If it is so wrong to cook meat and milk together, then what about cooking meat with milk-based products like yoghurt, cream, cheese, butter, etc?
I am not familiar with Judaism.
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All dairy products are forbidden with meat products. The law is that three aspects are forbidden.
Meat and milk products (including butter and cheese) may not be cooked together
Meat and milk products may not be eaten together
Mixed milk products and meat may not have any benefit (such as selling to a non-Jew)
I have shown two articles that discuss this along with a quick statement of the basic law. You can read the remainder of the articles which are too long to post here.
Mixtures of milk and meat (Hebrew: בשר בחלב, basar bechalav, literally "meat in milk") are forbidden according to Jewish law. This dietary law, basic to kashrut, is based on two verses in the Book of Exodus, which forbid "boiling a (kid) goat in its mother's milk" and a third repetition of this prohibition in Deuteronomy.
Judaism as we know it forbids Jews 1) to cook meat and milk together, 2) to eat meat that was cooked with milk, 3) to receive any benefit from such a concoction. Yet, there is no mention of these prohibitions in the Torah. Where did this prohibition come from? The Rabbis derive this prohibition from the phrase לא תבשל גדי בחלב אמו generally translated as “do not cook a kid in its mother’s milk,” which appears a total of 3 times in the Torah (Exodus 23:19, & 34:26 and Deuteronomy 14:21). But to be sure, there is little in the verse that would suggest anything beyond literally cooking a kid in its mother’s milk.
I don't know if it's allowed on this site but I think the OP needs an answer from an outsider.
If this is inappropriate here, please feel free to delete it. If any of it is factually incorrect, please correct it. If any of it is inadvertently offensive, please let me know. (In particular, Turk Hill's comment inspired this; I'm definitely not ridiculing it.)
An answer that is blatantly obvious to an insider (e.g. "Because cooking meat and milk together is not kosher.") sounds like it is begging the question and leaves an outsider none the wiser.
What seems like a simple question can sometimes require a great deal of background information to answer properly, sometimes even several thousand years of history. This is one of those questions.
Being Irish (and west Ireland too), you were almost certainly raised Catholic and are familiar with the term "Pharisee", though you might not know its real meaning:
The Pharisees were a social movement and a school of thought in the Holy Land during the time of Second Temple Judaism. After the destruction of the Second Temple in 70 CE, Pharisaic beliefs became the foundational, liturgical and ritualistic basis for Rabbinic Judaism. — Pharisees - Wikipedia
The Pharisees were not members of the priesthood, but were a social group with a strong belief in holy scripture and a strong feeling of what was right and wrong. They were respected by the general population as educated and moral people.
The Pharisees saw a pattern in the history of Israel, which had been chosen to present a moral example to the world ("the chosen people"). Whenever the nation's king (and as a result the people) followed the Lord's commandments and laws, the country prospered, easily defeating enemy attacks and living in peace and prosperity. Whenever the nation's king (and as a result the people) followed the ways of their pagan neighbours, the country eventually suffered, at times even being taken away in captivity as slaves.
They realized that the path to peace and prosperity, and to fulfill their purpose, was to ensure that the nation always behaved correctly.
But almost all rules can have fuzzy boundaries. Perhaps someone is following the rule but it looks like they aren't. And sometimes one can break a rule while not appearing to do so. Similarly one can follow the spirit of the law but not the letter, and vice versa.
So the Pharisees developed an expanded set of rules, which, if followed, would guarantee without any doubt that one was also following the Lord's rules. These rules were exaggerated, were overkill, but they accomplished their purpose.
Secular society does a similar thing today with many of its laws. The purpose of a traffic stop-sign is to inform those approaching it that cross traffic has right of way over them, and that they should wait until there is an appropriate break in traffic to allow them to safely cross or turn. But the law requires that each vehicle come to a full stop before proceeding; that the wheels can be seen to stop rotating. By the spirit of the law, there is often no need to actually stop, as the intersection can be clearly seen and there is no traffic in the area, but the letter of the law still requires that the vehicle's wheels stop rotating.
A similar but more extreme situation is with a traffic light. Many times a driver will sit waiting for what seems like a very long time, until a red light eventually changes to green, even though there would be no physical danger in simply driving through the empty intersection. But if they don't wait, the driver could receive a ticket for breaking the law even though what they did was perfectly safe and reasonable.
The Pharisees became what could be considered morality police, encouraging citizens to follow their rules to the letter, thereby ensuring that the underlying sacred laws would also, without the slightest doubt, be followed.
This system, with the priests proclaiming the sacred laws, and the Pharisees instructing people how to live their lives, worked well, with most people going along with it and believing it to be the right thing to do.
But, seeing the Jews as a potential threat, in the year 70CE the Roman military destroyed the great Temple and the priesthood, and deported most of the citizens throughout their empire (Jewish diaspora - Wikipedia).
Wherever the people resettled, they formed small communities to preserve their culture. The priesthood no longer existed, at least not in any form resembling what it had been. But there were still many Pharisees, with great historical and religious knowledge, who became the new religious leaders.
They were no longer known as Pharisees, simply retaining the title "rabbi", which meant "teacher".
Over the following centuries, the rabbis educated their people, and rabbinical schools were formed to train and educate the next generation of rabbis and to further study, understand, and develop the rules that governed Jewish life.
With that background, the original question can be answered.
In ancient Israel, one of the surrounding pagan nations practiced a fertility ritual:
It was a custom of the ancient heathens, when they had gathered in all their fruits, to take a kid and boil it in the milk of its dam; and then, in a magical way, to go about and besprinkle with it all their trees and fields, gardens and orchards; thinking by these means to make them fruitful, that they might bring forth more abundantly in the following year. — Clarke's Commentary
The Lord explicitly instructed the Israelites not to follow this practice: "Thou shalt not seethe a kid in his mother's milk.".
Judaism interprets this as a specific case of a more general biblical law forbidding cooking mammalian meat and milk together. Rabbinical tradition has extended this restriction to other similar meat, such as chicken, but not to obviously different flesh such as fish.
The Pharisees and later the Rabbis needed to ensure that no one would do this, so they developed what today forms the dairy/meat rules of kosher food. It's unarguably overkill, but if one can guarantee that no dairy product (including butter, cheese, etc.) and no meat product (even chicken) are ever cooked or even eaten together at the same meal, one can guarantee that the original commandment is not being broken.
There's no chance that someone could be cooking fish with milk and accidentally include a piece of goat meat. There's no chance that someone could be cooking goat and accidentally add a milk-based sauce to it.
These, and many other rules, have become an integral part of Jewish culture and religion.
Even knowing the original commandment, to an outsider the practice makes little sense; but to an insider "Because cooking meat and milk together is not kosher" makes perfect sense.