...the evidence is that science has slowed enormously per dollar or hour spent. That evidence demands a large-scale institutional response. It should be a major subject in public policy, and at grant agencies and universities. Better understanding the cause of this phenomenon is important, and identifying ways to reverse it is one of the greatest opportunities to improve our future.
Using spaced repetition systems to see through a piece of mathematics
You might suppose a great mathematician such as Kolmogorov would be writing about some very complicated piece of mathematics, but his subject was the humble equals sign: what made it a good piece of notation, and what its deficiencies were. even great mathematicians – perhaps, especially, great mathematicians – thought it worth their time to engage in such deepening. I’m still developing the heuristic, and my articulation will therefore be somewhat stumbling. Every piece should become a comfortable part of your mental furniture, ideally something you start to really feel. That means understanding every idea in multiple ways, People inexperienced at mathematics sometimes memorize proofs as linear lists of statements. A more useful way is to think of proofs is as interconnected networks of simple observations. For someone who has done a lot of linear algebra these are very natural observations to make, and questions to ask. But I’m not sure they would be so natural for everyone. The ability to ask the “right” questions – insight-generating questions – is a limiting part of this whole process, and requires some experience. Conventional words or other signs have to be sought for laboriously only in a secondary stage, when the mentioned associative play is sufficiently established and can be reproduced at will. So, my informal pop-psychology explanation is that when I’m doing mathematics really well, in the deeply internalized state I described earlier, I’m mostly using such higher-level chunks, and that’s why it no longer seems symbolic or verbal or even visual. I’m not entirely conscious of what’s going on – it’s more a sense of just playing around a lot with the various objects, trying things out, trying to find unexpected connections. But, presumably, what’s underlying the process is these chunked patterns.
There's also a useful version of such critical remarks which can be made. One wants two things: (a) remarks which help people understand their own potential better; and (b) remarks which help people better understand what's involved in working in a particular field.— Michael Nielsen (@michael_nielsen) November 24, 2018
On Volitional Philanthropy (a short essay!) T. E. Lawrence, the English soldier, diplomat and writer, possessed what one of his biographers called a capacity for enablement: he enabled others to make use of abilities they had always possessed but, until their acquaintance with him, had failed to realize. People would come into contact with Lawrence, sometimes for just a few minutes, and their lives would change, often dramatically, as they activated talents they did not know they had. Most of us have had similar experiences. A wise friend or acquaintance will look deeply into us, and see some latent aspiration, perhaps more clearly than we do ourselves. And they will see that we are capable of taking action to achieve that aspiration, and hold up a mirror showing us that capability in crystalline form. The usual self-doubts are silenced, and we realize with conviction: “yes, I can do this”. This is an instance of volitional philanthropy: helping expand the range of ways people can act on the world. I am fascinated by institutions which scale up this act of volitional philanthropy. Y Combinator is known as a startup incubator. When friends began participating in early batches, I noticed they often came back changed. Even if their company failed, they were more themselves, more confident, more capable of acting on the world. This was a gift of the program to participants [1]. And so I think of Y Combinator as volitional philanthropists. For a year I worked as a Research Fellow at the Recurse Center. It's a three-month long “writer's retreat for programmers”. It's unstructured: participants are not told what to do. Rather, they must pick projects for themselves, and structure their own path. This is challenging. But the floundering around and difficulty in picking a path is essential for growing one's sense of choice, and of responsibility for choice. And so creating that space is, again, a form of volitional philanthropy. There are institutions which think they're in the volitional philanthropy game, but which are not. Many educators believe they are. In non-compulsory education that's often true. But compulsory education is built around fundamental denials of volition: the student is denied choice about where they are, what they are doing, and who they are doing it with. With these choices denied, compulsory education shrinks and constrains a student's sense of volition, no matter how progressive it may appear in other ways. There is something paradoxical in the notion of helping someone develop their volition. By its nature, volition is not something which can be given; it must be taken. Nor do I think “rah-rah” encouragement helps much, since it does nothing to permanently expand the recipient's sense of self. Rather, I suspect the key lies in a kind of listening-for-enablement, as a way of helping people discover what they perhaps do not already know is in themselves. And then explaining honestly and realistically (and with an understanding that one may be in error) what it is one sees. It is interesting to ask both how to develop that ability in ourselves, and in institutions which can scale it up. [1] It is a median effect. I know people who start companies who become first consumed and then eventually diminished by the role. But most people I've known have been enlarged. Note, by the way, that I work at Y Combinator Research, which perhaps colours my impression. On the other hand, I've used YC as an example of volitional philanthropy since (I think) 2010, years before I started working for YCR.
