My name is Brad Goulter. I'm the owner of Longmont Flores. The business has been in my family for over 50 years, downtown Longmont. When you can't visit personally, you can always send flowers. We started doing no-contact deliveries where we ring the bell, we set the flowers down, step away and then call the recipient. And we're in the business of delivering smiles daily.


That's the perfect thing to do. I've seen classes doing Google. Google? Yeah. That's pretty good. Okay, if anybody's asked me on live stream, do we actually have it or so here? Yeah. Okay. Alright, we're going to get started. Anybody comes in? Well, that's just fine. That's just fine. Somebody must have hinted that we're going to be talking about some strange things tonight. And I'll grant you up front. Yeah, this is going to be a little unusual. And you're going to get four weeks of that because what we're going to talk about is going to be pretty foreign to the way we think in a number of ways. So the purpose of doing this is just, like it says, to help you try to think like an Israelite. This is going to be topical. So tonight we're going to talk about impurity and sin, you know, clean and unclean. And what I'm going to do is suggest to you or kind of toss out sort of where the state of best understanding is. This is a lot of strange stuff. We will have to think pretty abstractly to try to follow why in the world some of these, you know, laws and concepts are what they are in the Torah, in the Old Testament. And scholars have really kind of struggled to do that. So I'm going to present sort of what, again, what the consensus is currently as to what all this stuff means and why it is what it is. Next week we'll talk about sacred space, which is going to tie in this week. The week after that, we'll get into the actual sacrificial system which will tie into the previous two weeks. And then the last week is a bit of mop up and a bit sort of an appendix to the other three things. But if you are able to sort of grasp why, again, these concepts are what they are, they would actually help you through a lot of the Torah. Because a lot of the Torah is laws, it's sacrifices, it's rituals. And it's just kind of bizarre stuff. It carries through all the way to when you get a temple, tabernacle in a temple. And it will trickle into the New Testament in some kind of interesting ways. Again, if you can sort of understand and follow conceptually how it is they're thinking.  Now for tonight, when we talk about impurity and sin, we're talking about cleanness and uncleanness. So we're going to start with impurity. This is, if you've read through Torah, the first five books of the Old Testament, this is vocabulary that's going to be familiar. Don't do this because if you do that you'll be clean or unclean in all these regulations and what. There are actually two categories to an Israelite thinker. Someone who, again, would be a native experiencer of some of these things when you're living during a tabernacle period, where you're living during the time of the temples. And the only thing in biblical days is actually two parts to this. Ritual and moral. You can become ritually unclean. Ritual has to do with what you as either a person, just an average Israelite, or a priest, what you can do in certain places and what you can't do in certain places, where you're allowed to go and not go in the camp in relation to tabernacle and the temple. So ritual uncleanness is really about your participation, your eligibility or your ineligibility to do sacrificial stuff.  You know, you have to wait x number of days before you can participate in this or that festival. All of that is about ritual impurity, ritual uncleanness. Fundamentally it has nothing to do with a moral violation. There's no moral guilt. Having said that, there are a couple passages where if you do a moral transgression, lucky you, you're not put to death, but you may not be able to participate in the community. So there's a little bit of overlap, but for the most part, ritual uncleanness is about participation in the whole system, in the community life. Whether that occurs on sacred space, and by sacred space I mean space that only God is supposed to occupy or priests that are like God's substitutes. They're allowed to occupy that space or not. So again, some of these concepts we've already seen are related, sacred space, and the scholars will use terms like holy and common, holy and profane. Holy is that which is connected with God, his space, what he does, objects that are used in his space, and common or profane is stuff that everybody's doing. It doesn't matter. It's not connected with a specific piece of turf. Ritual impurity is about physical defilement that bars one from sacred space or participation. I'll mention that. Causes and examples. I have a quote here from Jonathan Clowans, who's sort of a specialist in ritual, and he's done some really good stuff on it. I really like his material. His line here is direct or indirect contact with any one of the number of natural processes and substances. The key word here is natural. He rendered ritually unclean by just living normal life. It's unavoidable. Specifics. Childbirth renders of a woman ritually impure, ritually unclean. A skin or scale disease. Genital discharges. Whether that is ill health or something normal doesn't matter. You are ritually impure. It's the darkesses of certain animals. If you touch them, you are ritually impure, unclean. Human corpses, you don't touch a corpse. Even if you have to bury your dead.  You're ritually unclean. And we're going to talk about what's the logic to some of this stuff in a moment. But again, you get the idea of how long your impure carries by whatever category it is, whatever circumstance it is, and how you are cleaning and cleansing from impurity means what are the procedures you do so that after a certain period of time you can participate in the community religiously again.  