from what I’d studied.

Ultimately, naming is pointing at clouds, but number theory IS older:


Afterthought: I wonder what caused the decline in the 80's (also true for other branches of math)

That's around the time public-key crypto was invented. Coincidence?

Isn't graph theory basically linear algebra, because every graph can be represented as https://en.wikipedia.org/wiki/Adjacency_matrix ?

The discovery was made by Aubrey de Grey, the same person who believes that humans will eventually live past 1,000. Weird!

> [de Grey] found his way to the chromatic number of the plane problem through a board game. Decades ago, de Grey was a competitive Othello player, and he fell in with some mathematicians who were also enthusiasts of the game. They introduced him to graph theory, and he comes back to it now and then. “Occasionally, when I need a rest from my real job, I’ll think about math,” he said. Over Christmas last year, he had a chance to do that.

Recreational math / CS is so much fun . I usually try thinking of knot theory. Haven’t really discovered anything yet but it’s still fun. So far the best question I have asked is https://cstheory.stackexchange.com/questions/32292/knot-reco... .

Yeah I did a spit-take when I read this. I remember him doing a TED talk ages ago...

Interesting tidbit, there was a bug in the computer assisted part of the proof (now fixed):

"Many thanks to Brendan McKay and Gordon Royle for letting me know overnight that they had successfully 4-coloured my 1567er; as a result I found a bug in the part of my code that implements the relaxation described in section 5.4 and now it agrees. "

Source: https://gilkalai.wordpress.com/2018/04/10/aubrey-de-grey-the...

Aubrey de Grey is much more than just an Amateur Mathematician, I highly recommend Rob Reid's podcast episode with him: https://after-on.com/episodes/020

This was great, thanks.

In my experience, being an amateur mathematician is more fun that doing it professionally, and potentially just as productive. You don't get nearly as much time for math, so you make the most of it and work only on the most interesting problems as opposed to just writing another paper. There's no time for beating your head against the wall, so you just do it when you're inspired, which is how problems get solved anyway.

People who only do something when inspired are reliably and overwhelmingly bested by people who do it consistently.

Is that a generic claim? I'm making a specific, counterintuitive, and empirical claim about productivity and work-style of mathematicians, having been on both sides. Even for professional mathematicians, the solution to a problem can come all at once, while one is not even actively pursuing it, which begs the question of the purpose of beating your head against the wall when doing math. For just one example amongst many, see the story of Yitang Zhang, who made his breakthrough in establishing the first finite bound on gaps between prime numbers, while out at a barbeque and having largely given up on the problem after two years of effort.

It seems likely that the two years of effort were essential in getting him to the point where he could have the breakthrough.

Are you confident the majority of the gains weren't made in the first year? It's exactly my point that knowing when to give it a rest is hard to estimate, and those who work on these problems professionally are likely to just keep pushing, potentially just spinning their wheels. Interestingly, the brain keeps working on these problems subconsciously, so you can actually get more done if you work on more things and tend to give up a bit earlier. I've solved hard problems before in an afternoon, a year after being hopelessly stuck on them, and my sincere estimate is that I would have made exactly zero progress had I kept pushing.

It seems unlikely that he would have had the breakthrough without those two years of effort. Step one of the Feynman algorithm is necessary for step two.

I think the counterintuitive aspect of problem solving at that level is how nondeterministic it is. It is no doubt that learning everything you can about a problem is critical to setting yourself up for being able to solve it, but this can happen a lot earlier than when someone burns out on a problem.

If that's so, why are most and the strongest results achieved by professionals?

Most mathematical results would be unreachable by amateurs due to the sheer amount of background knowledge, especially of existing literature, required to make progress on a problem (and in some cases to even understand the statement of the problem, e.g. https://en.m.wikipedia.org/wiki/Hodge_conjecture).

Elementary graph theory and combinatorics are somewhat outliers in this regard, as there is not so much "theory" per se that one has to build on or work with. Tim Gowers's "Two Cultures" essay is an interesting read on this topic: https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf

I would imagine there are more man-hours of professional mathematics research than amateur.

I think for a large number of people, myself included, the "Eureka!" moment often occurs when you're least focused on the problem at hand. I remember in college, studying EE, I was struggling to make sense of how a flip-flop worked (the basis of a register). My ah-ha moment was literally in the shower.

