"You know in my early twenties, let's say, people always thought that I would, you know, be a great mathematician. And be good at various things and so on. And in my late twenties, I hadn't achieved any of the things that people were predicting, and so I call it my black period. I started to wonder you know whether it was all nonsense. Whether I was not a good mathematician after all and so on, and then I made a certain discovery and was shot into international prominence.
As a mathematician when you become a prominent mathematician, in that sense, it doesn't mean that many people know your name. It means that many mathematicians know your name, and there aren't many mathematicians in the world anyway, you know. So it doesn't count very much, but it suddenly released me from feeling that I had to live up to my promise.
You know, I had lived up to my promise. I sort of made a vow to myself. It was so nice not worrying anymore that I thought I'm not going to worry anymore ever again, I was going to study whatever I thought was interesting and not worry whether this was serious enough. And most of the time I've kept to that."
Life, Death and the Monster (John Conway) - Numberphile
At 15 I said ”Welp, not gonna be a music prodigy, that’s okay”
At 25 I was like ”Damn, not a comp sci prodigy either that sucks”
Then at 30 it was ”Well crap, anything I achieve now will just be considered normal”
Pressure is still there but it feels lessened. I’m not gonna be famous for achieving great things before the curve. But I can still gain notoriety for doing cool things at a normal pace.
The key is to let the ego motivate but not crush you, I guess.
> To me, the question that springs to mind then is: does that kind of attitude work as well for “lesser” mathematicians/people?
My grades shot upwards when I stopped caring about them and decided to focus on what interested me (i.e. I like subject X and want to learn it well and will not use the grade as a metric of how well I learned it).
If work means to live a happy and meaningful life- absolutely.
If work means to gain the prestige of a man like Conway- probably not. But I can rest assured that I likely wouldnt achieve that with other attitudes either
They may not become famous in their fields, but they're not going to be worrying about that stuff anyway, so will probably be happier. Which, it turned out for me, was a better goal all along.
It's not necessarily about becoming famous.
If you're good enough, you can just work on what you enjoy doing, and it will probably yield something research worthy anyway, and your bosses will be happy, etc.
Otherwise, you might end up simply having fun on dead-end stuff, which are harder to sell, academically.
But that's just my impression… mind sharing your story?
I think a lot of us here will recognize this as the exact same sentiment that Feynman expressed pretty early in his career. It seems that while some great people got there by being hell-bent on greatness from early on (Schwarzenegger comes to mind), for others it’s ironically the release of that pressure that unlocks the capacity for greatness.
It's a bit tragic that he is remembered for something he himself wasn't very fond of. From his biography:
"Do you know something? I hate the Life game. I've really realized that I hate the damned Life game. ... I'm scared of the following thing happening. I'm scared of it becoming another one of these, 'John Conway, inventor of the celebrated Game of Life...' I told you, every time I turn to a book, a new math book, I look up the sacred name and it says: 'Conway, Game of life, pages 34 to 38.' And that's roughly all it says. And how can I say it---it's not character assassination, exactly, it's quality assassination. I regard Life as trash, and frankly. I mean, it was a real part of my life to have discovered it and so on. And I don't think it should be totally removed. But it seems to be all that I'm known for, among the general public. ... This was never a big deal as far as we were concerned. This was just a recreation, a game we played. Somehow it became a bit more important later on, or at least it did in the eyes of other people. I never thought of it as very important, I just thought of it as a bit of fun. In fact, in a way, I felt ashamed of it. I don't think it counts in the mathematical community, or at least in the serious mathematical community. I don't think any of my Princeton colleagues think this Life game is of any importance. I don't know. In a way it doesn't count for me."
I listened to the numberphile podcast on it yesterday where Brady interviewed Conway's biographer, and it looks like conway had come around to liking the game of life by the end. The real tragedy is that he never figured out the monster group
> I also recall Conway spending several weeks trying to construct a strange periscope-type device to try to help him visualize four-dimensional objects by giving his eyes vertical parallax in addition to the usual horizontal parallax, although he later told me that the only thing the device made him experience was a headache.
I'm quite curious what exactly Conway was trying to do here. The parallax we use for depth sensing is indeed horizontal in terms of the axes of the image we see, because our two eyes are separated horizontally. With a periscope you might be able to simulate two eyes separated diagonally or vertically. But how does that help you visualize four-dimensional objects?