It's about are you fit to come anywhere close to the presence. It's the presence of God in the Tabernacle Temple System. Ritual impurity is natural. It's unavoidable. Hey, guess what? You're going to have babies. You're going to have general discharges. You're going to get skin diseases. Hey, if you like to eat meat, chances are you're going to touch a carcass when you should. Human corpses. People die. You've got to bear it. You just can't avoid these things. There's no moral fault. The system is going to be designed to teach certain things. So it's unavoidable for humans in human circumstances. It usually doesn't arise from moral violation or sin. I mentioned that because if you read through Leviticus, they cure the ritual procedure for taking care of ritual uncleanness. It'll say things like take this bird, take this animal, do this and that, and you shall be cleansed from your sin. Okay, we have to realize that we have a translation problem. The Hebrew word for sin and sin often katat in many circumstances has nothing to do with moral violation. It actually refers to being decontaminate and it's about doing something to the person to decontaminate them so that they don't contaminate sacred space. Okay, so many times in uthora when you have this sin or sin offering language it isn't connected at all to moral violation. And although one step further to the end of the sacrificial system in a couple weeks you may not have noticed because I know we just read all the Leviticus that we can. When can I go back to Leviticus and read that again? The blood of a sacrifice is never applied to the person who brings the sacrifice evermore.  It never applied to a person. And even when it is applied to any person there's only two occasions for that. One of that was the ceremony to launch the priesthood system. They were just like okay, well let's like hit the reset button everybody's like clean then we can get started with this new priesthood thing that God sort of just gave us here at Sinai. It had nothing to do with moral violation. We tend to read words like sin and sin offering in line with the New Testament. No, no, no, no. Don't do that. If you do that it's not like you're going to commit some horrible heretical offense.  You can't read the Old Testament sacrificial system through the lens of New Testament talk about the blood of Jesus. That's one of the primary problems that we have. It's easy to do that because we're not Israelites. It is true that the refusal to purify yourself would constitute a transgression, a moral transgression. Let's say I'm the Israelite, I'm the originally impure and I don't want to do this. That's a heart problem. If you did that, then you got a moral problem going on. As would coming into contact with the sacred ball in the state of impurity. It's like okay, okay God. I stepped where I shouldn't do something. It's a heart problem. If you deal with those circumstances well there, then you're going to have a moral issue. But the whole system doesn't make being ritually impure sinful in itself. Moral impurity on the other hand, we talked about ritual impurity now, the moral side. There's this other category of moral impurity and some of these are kind of obvious. You can be morally impure sexual transgressions, various laws about sexuality, violating one of those idolatry, kind of elementary. Bloodshed, murder, something like that. These things we typically think of as big crimes. Those are moral violations very clearly and there is a category of moral uncleanness that operates separate from ritual uncleanness. Sometimes they overlap just a little bit but typically they're separate grounds. Now these are frequently described as abominations. It's not the only term that's used but if something's an abomination it's very clearly a moral offense. And again there's no sacrificial cleansing for a lot of them in the Torah. We'll get to that in a bit.  The land is defiled. In other words where these things are tolerated the land itself God will look at as though it is in a state of defilement itself it's just dirt. And then God's sanctuary this reference in Leviticus 20 is specifically about child sacrifice. And it says something I'm paraphrasing now but it says if you do this if you allow this to happen then God says my sanctuary will be defiled. What God means is I'm just not I'm not even going to visit you but my sanctuary itself is like polluted by stuff you're doing in the land if it's this bad. I'm out of here. This land, the whole land is no longer fit for my occupation. The land itself is defiled and the place that I told you to build so that I could visit you and commune with you and so on and so forth that's been made dirty by all of this and I'm just not going to be there. So the moral defilement was pretty serious and even extended to these other things. Corporately Leviticus 18 actually spends a lot of time talking about this and Leviticus of 26th as well that if there's enough of this going on God said I will expel you from the land. In other words, I'm not going to show up here because the whole country the whole land is polluted so the solution to that is to kick you out and if you're gone long enough then maybe I'll come back or maybe it will be fit for my occupation so there are all sorts of warnings about not persisting in this kind of thing because you'll be expelled of course when the exile happens the prophets go and they quote passages like this and say like duh didn't you people realize what was going to happen to you if you did this, that and the other thing here's kind of a summary you've got ritual and moral impurity source and natural things the effect is temporary defilement temporary impurity and the it's contagious ritual impurity is the one that if I'm