What made early 20th-century Vienna such an incubator for various intellectual activities?

The Viennese were considered decadent and frivolous, not great intellectuals prior to the the Fist War. Economically, the Empire was doing well, especially in new knowledge based industries, and the noveau riche created a market for Modernism, of a somewhat febrile type.

After the War, Vienna had a bloated tertiary sector with a tiny economic hinterland.This meant that its artists and thinkers reoriented themselves to export markets- primarily America- thus spreading the myth of a Viennese intellectual culture which never actually existed.

One final point, German was an important language for scholars a hundred years ago. Indeed, you couldn't get an Econ PhD from Harvard without knowing German prior to 1960. Vienna presented the Anglo Saxons German scholarship oppositional to that of Berlin.

Sorry, can't seem to edit the above or put paragraphs. Briefly, Vienna gained external economies of scope and scale in creative fields because the opportunity cost for creative types was lower because of the greatly diminished size of the state and opportunities in the professions. Property prices were depreciated and services- restaurants etc- were cheap in dollar terms. Vienna's 'febrile' intellectual culture of the pre War period- itself a reflection of rapid economic growth accompanied by radically increased internecine political conflict- was adaptive for Modernism whereas Berlin's intellectual mise en scene was not. The North Germans had turned pedagogy into a heavy industry- with philology and phenomenology and Annales type Historiography preponderating- and weren't really able to get back on track under much diminished financial circumstances. The Viennese had always been less freighted and thus could be more elastic in their response to economic decline. Other newly independent countries, by contrast, had plenty of 'jobs for the boys' and were doing well in the Twenties. Thus, Vienna's falling on hard times and the fact that its intellectual inheritance was insubstantial meant that people could be more daring more especially because there was no realistic prospect of getting a well paid professorship. One other point- sexual competition in Vienna (e.g. Bruno Betthelheim who had to take a PhD in Aesthetics to get his girl to marry him) involved intellectual or artistic merit- perhaps reflecting people's instinctive understanding that they might have to emigrate and their brains would be the only fungible asset they could take with them. To many- e.g. Robert Musil- Vienna's post War fate had been foreshadowed in the last years of the Hapsburgs. An impoverished hedonism was forced to nourish itself on an astringent intellectual diet- a good thing on balance because it broke down the traditional wall between 'Town and Gown'. A man might train to be a cabinet maker, as Spinoza was a lens grinder, while devoting himself to Logic and Epistemology.

Perhaps the most important answer is that Vienna had been, until 1918, the capital of the highly cosmopolitan Austro-Hungarian Empire, consisting of a score of nationalities, and people of a myriad of backgrounds.

It was also a city with a profound depth of history and culture - art, music, literature etc -, It had been the home of the Habsburg monarchy for centuries, with Austria the largest and most important state in the Holy Roman Empire. Over all the centuries of its being, I believe there was only one Emperor of the HRE who was not a Habsburg.

In a slightly similar sense London is today a place where 300 different languages are spoken, a reflection of Britain's imperial past - and where a depth of history ensures the continuation of a vibrant intellectual and artistic culture. Paris has similar claims.

I would not put Hitler, Trotsky and Stalin under the title "incubator of intellectual activities".

That said, it is indeed true that intellectual activities flourished in the beginning of 20s century in the Austro-Hungarian empire. (Budapest, Prague and Lemberg (now Lviv) also qualify for the surge of intellectual activities at the same time).

Your citation omits mathematics and exact sciences: Boltzmann worked in Viena, for example, and also E. Mach. There was a famous school of Logic, Vienna Circle. The so-called "Hungarian miracle" happened at the same time in Budapest. Suddenly it became a nursery of some of the future most famous physicists and mathematicians of 20s century. (Mathematicians from Hungary: John von Neumann, Riesz brothers, Polya, Szego, Fejer, Lanczos, Erdos, Turan; physicists Wigner, Teller and Szilard.)

There was also a revival of all kinds of intellectual activities in the East of the Empire, Galicia and Bukowina (modern Western Ukraine). Franz Kafka and Karel Capek lived and worked in Prague.

All this leaves no doubt that the late Austro-Hungarian empire had some very favorable climate for intellectual activities. One can only speculate what where the exact conditions which produced this climate. This was a good example of what can be called "enlightened monarchy".

I want to emphasize only one aspect of this empire: diversity and tolerance. I hope other aspects will be analysed in other answers. Most kinds of national, ethnic and religious discrimination were abolished. Earlier, in 19s century it transformed from Austrian Empire to Austro-Hungarian empire. Gradually many other "minorities" got equal rights (and representation). Unlike in Russian empire, for example discrimination of Jews was abolished: many positions in science opened to the Jews. Jews even became eligible for nobility titles (von Neumanns, for example).

Ukrainians were always complaining about Russian oppression, Polish oppression, but I have never heard or read them complaining about Austrian oppression. (I lived in the former Austrian part of Ukraine in my childhood, when some older people still remembered "Austrian times"). In the Russian empire, Ukrainian language was prohibited. Declared "non-existent". In Austro-Hungary, they published Ukrainian books and had a chair in Ukrainian history (in Lemberg/Lwow/Lviv University).

By the way, Germany of that time was also a very tolerant state, and this shows in the extraordinary intellectual achievements in all areas. But Germany was much less diverse in its population.

It is true, that the empire fell as a result of the world war and of the motion for independence of various nations. But it was not really so bad as some national independence propagandists tried to claim. In most of those new independent nations, the conditions for minorities became much worse than they were in the Empire. For example, most Hungarian mathematicians and physicists listed above had to leave in 1930s, and made their careers in the West.

Part of these careers was creation of nuclear (Szillard) and thermonuclear (Teller from Budapest and Ulam from Lwow) bombs, to mention the things best known to the general pubic.