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Magic Paper (cognitivemedium.com)
216 points by sebg on Nov 3, 2017 | hide | past | web | favorite | 30 comments

> An interesting project would be to create a single environment which can be used for both education and research.

This does happen sometimes, by accident. I wrote Quirk [1] as an educational tool for learning about quantum circuits. But then I started finding it useful as a tool for optimizing circuits to use fewer gates, just by dragging things around and seeing what happens (there's still enough low-hanging fruit for that to be viable). The paper "Halving the cost of quantum addition" [2] only exists because of a "that's strange..." moment while I was messing around with a decomposition of the Toffoli gate in Quirk.

(I bet Michael Nielsen, the author of the linked article, gets a particular kick out of this example. He's also the co-author of the de-facto standard textbook for quantum computing.)

1: http://algassert.com/quirk

2: https://arxiv.org/abs/1709.06648

This is excellent! Not only is it a great concept and a well-written article ("reify" is my new favourite word), but it completely succeeds in its goal of teaching a non-trivial concept. I'm very excited to see where this goes.

I've followed the work of some others in this space (like Bret Victor) for quite a while, and inspired by what I've seen and read (and also by frustration in college courses and elsewhere) I've started prototyping a new way to do derivations. It's effectively only a mock-up in its current stage but as it fits in the "new media for thinking" category I thought I'd share it anyway:


If you want to work on or talk about this open source project I'd love to hear from you! (Email is in bio)

I guess you are working on purely symbolic thinking? I think the approach works best when it’s just not equations being manipulated, but visualized instantiations of specific (and concrete) examples.

Yes, but I feel there is room for more than one approach. For something as specific as derivations--which tend to have a linear, step-by-step approach I think this makes sense. Similar to how different programming languages excel at different tasks I imagine different thinking media will also excel at different cognitive tasks.

There is also the terseness of mathematical notation to compete with--I can't think of a better way to replace the sum of something over the range 1 to 1000 than the current sigma notation, or even a for loop or some other construct like "sum(f(x), 1...1000)". I don't know if there is a good way to make something like a large summation visual but I am definitely open to any ideas. I wish that every mathematical concept or object could be visualised effectively but again I doubt this is the case with the hope of being proven wrong.

If your interest is future pedagogic utility in academia (funny etymology for pedagogue: 'leader of children') that's one thing. If your interest is in what can be called today the cognitive dynamics of creativity as experienced by highly creative individuals then who better than Albert Einstein to report on his experiences. In mathematician Jacques Hadamard's 'The Psychology of Invention in the Mathematical Field' (Dover (1954),ISBN 0486201074, pp. 142-3) Einstein reports his creative cognitive processes in response to a questionnaire titled 'An inquiry into the working methods of mathematicians' as follows:

A) The words or language as they are written or spoken do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be "voluntarily" reproduced and combined...

B) The above mentioned elements are, in my case, of visual and some of muscular type. Conventional words or other signs have to be sought for laboriously only in a secondary stage, when the mentioned associated play is sufficiently established and can be reproduced at will.

Or what might be called today visual/tactile qualial imagery manipulated in cognitive workspace (Baars). So yeah it would seem, 'a visual medium (can) be used to do serious mathematical exploration', at least as a lead-in to transcription from conceptual imagery to reporting.

This is an incredibly interesting comment! I think you've hit the nail on the head that this is better suited for pedagogy than for exploration. The extracts from the book are especially insightful and really something I had not considered before so thank you for that--I may just have to read the whole book (and it's by the same Hadamard of the Hadamard gate! It seems that anyone even slightly related to quantum computing has an interest in exploring and enhancing creative thought--that's 3 just in this thread).

I hope I didn't come across as suggesting that a visual medium couldn't be used to do serious mathematical exploration--I definitely think it can (and Strilanc's comment proves that it can). To clarify, I was more suggesting that it could be the case that not every mathematical concept is conducive to being explained or taught effectively using a visual medium, or even more loosely that in some cases a symbolic approach could be more effective than a visual one. However, I agree that for most concepts visual trumps symbolic and I currently have no concrete examples of the opposite.

I’ve heard even the most accomplished mathematicians build concrete visualizations in their head, they work with non-symbolic models when solving problems. I think that these mental models is what we should try to capture in the computer. So I guess it is a matter of what mental model is being used to make the derivation, and this would probably involve why the derivation is being made.

Anyways, I don’t want to discourage you, your work is definitely interesting. It’s just something you might want to keep in mind.

Yeah that definitely sounds right--at some point the brain will need to do the understanding as the paper won't. So I also agree that capturing more facets and the richness of thoughts is a much better goal than simply enhancing symbol-shunting. I suppose my idea could best fit in as either a transitionary model, or more likely as more of a replacement for the current method of teaching and learning derivations. In a sense just decreasing how bad the current worst case of teaching is rather than trying to increase the best case. Focusing on didactic rather than exploratory although I realise that both are possible with more expressive systems.

