The Afikei Mayim quotes a famous responsum from the Rivash (447):
Even though the master of blessed memory [Rambam] was wise in the wisdoms of medicine and science, and was an expert surgeon, we do not live from the mouths of nature and medicine. We rely on our Sages even when they tell us that right is left because they received the truth and the explanations of the mitzvah man from man directly from Moses. We do not believe the wise men of the Greeks and Arabs because they only spoke based on their conjectures and experimentation...This is a powerful quote that advocates tradition over science. However, is it normative? The Rivash, and the Rashba (Responsa 1:98), dealt with issues where contemporary science contradicts explicit Talmudic science. According to both of them, we ignore science and rely on our Sages. Either the scientists witnessed a miracle (cf. Shakh, Yoreh De'ah 57:48) or are lying (cf. Responsa Chakham Tzvi 77).
However, some have suggested (e.g. R. Yehuda Levi, The Science in Torah p. 100) that the Rashba and Rivash were only discussing laws that received from Sinai. On other issues, we might rely on science. Additionally, some suggest that they were only discussing scientific conclusions that are suspect (e.g. Iggeros Moshe, Choshen Mishpat 3:73). However, today experiments can be carefully controlled and many (but certainly not all) scientific conclusions are ironclad. The Rashba and Rivash would agree in such cases to follow science.
If that is the case, then one wonders what the point of the Afikei Mayim is in bringing the quote from the Rivash unqualified, as if circumstances have not changed and the quote is not all-encompassing. If not, then we have a further problem. If we accept the Rivash and Rashba at face value, then contemporary posekim have rejected their strident stance.
Terms like "nishtaneh ha-teva" (nature has changed) are frequently utilized to explain how contemporary conditions differ from Talmudic statements about nature. Thus, for example, the Bach (Responsa, 100) ruled that a woman does not menstruate from the day she becomes pregnant, rather than the third month of pregnancy as stated by the Gemara (Niddah 8b). More recently, the Shevet Ha-Levi (3:141:3) ruled that a baby born in the eighth month is viable. Do not tell me, he writes, that this disputes what the Sages said, because our eyes see that they can survive. The Sages must have spoken of their time but our times are different. According to a strict reading of the Rashba and the Rivash, we would have to say that doctors are lying to us about babies in the eighth month surviving. Similarly, R. Shlomo Zalman Auerbach (Minchas Shlomo 1:34) ruled that, despite the Gemara's conclusion that someone who is deaf but not mute is not functionally intelligent (bar da'as), the times have changed and he now is. He said this despite the Divrei Malkiel's statement that the Gemara's rule is a halakhah le-Moshe mi-Sinai that must even be followed today. The Divrei Malkiel, evidently, followed a strict interpretation of the Rashba and Rivash and concluded that we must say that those who contradict the Gemara are lying. R. Shlomo Zalman Auerbach disagree.
It is also worth noting that the Rambam in general disagreed with this position of the Rashba and Rivash. See the Iggeros Moshe cited above and R. Nachum L. Rabinovitch's essay in Fred Rosner and Samuel Kottek eds., Moses Maimonides: Physician, Scientist, and Philosopher.
(See also these posts: I, II)
UPDATE: See this article by R. David Horowitz about the Rashba's attitude to science.