Galileo’s abilities as a mathematician were far below that of many of his contemporaries. He made numerous technical mistakes — including several high-profile, mathematically erroneous applications of his own law of fall — that were swiftly spotted and corrected by the leading mathematicians of the day. Many aspects of Galileo’s work can be viewed as consequences of this limited technical proficiency in mathematics. For example, he ignores Kepler’s work and dismisses comets as a chimerical atmospheric phenomena: decisions that are difficult to justify on scientific grounds but which make sense if we grant that Galileo wanted to avoid technical mathematics at all costs. Instead he drops rocks, looks through tubes, rails against Aristotelian philosophers, and expounds at length about basic principles of scientific method: all of which can be seen as dwelling on precisely those parts of the mathematician’s worldview that do not require any actual mathematics.