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Finkelstein describes himself as a "red diaper baby", raised by leftist parents who encouraged him to develop an interest in radical politics. He began by protesting the Vietnam War in the 1960s, and has worked on many other activist causes since, including becoming an early member of the AIDS advocacy group ACT-UP. In 1986, Finkelstein was a member of the group Silence=Death Project, which created the "Silence=Death" anti-AIDS logo, later donated to ACT-UP. In 1994, in preparation for the Gay Games in New York City, he wrote a tract for ACT-UP, entitled "Welcome to New York", which asked gay men and lesbians who attended the games and other festivities surrounding the twenty-fifth anniversary of the Stonewall Riots to take action to stop the AIDS epidemic.
Finkelstein has written about art & culture, and art directed fashion and art features for Italian Vogue, Dazed and Confused, Visionaire, Genre, Van and Dune. With Gran Fury, he collaborated on public awareness campaigns and public art projects for publications, museums and foundations including The Whitney Museum of American Art, The Venice Biennale, ArtForum, The New Museum of Contemporary Art, Creative Time, and The Public Art Fund. As a surface designer, he created the Tom of Finland Toile for Dirty Linens. Finkelstein has been interviewed about art, activism and communication in the public sphere by publications including The New York Times, ArtForum, Bomb and Interview.