I think he was aiming to get a simultaneous perspective of an entire 3d object. A bit like taking a 3d viewing perspective of a gD shadow being better at helping understand the 3d shape than a viewing pespective of the same shadow from the 2d plane it rests on.
Perhaps by splitting each eye's vision in two or four. Sure, in a single instant you can only be looking in one pair, but vision has significant hysteresis effects.
I suspect something like this really requires you to grow up with it.
In "Death's End" of the Three Body Problem trilogy by Liu Cixin, there is a fantastical description of 4D space.
> kranner on Aug 4, 2018 | parent | favorite | on: 4D toys
This passage from Death's End by Cixin Liu really gave me pause to stop and wonder about what the experience of seeing extra dimensions might be like (here translated to English by Ken Liu): --
A person looking back upon the three-dimensional world from four-dimensional space for the first time realized this right away: He had never seen the world while he was in it. If the three-dimensional world were likened to a picture, all he had seen before was just a narrow view from the side: a line. Only from four-dimensional space could he see the picture as a whole. He would describe it this way: Nothing blocked whatever was placed behind it. Even the interiors of sealed spaces were laid open.
This seemed a simple change, but when the world was displayed this way, the visual effect was utterly stunning. When all barriers and concealments were stripped away, and everything was exposed, the amount of information entering the viewer’s eyes was hundreds of millions times greater than when he was in three-dimensional space. The brain could not even process so much information right away.
In Morovich and Guan’s eyes, Blue Space was a magnificent, immense painting that had just been unrolled. They could see all the way to the stern, and all the way to the bow; they could see the inside of every cabin and every sealed container in the ship; they could see the liquid flowing through the maze of tubes, and the fiery ball of fusion in the reactor at the stern....
Of course, the rules of perspective remained in operation, and objects far away appeared indistinct, but everything was visible.
Given this description, those who had never experienced four-dimensional space might get the wrong impression that they were seeing everything “through” the hull. But no, they were not seeing “through” anything. Everything was laid out in the open, just like when we look at a circle drawn on a piece of paper, we can see the inside of the circle without looking “through” anything.
This kind of openness extended to every level, and the hardest part was describing how it applied to solid objects. One could see the interior of solids, such as the bulkheads or a piece of metal or a rock—one could see all the cross sections at once!
Morovich and Guan were drowning in a sea of information—all the details of the universe were gathered around them and fighting for their attention in vivid colors.
Morovich and Guan had to learn to deal with an entirely novel visual phenomenon: unlimited details. In three-dimensional space, the human visual system dealt with limited details. No matter how complicated the environment or the object, the visible elements were limited.
Given enough time, it was always possible to take in most of the details one by one. But when one viewed the three-dimensional world from four-dimensional space, all concealed and hidden details were revealed simultaneously, since three-dimensional objects were laid open at every level. Take a sealed container as an example: One could see not only what was inside, but also the interiors of the objects inside. This boundless disclosure and exposure led to the unlimited details on display.
Everything in the ship lay exposed before Morovich and Guan, but even when observing some specific object, such as a cup or a pen, they saw infinite details, and the information received by their visual systems was incalculable. Even a lifetime would not be enough to take in the shape of any one of these objects in four-dimensional space. When an object was revealed at all levels in four-dimensional space, it created in the viewer a vertigo-inducing sensation of depth, like a set of Russian nesting dolls that went on without end. Bounded in a nutshell but counting oneself a king of infinite space was no longer merely a metaphor.
> they were not seeing “through” anything. Everything was laid out in the open, just like when we look at a circle drawn on a piece of paper, we can see the inside of the circle without looking “through” anything.
Brilliant, I had always imagined it as a sort of fuzzy MRI, but of course that's not how it would be at all.
I'm surprised by the next part though. My visual field already presents me with infinitely nested detail, which doesn't seem to depend on the dimensionality of the input. After all there are 1D, 2D, 3D fractals (and beyond!)
Interesting. A 2-D being in flatland would consider the inside of a closed, empty circle to be "dark", but from 3-D we can still see inside it. Maybe one could postulate a 4-D source of light.
The "old-school" exposition of live in 2D, 1D, and then 4D in fictional form is in the novel Flatland: A Romance of Many Dimensions by Edwin Abbott Abbott. An entertaining read, mixing maths and social critique of Victorian class society (women are line segments, men are polygons - the more vertices, the higher class...)
