
The Shocking State of Contemporary "Mathematics" - andreyf
http://www.math.rutgers.edu/~zeilberg/Opinion104.html
======
nrr
From the article: "Sure enough, the best invited talk was Michael Kiessling's
talk that used the ancient technology of _overhead projector_ , and it would
have been even better if he only used the _blackboard_ , and it would have
been better still if he didn't use _anything_ , just told us a _story_."

Exactly this. The fact that all of the mathematicians I've been around to date
(except for one, but he's a ham, so he doesn't really count in this statistic
since he already counts as a hacker :) have treated their overall field more
like a science than an art has really disenchanted me from it. Yes,
mathematics is regarded as the queen of all sciences, but I don't really buy
entirely into that. It has applications there, and that's probably as far as
it goes as far as science is concerned. (Disclaimer: I studied applied
mathematics at a small state university at the undergraduate level.)

Math is beautiful not because it is full of intricate logical machinery and
full of useful computational tools and full of pretty pictures; rather, math
is beautiful because the intricate logical machinery can take different forms
(how many different proofs of the FTIC are there? Pythagoras' theorem?) and
because it's an imaginary world inside one's head where there are arbitrary --
even infinite -- dimensions and fungible axioms.

The point of these meetings is to inform (and even pique the interest of) your
colleagues. If they have no idea what's going on and don't even understand the
fundamental notions, what's the point? You're wasting your time, keystrokes,
breath, and energy, not to mention the money provided to you by some grants.

~~~
dazmax
It would be nice if it were treated more as an art, but don't you think
whoever is paying for the grants would rather the effort be put into digging
deeper into specific research topics that may eventually be of use to science
rather than finding beautiful proofs and relationships between things we
already know to be true?

------
fh
This is essentially a critique of specialization, applied to the field of
mathematics. I think that this trend might be inevitable, not just in the
field of mathematics. The total amount of knowledge that humanity has
accumulated continues to grow. If the amount of science that a single person
can understand is finite, then individual scientists will understand an ever
smaller percentage of it. In some ways that makes me sad.

~~~
baddox
I don't see why two specialists couldn't link their specific knowledge when
necessary by explaining it in the context of the fundamental knowledge they
should all share. That is to say, as a person specializes more and more in
mathematics, there's no reason to think they should forget the earlier lessons
they've learned.

~~~
msluyter
I visualize the situation like a search tree. As you travel down the tree away
from the root, you become increasingly isolated from other branches of the
tree, and you can't even talk to distant leaf nodes because you lack the same
vocabulary and shared set of theorems.

Yes, we share some parent way up the tree, but we can't go all the way back to
that as a starting point because it'd simply take too long to progress from
there back down to the interesting leaf node.

This vision makes me wonder when mathematics will reach a point where
mastering the material required to understand a leaf node will take greater
than the average life span.

~~~
loup-vaillant
> I visualize the situation like a search tree.

I disagree. I observed that, until up to graduation, the situation is like a
search _graph_. When I studied, I noticed many interdisciplinary connections.
Fields like mechanic, electronic, logic, automatic, programming, often look
like two sides of the same coin.

However I also noticed an inability from others to see the damn connections.
If you present the same thing from another angle, they often fail to see it as
the same thing, while I factor the obvious pattern like you would code (I've
seen it in the case of 3D vision).

The consequences are quite catastrophic: the languages (jargon) used to
describe each discipline diverge, making it even more difficult to notice the
similarities. This convince even more people that there _isn't_ any
connection. That the situation is like a tree. At this point, they don't even
bother to seek the connections, and we end up with a disconnected mess.

> This vision makes me wonder when mathematics will reach a point where
> mastering the material required to understand a leaf node will take greater
> than the average life span.

That can't happen: more time to study relevant material means less time to
push further. It will take many geniuses to be able to learn and push fast
enough. But this is pointless. If it takes you a lifetime to learn a
particular field, that field will simply fall into oblivion. So, the geniuses
(at least the one worth mentioning) won't push further. They will simplify
their field, _reducing_ the time required to learn it.

------
jrockway
The programming world is like this too, only worse. Programming techniques
transfer between languages, but nobody seems to realize that. Instead, you see
each new language community "discover" something every other language
community had known for years. "Those languages suck, so we ignore everything
they do."

It is depressing, and I have nothing more to say about this.

~~~
andreyf
_The programming world is like this too, only worse_

Much worse. I think the problem is magnified by our obsession with languages -
overlapping subsets of syntax features that have highly intricate
relationships with programming techniques (making certain techniques easier to
implement, others-harder, regardless of problem domain).