For more on this subject, see the 1945 essay by French mathematician Jacques Hadamard, An Essay on the Psychology of Invention in the Mathematical Field. https://archive.org/details/eassayonthepsych006281mbp

Thank you for sharing. One thing that impressed me about Hamming's essay on how to produce world-class research [1] is that it suggests a necessary, but not sufficient prerequisite is to spend your waking life thinking hard about some problem. This is necessary in order to provoke the unconscious mind into doing the heavy-lifting required for a "Eureka!" moment, so I was pleasantly surprised when Hadamard's book has a chapter titled "The Unconscious and Discovery". It amazed me that even in the modern world we do not understand the processes that led to these breakthroughs, so we can only leave them in the realm of the otherworldly and the mystical.

[1] http://www.cs.virginia.edu/~robins/YouAndYourResearch.html

Another example: Kekule 'said that he had discovered the ring shape of the benzene molecule after having a reverie or day-dream of a snake seizing its own tail.'


Musicians commonly report waking up with song ideas fully formed. We shouldn't be too surprised, really; the complex sentences out of our mouths (more often than not) arrive without any conscious thinking. Jung said that our egos are like planets orbiting a Sun they're unaware of.

where do people have these discussions and can I join? I wake up almost every morning with music in my head from the dream that is (more often than not) original, and IMO sounded like any hit on the radio.

And then within a few minutes I quickly forget the music. And there’s no way to remember it because it’s not anything that I can just find and listen to.

If I've been working on a problem at work for too long, I'll often stop working on it and just work on something else, because I know that tomorrow when I take a shower or when I'm walking to work the next day I'll figure it out in my head.

What was it?

The paper has beautiful illustrations. https://arxiv.org/pdf/1804.02385.pdf

Animations showing how the larger graphs are built from smaller graphs would be amazing and illustrative for mortal non-mathematical-geniuses.

[Edit: because the drawing has unit edges. My mistake]

Why is the Moser Spindle drawn to make it look non-planar, when it obviously is planar? The drawing obscures the point of the Moser Spindle, which is how to construct a 4-color graph from 2 simple 3-color graphs.

It's also confusingly colored to make it not clear why it requires 4 colors


Better: https://en.wikipedia.org/wiki/Moser_spindle#Application_to_t...

You can embed the graph of the Moser spindle into the Euclidean plane without its edges crossing, but not in such a way that each edge is straight and of unit length.

If you haven't watched an Aubrey de Grey talk, you owe it to yourself to give the man a listen. Beyond the admittedly interesting topic, he's genuinely entertaining.

Better headline would be "Decades-old Graph Problem Yields to Amateur Mathematician and Straight-Up Genius Aubrey de Fucking Grey"

Is anyone else disturbed by the fact that the header image is incorrect? In that the distances between vertices are not all equal. See for example the blue node just beneath the outermost red node on the left side.

If you mean the blue node that's connected to the yellow node beneath it, I think it's actually connected to the yellow node beneath that one (the one that is distance 1 from the blue node :) ), and the edge happens to travel very close to the closer yellow node so that it looks like it intersects it.

If those three nodes (blue, yellow, yellow) were in fact connected by short edges, then another problem would be that the two yellow nodes were connected, which they couldn't be by the constraints of the original problem.

The length between any two vertices which share an edge is 1 https://en.wikipedia.org/wiki/Unit_distance_graph

Wow, impressive. I had heard some disparaging comments about de Gray, kind of like you hear disparaging comments about Kurzweil (which I mostly don't agree with). For some people, there appears to be a fundamental aversion to their ideas which goes beyond mere disagreement.

I don't know much about de Gray, but this piques my interest in what else he has done.

Well, many of their ideas challenge the traditional mental model we have about humans and humanity. I think people feeling particularly offended or uncomfortable with that shouldn't surprise us.

There's also the tiny issue that he hasn't produced anything of value.

This is precisely what I mean by "beyond mere disagreement". It's not enough to disagree; his character must be attacked as well.

de Grey seems like a great guy; he's a fantastic speaker, I generally agree with his vision of humanity, and this proof is impressive. I haven't attacked his character, nor do I plan to.

With respect to his work on resisting human aging, he hasn't produced anything of value.

what about this contribution to graph theory that we’re commenting on?

“Occasionally, when I need a rest from my real job, I’ll think about math,” Signature material.

I'm so glad Aubrey de Grey got a nice win here. I feel like he doesn't get enough credit for his other endeavours.

Is it just me or does the image of 1581 vertex graph contain a face that looks a bit like krusty the clown?

Oh good! I'm not the only one who saw that.

The huge, flashing "Vote for us" button on the bottom right corner of the screen (at least on mobile) is incredibly distracting.

“The guy” is Aubrey de Grey, who used to get more discussion on HN:


Curing death doesn’t get much buzz these days. Maybe we’ll give it another look?

Anti-aging research is stronger than ever, with some very impressive results recently coming out of David Sinclair's lab at Harvard for example.

