mildly relevant: startup $\perp$ research?

Beware of research. If an undergrad writes something all his friends start using, it's quite likely to represent a good startup idea. Whereas a PhD dissertation is extremely unlikely to. For some reason, the more a project has to count as research, the less likely it is to be something that could be turned into a startup. I think the reason is that the subset of ideas that count as research is so narrow that it's unlikely that a project that satisfied that constraint would also satisfy the orthogonal constraint of solving users' problems.

~ from "how to get startup ideas" by paul graham.

well, in that case i'm set:
being a researcher, that means it's easy to leave the money-making to the mercenaries.

---

maybe i'm channeling cayley and hardy overmuch [1], but there is something relieving in doing work that is not immediately applicable to anything. for instance,

• you're not arming militaries with new, more dangerous weapons,
• you're not making poor people poorer and rich people richer,
• you're not coding an iphone app that just makes it more likely for tech-addicts to further ignore each other at the dinner table.

to do useless work is to do no harm; i can get behind that!


[1] “I have the highest admiration for the notion of a quaternion; but, as I consider the full moon far more beautiful than any moonlit view, so I regard the notion of a quaternion as far more beautiful than any of its applications.”

mildly relevant: an unexpected formation, some unsolicited advice.

~ from "topοlogy: the secret ingredient in the latest theory of everything" @techreview

For example, certain quantum particles cannot form pairs but do form triplets called Efimov states. That's curious--surely the bonds that allow three particles to bond together should also allow two to become linked?

Actually, no and topοlogy explains why. The reason is that the mathematical connection between these quantum particles takes the form of a Borrοmean ring: three circles intertwined in such a way that cutting one releases the other two. Only three rings can be connected in this way, not two. Voila!

~ from "you and your research" by richard hamming ('86) @paulgraham:

One of the characteristics of successful scientists is having courage. Once you get your courage up and believe that you can do important problems, then you can. If you think you can't, almost surely you are not going to. Courage is one of the things that Shannon had supremely. You have only to think of his major theorem. He wants to create a method of coding, but he doesn't know what to do so he makes a random code. Then he is stuck. And then he asks the impossible question, ``What would the average random code do?'' He then proves that the average code is arbitrarily good, and that therefore there must be at least one good code. Who but a man of infinite courage could have dared to think those thoughts? That is the characteristic of great scientists; they have courage. They will go forward under incredible circumstances; they think and continue to think.

.. There's another trait on the side which I want to talk about; that trait is ambiguity. It took me a while to discover its importance. Most people like to believe something is or is not true. Great scientists tolerate ambiguity very well. They believe the theory enough to go ahead; they doubt it enough to notice the errors and faults so they can step forward and create the new replacement theory. If you believe too much you'll never notice the flaws; if you doubt too much you won't get started. It requires a lovely balance. But most great scientists are well aware of why their theories are true and they are also well aware of some slight misfits which don't quite fit and they don't forget it. Darwin writes in his autobiography that he found it necessary to write down every piece of evidence which appeared to contradict his beliefs because otherwise they would disappear from his mind.

~ from "why does it take so long to learn mathematics" @tonysmathsblog

I don't think I'm a better mathematician than I was 35 years ago. In terms of solving exam questions, I would not perform as I did when I was twenty. Even with practice, I am sure I could not get back to that level, and not only because I no longer value that kind of cleverness enough to put the effort in. I now have a much better general understanding of mathematics and how it all fits together, but I no longer have the ability to master detail that I once did.

Perhaps I am misremembering my difficulties as a student: perhaps I didn't find it as difficult as I now remember it.  Certainly I had little understanding of how an area of mathematics fitted together: my learning at University consisted of reading strings of definitions and theorems, with little idea where it was all going, making sure I understood each result before going on to the next one, until, perhaps, in the last lecture of the course the lecturer would say something like "and so we have now classified all Lie algebras" and I would suddenly find out what the point of it all had been.  I now feel that I would have been a much more effective mathematician if I had read more superficially, skipping proofs until I understood the context, but since got good marks as an undergraduate I had no incentive to adopt what I now feel would have been a much better strategy..

mildly relevant: growth rates, as mentioned by pauΙ graham.

~from "startup = growth" by pauΙ graham

The phase whose growth defines the startup is the second one, the ascent. Its length and slope determine how big the company will be.

The slope is the company's growth rate. If there's one number every founder should always know, it's the company's growth rate. That's the measure of a startup. If you don't know that number, you don't even know if you're doing well or badly.