ritually impure I touch a dead body I'm in a state of ritual impureness if I touch you the impurity's just got passed on you're ritually impure now that isn't the case if I'm just I go out and kill somebody and I can touch you all I want you're not rendered unclean by now but if it's ritual impurity you are again it's a totally different kind of system the effect of the moral is frankly worse desecration of a person a land, a sanctuary, a whole book and then the resolution there are different rituals that involve bathing and then waiting periods here, clowns lists atonement or punishment and ultimately exile by atonement he's referring to restitution which we'll get to in another week but moral defilement defiles the center land and sanctuary not anyone with which he comes into contact ritual impurity is like a contagion, moral impurity is not there's nothing if you commit adultery commit murder I didn't go in a way by taking a bath okay there's a different set of penalties for that that's not going to work the solution for that kind of impurity is punishment you either are put to death or you're exiled from the community the word for that in Torah is the person shall be cut off from his people and scholars fight over well is that also the death penalty or is that just you're kicked out of the community and you'll never come back it's probably the latter but you'll see that some crimes, some moral crimes involve restitution if I steal again that ain't going away with a little water bath and a waiting period what I have to do is I do have to bring a sacrifice so I can be purged I can be decontaminated from what I've done but I also have to pay restitution to the person who suffered loss basically we're going to overlap between the ritual and the moral there and then abstinence I mean you can cure this by just not doing that stuff which of course is what the leadership would want by way of summary since moral impurity does not produce ritual defilement sinners in contrast to those who are ritual impure are not excluded from the sanctuary isn't that odd so if I'm a murderer if I'm still alive if I went on a holy ground I didn't defile because I'm not my problem as a ritual impurity it's moral impurity so you actually had like murderers run into the sanctuary there are a couple of accounts in this in the old testament they didn't defile the sanctuary because the Torah said that their moral crime doesn't defile ritually it's just a moral problem and they were trying to they're not going to get me in here because that would be shedding of blood of course that didn't work out too well for some of the people who tried it but that was the thinking Numbers 5 the so-called test for adultery they bring the woman into the sanctuary well if they assumed and some of them were thinking why would they bring her into the sanctuary well the answer is because that's not going to pollute the sanctuary it's a moral problem, it's not a ritual problem so again this seems kind of odd to us but they have these two categories and they know what to plunk in each category on those occasions where moral impurity does defile the sanctuary like some of that stuff we talked about it's not because a sinner is brought into the sanctuary it's because of what happens in there the whole like case would be idolatry so that's more of an issue now there's a long quote and I want to read it to you because it's kind of interesting Clowns again he says we're dealing with perceived effects that result from actual physical processes in the case of ritual impurity a real physical process or event such as somebody dying or menstruation loss of blood or whatever has a perceived effect it's an impermanent contagion that affects people in certain objects within their reach in the case of moral impurity a real physical process or event such as child sacrifice or adultery has a different perceived effect a non-contagious defilement that affects persons the land and the sanctuary in both cases the impurity can be conveyed ritual impurities conveyed by direct human contact moral impurity is conveyed to the land by sins that take place on it and through the land the impurity is conveyed to the sanctuary that's why God has to leave in both cases moreover there are practical legal ramifications that result from the impurity the ritual impure person must keep away from sacred things and in some cases they're barred from certain precepts sacred space the morally impure person may be subject to capital punishment or in the case of unwitting female partners to sexual misconduct permanent degradation and fewer options for marriage where the innocent party wouldn't be under the death penalty but again in the system there'd be consequences to that when the land has been defiled to a great extent people are ends up so that's kind of a summary of the two categories now here's one you kind of want to count the logic to all of this why do you have the two categories why why in the one flight let's just take one of them the ritual impure why do some of those things why were they considered to contaminate someone to make them ineligible to be near God or occupy sacred space what's the point bodily flows the conduct of the dead child what's the point now I'm going to give you I think it's either four or five these are sort of the best the best attempts at consensus there's no one approach that really explains everything in the Torah in these two categories taken collectively you can pretty much account for everything there are a few outliers and scholars are typically like we got like 98% of this sort of explain there's this other thing over here you're still up against that there are some broad principles that really sort of capture the logic all of them are going to have to do with binary opposition in other words they are what they are to teach binary opposition