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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 observatio

by xamuel 8y ago

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?"

Ericson2314 8y ago
If the edges go to parent you mean out degree one.
maxander 8y ago
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.
- 0-_-0 8y ago
  I definitely have the impression that many of the "low hanging fruits" are found at the intersection of disparate fields. There are much more people highly proficient in a single field than mediocre in two different fields, so if you want to discover something new and important, look at a new field through the lens of one you're proficient in.
- bookofjoe 8y ago
  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 8y ago
    Would Google translate be able to help or has German moved significantly in the meantime?
semi-extrinsic 8y ago
I would assert more strongly that in general, essentially nobody actually goes back and reads the classic papers in any field, and those few who do usually end up with new results.
ajuc 8y ago
> there is no infinite directed path through G ... I'm out of my depths here, but isn't this trivially true regardless of means of reproduction? You can't be your own parent so there's no directed cycles, and without directed cycles you can't have infinite directed paths in a finite graph. Maybe he meant there's no path that starts from a root and ends in the leaf that has these properties? The leaf being the first life on earth in this example. I know the tree of life isn't a tree, but it surely is acyclic, right?
- xamuel 8y ago
  You're right, the claim is trivially true if G is finite. In the paper, I argue that Darwin must have implicitly allowed at least the possibility of G being infinite (i.e. life going on forever), or else, as you rightly point out, the Knight-Darwin Law would be trivial. Even if life will eventually die out, we can pretend G is infinite, as a simplifying assumption, similarly to how physicists and chemists pretend crystals fill up infinite space in order to simplify talking about their translation symmetries. Under this simplifying assumption, "finite" and "infinite" can serve as concrete alternatives to vague slippery predicates like "small" and "large". The following quote from Origin of Species reinforces that Darwin definitely didn't take for granted that life will someday end: "Nevertheless so profound is our ignorance, and so high our presumption, that we marvel when we hear of the extinction of an organic being; and as we do not see the cause, we invoke cataclysms to desolate the world, or invent laws on the duration of the forms of life!"
deleted 8y ago
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LolWolf 8y ago
Is this true? I mean, consider the infinite directed graph generated by the total ordering of ℤ, then all vertices have indegree and outdegree one (by definition of a total ordering), so there is such an infinite path. Are there some further restrictions on this graph?
- LolWolf 8y ago
  I just read you paper, it’s quite fun! This comment was made with the belief that you claimed that all infinite graphs have this property. Anyways, I know it’s an old thread, but if anyone is reading this, I highly recommend reading it. Here is the paper: https://arxiv.org/pdf/1201.2869.pdf https://arxiv.org/pdf/1201.2869.pdf

by xamuel 8y ago

Ericson2314 8y ago

If the edges go to parent you mean out degree one.

maxander 8y ago

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.

0-_-0 8y ago
I definitely have the impression that many of the "low hanging fruits" are found at the intersection of disparate fields. There are much more people highly proficient in a single field than mediocre in two different fields, so if you want to discover something new and important, look at a new field through the lens of one you're proficient in.
bookofjoe 8y ago
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 8y ago
  Would Google translate be able to help or has German moved significantly in the meantime?

semi-extrinsic 8y ago

I would assert more strongly that in general, essentially nobody actually goes back and reads the classic papers in any field, and those few who do usually end up with new results.

ajuc 8y ago

> there is no infinite directed path through G ... I'm out of my depths here, but isn't this trivially true regardless of means of reproduction? You can't be your own parent so there's no directed cycles, and without directed cycles you can't have infinite directed paths in a finite graph. Maybe he meant there's no path that starts from a root and ends in the leaf that has these properties? The leaf being the first life on earth in this example. I know the tree of life isn't a tree, but it surely is acyclic, right?

xamuel 8y ago
You're right, the claim is trivially true if G is finite. In the paper, I argue that Darwin must have implicitly allowed at least the possibility of G being infinite (i.e. life going on forever), or else, as you rightly point out, the Knight-Darwin Law would be trivial. Even if life will eventually die out, we can pretend G is infinite, as a simplifying assumption, similarly to how physicists and chemists pretend crystals fill up infinite space in order to simplify talking about their translation symmetries. Under this simplifying assumption, "finite" and "infinite" can serve as concrete alternatives to vague slippery predicates like "small" and "large". The following quote from Origin of Species reinforces that Darwin definitely didn't take for granted that life will someday end: "Nevertheless so profound is our ignorance, and so high our presumption, that we marvel when we hear of the extinction of an organic being; and as we do not see the cause, we invoke cataclysms to desolate the world, or invent laws on the duration of the forms of life!"

deleted 8y ago

[deleted]

LolWolf 8y ago

Is this true? I mean, consider the infinite directed graph generated by the total ordering of ℤ, then all vertices have indegree and outdegree one (by definition of a total ordering), so there is such an infinite path. Are there some further restrictions on this graph?

LolWolf 8y ago
I just read you paper, it’s quite fun! This comment was made with the belief that you claimed that all infinite graphs have this property. Anyways, I know it’s an old thread, but if anyone is reading this, I highly recommend reading it. Here is the paper: https://arxiv.org/pdf/1201.2869.pdf https://arxiv.org/pdf/1201.2869.pdf