When I first meet founders and ask what their growth rate is, sometimes they tell me "we get about a hundred new customers a month." That's not a rate. What matters is not the abolute number of new customers, but the ratio of new customers to existing ones. If you're really getting a constant number of new customers every month, you're in trouble, because that means your growth rate is decreasing.

yes, a basic notion, but one not immediately obvious to the unambitious. a constant growth rate implies an exponential growth, which is more than many people expect out of their endeavors.

in general, i like how graham is willing to quantify his conclusions. a rough computation, for example, explains why what we observe about startups seems different from why they should exist:

If you judge by the median startup, the whole concept of a startup seems like a fraud. You have to invent a bubble to explain why founders want to start them or investors want to fund them. But it's a mistake to use the median in a domain with so much variation. If you look at the average outcome rather than the median, you can understand why investors like them, and why, if they aren't median people, it's a rational choice for founders to start them.

---

i also noticed this observation of graham's, though meant for startups, might also be relevant for us mathematicians to revisit:

You'll generally do best to follow that constraint wherever it leads rather than being influenced by some initial vision, just as a scientist is better off following the truth wherever it leads rather than being influenced by what he wishes were the case. When Richard Feynman said that the imagination of nature was greater than the imagination of man, he meant that if you just keep following the truth you'll discover cooler things than you could ever have made up.
..
What you're looking for initially is not so much a great idea as an idea that could evolve into a great one. The danger is that promising ideas are not merely blurry versions of great ones. They're often different in kind, because the early adopters you evolve the idea upon have different needs from the rest of the market. For example, the idea that evolves into Facebook isn't merely a subset of Facebook; the idea that evolves into Facebook is a site for Harvard undergrads.

on a related note, my prejudice of startups is generally that of hyper-capitalists, fueled by powerful greed, and willing to use science and technology to do so (i.e. the dark side of the force).

the feeling i've been getting though, from the quants and startup acquaintances i've met, is that these just happen to be hard-working, quantitatively-minded people who are making usable products efficiently. they could have easily have been scientists or engineers, but working in this kind of business offers better rewards with the same interesting (and fulfilling) kind of work ..

.. or, as graham writes:

Starting a startup is thus very much like deciding to be research scientist: you're not committing to solve any specific problem; you don't know for sure which problems are soluble; but you're committing to try to discover something no one knew before. A startup founder is in effect an economic research scientist.

sometimes, my first instincts are mathematical.

immediately after reading this ..

These smaller groups are always arranged in a tree structure. Your boss is the point where your group attaches to the tree. But when you use this trick for dividing a large group into smaller ones, something strange happens that I've never heard anyone mention explicitly. In the group one level up from yours, your boss represents your entire group. A group of 10 managers is not merely a group of 10 people working together in the usual way. It's really a group of groups. Which means for a group of 10 managers to work together as if they were simply a group of 10 individuals, the group working for each manager would have to work as if they were a single person—the workers and manager would each share only one person's worth of freedom between them. [1]

.. i started wondering if this could be viewed as something like a sub-martιngale or a variant of a hardy-littlewoοd maximaΙ function.


[1] from "You Weren't Meant to Have a Boss" by PauΙ Graham

more from ρaul 9raham: on writing.

maybe i've spent too many hours awake, typing away on an N$F grant proposal, but this recent essay of paul gr@ham struck a chord in me.

That's not even the worst danger. I think the goal of an essay should be to discover surprising things. That's my goal, at least. And most surprising means most different from what people currently believe. So writing to persuade and writing to discover are diametrically opposed. The more your conclusions disagree with readers' present beliefs, the more effort you'll have to expend on selling your ideas rather than having them. As you accelerate, this drag increases, till eventually you reach a point where 100% of your energy is devoted to overcoming it and you can't go any faster.

lately, i've been writing to persuade. specifically, i want the government to give me money. then again, i look at my writeup and i realise,

it looks too much like a paper or a lecture.

there's a bad habit of "polishing" a research article. as a general rule, technical writing is concise writing. i'm guilty of this. often i ask myself, even in casual conversations,

what is the best (read: most efficient) way to say this?

as a result, often reading a research article is like decoding, trying to figure out what the author was really thinking.

it's rare that someone writes in roughly the same order in which one discovers the proof of a theorem. that sort of exposition is left for talks and lectures.

on schedules (according to paul graham)

one reason why i don't love teaching has nothing to do with the actual practice of teaching. writing lectures is fine. the frustrating thing is the scheduling.

despite a few years' of experience, on my lecture days i still find it difficult to get anything other than teaching done.

when i count the hours, they are few:

2 hours of actual lecture,
1-2 hours of preparation,
1 office hour [1]

even if i were to work an 8-hour day, that leaves 3 hours left for researching, reading, or writing .. that is, if i actually have something worth writing up \-: ]

but does it ever happen, that i have 3 good research hours? occasionally, at most half the time. usually it takes an hour or two until i'm out of teaching mode and back into my crazy research self.

