[hrkucuk]

>“There is a grave danger that the subject will develop along the line of least resistance.”

What does von Neumann mean here? Why is it bad that it will develop along the line of least resistance? Does von Neumann advice that working on "harder" problems is more beneficial for TCS? Could not one argue that we should be solving away low hanging fruits first?

I am not sure if I am understanding von Neumann's quote nor this article properly. I would love to hear some simpler explanation (I am a new BSc. CS graduate).

[youoy]

This is how I see it: you can work on mathematics from two different levels. There is more direct level, which is the formal system, and there is more indirect level which are the intuitions behind the formal system.

If you look at new theories (let's say you are starting to study topology, or group theory) they start from some definitions/axioms that seem to come from "nowhere", but they are in fact a product of working and perfectioning a language for the intuitions that we have in mind. Once we set for the correct descriptions, then there are a lot of consequences and new results that come from interaction almost entirely with the formal system.

The interactions with the formal system is the path of least resistance.

The power of mathematics is that once you figure out a correct formalization of the intuitions, using just the formal system allows you to get a lot of information. That is why sometimes people identify mathematics with just the formal system.

[tightbookkeeper]

To take it one step further, mathematical ideas which do not nicely fit within one of the highly developed theories then feels underbaked and less attractive to other mathematicians.

Knuth thinking about algorithms led to all these research questions about combinatorics, that have turned out to be very interesting, but are much more messy and disjointed results.

[throw_pm23]

It becomes clearer if you check out the entire quote from von Neumann:

.. mathematical ideas originate in empirics, although the genealogy is sometimes long and obscure. But, once they are so conceived, the subject begins to live a peculiar life of its own and is better compared to a creative one, governed by almost entirely aesthetical motivations, than to anything else and, in particular, to an empirical science. There is, however, a further point which, I believe, needs stressing.

As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired by ideas coming from "reality," it is beset with very grave dangers. It becomes more and more purely aestheticizing more and more purely l'art pour l'art. This need not be bad, if the field is surrounded by correlated subjects, which still have closer empirical connections, or if the discipline is under the influence of men with an exceptionally well-developed taste. But there is a grave danger that the subject will develop along the line of least resistance, that the stream, so far from its source, will separate into a multitude of insignificant branches, and that the discipline will become a disorganized mass of details and complexities. In other words, at a great distance from its empirical source, or after much "abstract" inbreeding, a mathematical subject is in danger of degeneration. At the inception the style is usually classical; when it shows signs of becoming baroque, then the danger signal is up. It would be easy to give examples, to trace specific evolutions into the baroque and the very high baroque, but this, again, would be too technical.

In any event, whenever this stage is reached, the only remedy seems to me to be rejuvenating return to the source: the reinjection of more or less directly empirical ideas. I am convinced that this was a necessary condition to conserve the freshness and the vitality of the subject and that this will remain equally true in the future.

[cubefox]

My guess is that von Neumann worried that it develops in directions that people happen to find interesting, instead of in directions that are important by some objective external measure.

[mannyv]

There is an argument to be made that this is what's happened to the liberal arts.