Kepler's idea of the universe was based on his study of the ancient theorists, which, combined with years of indoctrination, had instilled in him a sense of mysticism. He was convinced that the arrangement of the universe must correspond with some concept of geometric harmony and beauty. Thus he sought a simple mathematical solution to the problem of mapping the universe, trusting that the laws of nature would provide one. This was a regression in the attitudes of the seventeenth century astronomers, most of who were prepared to accept an irregularly shaped universe, or at least one that did not fit any simple geometric scheme. However, Kepler persisted in his efforts to discover a universe that worked under a unified grand scheme. Though he attempted to map the solar system in this method, he repeatedly failed due to his convictions. Kepler expressed his astronomical goal as to replace arbitrary hypotheses with mathematical explanation. He was hindered in this task by his belief in mysticism and astrology. Astrology was widely practiced and widely believed to be accurate by most Europeans in the seventeenth century, and Kepler was no exception. He attempted, through his studies of the solar system, to confirm with reason the astrological influence of heavenly bodies, something neither he nor anyone has ever been able to do conclusively.
Galileo, devoid of any adherence to mysticism, posed a sharp contrast to Kepler. Kepler was a German Protestant, a mystic and a dreamer, while Galileo was an Italian Catholic who sought evidence, explicable facts, and was unrivaled in his time in experimental acumen. However, in their genius, the two astronomers were peers, and between the two of them, created the modern view of a mathematical universe. Galileo, especially, contributed to this progress, confirming with observation the fact that the ancients had not known enough to formulate theories on the universe, and combining his in depth knowledge of mathematics and physics with that observation to begin to formulate a new view of the universe.
The discovery of Jupiter's moons was significant because it provided, in effect, a small model of the solar system. Galileo used the model of Jupiter and its moons to explore the manner in which the planets might orbit the sun. The implication of his first observations was to call the traditional Aristotelian system, advocated by the Church, into question. Thus a number of forces joined together to oppose Galileo. Academic Aristotelians had long despised him. Added to these were Jesuits, who were actively engaged in teaching the beliefs of the church. Many pious Europeans joined their ranks, and a mass of common citizens unwilling to entertain novel ideas united with this oppositional group as well. Thus Galileo was advised to be cautious if he valued his life and his work, and he resigned himself to remain somewhat reticent until he had gathered enough evidence and built up strong enough theories to fully unleash his view of the universe upon civilization. This moment came in 1630, and upon the release of his theories he was soundly punished by his opponents.
Galileo, more than any other scientist of the era, introduced the change in thought that broke with the ancients and led to modern science. The Dialogue presents a character convinced of the Aristotelian system, who is made to look hopelessly stupid, in a symbol of Galileo's conviction that the Aristotelian system, based on arbitrary and mystical hypotheses, was outdated in the world of exact science. Perhaps his greatest theoretical contribution was the argument that the laws of physics operated equally everywhere, a conclusion that vastly expanded the possibility for better understanding of astronomy through terrestrial experimentation. Further, Galileo developed tools, both mathematical and physical, to explore the universe on all of its levels. The telescope allowed for magnification and better resolution of objects at a great distance, and the microscope allowed scientists to observe the complexity of nature on a smaller scale than ever before. However, there was still much more work to be done in the exploration of the heavens. Galileo had not fully integrated his physics with Kepler's theories on the motions of celestial bodies. That was left to Isaac Newton.