maybe i'm just weird.

then again, i can appreciate paul graham's most recent essay because it suggests that .. maybe i'm not so weird:

"When you're operating on the maker's schedule, meetings are a disaster. A single meeting can blow a whole afternoon, by breaking it into two pieces each too small to do anything hard in. Plus you have to remember to go to the meeting. That's no problem for someone on the manager's schedule. There's always something coming on the next hour; the only question is what. But when someone on the maker's schedule has a meeting, they have to think about it.

i never thought about it, but having to remember to show up to class on time is already bothersome. it's not unlike:

being able to think about maths, when walking to work,
vs. not being as able to think, when waiting for the bus.

again, it could be my own idiosyncrasies. there's more:

For someone on the maker's schedule, having a meeting is like throwing an exception. It doesn't merely cause you to switch from one task to another; it changes the mode in which you work."

if this were true, then i'd feel slightly better about my shortcomings. among the mathematicians that i know, my own work stamina seems the shortest.

[1] this is provided that office hours occur only on non-teaching days. almost every semester and against my better judgment, i schedule one office hour when i don't have to be in the office at all.

almost always, nobody shows up. i should have my head examined.

about web writers, techies, and philosophical discussions.

sometimes i really like reading this one writer, paul graham. he may be embedded in the real world of money and companies and building the internet and whatnot, but he writes as if he is a techie and understands techies. [1]

i might have written this before, but one of my reasons for liking his essays is that he's willing to take a chance on philosophy. never mind if he's right or not; he's giving it a shot.

as for why i'm writing this, he has a new essay up, called "how to disagree." here is how it begins:

The result is there's a lot more disagreeing going on, especially measured by the word. That doesn't mean people are getting angrier. The structural change in the way we communicate is enough to account for it. But though it's not anger that's driving the increase in disagreement, there's a danger that the increase in disagreement will make people angrier. Particularly online, where it's easy to say things you'd never say face to face.

If we're all going to be disagreeing more, we should be careful to do it well. What does it mean to disagree well? Most readers can tell the difference between mere name-calling and a carefully reasoned refutation, but I think it would help to put names on the intermediate stages. So here's an attempt at a disagreement hierarchy:

perhaps there is not much new in this, and perhaps other web writers have written such things already. also, it's short.

perhaps i should stop here, in my attempts to explain why i like this or that. these discussions go the way most philosophical arguments tend to go:

after a while, the words you use don't mean anything. they're used colloquially and not rigorously. if your company is persistent enough, then this might reduce to a first-order logic problem. then your discussion becomes a standard exercise.

more often than not, it will also lose most of its original feeling and humanness, though you can't really explain why.

this is the curse of arguing with mathematicians and techies; everything must be well-defined.

in other words, humanity is often sacrificed for the sake of rigor.

[1] i didn't make up the word "techie," but having tried several times before, i won't define it here. the term is too vague, but let me give an example or two:

if you've ever done any programming in your spare time, you're a techie;

if you've talked math at a bar before because it was natural to do so (and not because you intended to annoy someone) then you're a techie;

if you played with legos as a child, still like playing legos with your little cousins, but are careful not to build too extravagantly (so that you won't be found out), then you're a techie.

flattery will get you everywhere.

man, sometimes i love reading paul graham:

      Outside of math there's a limit to how far you can push words; in fact, it would not be a bad definition of math to call it the study of terms that have precise meanings.

but sometimes..

      [4] Philosophy is like math's ne'er-do-well brother. It was born when Plato and Aristotle looked at the works of their predecessors and said in effect "why can't you be more like your brother?" Russell was still saying the same thing 2300 years later.

Math is the precise half of the most abstract ideas, and philosophy the imprecise half. It's probably inevitable that philosophy will suffer by comparison, because there's no lower bound to its precision. Bad math is merely boring, whereas bad philosophy is nonsense. And yet there are some good ideas in the imprecise half.

ouch. i may have said similar things myself, but usually when i'm angry or trying to exhibit wry wit. myself, i tend to take the historical perspective and view math as a subset of philosophy.

it has an added bonus, too: if mathematics is philosophy, then there is no reason why it should be useful, except to prepare the mind for other processes of thought. if it is useful, fine, but that's not why we do it.

suggested reading, for a techie.

i must have mentioned it before, but i really enjoy reading paul graham's essays.

maybe it's because he flatters mathematicians and other techies, or maybe it's because i see in his work what i would have liked to have done, but wasn't brave or deft enough.

for example, his latest essay is "Is It Worth Being Wise?" and weighs wisdom vs. intelligence in a way which i couldn't scoff at. he's willing to approach issues in philosophy, but in a clean way; his reasoning reminds me of a mathematician's.

then, maybe i like him because admittedly, he writes things that i would like to believe. this is different from flattery, but a related matter.

for example, here's an excerpt.

