[Galileo] was the first to show that the language of mathematics could be used to describe the behaviour of material objects, such as falling bodies and projectiles. To us this see... [Galileo] was the first to show that the language of mathematics could be used to describe the behaviour of material objects, such as falling bodies and projectiles. To us this seems obvious—today’s scientific theories are routinely formulated in mathematical language, not only in physics but also in the biological and social sciences. But in Galileo’s day it was not obvious: mathematics was widely regarded as dealing with purely abstract entities, hence inapplicable to physical reality. Another innovative aspect was Galileo’s emphasis on testing hypotheses experimentally. To the modern scientist this may again seem obvious. But in Galileo’s day experimentation was not generally regarded as a reliable means of gaining knowledge. Galileo’s emphasis on experiment marks the beginning of an empirical approach to studying nature that continues to this day. Read more

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The question appears to be discussing Galileo's contributions to the use of mathematics in describing physical phenomena and the emphasis on empirical experimentation, highlighting its significance in the development of modern scientific inquiry.

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Galileo's work was revolutionary in integrating mathematics with physical sciences and initiating the empirical method, laying the foundation for modern scientific inquiry.

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Common mistakes include underestimating the historical impact of combining math with physics and ignoring the novelty of Galileo's experimental methods.

Sources

• Galileo was the first to … what exactly? - Intellectual Mathematics - intellectualmathematics.com
• 7 Galileo and the Law of Fall - Oxford Academic - academic.oup.com