Basically science is about trying to discern the facts of the world and understand them. For that you need "judgement" - and judgement is mostly what many of the Jewish religious texts - and the Jewish religious institutes were about. So the two are tied in tightly.
The book of Koheleth in the bible attributed to king Solomon - from 900bc (while some say it is from 200bc) discusses knowledge and understanding of the world - being pessimistic about the ability to ever truly grasp it fully, but stating the importance of knowledge - "the knowledgeable has eyes in his head, while the stupefied one walks in the darkness".
About the year 200 The Babylonian Talmud discusses medicine (which is a type of science) trying to decide if eating a dog that bit you is a valid treatment against rabies (not rabbis) or is it "Darkei Ha-Emori" the way of Emorites - in other words a superstition only, with ties to idol worshiping. The Talmud is full of both myths and superstitions on the one hand, and "scientific" discussions of facts like this, on the other.
About year 900 Rav Sa'adia Gaon was the first to write a full fledged treatise as to the Jewish religion and "science" of the day - mostly Greek philosophy, anatomy and medicine, and Babylonian knowledge of trigonometry, algebra, theory of music, with the most important of all: astronomy.
Around 1100, he was followed by the Rambam - Moses Maimonides who, aside from being a doctor, wrote many books in Arabic about science topics. His famous (and only) Hebrew book begins with "The book of Science" and begins by describing the four elements (Fire, Water, Earth and Air).
Haim Ibn Zakuto was a Geographer and Cartographer . He wrote a map for Colone (Christofer Columbus) and his team, funded by Jewish merchants. According to the map, based on a "printing" mistake (actually a copying mistake) which said that the world circumference was only 15,000 miles, it would be quick and easy to reach India by going west rather than east. Cheaper too because you don't have to pay all the stations and pirates on the way, or meet the enemy (Spain was the enemy, when the idea was proposed to the Portuguese king, Portugal was the enemy when the idea was proposed to the Spanish king, who later accepted it). India had spices which could be sold in Europe for a lot of money, and it was worth it.
They set sail accordingly, on the day of expulsion of Jews from Spain, Tisha Beav 1492. Many on the ship were in fact Jews (forced into Christianity) or their descendants and there are various indications that Colone himself was the same. They almost turned back because of the extra 9000 miles not accounted for, but 3 days off schedule and after a mutiny, when the sailors were afraid they would fall off the end of the world, they finally saw India - actually Cuba.
Baruch Spinoza in Amsterdam was deeply affected by his scientific studies, and wrote a whole new idea of living in a Godless world. Taking on this way of thought became prevalent in Germany under Moses Mendelsohn and the Haskala movement, and later on by many secular Jews around the world. Jonathan Miller in a BBC documentary said: The Jewish religion is about eliminating false gods. We just took the Jewish idea one god further.
Many contemporary Jewish thinkers disagree. In Israel, 90 years ago, at the beginning of world war I, Rabbi A.I. Hacohen Kook, began writing his book on science and especially evolution theory (as he understood it then, as a teleological theory - that is: a theory that claims the world is moving and developing towards a goal) as an indicator of a foundation for Jewish ethics. He was following two major thought trends in the Jewish tradition. The first - around year 1700, that of Rabbi Dov Bear of Mezritch, the founder of Hassidism as a movement (he was the student and appointed successor of the Baal Shem Tov), who spoke of studying nature and science as the "Truth" which one must seek, since "God's stamp is Truth". And the second which Rabbi Kook was following was that of the German "Hochmat Israel" (Jewish Studies) - a trend under the Haskala movement, using philology (comparative study of texts), archaeology, and scientific history studies, to understand the original meaning of texts and content.
At the same time there was a strong tendency by religious Jews to move away from science, sometimes forced on them since the Napoleonic era, when Napoleon captured most of eastern Europe and created the "emancipation" laws - where Jews had equal rights. And so we find today many Orthodox and especially ultra-orthodox rabbis with an "anti-scientific" way of thought, or at most with an attitude of reconciliation between the two, with the Jewish traditions especially about the creation of the world, usually prevailing while the scientific view being skeptically scuffed at. - in a similar was as Christian creationists.