> Imagine, he said, a graph — a collection of points connected by lines. Ensure that all of the lines are exactly the same length, and that everything lies on the plane.

I do not understand what 'everything lies on the plane' means. I take it as meaning 'everything is laid out in two dimensions' but that doesn't make sense. Can HN help?

That is what it means. Note that the lines ("edges" in graph theory terms) can cross each other.

Nelson asked: What is the smallest number of colors that you’d need to color any such graph, even one formed by linking an infinite number of vertices?

The Wikipedia page describes the infinite-vertices version of this graph as

an infinite graph with all points of the plane as vertices and with an edge between two vertices if and only if the distance between the two points is 1.

This of course is impossible to draw but Wikipedia shows seven-vertex and ten-vertex subgraphs of it:


Math is terse and biology is verbose. Normally a person good in biology dislikes math and a person good in math dislikes biology. He has done extremely well in both! How can one be so good in both?!

the fact that Biology is verbose just means that we haven’t quite figured it out yet. And he’s over here trying to figure it out.

A part omitted from the article is that de Gray is a genius. That partly explains his other achievements in other fields.


I don't think a comment on HN has ever made me as viscerally upset as this one.

It's just the epitome of hubris. People have been trying to find counterexamples since at least the 19th century, without success, and now that a formal proof has come around, none of the thousands of professional research mathematicians alive has found any flaw with it.

But no, they must all be wrong, because of your (trivial and very easy to come up with) counterexample! Nope, the counterexample was shown wrong in minutes.

The worst part isn't that you didn't see how your example could be 4-colored. That's fine, everyone makes mistakes in mathematics. The worst part is that you have such a low opinion of everyone else's intellectual abilities that instead of asking "am I understanding the problem statement wrong?" or "what's the four-coloring of this graph that I'm missing?" you simply assert that you're right and the entire field of specialist mathematicians is wrong. Do you really think your counterexample is so shockingly clever that you're the only person capable of coming up with it in 150 years?

Your comment made me feel a little better.

I'm endlessly amused by him confusing the words "theory" and "theorm"

You're horribly mistaken, but rather than just say that you're an idiot because there's a proof that every planar graph is 4-colorable, let me explain your confusion.

You are trying to construct a complete graph on N vertices in the plane, giving an example for N=5. This cannot be done. Indeed, every graph is either planar xor contains either the complete graph on 5 vertices or the bipartite graph of 3 vertices in each group as an implied subgraph.

You can embed a complete graph on N vertices in 3 dimensions easily, with thin rods. But this doesn't work for 2 dimensions because that connecting line divides 2-d space, so you can't cross it with another line. This property imposes some sharp constraints on what planar graphs have to look like, with implications for its colorability. The Four-Color theorem amounts to an exhaustive enumeration of the possible situations arising from these constraints and then showing that all of them can be colored with only 4 colors.

Here's a chance for you to make some easy money: I'l bet you any amount you care to wager that I can color any planar map you can produce an actual drawing of with four colors.

Be careful in how you phrase that. I would say graph where you say map.

Using map opens you to claims such as that a map containing the North Sea, the kingdom of the Netherlands, the kingdom of Belgium, the French Republic and the Federal Republic of Germany can’t be four-colored because they all border each other (the kingdom of the Netherlands and the French Republic on Saint Martin in the Caribbean)

Historically, other examples likely existed, as land ownership tended to be patchy in feudal Europe, and, later, colonies added borders between various European countries.

For convex shapes, isn’t it?

Huh??? Are you asking whether the 4-color theorem only applies to maps with only convex shapes? If so, the answer is no, the theorem applies to any planar map.

I think he meant connex.

Connex isn't a word, so that's unlikely. Convex is, but that's also not what he meant.


Theories are not laws; only takes one counter-example to disprove them.

Is this a joke? The four color theorem is a proven theorem and not some crackpot theory.


You didn't even try it, did you? Took 10 seconds.


I love that OP was at least willing to provide an example of a planar map that he believed could not be 4-colored. Credit for sticking his neck out.

Red, blue, yellow, green. And white. That's five colors! ;-)

This is like a textbook case of the Dunning-Kruger effect

I'm really glad I'm almost never this cocksure of myself. I can't imagine the embarrassment.

There's a difference between a "theory" (in popular usage) and a "theorem" (in a field like mathematics).

A theory is a hypothesis. It's a plausible-but-unproven guess.

A theorem is a fact which is known because it has been proven.

Just to be clear: "theory" has a lot of different meanings, depending on the context. It can mean "hypothesis", but I can think of at least three other meanings, with example phrases: "Marxist theory"[1], "the theory of evolution"[2], "group theory"[3]...

