Not long ago, Michael Tilson Thomas was a whippersnapper, conducting the Young People’s Concerts of the New York Philharmonic. Now he is a whippersnapper of 63, leading the San Francisco Symphony. He has been music director since 1995. And he brought the orchestra to Carnegie Hall for a two-concert stand, Tuesday and Wednesday.

Tuesday night’s program consisted of Schuman and Beethoven — no, not Schumann: Schuman. That is William Schuman, the American composer whose dates are 1910 to 1992. He had several major roles in New York, including president of the Juilliard School and president of Lincoln Center.

Time was, we heard his “New England Triptych” quite a bit. It was almost a staple of the orchestral repertory. But it seems to have disappeared. What Mr. Tilson Thomas and the San Francisco Symphony brought on Tuesday night was the Schuman Violin Concerto, composed in 1947, but revised in 1959. The soloist was Gil Shaham, the Israeli-American violinist.

This concerto has two movements, but those movements have many sections — movements within movements, if you will. This is a moody piece, and those moods are fast-changing. Sections have markings such as “Allegro risoluto,” “Molto tranquillo,” and “Agitato fervente.” And, in Tuesday night’s performance, the sections sounded like their markings.

Also, the music tends to be angular yet tonal, spiky yet lyrical. These qualities are redolent of a school: a mid-century American school, of which Schuman is an exemplar. And Messrs. Tilson Thomas and Shaham treated the concerto with understanding and sympathy.

The violinist was absolutely on top of his game. Daunting technical challenges were met with ease; Mr. Shaham hardly broke a sweat. Tricky rhythm was handled astutely, too. I believe I could actually see Mr. Shaham count at one point — but he was not unnatural. Over and over, he contributed liquid lines. But he was incisive and gutsy where the music asks him to be. He varied his colors impressively, reflecting the many and quick-changing moods of the piece. He subtracted or added vibrato intelligently. In the high altitudes of his instrument, he played with remarkable sweetness.

I have often made a peculiar criticism of this violinist: He can be too relaxed, too laid-back — Joe Cool. But, in the Schuman Concerto, he displayed a wonderful and necessary seriousness of purpose. Somewhat surprisingly, he had not memorized the piece; he looked at the music on the stand. But, as mentioned earlier, he was not unnatural.

Mr. Tilson Thomas led the proceedings with discipline and flair. The music had propulsiveness and beauty, a dose of mystery, a bit of a thrill. The San Francisco brass are to be particularly commended: They played pointedly, but never obnoxiously. And a woodwind must be mentioned, too: the clarinetist, who played soulfully and skillfully. When he took his solo bow at the end, he looked about 12.

And we should congratulate Mr. Shaham and his conductor for bringing a “new” concerto to the stage — a new concerto completed in the 1950s. It is not a masterpiece. But man, violinist or not, cannot live on the Beethoven Concerto alone.

And the music world has stupidly ignored American composers such as Schuman — the “neo-Romantics,” they’re sometimes called. Others are Vincent Persichetti, Peter Mennin, Roy Harris, and Walter Piston. At some, sad point, they were all shoved aside by Elliott Carter, Pierre Boulez, and that crowd. Let there be room for all!

I’ll tell you something interesting (at least I think so): The Piston Symphonies Nos. 4–6 have long been favorites of mine. We know them through recordings. But I have never heard a Piston symphony — any of them — in a concert hall. In decades of steady concertgoing.

After intermission, Mr. Tilson Thomas conducted a Beethoven symphony — No. 3 in E flat, known as the “Eroica.” The beginning was not really together. But the performance was far better thereafter.

Mr. Tilson Thomas took the first movement briskly, but not absurdly. He let the music have its basic shape, with his knee bends and body swirls. This movement was admirably intense.

But the beginning of the next movement, the Funeral March, was problematic. It was kind of fussed over, toyed with, when a straighter approach would have been more effective. If you want to manipulate rhythm, try it later, when the music has been established. Also, the fugue at the end was somewhat effortful, labored.

But the third movement, the Scherzo, was a joy. It was correct and invigorating, energized and energizing. This was the best playing the San Franciscans did all night. The horns were outstanding, burnished and unflubbing. Yes, the instrument is hard: but it can be played by professionals.

Beethoven’s Finale did not have its maximum impact. Mr. Tilson Thomas went in for unusual fluctuations and pauses, not all of which were successful. Fast sections were too fast, in my judgment; and slow sections too slow. Overall, the symphony seemed to run out of gas. But Mr. Tilson Thomas and the SFS gave us a satisfying performance of a great and pathbreaking work.

The audience was very enthusiastic, and Mr. Tilson Thomas provided an encore: an item from Schubert’s “Rosamunde” music. This was the Entr’acte No. 3, and, from these forces, it was smooth, insinuating, graceful, and lovely. A little long for an encore — but nice.