
The Erdos Paradox - godelmachine
https://arxiv.org/abs/1812.11935
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xamuel
The paper mentions how Erdos accomplished so much while knowing relatively
little machinery. This is, I think, a more general paradox of mathematical
research, not limited to Erdos: it's a mis-conception that to do important
original mathematics you need to be lightyears out there. The reality is,
there's so much low-hanging fruit it's almost silly. But if you want to make a
career picking low-hanging fruit, you'd better be financially independent,
because as the paper also points out, mathematical power brokers don't
appreciate low-hanging fruit-pickers!

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cjohnson318
What's some low-hanging fruit in mathematics?

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xamuel
Here's one example. In Darwin's "Origin of Species", a key principle which
Darwin talks about at great length is the so-called Knight-Darwin Law, the
observation that even in apparently asexual species, sexual reproduction still
occasionally occurs, if only after long spans of time. In an article in Acta
Biotheoretica, I argued this is in fact an infinite graph-theoretical
principle. If G is the directed graph of all organisms, where an edge from x
to y means that x is a parent of y, then (Darwin theorizes that) there is no
infinite directed path through G all of whose vertices have indegree 1. (A
vertex has indegree 1 if it has exactly 1 parent.) The low-hanging fruit (some
of which I picked in my paper): Investigate the mathematical implications of
the above property (esp. if G also has other biologically motivated
properties).

The Knight-Darwin law seems to have vanished completely from the literature
around the start of the 20th century, until I cited it. Almost as if no-one
has actually been reading Darwin in all that time.

To generalize: If you want to find low-hanging fruit, read non-fiction
classics like Darwin and constantly be asking, "Is this some unrecognized
mathematics in disguise?"

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maxander
I would further generalize; if you want to find low-hanging fruit, know about
relevant subjects that aren't presently very popular (such as the Knight-
Darwin law) and find relations between disparate fields (such as evolutionary
biology and graph theory) where few others are well-versed in both.

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bookofjoe
In this vein of low-hanging fruit, my former anesthesiology chair once
remarked that you could build a first-class academic career in the specialty
if you knew German and simply went through the 1900-1920 German physiology
literature and gleaned significant findings unknown even now in the non-
German-speaking world. Alas, my other language was Japanese.

~~~
distant_hat
Would Google translate be able to help or has German moved significantly in
the meantime?

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sn41
A great article. One of the prime examples of the disdain that mainstream
mathematicians had for "Hungarian" style mathematics is the fact that
Szemeredi did not get the Fields medal (apparently Erdos was very angry at
this), despite the fact that a lot of deep mathematics, at least two Fields
medals (Gowers and Tao), a Wolf prize (Furstenberg) have later resulted from
Szemeredi's work.

After the wider acceptance of combinatorics, Szemeredi was awarded the Abel
Prize.

Another area of mathematics generally ignored by mathematicans is mathematical
logic. I've often heard an opinion is that it is not really "relevant" to
general mathematical practice.

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StefanKarpinski
There's an age limit on the Fields medal (40), so it's quite common for the
importance of someone's work to be recognized too late for them to be
eligible. This is somewhat intentional as the medal is meant to be given to
mathematicians early enough in their careers to have an impact (c.f. Nobel
prizes, which are typically only awarded when someone is famous enough that
getting a Nobel prize hardly matters).

~~~
sn41
Sure. But this was not the case with Szemeredi's work. It was recognised as
great immediately, and Szemeredi was below 40 at the time. The problem was
that his combinatorial approach was not considered to be deep enough to merit
the medal.

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Pietertje
For those of you who wish to read more on Erdos personal life I can recommend
the book: The Man Who Loved Only Numbers [0]. Highly entertaining.

[0]:
[https://www.goodreads.com/book/show/714583.The_Man_Who_Loved...](https://www.goodreads.com/book/show/714583.The_Man_Who_Loved_Only_Numbers)

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svat
It's a great book, probably the best (or most delightful) biography I've read
of anyone, in any field. It not only gives a picture of the subject as a
person (as any biography should I guess), but actually engages with the
person's work and gives a flavour of it — what's the point of a biography of
an artist or poet that does not show their art or poetry? The "love" in the
title of this book is very apt: not only did Erdős love numbers, but this a
biography written with love, and the reader will love it too.

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Topolomancer
The article mentions two paradoxes that are worth considering:

> How could a great mathematician not want to study these things? [these
> things = Lie groups, Riemannian manifolds, algebraic geometry, algebraic
> topology, global analysis, or the deep ocean of mathematics connected with
> quantum mechanics and relativity theory] > > This suggests the fundamental
> question: How much, or how little, must one know in order to do great
> mathematics? > [...] > The second Erdos paradox is that his methods and
> results, > considered marginal in the twentieth century, have become >
> central in twenty-first century mathematics.

I never had the feeling that his results were 'marginal', but the fact that he
never got a full position anywhere got me thinking---maybe Erdos was just
interested in these positions (which, given his personality, might be likely),
or maybe he did not 'sell' his work well enough. As someone who hates
advertising their own work, I can see how tragic this would be.

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webmaven
_> maybe Erdos was just interested in these positions_

Did you mean, "Just _not_ interested"?

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MrXOR
N is a Number and E is for Erdos

