
Magic Paper - sebg
http://cognitivemedium.com/magic_paper/
======
Strilanc
> _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](http://algassert.com/quirk)

2: [https://arxiv.org/abs/1709.06648](https://arxiv.org/abs/1709.06648)

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OisinMoran
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:

[https://oisinmoran.com/projects/derivation.html](https://oisinmoran.com/projects/derivation.html)

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

~~~
seanmcdirmid
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.

~~~
OisinMoran
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.

~~~
escherplex
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.

~~~
OisinMoran
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.

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fasteo
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](http://dle.rae.es/?id=B58Iitz)

~~~
harperlee
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](https://es.wikipedia.org/wiki/Reificaci%C3%B3n)

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mlevental
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."

[https://en.wikipedia.org/wiki/Reification_(fallacy)](https://en.wikipedia.org/wiki/Reification_\(fallacy\))

~~~
seanmcdirmid
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).

~~~
posterboy
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.

~~~
abecedarius
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. :)

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kbenson
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/](http://worrydream.com/MagicInk/)

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

[https://www.youtube.com/watch?v=QQhVQ1UG6aM](https://www.youtube.com/watch?v=QQhVQ1UG6aM)

Pretty amazing stuff, even today.

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Bjartr
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.

~~~
jeffhuys
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.

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

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

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agumonkey
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..

~~~
yters
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.

~~~
agumonkey
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.

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raj2552
The project will be more useful for education and research.

