Alan Lightman is a writer, physicist, and professor of the practice of the humanities at MIT. His next book, “Searching for Stars on an Island in Maine,” about religion and science, will be published in March.

When I was a physics graduate student, I was introduced to the philosophical ideas of Thomas Kuhn and Karl Popper. (Regrettably, such books are not often taught in the education of a science student.) Kuhn, in his landmark “The Structure of Scientific Revolutions” (1962), suggested that major advances in science do not result from small, incremental steps but instead from sudden and occasional sea changes, “paradigm shifts,” in which a completely new view is proposed and finally accepted by the community of scientists, often a younger generation who must shove aside the old farts or wait until they die.

The second book of my science salad days, Popper’s “The Logic of Scientific Discovery” (1934), proposed that science progresses by falsifying theories rather than proving them. We can never prove a scientific theory true because we can never be sure that a new experiment tomorrow might not contradict the theory. The theories of science are always provisional. Science is not an enterprise of complete verification of beliefs. Theories currently subscribed to are those that have not yet been shown false. Still, one wouldn’t claim that our “provisional” theories haven’t been useful. They’ve led to such conveniences as antibiotics and smartphones.

Popper was born in Vienna in 1902. He became an associate (but not a full member) of the Vienna Circle, an extraordinary group of mathematicians, physicists and philosophers portrayed in Karl Sigmund’s delightful new book, “Exact Thinking in Demented Times.” Much of what we think today about the nature and boundaries of science emerged from this group. The men (and they were all men) met every other Thursday evening at 6 o’clock in a small lecture room of the mathematics institute of the University of Vienna. The years were 1924 to 1936, between the great wars, against a backdrop of hyperinflation, political turmoil and the growing drumbeat of the Nazis in nearby Germany. Few such gatherings have produced so many profound ideas and new works. One might think of the Lunar Society, which met in Birmingham, England, between 1765 and 1813, and the Transcendentalists, who met in the Boston area in the 1820s and 1830s.

“Exact Thinking in Demented Times,” by Karl Sigmund (Basic Books)

The central members of the Vienna Circle were physicist and philosopher Moritz Schlick, mathematician and philosopher Hans Hahn, philosopher Rudolf Carnap, and social scientist and philosopher Otto Neurath. Other luminaries who regularly or occasionally attended the meetings included Albert Einstein, Bertrand Russell, Kurt Gödel and Ludwig Wittgenstein. (Sigmund Freud was working in Vienna during the same period. His ideas were, of course, known by the Circle.) Over the years, the size of the Circle ranged between 10 and 20. The distinguished members brooded over such questions as the nature of science, the nature of philosophy and the dividing line between the two — said Schlick: “The scientist seeks the truth (the correct answers) and the philosopher attempts to clarify the meaning (of the questions)”; what meaningful statements can be made about the world; and the challenges of language itself in describing the world. Is mathematics merely the logical manipulation of symbols, or does it create new knowledge about real objects in the world of ideas? What is the difference between reality and the representation of reality? What is the role, if any, of human perception and intuition in scientific inquiry? Are all meaningful propositions and beliefs subject to experimental verification? And what, anyway, is “meaning?” The members of the Vienna Circle were not shy about asking the big questions, nor giving their answers. We still struggle with the aftermath.

Although the members of the Circle frequently argued with one another, they were unified in rejecting the abstractions of the great German philosopher Immanuel Kant, who proposed that there were things in themselves outside perception, and knowledge independent of human experience. Schlick and Hahn and Carnap proclaimed, instead, that all our beliefs should be testable and verified, a philosophical theory that became known as “logical positivism.” (Such a philosophy, of course, has trouble with morality and ethics, God, and other clearly important beliefs that cannot be tested.) The Vienna Circle considered itself on a crusade against all metaphysical (abstract) and theological doctrine.

Sigmund, a distinguished mathematician himself and a professor emeritus at the University of Vienna, has produced a stimulating account of the Circle, not only stating with clarity its ideas but also giving colorful portraits of and personal stories about its members — altogether a more accessible and entertaining work than an older book on the subject by Victor Kraft, published in 1953. At the same time, Sigmund has thoroughly researched his subject, with many quotations from the journals, papers and books of the people concerned and a bibliography citing more than 350 original sources, many of which the author read in the original German.

Again, these were not timid folks. At the age of 25, Schlick published a book titled “The Wisdom of Life: An Essay on the Theory of Bliss.” Other books published by members of the group had titles such as “The General Theory of Knowledge” and “The Logical Syntax of Language” and “The Scientific Worldview.” At age 29, Wittgenstein wrote to Russell (perhaps the greatest living Western philosopher at the time), “I think I have solved the problems once and for all.” By “problems,” the young Wittgenstein meant all the problems of philosophy.

Some of these deep thinkers thought themselves into a state of confusion and even despair. Gödel, considered by many to be the most brilliant living mathematician, the one person Einstein went walking with later at the Institute for Advanced Study, a person who had demons in his head and constant visions of being poisoned, once confessed to another Circle member, “The more I think about language, the more it amazes me that people ever understand each other.”

Schlick became the ringleader of the group. Born in Berlin to a family that traced back to old Bohemian nobility, he wrote an essay on the philosophy of Einstein’s relativity principle early on, exchanged correspondence with the great physicist and, according to Sigmund, became the “philosophical mouthpiece of Einstein.” According to Circle member and mathematician Karl Menger: “People would mill around in informal groups until Schlick clapped his hands. Then all conversations would stop, everyone would take their seat, and Schlick, who usually sat at one end of the table near the blackboard, would announce the topic of the paper or the report or the discussion of the evening.”

Sigmund’s book is full of vivid descriptions of people and places. Schlick’s route from his apartment on Prinz-Eugen-Strasse to the mathematical institute: “The D tram line glided by the Belvedere and Schwarzenberg Palaces, with their baroque parks, before turning into the Ring Boulevard with its shade trees and its showcase buildings. . . . Schlick’s lectures in Vienna proved an instant hit. . . . [His] popularity, however, did not go to his head.” There is a gripping description of Neurath and his wife escaping the Nazis in a boat, and many others.

My main quibble with Sigmund’s book is its frequent digressions, breaking the narrative flow to give us several pages of biography of a new character — and there are far too many characters to follow — or to provide historical or intellectual background, often jumping backward and forward in time. While many of these digressions are interesting, they could be woven better into the text.

In the mid-1930s, as Hitler was rising to power but had not yet annexed Austria, Schlick suggested a PhD thesis topic about the “agreeable and the beautiful in philosophy” to a student named Sylvia Borowicka, the daughter of a prosperous Viennese family. Another of Schlick’s students, Johann Nelböck, was obsessed with Borowicka. One day, Borowicka told the young Nelböck that “Professor Schlick was interested in her, and that she reciprocated the interest.” At which point, Nelböck went berserk. After getting his doctorate, the young philosopher took to stalking Schlick in the hallways and shouting out to anyone who would listen that Schlick had engaged in “immoral games” with Borowicka. (There is no evidence that the relationship was intimate.) At times, Nelböck waved a gun. He was arrested and sent to a mental hospital for a stay. This happened twice. On June 22, 1936, Nelböck caught up with Schlick on the Philosophers’ Staircase in the university district and fatally shot him four times. It was an illogical act but one we can understand.

Exact Thinking in Demented Times
The Vienna Circle and the Epic Quest for the Foundations of Science

By Karl Sigmund

Basic. 449 pp. $32