Sometime around 1912, painting changed. Artists from Moscow to Westport, by way of Munich and Paris, began making abstract works. “Observers spoke of the exhilaration and terror of leaping into unknown territory,” Leah Dickerman writes in the catalog for “Inventing Abstraction, 1910–1925: How a Radical Idea Changed Modern Art,” the monumental exhibition she has curated at the Museum of Modern Art in New York City, on view through April 15. This saut dans le vide, she notes, was “accompanied by a shower of celebratory manifestos, lectures, and criticism, a flood of words flung forth perhaps in compensation for their makers’ worry about how the meaning of these pictures might be established.” It also brought a deluge of labels: “pure painting,” “nonobjective painting” and many more, with “abstraction” being merely the stickiest. In the century since then, the squalls of talk haven’t stopped, with art historians and cultural critics joining artists, their promoters and detractors in worrying at the meaning of abstraction. That so much has been said about abstraction has itself become a topic of discussion. In his 1975 book The Painted Word, Tom Wolfe dismissed contemporary art as mere illustrations of art theory—which some of it has been. What’s more striking to me, though, is that after 100 years of abstraction, no one has been able to state conclusively just what it is.

Sometimes I think that indefinability is the defining feature of abstraction: if you can identify what a painting depicts, then it’s not abstract. The problem is that this notion excludes a good deal of the art normally categorized as abstract. I can say that a Josef Albers painting depicts some squares or a Gene Davis painting shows some stripes, and this ought to disqualify them from being called abstract—just as much as my being able to identify a Philip Pearlstein painting as showing some nude models or one by Rackstraw Downes as depicting an industrial landscape would rule out those works.

At other times, I think the opposite. Although abstraction may have been a thrilling venture into the unknown, it could not remain so. Falling in love leads to marriage; there are no permanent revolutions. In the long run, far from being ineffable, abstraction is an artistic genre like any other, like still life or history painting. If a definition is hard to come by, the general parameters are not: abstraction means paintings of things like squares and stripes, brushstrokes and drips, the basic elements of pictorial form and painterly activity.

* * *

I don’t much care for this second definition, but it’s hard to avoid. The virtue of ”Inventing Abstraction” is that it seductively reminds us of the time when abstraction was still a leap, when it affected certain people like love or revolution. And more like revolution than love, for it was a group effort instigated by a determined, committed few, a pivotal fact that “Inventing Abstraction” gets wrong. “Abstraction was not the inspiration of a solitary genius but the product of network thinking,” announces the opening text panel. This seems to promise a new outlook on what is, after all, a pretty familiar history, of which MoMA has been the central proponent for many years. The problem lies in trying to realize it through the evocation of “network thinking,” a trendy concept that tends to downplay the importance of agency—and not only of individuals, whether or not they are “solitary geniuses,” but also of organized groups, movements, coteries. Many of the key nodes of Dickerman’s proposed network are, as she says, “editors of little reviews,” thanks to their ability to connect far-flung artists and writers. Boosters of networking seem to assume that it is always advantageous to accumulate more and more “weak ties,” as they are called—but the classic avant-gardes who contributed to the invention of abstraction valued intense connections and decisive action. As Renato Poggioli pointed out long ago, “the avant-garde periodical functions as an independent and isolated military unit, completely and sharply detached from the public, quick to act, not only to explore but also to battle.”

For all the trendy lingo, MoMA is repeating the story about abstraction it has always told, only with a few of the details filled in differently, and with a concerted effort to point out connections to parallel developments in music, poetry and dance rather than cordoning off the visual arts as a self-contained realm. Dickerman’s appeals to “network thinking,” and her borrowing of terminology from the likes of Malcolm Gladwell—Guillaume Apollinaire as “connector”—are more decorative than transformative. And despite abjuring “solitary genius,” Dickerman begins the story with Picasso, Mr. Genius himself, where MoMA’s tales of modern art so often begin. Picasso, as is well-known, periodically flirted with something like abstraction but consistently pushed it away, even denying its existence: “There is no abstract art,” he once said. “You must always start with something. Afterwards you can remove all appearance of reality; there is no longer any danger, because the idea of the object left an indelible mark.”

Yet beginning with Picasso does make sense, and especially with one of his early Cubist paintings from which the “appearance of reality” has been so successfully effaced that, if not for its title, Femme à la mandoline (Woman With Mandolin; 1910), we could not make out its subject—or is it just that we imagine we can discern it? In any case, our effort to reconstruct the image helps us see the painting as a whole. Such paintings, as well as slightly later ones like “Ma Jolie” (1911–12), which Picasso endowed with a few more visual cues about the possible subject, are still amazing: solidly constructed and entirely evanescent. As Yve-Alain Bois explains in the exhibition catalog, “Each facet, each plane, whether included in the grid or contravening it, is lit and shaded independently, to the effect that no solid is depicted in the round yet at the same time a sense of depth”—and, I would add, a sense of concreteness—“is conveyed.” The wonder of these paintings is not just that the real-world referent has been nearly expunged, but that the painting itself has been endowed with such a distinct sense of presence.

