I know this isn’t written as a typical obituary but I can’t imagine a better way to say goodbye and thank you a true scientist. The entire blog post is beautiful- not just the presentation but also the content and sentiment.
Mitchell Feigenbaum has, for the last ~17 years, been my vote for the next Nobel Laureate in Physics. It is a rare thing to derive an irrational fundamental mathematical constant that appears in nature. The Feigenbaum constants are as fundamental as e and \pi, and they are readily observed experimentally (though, necessarily, not to full precision :) ).
Yes, it's kind of hard to get more fundamental than waves and circles.
Although, as I was just about to say what math could you do without them on second thought, the answer seems to be quite a lot.
For example I couldn't say without researching it but how much did Gödel or Turing rely on them, at least for their most influential work?
That question will probably come back to bite hard given their footprints.
Another conjecture, most software developers will never have to use them extensively. Yes gaming, computer graphics, and I guess all signal processing would become suddenly more challenging but corporate IT and devops ought to be pretty safe.
Maybe it depends on a personal definition of fundamental. I think you could make an argument that Newton's results were more fundamental than Einstein's, however staggeringly less complete they might seem hundreds of years in the future.
Agreed -- I used the word "fundamental" to suggest that it emerges from pure mathematics. In that sense, Feigenbaum's constants are in the same class as \pi and e, which is distinct from the traditional "fundamental constants" of physics like c, \hbar, fundamental charge, etcetera.
The fundamental constants of physics have, as yet, no known mathematical origin. Feigenbaum's constants, on the other hand, emerge from pure mathematics and also appear in nature. That they are expected to be irrational, like \pi and e, only adds to the charm.
Fundamental is kind of binary, something is either fundamental or it isn't. This is like arguing what element is more fundamental to human existence oxygen (65% by mass) or nitrogen (just 3% by mass); we die without both.
Now, like oxygen, pi and e are certainly more prevalent ...
Just because the Feigenbaum constant does not pop everywhere in our formulas, it does not mean that it's not as fundamental. I would bet that if it were different, the universe would not be able to exist as it does now (akin to making pi different).
I guess I'm wrong, but I read that they try to give it to people just starting out in their field, to encourage them to continue, rather than as a prize for finishing.
Update: user "ISL" says "that was Nobel's original intent, but the reality of the prize has changed with time."
Yes, Nobel Prize is only given by living people to encourage innovation rather than remembering the death, yes. However, it's very common for the Nobel Prize to aware scientists after a discovery has been made. Nobel Prize of Literature is the same, I believe.
For example, Albert Einstein won the Nobel Prize in 1921 for "for his services to theoretical physics, and especially for his discovery of the law of the photoelectric effect", but Einstein's relevant papers were published in 1905 on Annalen der Physik, it's 16 years later after the discovery. Another recent example is the Nobel Prize of Physics of 2014, given to Isamu Akasaki, Hiroshi Amano, and Shuji Nakamura for blue LEDs, it's also 20 years later.
Due to this very reason, The Breakthrough Prize was established in 2012 by multiple entrepreneurs, including Mark Zuckerberg, as a prize, which its proponents claim to be that they try to give it to people just starting out in their field, to encourage them to continue, rather than as a prize for finishing. Although it has received some criticisms of being a rockstar award as well.
Is it, by any chance, the prize you were thinking about?
I was an undergrad CS intern (and Chaos enthusiast) assisting the physicists at Los Alamos Natl. Labs Center for Nonlinear Studies when he visited to give the monthly “Tea Talk” on some subject in nonlinear mathematics/ chaos theory back in ‘88. I didn’t understand most of what he said. I asked the Deputy Director, a long-time PhD Theoretical Physicist, about the gist of the talk afterwards. He said “I didn’t understand most of what he said. Maybe if I had been collaborating with him for a few months...”
Feigenbaum was a fascinating man and some of his writing is what made me want to get an internship at the Center. I never dreamed I meet the man over tea.
I was there as well, also as an intern. Around the same time, Mandelbrot also stopped by CNLS to give a talk. CNLS in the late 80s was truly a magical place.
It was pretty amazing. I recall Doyne Farmer was there too - he had this crazy custom coprocessor card for his cellular automata explorations. I got to write some stuff to run on a CRAY Y-MP. Got to to play with NeXT black box serial no. 36. (LANL had the first 50 I think). Lots of crazy cool people and tech!