To me it was a relief just to realize it might be ok to be discontented. The idea that a successful person should be happy has thousands of years of momentum behind it. If I was any good, why didn't I have the easy confidence winners are supposed to have? But that, I now believe, is like a runner asking "If I'm such a good athlete, why do I feel so tired?" Good runners still get tired; they just get tired at higher speeds.

People whose work is to invent or discover things are in the same position as the runner. There's no way for them to do the best they can, because there's no limit to what they could do. The closest you can come is to compare yourself to other people. But the better you do, the less this matters. An undergrad who gets something published feels like a star. But for someone at the top of the field, what's the test of doing well? Runners can at least compare themselves to others doing exactly the same thing; if you win an Olympic gold medal, you can be fairly content, even if you think you could have run a bit faster. But what is a novelist to do?

and here is another:

The path to intelligence seems to be through working on hard problems. You develop intelligence as you might develop muscles, through exercise. But there can't be too much compulsion here. No amount of discipline can replace genuine curiosity. So cultivating intelligence seems to be a matter of identifying some bias in one's character—some tendency to be interested in certain types of things—and nurturing it. Instead of obliterating your idiosyncrasies in an effort to make yourself a neutral vessel for the truth, you select one and try to grow it from a seedling into a tree.

graham inspires. he inspires me, at least; few people can do that anymore, and at this point of my life, i need all the inspiration i can muster.

something from computer science.

i like reading essays from a particular author/programmer/"startupper" by the name of paul graham.

irrationally, i distrust capitalists and the rich, but any hacker that can use the word "isomorphic" colloquially and correctly is all right in my book; in his essays, paul graham has, so he is all right.

it could also be that he writes essays that i want to believe, like "why nerds are unpopular" (an uplifting piece) and "what you'll wish you'd known" (something like a speech).

anyways, here are two excerpts from his essay, "the hundred-year language" :

Any programming language can be divided into two parts: some set of fundamental operators that play the role of axioms, and the rest of the language, which could in principle be written in terms of these fundamental operators.

I think the fundamental operators are the most important factor in a language's long term survival. The rest you can change. It's like the rule that in buying a house you should consider location first of all. Everything else you can fix later, but you can't fix the location.

I think it's important not just that the axioms be well chosen, but that there be few of them. Mathematicians have always felt this way about axioms-- the fewer, the better-- and I think they're onto something.

so paul knows axioms! dandy! but this second bit seems to me a little insightful.

Languages evolve slowly because they're not really technologies. Languages are notation. A program is a formal description of the problem you want a computer to solve for you. So the rate of evolution in programming languages is more like the rate of evolution in mathematical notation than, say, transportation or communications. Mathematical notation does evolve, but not with the giant leaps you see in technology.

i can't resist another excerpt. as you can imagine, i didn't finish the essay before posting this .. but this is the last, i promise.

it's worth reading because as mathematicians, numbers are numbers, and what paul describes is something like ordinals from set theory. but from a programmer's perspective .. mmmmrrrppgh.

There are more shocking prospects even than that. The Lisp that McCarthy described in 1960, for example, didn't have numbers. Logically, you don't need to have a separate notion of numbers, because you can represent them as lists: the integer n could be represented as a list of n elements. You can do math this way. It's just unbearably inefficient.

another teaching stint.

in two weeks i've subbed for two of my g.s.i. friends and i think i've become more boring than i used to be. it's either that or more likely, students really don't find single-variable differential calculus very interesting, at all.

i can't blame them. a basic calculus course is boring, in the same way that most courses in basic methods are boring. there are better words to say this, so here are some of paul graham's:

"I'm not saying we should let little kids do whatever they want. They may have to be made to work on certain things. But if we make kids work on dull stuff, it might be wise to tell them that tediousness is not the defining quality of work, and indeed that the reason they have to work on dull stuff now is so they can work on more interesting stuff later. [1]"

it's assuming, of course, that there is a "later."

i can't quite perceive the reason or cause, but somehow the calculus of Newton and Leibniz seems to take a hallowed role in being "educated," if only to use the right keyword.

for quite a few disciplines, i believe that calculus is quite useless. in a greater sense, mathematics as it exists is often little in worth, and so is the nature of philosophy. As a curiosity, it disciplines the mind without actually being inherently useful for anything.

so it stands. for many calculus is another hoop to jump, through the obstacle course that is college. add the psychological effects of math anxiety and the societal norm of innumeracy, and it's a wonder i can teach calculus with any conviction at all.

[1] short for 'graduate student instructor.' in the mathematics department, almost all the calculus classes (save the honors courses) are taught by g.s.i.'s in exchange for funding and tuition privileges.