For painters across Europe, Picasso proved that a different kind of painting was possible, one that would no longer have to “start with something” other than the gestures and materials of painting. Even Arthur Dove, in relative isolation in Westport, had seen in New York City a Picasso drawing that Edward Steichen described at the time as “certainly ‘abstract’ nothing but angles and lines that has got [to be] the wildest thing you ever saw.” Yet these painters continued to look to real things as inspiration for paintings that would no longer depict real things. Consider a painting from 1911 by Vasily Kandinsky, who knew Picasso’s work from photographs; he thought that something about the Spaniard’s paintings was “frankly false,” but also constituted a “sign of the enormous struggle toward the immaterial.” Kandinsky’s Impression III (Konzert) (Impression III [Concert]) announces its subject in its subtitle, yet that clue proves insufficient. It takes a comparison with a sketch Kandinsky made that year at a performance of Schöenberg’s music to see how literal a transcription the work really is: the large black shape that rises toward the upper right is nothing other than a piano lid; the oval blobs and fingerlike forms below it are members of the audience. Unlike Picasso’s painting, Kandinsky’s gains little from being sourced; on the contrary, it seems stronger if seen entirely as an implacable assertion of the force of color and texture. Kandinsky needed an abstraction that would no longer have to “start with something,” and having gone this far, he would reach that goal soon enough, for instance in his Komposition V (Composition V), also from 1911. Note not only the lack of subtitle, but also that the musical reference (this is not a depiction of a concert) is conveyed not visually but structurally. Just as, in the nineteenth century, Whistler had given his paintings titles incorporating words like “symphony” and “nocturne” to suggest that his real subject was not the depicted scene but pure form, Kandinsky invokes a musical analogy to tell the viewer not to look for a depicted subject, but rather the relations between the various “notes” of color.

The pairing of Picasso and Kandinsky presents in a nutshell all the dilemmas of abstraction. Whether starting from something already seen was better than starting with the materials of painting was a new problem for painters, but it didn’t spare them the old ones, such as the age-old conflict between the primacy of drawing (as seen in Picasso’s early cubism) or of color (as with Kandinsky). And that’s only the start. Does abstraction, by eschewing pre-established conventions, offer an expression and celebration of “those things that make us individually different and separate from each other,” as MoMA’s former chief curator Kirk Varnedoe once claimed? Picasso might have been pleased to father an art of this sort, just as he would probably have smiled on hearing Vanessa Bell describe a visit to him, in 1914, as one in which “the whole studio seemed to be bristling with Picasso”—where each thing, however unfinished, presented its maker back to himself. But abstraction can also be the herald of whatever is common and universal, as Kandinsky must have believed when he later threw in his lot with the Bauhaus. On this view, the point of abstraction was not just to level the old academies but to supplant them with a new one propagating the new shared values.

In an all-too-contemporary fashion, the metaphor of the “network” allows Dickerman to finesse such disagreements. A network is not an individual, but it’s not a collective either. It is a function neither of inner will or insight nor of shared decision-making. And it lacks discrimination, tending to accept far more than it rejects. But by the same token, Dickerman’s LinkedIn approach makes the exhibition—as Willem de Kooning once said of art itself—seem “like a big bowl of soup,” because “everything is in there already, and you stick your hand in and you find something for you.” At the same time, through its density and the way so many unexpected differences and similarities are provoked, the exhibition communicates something of the giddiness that artists must have felt upon realizing that the rule book was being torn up and would possibly be pieced back together differently. The galleries hum with the feverish mood of a gold rush.

* * *

All the marquee names are here: not only Picasso and Kandinsky, but also Malevich and Mondrian, Duchamp and Léger, Arp and Schwitters, Albers and Lissitzky. They may not have been solitary artists, but that’s no proof they weren’t geniuses. Some play a bigger role than might be expected. Because Francis Picabia gets routinely associated with Dada and Giacomo Balla with Futurism, we may not remember them as great proponents of abstraction. This exhibition tells us otherwise. It also cogently charts the way abstract painting gave birth to abstract sculpture—not so much because sculptors imitated what painters were doing, but because abstraction drew the attention of painters toward the tactile substance of their materials, which turned many of them into sculptors.

But as an exhibition on this scale should do, it also offers surprises. I didn’t know that abstraction had found a toehold in Bloomsbury as early as 1914, when Duncan Grant created a long, scroll-like Abstract Kinetic Collage Painting With Sound and Vanessa Bell made several abstract paintings—including one, with floating rectangles of various colors against a yellow surround (now in the collection of the Tate), that is far more thoroughly reduced, flat and frontal than anything anyone else, even Mondrian, had made at that time. Yet Grant and Bell must have found these experiments unsatisfactory (I certainly do), because they soon returned to making figurative art.