I'm amazed by the observation that even the Deputy Director, a seasoned theoretician, did not expect to understand a peer researcher's work without several months of collaboration. What chance does a mere undergrad or practitioner stand ! :)
Usually, after a certain level math problems become very narrow, and a small subfield develops around them. They are "easy" to get into, to catch up, read the important papers, sketch out the proofs for yourself, get a feel for the concepts and for the conjectures that lurk among them.
And usually that's it. Usually a few folks of that very particular topic/field have some ideas that they are working on, usually for at least a year, they try to formalize them, conceptualize them, communicate them, and naturally to prove them.
And sometimes draft/working papers are all we get. Sometimes someone becomes a "household" name after many persistent years like Wiles.
Feigenbaum barely ever showed up in the physics building at Rockefeller while I was there. He had an office. I saw his retreating back once, but he chose not to engage with the community there, despite it being full of people who wandered as widely as he did.
One interesting place that period doubling shows up that the article doesn't mention is in population ecology. I'm most familiar with the Tribolium work of Constantino, Dennis, Cushing, and some other folks, which you can read an example of here[0]. Basically, the beetles have three life stages (four, actually - egg, larva, pupa, adult - but adding eggs in the model doesn't increase accuracy any so they leave it out for simplicity), and if you model the respective numbers of adults for a starting number of larvae, you can see truly chaotic behavior, including period doubling. It's fascinating work.
There were several references to fish populations and and even "population biology". I wouldn't have thought 'population ecology' was that different but a cursory Google search leads me to believe I was wrong.
Chaos : Making a New Science by James Gleick https://www.amazon.com/dp/B003YL4KOO/ contextualizes the period-doubling thing and talks about some of the other related stuff. Good book.
I first saw that beautiful bifurcation diagram as a teenager in this book called "Fractals" that showed a bunch of fractals as well as programs to draw them. I was amazed that such simple arithmetic could lead to such weird, organic results. The code to draw it is astonishingly simple. Here's an implementation in Dart that prints it to your console:
import 'dart:math' as math;
const min = 2.4;
const max = 4.0;
const width = 120;
const height = 100;
main() {
for (var r = min; r < max; r += (max - min) / height ) {
var x = 0.5;
var counts = [for (var i = 0; i < width; i++) 0];
var greatest = 0;
for (var i = 0; i < 200; i++) {
x = r * x * (1 - x);
var i = (x * width).toInt();
counts[i]++;
greatest = math.max(greatest, counts[i]);
}
print(counts.map((n) => n > greatest / 10 ? "*" : " ").join(""));
}
}
You might enjoy his "Idea Makers: Personal Perspectives on the Lives & Ideas of Some Notable People", which consists of short essays on a few interesting people, some of which he knew personally.
A completely frivolous and useless note: the photo at the bottom of the article was taken at a table in the restaurant called Mediterraneo on the corner of 2nd and 66th, and behind him is the block on which one of the scenes in Annie Hall was shot.
I normally find Stephen Wolfram insufferable (and he still is, in this article) but boy what a privilege it is to have a friend who will eulogize us like this. If I was an intellectual associate of Wolfram’s I’d ask him to write this article for me to read before I died.
I think he reined it in pretty well for this article, there were a few personal asides but it didn't throw me out of the narrative like it usually does.
The first thing I did was to start scrolling to see where he would either inject himself into the story, or go off touting his own achievements. I was then going to come here to post a snarky comment here about it.
But as I scrolled, I did see quite a bit of really interesting content in the article, and thought such a comment may turn people away from the article, which would be a shame, so I decided not to.
> Can someone elaborate as to why Wolfram’s insufferable?
My own belief, based on work I did in the 1980s, is that it’s actually an intrinsic youthful genius phenomenon—analogous to the insufferability one sees in my rule 34 cellular automaton.
I have a lot of respect for the man despite his personality problems. He knew what he wanted at a young age and pursued it with determination and persistence and achieved it. Mathematica and the business he built to support it are both great accomplishments. I'm sometimes surprised he's not more popular on HN than he is.
I mention all that so it won't seem too mean-spirited when I repeat this:
I've heard a joke, "Which will achieve self-awareness first, Wolfram or Wolfram Alpha?"