This is fantastic, but one thing it seems to be missing is a projection of how objects change and interact over time, eg how revolving bodies form a helix in 4D, etc.
Or is that described elsewhere? I've not read the book (one more for the quarantine list!)
I believe this is about 4 space-like dimensions, not spacetime. Imagine perceiving a 2d image from within the plane: you'd only see width, not height, and lines would be hidden behind other lines. Only if you're in 3d, you can look at the plane from atop and see everything. Similar for looking at a 3d scene from 4 dimensions.
In Conway's game of Life, if you start with a cross as the symbol of death (5x7 grid 00100/00100/11111/00100/00100/00100/00100), you get first a "gravestone", then an egg which is traditional symbol of new life and rebirth. The egg stays around forever, unchanging, perhaps waiting for the right moment (or for you to set one of the two interior pixels).
and if you then fill in the top cell inside the egg you get a (small) exploder that will create four eggs. And if you fill in again the top egg (and only the top egg), you will get a more complex pattern result in a four gliders walking off scene.
Terry always amazes me with his gift of graceful and insightful writing. This post was much more interesting and meaningful to me than 90% of the Conway tribute posts I've come across.
If you're going to be so disparaging of someone with such great contributions and intellect, it'd be nice if you could at least get the facts straight and not muddy everything in your response. It's pretty impossible to comprehend what you're talking about and what's been so "misleading" about Tao's image, especially considering the "herd immunity" strategy (UK) is in many ways the diametric opposite of the "flattening the curve" approach (that of much of the rest of the world).
I'm also confused about the notion of John Conway's COVID-19 case being a UK case. Although Conway was originally from the UK, the press and Wikipedia reports refer to his getting sick and dying in New Jersey, where he was a professor at Princeton.
A few days after Joshua Bach posted the above critique, the original "flatten the curve" blogger, Thomas Pueyo, basically echoed Bach's points and called it "the hammer and the dance."
Wasn't the issue with the UK's strategy that they didn't try to flatten the curve and only isolated old people in order to build up herd immunity in the rest of the population as quickly as possible? I think they abandoned that strategy when they realised that young people can still die from COVID-19.
> Herd immunity: (noun) the resistance to the spread of a contagious disease within a population that results if a sufficiently high proportion of individuals are immune to the disease, especially through vaccination.
We have to have a vaccine first!
Treating our populations like a herd means that the ones most susceptible die... by the millions. When I hear the term thrown about like this I can't help thinking of armchair quarterbacks.
We really don't want a policy saying get everyone together and let the ones that can't hack it die, then declare success. That's awful.
Flattening the curve is the opposite of the herd immunity approach. And conway died in New Jersey, not the UK.
Clearly, whatever field you are an expert in, it isn't epidemiology, and and its unfortunately not the case that your expertise in whatever has given your opinions in disease control any more weight.
What you mean by FTC, and what you mean by herd immunity is not relevant. The graph on Tao's blog is misleading and that's all I was pointing out.
Yes I was mistaken about where he died, and can't edit the post because it was too heavily down-voted. The point stands that the graph posted by Tao has cost lives.
So the analogy with the Krein–Milman theorem would be this: Any proof is (approximately, at least) a combination of the extreme proofs of the same result?
And Sir, you win the Internet for not mentioning GOL there. Kudos well deserved, Conway was a great mathematician (among other 'skills'). He'll be missed.
"You know in my early twenties, let's say, people always thought that I would, you know, be a great mathematician. And be good at various things and so on. And in my late twenties, I hadn't achieved any of the things that people were predicting, and so I call it my black period. I started to wonder you know whether it was all nonsense. Whether I was not a good mathematician after all and so on, and then I made a certain discovery and was shot into international prominence.
As a mathematician when you become a prominent mathematician, in that sense, it doesn't mean that many people know your name. It means that many mathematicians know your name, and there aren't many mathematicians in the world anyway, you know. So it doesn't count very much, but it suddenly released me from feeling that I had to live up to my promise.
You know, I had lived up to my promise. I sort of made a vow to myself. It was so nice not worrying anymore that I thought I'm not going to worry anymore ever again, I was going to study whatever I thought was interesting and not worry whether this was serious enough. And most of the time I've kept to that."
Life, Death and the Monster (John Conway) - Numberphile
https://www.youtube.com/watch?v=xOCe5HUObD4
I take great solace from these words.