Greg Egan thinks that there being no royal road to understanding, of the sort that enables original creative thinking, will continue to be true even in a future as posthuman as Diaspora where you can insta-mind-graft others’ hard-won understanding:
In one sense, the Truth Mines were just another indexscape. Hundreds of thousands of specialized selections of the library’s contents were accessible in similar ways—and Yatima had climbed the Evolutionary Tree, hopscotched the Periodic Table, walked the avenue-like Timelines for the histories of fleshers, gleisners, and citizens. Half a megatau before, ve’d swum through the Eukaryotic Cell; every protein, every nucleotide, even carbohydrate drifting through the cytoplasm had broadcast gestalt tags with references to everything the library had to say about the molecule in question. In the Truth Mines, though, the tags weren’t just references; they included complete statements of the particular definitions, axioms, or theorems the objects represented. The Mines were self-contained: every mathematical result that fleshers and their descendants had ever proven was on display in its entirety. The library’s exegesis was helpful-but the truths themselves were all here. …
Learning was a strange business. Ve could have had vis exoself wire all this raw information straight into vis mind, in an instant ve could have engulfed a complete copy of the Truth Mines, like an amoeba ingesting a planet—but the facts would have become barely more accessible than they already were, and it would have done nothing to increase vis understanding. The only way to grasp a mathematical concept was to see it in a multitude of different contexts, think through dozens of specific examples, and find at least two or three metaphors to power intuitive speculations. Curvature means the angles of a triangle might not add up to 180 degrees. Curvature means you have to stretch or shrink a plane non-uniformly to make it wrap a surface. Curvature means no room for parallel lines-or room for far more than Euclid ever dreamt of. Understanding an idea meant entangling it so thoroughly with all the other symbols in your mind that it changed the way you thought about everything.
Still, the library was full of the ways past miners had fleshed out the theorems, and Yatima could have had those details grafted in alongside the raw data, granting ver the archived understanding of thousands of Konishi citizens who’d traveled this route before. The right mind grafts would have enabled ver effortlessly to catch up with all the living miners who were pushing the coal face ever deeper in their own inspired directions . . . at the cost of making ver, mathematically speaking, little more than a patchwork clone of them, capable only of following in their shadows.
If ve ever wanted to be a miner in vis own right making and testing vis own conjectures at the coal face, like Gauss and Euler, Riemann and Levi-Civita, deRham and Cartan, Radiya and Blanca then Yatima knew there were no shortcuts, no alternatives to exploring the Mines firsthand. Ve couldn’t hope to strike out in a fresh direction, a route no one had ever chosen before, without a new take on the old results. Only once ve’d constructed vis own map of the Mines-idiosyncratically crumpled and stained, adorned and annotated like no one else’s could ve begin to guess where the next rich vein of undiscovered truths lay buried.
(I’m taking as given the assumption that this passage is an authorial insert, like a lot of his fiction.)
Egan’s claim that the only way to grasp a mathematical concept was to see it in a multitude of different contexts reminds me of Bill Thurston’s musings on the notion of the derivative:
And his claim that “understanding an idea meant entangling it so thoroughly with all the other symbols in your mind that it changed the way you thought about everything” seems related to the advice to “strive to make yourself the source of every thought worth thinking”.
Yeah, I’d say that the knowing a thing from multiple perspectives instead of only one is an improvement similar to knowing the thing from one perspective instead of just repeating the teacher’s password.
That does not imply that we should maximize the number of perspectives, because some of them may be more difficult than the original concept—e.g. in your example, I know what is a derivative, but not what is a Lagrangian section, so explaining the former using the latter is not useful to me. But for someone familiar with both it could be useful. So I guess it is useful to see many relations between the things you already know.
And this is sometimes a problem in education, that in the effort to cram as many concepts as possible, we explain each of them from one perspective only, and then hurry to learn the next concept. It may be better to spend more time, exploring the new concepts from different angles.
(That said, the number of possible relation might grow quadratically with the number of concepts, so perhaps trying to fully explore the graph would slow you down too much? The time is limited, you need to prioritize. People can explore more collectively if everyone prioritizes something else? Then again, some things may be generally more useful than others...)
Yeah. I’m a fan of multiple learning passes over important concepts covering various perspectives, although not necessarily of full graph exploration for its own sake except in certain sandboxed settings. When to do the passes and what perspectives would be determined by some mix of internal signals (interest/burgeoning taste and learning-by-writing, whose step 3 forcing function seems orthogonal to interest) and external signals (e.g. what end users and collaborators find useful). I do think the right amount of cramming is not zero, although not necessarily maximising concept count but rather something like forcing a viewpoint phase shift.
I think this is royghly correct for neural network based minds. Integrating new knowledge so that it becomes usable in the way we call understanding must be done by either thinking, or a process so sophisticated in its understanding of that mind that it would require substantial work from a superintelligence. It would have to anticipate many of the trajectories those weights could cause through complex wanting space. The easiest rout would be simulating that thinking; other routes might have similar implications of creating a duplicate of that mind.
To put it another way, understanding is a process happening over time in both the learning and expression, making it much more complex than just adding some weights like a set of training example does.
Greg Egan thinks that there being no royal road to understanding, of the sort that enables original creative thinking, will continue to be true even in a future as posthuman as Diaspora where you can insta-mind-graft others’ hard-won understanding:
(I’m taking as given the assumption that this passage is an authorial insert, like a lot of his fiction.)
Egan’s claim that the only way to grasp a mathematical concept was to see it in a multitude of different contexts reminds me of Bill Thurston’s musings on the notion of the derivative:
And his claim that “understanding an idea meant entangling it so thoroughly with all the other symbols in your mind that it changed the way you thought about everything” seems related to the advice to “strive to make yourself the source of every thought worth thinking”.
Yeah, I’d say that the knowing a thing from multiple perspectives instead of only one is an improvement similar to knowing the thing from one perspective instead of just repeating the teacher’s password.
That does not imply that we should maximize the number of perspectives, because some of them may be more difficult than the original concept—e.g. in your example, I know what is a derivative, but not what is a Lagrangian section, so explaining the former using the latter is not useful to me. But for someone familiar with both it could be useful. So I guess it is useful to see many relations between the things you already know.
And this is sometimes a problem in education, that in the effort to cram as many concepts as possible, we explain each of them from one perspective only, and then hurry to learn the next concept. It may be better to spend more time, exploring the new concepts from different angles.
(That said, the number of possible relation might grow quadratically with the number of concepts, so perhaps trying to fully explore the graph would slow you down too much? The time is limited, you need to prioritize. People can explore more collectively if everyone prioritizes something else? Then again, some things may be generally more useful than others...)
Yeah. I’m a fan of multiple learning passes over important concepts covering various perspectives, although not necessarily of full graph exploration for its own sake except in certain sandboxed settings. When to do the passes and what perspectives would be determined by some mix of internal signals (interest/burgeoning taste and learning-by-writing, whose step 3 forcing function seems orthogonal to interest) and external signals (e.g. what end users and collaborators find useful). I do think the right amount of cramming is not zero, although not necessarily maximising concept count but rather something like forcing a viewpoint phase shift.
I think this is royghly correct for neural network based minds. Integrating new knowledge so that it becomes usable in the way we call understanding must be done by either thinking, or a process so sophisticated in its understanding of that mind that it would require substantial work from a superintelligence. It would have to anticipate many of the trajectories those weights could cause through complex wanting space. The easiest rout would be simulating that thinking; other routes might have similar implications of creating a duplicate of that mind.
To put it another way, understanding is a process happening over time in both the learning and expression, making it much more complex than just adding some weights like a set of training example does.