No, don't worry you have not done any discouraging. I really appreciate all your thoughts on the matter and will definitely think more about what (or if) this should be. I currently don't have much time to work on it but definitely have time for some good mulling.

For all Spanish speaking folks here in HN, I just learned that "reify" means "cosificar". And I just learned that "cosificar" is an actual Spanish word[1] !

1. tr. Convertir algo abstracto en una cosa concreta.

2. tr. Reducir a la condición de cosa a una persona.

[1] http://dle.rae.es/?id=B58Iitz

Reificar is also a spanish word :) the Real Academia does not register it but you can see its use going at least as back as Marx translations through https://es.wikipedia.org/wiki/Reificaci%C3%B3n

to reify means to make an abstract concept concrete (reified abstract classes...) but it's interesting that reification is actually the name of a formal logical fallacy: "Reification (also known as concretism, hypostatization, or the fallacy of misplaced concreteness) is a fallacy of ambiguity, when an abstraction (abstract belief or hypothetical construct) is treated as if it were a concrete real event or physical entity."


Reification of abstract concepts is an important technique in thinking. Few of us, if any, are pure abstract thinkers, and we will form examples in our head when presented with an abstraction (e.g. we will see an some kind of real apple when we hear the word apple).

seems to me to be related to 'ripe', because in german that's "reif", which in "ausgereifte idee" has almost the same meaning and it's probably from "Reifen" ('tire') cognate to ring, so round or rather whole.

The semantic distinction between abstract and concrete idea is still virtual, though. The ripe apple you think about is still a virtual image. Or rather original, because even scientifically in e.g. biology, specific specimen are used as arch example to define clades and such. We care very much about relational structure, after all. An abstraction thus is something loosely or not at all connected to the currently favoured theory of the overall structure of life.

It's from Latin 'res' meaning 'thing'. If that's cognate to 'reif' I wouldn't know. I'd be happy with 'thingify' but that's at least less established. :)

Like instantiating an object from a class.

Bret Victor has done a lot of work on this, as well, and his paper Magic Ink[1] goes into a quite a bit of details. His work is referenced by Magic Paper, and like most things from Bret, it's probably worth looking at.

It's nice to see more stuff coming out of this.

1: http://worrydream.com/MagicInk/

This line of thinking goes back even further. Back to 1968 when GRAIL was developed:


Pretty amazing stuff, even today.

I don't have a lot to add the the discussion, but I am excited that some people are still working to find new and better ways to think, communicate, and iterate.

Me, too! Humans are very good at instinctively making decisions, so if you can present a problem in the simplest possible manner, intuition can take over and solve a problem much easier than when the problem is defined using, for instance, algebra.

The proof looks similar to the proof that every continuous function can be integrated over a finite interval.

Wow this looks awesome. I think I might use this for some notes on topological quantum computing or something .

I'm less and less into technology.

I really believe the matter is just human. We just need to take time to speak to each others peacefully. Instead we delegate to hurried teachers in potentially shamefull classes, or technology to do exercises..

Do you not think that maybe, many millennia ago, some caveman may have thought the same?

"I'm less and less into speech.

I really believe the matter is just human. We just need to take time to grunt and mime to each other peacefully. Instead we delegate to sitting lazily and screeching loud sounds and noises"

Just as the spoken and written word opened up the world to us in new ways in the past, I feel that these new ideas may also lead us to new understandings.

that caveman never went to school, he'd accept my argument

I don't dismiss tech, it's just that it never address the issues to me. So it's not the solution, it's a tail chase.

I agree, technology has not been a significant help. Even being able to knock out quick test scripts in python has never been the pivotal issue. The real issue is understanding the material, and a human teacher who really understands what is going on and can ask the right motivating questions to spur the answer is most helpful. I've only encountered a handful of teachers like this, who can really get to the heart of something and not get lost in the weeds.

I'd add a few things:

We're humans, we like human relationships, sharing beauty/knowledge is part of that, that's why hurried teachers and classrooms are subpar, they turn it into a domination scheme (harsh words but I stand by them). On the other hand someone who shares the beauty he feels in a subject, his passion, will drive a pupil in the deepest of ways.

It will also remove some imaginary need for "school" to understand the world.

Lastly it will make people sense that sharing and exchanging is key. I'd like to see more of that.

pencils are a technology, as are paper, chalkboards. Printing-presses, etc etc. Technology enables all of our ability to educate people.

When it comes to more modern technologies, like computers, I think we need to remember that, in historical terms, we're only just starting to make use of them. I would bet that in 100 years from now, there'll be much more beneficial applications of them in education than now.

I can't help feeling that you're operating on a very narrow notion of "technology". Without technology we wouldn't have societies where we could have schools, nor writing, paper, printed materials, the free time to have education, mathematics, microscopes, the global sharing and dissemination of knowledge, etc etc.

Well, the technology showcased here is designed specifically to enable clearer communication and explanation, i.e. better speaking.

The project will be more useful for education and research.

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