Interestingly, there is also a strong connection the other way - from the Jewish religion to science.
Religious and secular Jewish scientists have made a major impact on scientific theory and discovery, many times with a strong relation to the Jewish religion - for instance Jewish medical researchers were dominant in the discovery of Insulin, a cure (- so to speak) for diabetes, the development of streptomycin and other antibiotics, the understanding of the Mitochondrea and much more - due to their seeking for a unified understanding of the world, and viewing medicine as an important deed that humans must participate in. The Jewish notion of cleanliness and especially the ritual washing of hands before meals and after using the toilet, and the ritual baths were instrumental in the adaptation and use of hygiene and sterilization in medicine (as well as saving many Jews lives during the black plague). There were many non Jewish "scientists" who opposed this at first. Read about the ordeal of Zemelweis.
Albert Einstein was only one of many Jewish physicists and mathematicians who made a strong impression on science 70 years ago just after world war I. His whole life he was caught up with the notion of the world being actually some kind of unity, one single rule, one single particle (the proton or quark). This was strongly related to his studies as a young child about the Jewish religion. Many Jewish great mathematicians and developers of statistics and game theory have a deep association with math in the Jewish tradition, such as Gimatria and the "Remez" field of study.
The "big bang" theory caught on strongly and quickly, because of Jewish scientists who were happy to re-conciliate Biblical texts with scientific fact. - Well, how long was the first day, when there were no planets not even heaven and earth, just light and darkness?
Archaeology, scientific text studies and much of linguistics (even anti-Israeli Noam Chomsky's father was into it due to his Hebrew studies with his father Zeev) have a strong following by Jewish scientists, who have a feeling of membership in the field - "this is ours". You can trace the seek for Adam and Eve in paleo-anthropology theories, where early huminids passed through Israel from Africa before dispersing to the rest of the world. Or "The four mothers" study, which of course refers to the Jewish four mothers (Sarah, Rebecca, Rachel and Leah)
The same is true in political sciences where Tikkun Olam (correction or fixing of the world) has had a strong Jewish influence. Professor Leo Strauss' ideas in philosophy are strongly connected to Jewish texts and the Jewish religion in his Jerusalem and Athens theories. Even Chomsky's extreme ideas on ethics can be shown to be closely related to the Jewish religion (which he opposes).
At the same time, this also works the other way - the big political movements had a strong influence on the Jewish religion. Communism was invented by the Jews Carl Marx and Engels, writing in what they figured was a scientific way and the ideas of Capitalism in Jewish "Ein Rand's" books, were spread to Jewish Milton Friedman who is a known Professor of Economic sciences.
Feminism, and feminist studies were strongly advanced in the US by prominent Jewish women, many of them discussing their ties to the Jewish religion and its ideas.
A much more boring discussion would be about science and technology in Halacha. Jewish scientists who are also rabbis are prominent in the decisions about the moment of death, science has been used to determine if there are "worms" in lettuce, and to create cherry tomatoes (invented and developed by Orthodox Jew) so that they can grow disconnected from the ground, and thus be permitted to be grown on the Shemitta - seventh land-rest year.
Advanced timing clocks were developed for turning electricity on and off on the Sabbath day, and other quirks have developed, so that the Jewish tradition can continue, while living in the modern world. For example, today there are medical procedures added to food rituals, and even to circumcision.
Science today strongly affects religion in many subtle ways. The large reform Jewish movement is lead by scientists, and the same was true for the first Jewish Conservative Movement leadership - who were all "Rabbi Professor" or "Rabbi Dr."
Here is a summary of points directly answering your question, I hope you find it instructive:
I showed that science influencing Kashrut is reciprocal, having an effect back on science. I noted that this reciprocal influence of Kashrut and other Jewish religious ritual rules on science is far more interesting, and gave several examples. First and foremost the washing of hands before a meal and after the bathroom, which is part of the Kashrut field (Kosher food in biblical terms is ritually-clean "Tahore" food).
