
Computational Knowledge and the Future of Pure Mathematics - kylemaxwell
http://blog.stephenwolfram.com/2014/08/computational-knowledge-and-the-future-of-pure-mathematics/
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fiatmoney
This is very similar to Doug Lenat's work on Automated Mathematician & later
on Eurisko, and later Ken Haase's follow up work on representation languages.

[http://oai.dtic.mil/oai/oai?verb=getRecord&metadataPrefix=ht...](http://oai.dtic.mil/oai/oai?verb=getRecord&metadataPrefix=html&identifier=ADA155378)

There were severe sticking points around the cultivation of an idea of
"interesting" properties and the performance issues around evaluating a
combinatoric space of possible manipulations. There hasn't been serious work
along those lines since the early 90s or so.

It's annoying because especially Haase's work has some very practical
insights, but Wolfram seems to be loathe to ever admit he's building off of
someone else's work.

~~~
mjn
Yes! I wish there was more work in that line. Very interesting work, but also
hit some major problems. This 1984 postmortem paper by Lenat and a
collaborator is also thought-provoking:
[http://eksl.isi.edu/files/library/Lenat_Brown-1984-why-AM-
an...](http://eksl.isi.edu/files/library/Lenat_Brown-1984-why-AM-and-EURISKO-
work.pdf)

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pnut
Can some bored billionaire please throw $100M at this project?

Talk about revolutionary, true automated pure math would be a human milestone
on par with very few developments in history.

~~~
JadeNB
> Talk about revolutionary, true automated pure math would be a human
> milestone on par with very few developments in history.

Hilbert thought so too, but it is proveably not to be
([http://en.wikipedia.org/wiki/Entscheidungsproblem](http://en.wikipedia.org/wiki/Entscheidungsproblem)),
no matter how much money is thrown at it by how many bored billionaires.

(EDIT: To be clear, I am not claiming that there is no room for automated
_assistance_ of pure math, only that it can never be _wholly_ automated.)

(Second, important EDIT: As Khaki points out
([https://news.ycombinator.com/item?id=8170062](https://news.ycombinator.com/item?id=8170062)),
I should make it clearer that many objections about what computers can't do
apply equally well to show what humans also can't do.)

~~~
diakopter
Wolfram's essay addresses this point...

~~~
JadeNB
Life's too short to read Wolfram when he gets going, so I only skimmed it;
but: where? I see:

> In a sense an axiom system is a way of giving constraints too: it doesn’t
> say that such-and-such an operator “is Nand”; it just says that the operator
> must satisfy certain constraints. And even for something like standard Peano
> arithmetic, we know from Gödel’s Theorem that we can never ultimately
> resolve the constraints–we can never nail down that the thing we denote by
> “+” in the axioms is the particular operation of ordinary integer addition.
> Of course, we can still prove plenty of theorems about “+”, and those are
> what we choose from for our report.

and:

> But there will inevitably be some limitations—resulting in fact from
> features of mathematics itself. For example, it won’t necessarily be easy to
> tell what theorem might apply to what, or even what theorems might be
> equivalent. Ultimately these are classic theoretically undecidable
> problems—and I suspect that they will often actually be difficult in
> practical cases too. And at the very least, all of them involve the same
> kind of basic process as automated theorem proving.

Are these what you mean? Both of these seem to amount to, "Sure,
mathematicians _say_ you can't do it, but I hold out hope", an argument which
is no more persuasive than one would expect to find from various circle-
squarers. Less subjectively, they both seem to miss the point; the
incompleteness results, for example, do not just say that certain theorems
aren't clearly specified, or are equivalent to unknown other theorems, or
vague things like that, but specifically that there are _true_ but
_unproveable_ theorems—putting paid to any attempt at _complete_ automation.

I emphasise again that this is only a knock if you dream grandiosely of
capturing _all_ of mathematics in an automated (or, as Khaki pointed out above
([https://news.ycombinator.com/item?id=8170062](https://news.ycombinator.com/item?id=8170062))
that I should really be saying, even formal but human-constructed) framework;
it says nothing about the feasibility of automated theorem-proving for _some_
results, and, indeed, we have a success story for one such result on the front
page even now:
[https://news.ycombinator.com/item?id=8169686](https://news.ycombinator.com/item?id=8169686)
.

~~~
kazagistar
Automation is doing automatically what humans manually. So your point misses
the point.

~~~
JadeNB
> Automation is doing automatically what humans manually. So your point misses
> the point.

Could you clarify what point I'm missing? It's true that I am, intentionally,
conflating automation and formalisation of mathematics, activities which can
be meaningfully distinguished. Is it to that that you object?

On the other hand, I think that it is also meaningful to point out that,
although automation is just a version of human activity carried out by a
computer, not all human activity is subject to automation—at least, not yet.
The two bad examples that spring to mind are chess-playing and facial
recognition (bad because both have recently come into the reach of computers).

In principle, any human activity _that we understand_ can be automated simply
by giving a computer sufficiently precise instructions; but (1) if those
instructions are too detailed and _ad hoc_ , then it may be just as hard to
give them as to carry out the task, and (2) there are many human activities
that we do _not_ understand.

~~~
proveit123
> In principle

What principle is that? I've never heard of such a thing.

> any human activity that we understand can be automated simply by giving a
> computer sufficiently precise instructions

Care to back up this clearly false statement? I've heard only the opposite
regarding computers. Also, hard AI being nothing more than an impossibility
and (political)fantasy of know-nothings is a very real option. The peak of
automation will be reached within this century and it won't be ``great'' in
its ability.

> (2) there are many human activities that we do not understand.

You're just being pseudo-intellectual and making up false principles.

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bkirwi
I find it odd that Wolfram talks about all the thousands of things that will
need to be 'built in' to Mathematica for this project to work -- shouldn't you
be able to implement these things in the language itself?

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jaan
Very cool - I'm working on a related project: [https://www.google-
melange.com/gsoc/project/details/google/g...](https://www.google-
melange.com/gsoc/project/details/google/gsoc2014/jaanaltosaar/5741031244955648)

I'll put up a blog post soon on this!

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diakopter
some related and well-reasoned, well-written essays:

[http://monasandnomos.org/2012/12/05/the-idea-of-a-
characteri...](http://monasandnomos.org/2012/12/05/the-idea-of-a-
characteristica-universalis-between-leibniz-and-russell-and-its-relevancy-
today/)

[http://vanemden.wordpress.com/2012/04/08/flowcharts-the-
once...](http://vanemden.wordpress.com/2012/04/08/flowcharts-the-once-and-
future-programming-language/)

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kevinwang
Absolutely fascinating. Stoked to see where this'll go!