As you're probably aware, this confusing patchwork of similar and overlapping yet distinct meanings is ripe for abuse: it's what allows people to get away with saying things like "evolution is just a theory".

[1] Here it means "the collective body of work of some school of thought".

[2] "A widely-accepted, well-tested scientific explanation for a phenomenon".

[3] "A subfield of mathematics; the set of definitions, theorems, and techniques related to a particular type of abstract object" (in this case, groups).

Yeah, the word "theory" is heavily overloaded and I wanted to keep it snappy so I just specified "general usage" and "in mathematics" to try and keep it on track. :)

I would correct that, for physics at least -- "theory" is just an explanation for something. "Law" is an experimental observation that holds true, but doesn't provide an explanation. E.g. Newton's laws describe macroscopic non-relativistic motion pretty well, but do not explain motion. The theory of relativity, on the other hand, explains the description of a bunch of physical phenomena, and also has lots of evidence behind it.

Theories are not laws; only takes one counter-example to disprove them

I agree, apparently it only took a couple of minutes in mspaint to disprove yours :)

On a serious note, what hermitdev did we've probably all done at some point in our past, especially in our youth (disparaging long-standing theories about life, science, philosophy, etc.), no need to be mean.

Few times this leads to groundbreaking revelations that help humanity progress, but most times it leads to the embarrassing conclusion that past generations were not idiots.

When I was in high school I was convinced I could trisect an angle using compass and straightedge, and I told my math teacher so. He just smiled and said "Prove it."

He was a good guy. Took me several days to finally understand my mistake, but afterward I had a much better understanding of the problem

So put your money where your mouth is. How much are you willing to wager?

there are more than 7 billion people in the world. every 4.2 seconds, we collectively live more than 100 years. there's a lot of clever-thinking time in a day, it's always nice to keep that in mind

I guess that's technically true, since 931 years is more than 100 years.

yeah, sorry, that was a typo, it's 1000, not 100. I calculated it a while ago. now population estimations are around 7.61 billions, which is around 1014 years every 4.2 seconds. it's an impressive amount of time

upboated for providing a pic. Thanks man but sorry man.

I can't understand your description, but if you can connect any number of nodes to all other nodes in a single plane, you may have a career in electronic circuit design.

> It's a rather easy formula

So what are you waiting for? Draw it up. You'd shake the very foundations of mathematics.

... I guarantee that if you draw that out for us, someone will 4-colour it for you. That's what "theorem" means.

Your construction is a bit hard to understand. Would you mind describing it as an explicit plane graph, rather than in terms of a "map"?

"This mathematical theory that has been proven is bunk"

Right... Either you're misunderstanding what the theory says or your crazy concentric countries idea doesn't work (hard to tell from your description).

Like I said, describing it is hard, but showing it is not See this: https://imgur.com/iU7oc8d There's no way to color that with 4 colors without a color hitting itself. Yes, it's contrived, but so are borders. Also, the 4-color theorem didn't limit itself to established borders, it claims to be for an arbitrary map. This image, is well, arbitrary and drawn up in paint in a few minutes, but it shows the point.

You're mistaken about that not being 4-colorable. Color the "rings", from outside to inside: red, blue, green, blue, yellow.

Again, as mentioned in my earlier comment, I would suggest thinking of things / describing things in terms of plane graphs rather than "maps". It'll make everything easier.

I might also suggest learning some of the relevant graph theory? It sounds like you're trying and failing to embed a K_5, or perhaps an arbitrary K_n, into the plane. That can't be done, and the proof that this is impossible is much easier than the full 4-color theorem. (There are also the 5-color and 6-color theorems, which again have much easier proofs than the 4-color theorem, and which are another reason you can't embed an arbitrary K_n in the plane.)

You can do R G B Y B from the inside working out.

The third and fifth rings (counting from the inside) don’t touch each other so, you can give them the same color.

"It's a rather easy formula to produce a "map" that requires N colors if no two colors are to touch."

Then please do, we'd all love to see it.

Although you can start with a bulls-eye of concentric ring-countries; once you start making enough of these isthmuses into the centre, then some of those countries must be broken up into "arc-islands".

You are correct that O(N) colours are needed if all the islands of one ring-country must have the same colour. But to encode that constraint graph-theoretically you would need "bridges" (edges) over isthmuses, which would make the thing non-planar.

Dude, my whiskey is almost over, still waiting for you to shatter centuries of mathematical foundations. Don't make me wait too long please. Getting drunf af.

You know you have a giant prize and glory awaiting you right?

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