Science influences Kashrut, for medicine. In the case of non-Kosher medication, there is a need to discern between viable medicine and superstition. (With the example of dogs and diabetes - in ancient science. Of course today's science can decide these questions)
As an aside, the reciprocal effect of Kashrut on science may have added to the research and discovery of Insulin and further interest in diabetes, by religious and secular Jews.
Science affected Kashrut, since scientific methods for growing foods have been used for developing new types of foods and new types of growing methods due to Kashrut restrictions: i.e. Cherry tomatoes due to the Shemita year, or the worry of eating "worms" (discovered in the microscope).
Technology (a branch of science, and definitely affected by science) is used to automatically milk cows on the Sabbath day, to watch over secular kibbutz farms, and to develop new types of produce from known foods or from non-foods. So there have been Halachic discussions about algae (sea weeds) which biologically are not defined as plants, and there are various "levels" of Kashrut dealing with milk-powder or glycerin from non-kosher sources, the effect of trace materials on production lines, and the Kashrut of new types of foods such as artificial meat (from animal cell sources created in the lab with no living animal). So in all these ways science is affecting and has affected Kashrut.
In several cases Kashrut laws, especially one's that are "only" traditions (Minhagim) are contrary or even contradictory to scientific knowledge, or are regarded as being scientifically questionable. Among others are the issue of "water memory" after boiling water once on the Sabbath, cooking with fire started by a match on holidays (allowed by early rabbis of Morocco), defining "Kitniyot" - (literally kidneys - meaning legumes), and forbidding new types of oils on Passover such as CANOLA produced in an industrial procedure.
As a side note, if not for the explanation that water "remembers its heat" and
therefore is easier to boil a second time - independent of its current temperature, it could easily be explained that boiling water once gets rid of most materials and germs in the water).
I'm adding another side note here, because the comments can only be concise and short. I was asked where the Rambam specifically refers to science and why I think that Sefer Hamada should be translated as "The book of Science".
In his explanation of the Mishnah, tractate Berachot - blessings in Hebrew, written in Arabic translated to Hebrew by Rabbi Yehuda El-Harizzi and can be found in any Vilna edition of the Gemara, or online here: https://he.wikisource.org/wiki/הקדמת_הרמב"ם_למשנה_(אלחריזי) while writing about why Torah learning needs to be preceded by a long and deep preceding learning period says thus: (search for the word תשבורת)
"Now lets imagine an expert of medicine, algebra and the physics of sound, who he is a learned man and quick to comprehend, but who has no knowledge of trigonometry or astronomy, and we ask him the following question: What would you say about someone who claims that the sun which seems to us as a small circle, is in fact the shape of a ball (a sphere), and the size of that sphere is 166 times and 3/8ths the size of the sphere of the earth. And that the earth by which we are measuring, is itself of a spherical shape with a circumference of 24,000 miles. So in this way he could figure the exact miles of the sun's dimensions."