~~~
silentbicycle
Would you say that focusing on programming _paradigms_ , rather than language
implementations, would help to counteract this?

~~~
andreyf
I think so. The big ones I know of being:

\- dynamic object oriented (Smalltalk)

\- homoiconic for compile-time metaprogramming (Scheme, Lisp)

\- dynamic for runtime metaprogramming, like continuations (Scheme, Ruby,
Smalltalk)

\- functional (Scheme, Haskell)

\- strongly typed (Haskell, Scala)

\- logical (Prolog)

These are the ones I've come across, but there are more on wikipedia:
<http://en.wikipedia.org/wiki/Programming_paradigm>

~~~
silentbicycle
There's some overlap, as well as actor/message-passing concurrency (Erlang),
dataflow (Prolog, Oz, etc.), constraint, vector-oriented (APL), etc. Probably
a dozen more, depending on where you draw some fine lines. (See CTM for a good
overview.) FWIW, Prolog is just as homoiconic as Lisp, and has compile and
runtime macros.

Still, I was more wondering about _teaching methodology_ , not enumerating
paradigms themselves. What could be done to counteract the "standing on the
toes of giants" effect?

~~~
andreyf
_I was more wondering about teaching methodology_

If I understand correctly, I'd imagine teaching everyone to implement
languages using a system like Ian Piumarta's COLA might do the trick. The
point is to break open these black boxes of abstraction (even though black
boxes are good sometimes).

------
wtallis
My general understanding is that the last person to be well-versed in
basically all of mathematics was Euler. When he got through with it, the field
was too broad for any mortal to obtain a working knowledge of most of it. It
seems that now, given the above, hardly anybody tries to broaden their
horizons after a certain point in their career. (It feels like every course I
take these days, I end up making a connection between a topic and another
branch of math or computer science, but my instructor doesn't know enough
about the other field to appreciate what I bring up.)

While I don't think the problem is quite as severe as Zeilberger claims, I do
feel like conference talks should strive to be more accessible to specialists
of other fields and to students. Particularly for a conference like the Joint
Mathematics Meetings, it would be cool if speakers prepared their talks to be
more like TED talks: just technical enough to make sure that the audience can
understand the really interesting aspect of the research.

~~~
shib71
One of my college professors put that suggestion like this: Write to an
audience of intelligent 17 year olds. Ignorant of your topic and unwilling to
put up with confusing writing, but curious and capable of understanding
complex ideas.

When I followed that advice my writing was clearer and better balanced. I also
found that the narrative in every piece (even technically orientated stuff)
became stronger.

I don't write essay's any more, but it's still the standard by which I measure
every email and comment I write.

~~~
maximilian
I'm in the process of writing my thesis and one of my advisors asked me: "Who
are you writing this for?". I had a few ideas, but I answered, "For him and
our collaborators."

"Wrong! - You should be writing for the graduate student who will be picking
up your work."

Any graduate student or even many professors will be on a similar level when
exposed to something new. By explaining stuff at a relatively simple level,
with enough intermediate steps to outline the method, most people can grasp
how you did what you did.

Maybe not 17 year olds, but aimed at people well versed in mathematics, but
not necessarily in that field.

------
carterschonwald
Doron Zeilberger is a pretty clever mathematician, so at the very least, any
of his opinions are certainly worth thinking about. I think as a comment about
the state of modern research culture, he also recently had one of his paper's
published in Rejecta Mathematica

------
amichail
_For the good of future mathematics we need generalists and strategians who
can see the big picture._

Older mathematicians probably already see the big picture -- well enough
anyway.

Younger mathematicians need to worry about tenure and can't waste their time
on big picture things that are not likely to pay off.

~~~
nrr
I would go as far as to say that we need _generalists_ who will take the time
to _chronicle_ and _revive_ works from the past to make them relevant again.

Better still, we need the mathematics community at large to become more
accustomed to seeing papers published in journals that do this so that a
larger contingent can be made aware that these
papers/ideas/theorems/corollaries/lemmata/definitions/people/pictures/etc.
exist.

Note also that this doesn't necessarily preclude anyone in mathematics from
adopting a specialization. In my eyes, the younger mathematician is in the
same place as the startup founder: in the position to take a big risk and do
something that sounds like a stupid idea (e.g., avoid publishing results
anywhere but freely and openly on the WWW; see also: Daniel Bernstein) in
order to have it pay off (e.g., earn tenure).

------
tybris
Math has always relied on a few geniuses to set things straight, the rest is
just noise.

~~~
eru
Sources?

------
buugs
Some or most of this may be related to how complicated modern mathematics has
become, I know at my college there are 6 undergraduate mathematics degrees to
choose from (in regards to specialty) and only grows when you move into
graduate school.

------
sramam
It seems what the 'Kingdom of Mathematics' really needs, is a good
toastmasters club!

~~~
tome
Yes, very much so. Mathematical expository talks are, in my experience, of
appallingly low quality.