He's a great man, but flawed by too-great appreciation of his own greatness.
He’s one of the great minds of his generation but his insecurity complex is so severe that he constantly needs to boast about himself and his achievements to an extent that is unbecoming even in a professional athlete. Tilting his (extremely interesting) book “A New Kind of Science” and saying that it would take a place next to Chemistry and Physics as a discipline is one such example. Even in this eulogy to a dead friend he is content marketing Mathematica’s numerical precision.
> Tilting his (extremely interesting) book “A New Kind of Science” and saying that it would take a place next to Chemistry and Physics as a discipline is one such example.
There are a couple transcribed lectures of his on amazon, one of them "Computation and the Future of the Human Condition", they are short and deep and I find them very interesting, I read them maybe a couple years ago... somehow within that little context I was so entranced by the underlying concepts and material which just clicked with me, that this aspect of Wolfram's personality completely escaped me... My partner tried to read them and immediately picked up on it, now identified - I find everything I read from him slightly tainted. I've yet to pluck up the courage to read NKS, i'm worried I will not be able to get through it.
It's a shame because I believe some of the work he has produced and ideas he has explored to be a truly fundamental and under-explored part of nature, hopefully enough of us will appreciate that without letting his personality get in the way.
NKS has a lot of very interesting bits of information, ideas and experiments. Some of it is truly novel. At least, I think some are truly novel, because there are also quite a few ideas that aren't novel, but he doesn't mention that fact or cite anyone. And he kind of speaks down about biology and other sciences, about things they are supposedly not researching, except they do.
It would have been a much nicer book if he had connected his ideas in universality, fundamental computing, math, physics and biology, to the rest of those fields. He mentions many of these connections/theories, but does not cite them or even attribute them to their respective fields of science.
Otherwise if you can read past all that, it's super interesting. It just leaves me with the inclination that, if I want to find out more about these new subjects, I'd be best of starting out looking for information not by Wolfram, if only just to get a good view/idea of the field right now, since Wolfram isn't citing anyone, but others probably are.
Are you including the extensive notes in the book when you claim that he doesn't mention any connections nor cite any existing research? I think he wrote the main book much like a textbook, i.e. a straightforward presentation of the ideas, and not like a typical academic publication. IIRC, the extensive notes do mention many connections and include a lot of citations.
The entire book is available online for free. Here's the first section of the notes:
... but, after looking at some more sections of the notes, I think now I remembered very much incorrectly – there are in fact very few citations or specific references. There are, in my opinion, very many connections noted tho.
In developing the ideas described in this book I have looked at many thousands of books, papers and websites—and have interacted with hundreds of people (see page xiii). But rather than trying to give a huge list of specific references, I have instead included in these notes historical information tracing key contributions. From the names of concepts and people that I mention, it is straightforward to do web or database searches that give a vastly more complete picture of available references than could possibly fit in a book of manageable size—or than could be created correctly without immense scholarship. Note that while most current works of science tend to refer mainly just to very recent material, this book often refers to material that is centuries or even millennia old—in some ways more in the tradition of fields like philosophy.
If you can overlook his immodesty, or choose to ignore it, I think you might enjoy reading NKS. It's very interesting, even if you don't agree that he deserves all of the accolades he claims he's owed.
>(Confusingly enough, with typical “machine precision” computer arithmetic, this doesn’t work correctly, because even though one “runs out of precision”, the IEEE Floating Point standard says to keep on delivering digits, even though they are completely wrong. Arbitrary precision in the Wolfram Language gets it right.)
The man can't help but promote himself. He worded this in a way that makes it look to a lay person like he invented arbitrary precision arithmetic, but in a way that is still technically correct. It's like he's actively trying to peddle something to people who don't know any better.
> makes it look to a lay person like he invented arbitrary precision arithmetic
I think that's pretty uncharitable. If what you claimed were true, particularly of his intentions, I'd expect him to have written something like "Arbitrary precision in the Wolfram Language got it right.".
But then I can't imagine a layperson having any idea what almost any of the sentence you quoted means anyways so your point seems pretty moot.
For anyone that already understands or is at least familiar with 'machine precision', 'floating point numbers', or 'arbitrary precision' (computer arithmetic), I'd think his claim that "Wolfram Language gets it right" would seem to be a pretty innocuous plug. Yes, the man regularly promotes the products of the company that bears his name on his own blog. I think it's fine.