Also from 1914 is a striking Chromatische Phantasie (Chromatic Fantasy) by Augusto Giacometti, cousin of the far more famous Alberto Giacometti. The very few of his works I’ve seen before have been landscapes and still lifes of a broadly post-Impressionist stamp, and no more abstract than a work by Gauguin or Bonnard. But this piece—made, it seems, by roughly dabbing colors onto the canvas with a palette knife—is not only resolutely nonrepresentational but also an abstraction of a sort that seems out of place with anything else in the show, and out of time. With its confident formlessness, and the way touch and color become one, I’d have guessed it to be the work of a tachiste of the 1950s.

For a contrast to Giacometti’s cultivation of the near-random-seeming placement of quite physically distinct bits of paint, there are three drawings by Wacław Szpakowski. Made in 1924, they describe patterns formed by continuous black lines undergoing incessant movement, though always at right angles: the line is always moving either horizontally or vertically, but the patterns created include diagonals. If Giacometti is an unheralded precursor of tachisme, then I suppose Szpakowski plays the same role in relation to Op Art, which makes much of similar optical effects. But as with Giacometti, what’s exciting is not that Szpakowski anticipated a later development; it’s that even within his own time, there is something inexplicable about his having done what he’s done. Using ideas and information similar to those of his peers, he’s arrived at something that is abstract in the strong sense of remaining somehow uncategorizable and even, in a deep sense, unknowable—abstract in a way unlike anything else in “Inventing Abstraction.”

Unfortunately for an exhibition goer who wants to know how Giacometti came to make his Chromatische Phantasie or why he didn’t continue along this line, there’s not a word about him in the catalog. In Szpakowski’s case, one can learn from Jaroslaw Suchan’s contribution that he “was drawn to abstraction by his fascination with the mathematical laws observable in nature” and that “he developed his work not just in isolation from the Polish avant-garde but in complete indifference to the art of the time.” You might find his drawings difficult to distinguish from the kinds of mathematical, scientific or even spiritualistic images that Dickerman insists “are not art at all” because “they were intended to produce meaning in other discursive frameworks.” But that is part of what makes his drawings unsettling and strong. Szpakowski died in 1973, and his works were first exhibited in 1978. The network isn’t everything, and isolation can be necessary even to those who may not quite be geniuses. Szpakowski wasn’t concerned, as Picasso was, with expressing his own anxiety; he was searching for impersonal patterns of universal order. Yet his art was distinctly personal, with a flavor peculiar to itself. Perhaps this is the great lesson of abstraction: that sometimes it can overcome its own antinomies.

* * *

For curators, the inconsistencies between an exhibition and its catalog can be hard to overcome. Anyone who has seen Dickerman’s previous blockbusters for MoMA—on the Bauhaus in 2009 and on Dada in 2006—knows that she is adept at organizing complex exhibitions with scads of material in a lucid way. The same is mostly true here: only the attempt to integrate music into the story falls flat. However, such exhibitions have a particularly symbiotic relation with their catalogs, which need to fill in and give perspective to the historical narrative. In this respect, “Inventing Abstraction” is a disappointment. Perhaps in deference to her fascination with networks, Dickerman’s substantial but fairly succinct introductory essay is followed by thirty-six brief texts on various topics by twenty-four authors (including herself)—not only art historians but luminaries from other fields, such as the composer David Lang and the historian of science Peter Galison. As a result, there is insufficient mediation between her overview and the multitude of details it ought to encompass, and which have been parceled out to the various contributors, who do not always agree with each other or with her.

In her introductory essay, Dickerman seems to take at face value Picasso’s assertion that his first Cubist paintings were done more or less as “pure painting, and the composition was done as composition,” with any identifying “attributes” added only as an afterthought. But in his entry, Yve-Alain Bois refutes this, concluding that Picasso’s interlocutor, Françoise Gilot, had either misunderstood the painter or that he had been indulging in some kind of “convenient” fib. At times, for that matter, Dickerman’s introduction doesn’t even agree with the exhibition. She ends her essay with a brilliant stroke, by claiming Duchamp’s readymades as products of abstraction, and she’s right—but then why isn’t one of them on view? I don’t normally think of Duchamp as a great painter, but really, it’s good to be reminded that Le passage de la vierge à la mariée (The Passage From Virgin to Bride; 1912) is as ravishingly painted as anything in the show. Even so, the inclusion of his Bottle Rack (1914) or his snow shovel, In Advance of the Broken Arm (1915), would have shown another outcome of his interest in abstraction altogether. Like much of the best abstraction, those works are at once paradigmatic and almost inscrutably idiosyncratic.

In our July 16-23, 2012, issue—shortly after the opening of the new site of the Barnes Foundation— Barry Schwabsky looked at the pioneering collectors of modern art, in “Extreme Eccentrics.”