"Of course that smart and learned person would studied all the subjects we referred to earlier, would not find any way to accept this belief, and it would be indefinitely far from the truth in his eyes. And a primary intellectual assertion of his would be that this claim must be false, since no man can remove himself even more than a pinky finger from earth, and therefor how could he possibly know the exact shape, size and area of the sun, measuring it in the same way one measures the earth?! And besides, the sun is so far away, that we can only see its corona. So how could someone possibly measure it, let alone get the precision of 3/8ths. Obviously this is a total fallacy! And he will have no doubt that this claim is void, and cannot be. "
" But then when he regularly studies books about measurements and the understanding of spheres or other shapes, and value comparisons, and then studies a basic book on the subject, such as the famous Book of Spheres known as 'Al Magesty' or a similar book, then this claim will be clarified, and he will change his mind, this time understanding it as a correct claim with no doubt, and one with proof. And there will not be a difference in his eyes between the claim that the sun is of these dimensions, or that that there actually is a sun. And he will train his mind to believe in this thing which at first was the farthest from comprehension, to come to believe in it fully. "
" This (change of mind as you see), is possible. And we did not require that the person be lacking any other scientific knowledge, we only required that he be sharp minded, clear thinking and smart. And the question we asked him was a studious question, one which is a step in reaching divinity. "
" That would be nothing in comparison to someone who has no knowledge at all... if we would ask them a question about divinity hidden between the lines of the Midrash texts, those would undoubtedly be far in their eyes as the heaven is from earth... "
In his famous "8 chapters" prefix to Pirkei Avot he writes:
" Note that the things I write in these chapters and in the commentary (on the Mishna text) are not my own fiction, and not innovative commentary of mine, but rather are an anthology and collection of the words of our sages (lit: the learned men) in the Midrash, the Talmud and other writings of theirs, and of the words of early and current philosophers, and of writings by many people. "
" And hear the truth from whoever speaks it."
" And it is possible that sometimes I bring a complete section from some famous book in its entirety as is, there is nothing wrong with that. I am not plagiarizing or boasting about words of my predecessors, since I hereby confessed about that, but I won't quote: 'This person said...', because it makes the reading cumbersome and is useless information, besides the likelihood that someone with bad taste may find the content of the words damaging or harmful, when in fact they simply lack comprehension. For this reason I chose not to name the quotee, since my intention is to assist the reader and explain the deep understandings to be found in this tractate. "
The book of Mada - what you call knowledge and the Rambam calls truth, I can easily show that he was taking from the science of his day. Here's what he writes about scientific proof (in the Moreh Nevuchim - way for perplexed):
" And you should know that our break away from accepting the theory of an ever existing world is not due to the writings of the Torah that the world was created, since those verses pointing to a generated world are no more in quantity than those portraying God with a body. "
" And also the ways of interpretation are not locked for us and we are not limited in any way from using them for the issue of world creation. Rather, we could have interpreted those verses in the same way we did when denying the material realization (of God), and it probably would be much easier to accept the ever existing world theory explaining the verses in a similar way we did interpreting the verses while denying the materialization of his highness. "
" But the reasons for us not doing so where two: First: The fact that God cannot be materially realized has been proven, but the claim of an ever existing earth has never been proven... And secondly: Our conjecture that God has no body did not contradict the basics of the Torah teachings...
" ...whereas Aristotle's theory that the world always existed, and that everything is according to a constant and never-changing nature, that flatly contradicts the Torah... "
" ...and so if creation could be proven, even only according to Plato's explanation (that the universe including the stars is material and has an ending), all the claims against us would fall, and if an ever-existing world could be proven, according to Aristotle's concept, (our understanding of) the Torah's concept would fall, and we would need to move on to some other concept... "
Finally: I translate the Sefer Hamada (Rules of Torah Foundations)
These four bodies which are fire, wind, water and earth are the fundumental elements of all created substance under the heaven, and all that are human or from animals, birds, insects, fish and plants, metals and precious rocks, pearls and building rocks, mountains and clumps of mud, everything consists of these four elements combined. And thus all bodies under the heaven are built of substance and shape, and their substance consists of a combination of these four elements, but each of the elements is only consists of an elementary substance and shape (meaning: appearance / phase).
Here is the same in the scientific writings of its time in Plato's words:
The world's body is composed of fire (for visibility) and earth (for tangibility), but these so-called elements require the mediation of air and water in a progression of proportion to bind them together into a unified, concordant whole. The shape of the universe's body and the characteristics it possesses or lacks are all explained in terms of their various purposes. The composition of the world's soul out of a harmonically proportionate series of portions of a mixture of both divisible and indivisible Sameness, Difference and Being, and the division of these portions into two intersecting circles (of the Same and of the Different) explain the cognitive powers of the soul in relation to the different types of objects of cognition: those that are and those that become.