The philosopher and the mathematician were naturally very averse to Galileo’s suggestions for a new theory. Throughout their lives, the philosopher and the mathematician had probably come to understand the world through the Ptolemaic system (the theory that is in question). They had been taught this system many years ago by their own teachers, and they had probably pursued their own science with this theory in mind. A change in the Ptolemaic theory would mean that all they may have pursued in the last several years amounted to nothing – that their own theories were based on unsubstantiated claims and would therefore need to be revised or scrapped. Naturally, the philosopher and the mathematician would wish to believe their original theory, even if they carried some sense of doubt.
Therefore, if I were Galileo, I would doubt that showing my telescope to the philosopher and the mathematician would have any major effect in getting them to profess any change in their beliefs. The only way for either of them to experience their own paradigm shift is by observing (on their own terms and in their own experiment) something that refutes their own sacred theory or something that proves Galileo’s new one. Conversations with powerful rhetoric and attacks on Aristotle would most likely be ineffective.
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