The next day Peter started out with them, and some of the believers from Joppa went along. 24 The following day he arrived in Caesarea. Cornelius was expecting them and had called together his relatives and close friends. 25 As Peter entered the house, Cornelius met him and fell at his feet in reverence. 26 But Peter made him get up. “Stand up,” he said, “I am only a man myself.”27 While talking with him, Peter went inside and found a large gathering of people. 28 He said to them: “You are well aware that it is against our law for a Jew to associate with or visit a Gentile. But God has shown me that I should not call anyone impure or unclean. 29 So when I was sent for, I came without raising any objection. May I ask why you sent for me?”30 Cornelius answered: “Three days ago I was in my house praying at this hour, at three in the afternoon. Suddenly a man in shining clothes stood before me 31 and said, ‘Cornelius, God has heard your prayer and remembered your gifts to the poor. 32 Send to Joppa for Simon who is called Peter. He is a guest in the home of Simon the tanner, who lives by the sea.’ 33 So I sent for you immediately, and it was good of you to come. Now we are all here in the presence of God to listen to everything the Lord has commanded you to tell us.” Acts 10:23-33
What's the deal with clean and unclean? In this passage we see both clean and unclean foods and clean and unclean people and both of these need to be addressed in order that the gospel can go out to all mankind.
First of all, why unclean food? There are several reasons but first in a book called None of these Diseases, the author quotes the verse in Exodus where God tells the Jews that if they obey His law they will have none of those diseases that the Egyptians have. So, health wise, unclean food generally was unhealthy food from pork with trichinosis to roadkill carrying bacteria to bottom feeding fish eating refuse, to blood with viruses, God was protecting them from diseases. Secondly, and this refers to clean and unclean people also, God wanted to use something that would inherently separate the Jews from other nations and keep them a pure race so that millennia later after they have been dispersed, He could bring them back in 1948 and they would still be a people group distinct and not absorbed into the rest of civilization. I mean, how can you go out on a date when you eat different food? Lastly, just like menses, mildew, death, discharges, etc., God is reminding them that we are in an unclean world because of sin causing death and dying. Having unclean food in your mind on a daily basis reminded them of sin and their need for a sacrifice to make things clean. Things were not how they were meant to be in this world and they were in need of a "cleanser".
Second, why unclean people? Besides the need for a way to keep Jews distinct so that God could bring them back in the last days showing He's not done with them, the second reason is that darkness and light cannot coexist. I am made acutely aware of this even this morning as I had breakfast with the residents - they have a completely different worldview than me. You can see it happening with youth group kids going to college. Unless they have a very strong family of believers that are their core, they will start being and thinking like the godless peer group around them who live for gratifying the flesh. This is why Paul urges the Corinthians not to be unequally "yoked" and to come out from among them and be separate. Soon, God knew that if allowed to associate with the "unclean", they would forget Him and go after other gods. Lastly it reminds us that everyone is not all right. We are not all by birth and common humanity children of God. The world is divided into clean (forgiven righteous justified sanctified followers of Christ) and unclean (basically everyone else). It's our job to make as many clean as possible, not us but the Holy Spirit who lives in us.
Getting back to the passage, Cornelius, who by looking at him with human eyes appeared clean, wasn't. The uncleanliness of his "gentileness" pointed to a deeper need which far surpassed his ceremonial uncleanness. He had an unclean inside which needed to be cleansed by the blood of Jesus Christ. Jesus fulfilled the law by cleaning "the inside of the cup versus the Pharisees being more concerned with the outside. From this point on, the ceremonial uncleanness of food and people was superseded by the uncleanness of the heart that we as believers should be focused on.
So, mix all you want for one reason only - not to become like them but to offer the internal cleansing of the blood of Jesus Christ.