Out of all the ways he could get his point across, you think he just landed within a hamming distance of 2 of a misleading one on accident?
>But then I can't imagine a layperson having any idea what almost any of the sentence you quoted means anyways so your point seems pretty moot.
You're not being honest here. If you can't imagine it, it is because you're not trying. People walk away with impressions, even if they don't retain the facts. He always writes to give an impression, not to give facts. Facts are always the cover for the impression.
Here's another example from the article:
>But what is actually going on in a phase transition? I think the clearest way to see this is by looking at an analog in cellular automata.
Is he being honest here? Is this actually the best way to understand a phase transition? And is this cellular automata the one he should be using?
Of course! It's His Holy Cellular Automata. It must be introduced, even if only in passing.
Yeah, "it's fine." He knows what he knows, and you can't fault someone for speaking to what they know. But it is a grating and self-serving stylistic choice, and it does undermine his message. But if you're an investor or someone looking for a savior to worship or a bandwagon to jump on, he's certainly got all the answers for you.
> You're not being honest here. If you can't imagine it, it is because you're not trying.
I was being honest. Please don't tell me I'm lying; assume good faith. I still claim that his wording 'Wolfram Language gets it right' versus 'Wolfram Language got it right' would leave a layperson with the impression different than what you claim.
I could be wrong. I'd be willing to make a (nominal) bet about the outcome of a reasonable proxy if you're interested in testing our disagreement. I'm not claiming near-certainty tho. I don't think you're 'definitely' or 'obviously' wrong.
For the second example, I think his cellular automata is a pretty good way to see a phase transition. I'm also interpreting his claim pretty charitably, e.g. not that it's literally the clearest way for anyone to see, let alone understand generally, for whatever reference class of 'people' would possibly be the best definition to use in this case. I think you're splitting hairs pretty finely here.
I do agree that his writing style is "grating and self-serving". There's lots of evidence that that's the case, just in the comments on this post alone. I think part of that interpretation is warranted and valid. I think another, significant, tho lesser part is confirmation bias.
Stephen Wolfram definitely doesn't have all the answers for me, or likely anyone. I'm pretty confused why you wrote the last sentence in your reply to me. I think my comment was pretty anodyne. You seem to have pretty strong feelings about it.
I'm also unsure why you'd think an "investor" would be particularly interested in him or his companies. Wolram basically sells enterprise software and related services. It doesn't seem like a likely undervalued opportunity for significant growth.
I'm also unsure why you think anyone would "worship" him as a "savior". I don't regardless.
I'm happy enough to wish him and anyone that's jumped on his bandwagon well tho. I think he's got a very good point about the likely benefits of engaging in what I'd call 'exploratory computational discovery', as he describes in NKS. Yes, he uses much stronger and strident language to describe the same thing, and I think the magnitude of his claims are too large, but I do think his point is underrated, even significantly. I'm more skeptical of how much low hanging fruit there's likely to be from applying methods like what he advocates, but not radically so.
pro.
Used to refer to an indefinitely specified person; one.
The word is ambiguous, but don't ask me to "assume good faith" when you're choosing the more offensive or questionable interpretation of my words over the most fitting one.
In any case, this issue is not important enough for me to be spending any more time on it, so if you don't understand me, we'll just have to live with that.
If the only reason Wolfram is writing about Feigenbaum in the first place is because it promotes his product, that makes it seem even more self-serving.
This guy manages to, within three sentences, turn a eulogy into a diss of IEEE floating point calculations, only to boast about how his shitty proprietary math toolbox does arbitrary precision better.
Just one of many grating examples of how he communicates. He's a genius though, so he has that going for him...
There's a place, a time, and a style for everything though.
Were he to include a WolframAlpha link to a working snippet, and mention that you need true decimals, because IEEE floats won't cut it would have probably conveyed the same without what the other commenters perceive as useless boasting.
Not being wrong doesn't absolve you from being deceptive. ("Save up to $1000!") A scientist doesn't hide the truth in fine print and technicalities, they expose it.
"Chaos and Fractals", a huge tome of a book , from the 1990s, is an incredible book to learn about Feigenbaum's work and others. The programming language it uses is rather obsolete, but it's such a wonderful book for a motivated high school or college student.