between us and God there's a clear differentiation life is associated with God death is not blood, it's seen in skin diseases if we take life versus death this is a broad category if we look at the bodily fluids that render someone unfit for sacred space what do blood and it's seen in common they are associated with life if you lose enough blood you're going to die so they don't have to be medical scholars to know that hey that red stuff that comes out of your body you got to keep that in your body or you're going to die so the loss of blood is sort of like moving toward death and to teach the idea that God is associated with life death is not associated with God the loss of blood renders you unfit for sacred space and that is supposed to reinforce that lesson wholeness having all your blood that is associated with life and life is associated with God so if I lose blood then I have to do this ritual I got to wait a certain number of days nobody's wanting there's no Israelite guys going around taking your blood care we're not talking about any technological it's just to reinforce an idea if you have demographic you have a seminal mission at night or you have sex as a man you have lost the life giving fluid that thing which produces life that makes you ritually impure for a while not so that somebody the Israelite doctor can take a sperm count later on with his fancy medical instruments that has nothing to do with it the idea is that you need to realize that any loss of life any loss of the life force cannot be associated with the presence of God because God is life it's a simple object lesson sounds kind of weird to our ear but you know you do that often enough people are going to get the point this is associated with life life is associated with God if I lose the life force then I am not fit to be on sacred space so you have a broad sort of categorization skin diseases there's a passage in Leviticus that describes skin disease as that the person is their skin is reminiscent of a corpse okay and so the association the visual association of that skin disease with a dead body is enough to make it a teaching point you have a skin disease it looks like you're deteriorating deterioration death we can't have that on sacred space because God is not associated with deterioration of death so we have to take care of that different procedures for doing that it's numbers 1212 let her not be like a corpse this is when Miriam gets punished with leprosy let her not be like a corpse it was associated with she's decaying dying they hurt unclean another idea is imitation of God's nature so one rational is binary opposition of life and death that explains some of the ideas number two idea is if the person who undergoes them is sort of viewed as imitating God in some way that sounds really weird this one's actually kind of interesting sex is one of the places that scholars camp on for this idea why does sex make us unfit for sacred space the answer is because it's being unlike God does that mean sex is bad oh God made it and said be fruitful multiply it's unlike God because God is eternal he doesn't need to reproduce he has no beginning he has no end because God has no consort he has no wife God cannot have sex therefore Fran Mikenski she's passed away now but she was a scholar of ritual studies in order to approach God one has to leave the sexual realm by separating sex from death is by following the ritual purity regulations ancient Israelites and especially ancient Israelite priests and Levites separated themselves from what made them the least God-like in other words the point of following these regulations is nothing other than the theological underpinning of the entire holiness curve the imitation of God so knowledgeable to enter the sanctuary if you are in a God-like state you have sex you disqualify it for two reasons you've lost the life force and you're not imitating God because God doesn't need to reproduce so all the correct sexual behaviors even those will render you ritually impure because of this idea God does not he does not need to be produced in his presence as life God does not have sex because he is eternal so if you want to serve as a priest you can't have sex for X number of days before you job out of auditions if you do then you can't do that because to enter sacred space and perform your duty you have to be like a divine proxy you're a substitute for God's presence you were selected in this case because you're from a certain tribe you were selected to be a stand-in to occupy sacred space to do certain things in the presence of God and to do that you have to be God-like so again it sounds really weird to our ear think about how I'm sure Dax is like the way we go through the rest of the list because it's like do I really have to jump through all these hoops just to have church thankfully now but if you're in the Israelite system there are all these ways of circumscribing your behavior good behavior, normal behavior because by doing that you teach people what God is like this space over here is different from the space I'm talking about God's presence is over here it's not here, I'm okay over here but to go over there I have to be in special circumstance in special condition in a special person from a specific tribe I can't have done this, that, and the other thing for X number of days is to distinguish between who God is and who you are and you do that by rendering certain places and objects, sacred you don't touch them you don't even go near them and even if you're allowed to go near them let's go down the checklist and see if you're eligible and you have all these hoops you have to jump through third category, some have argued and let me give you an example again it's kind of gross or weird but it's still part of Torah that there's an issue of controllable versus uncontrollable Collins is quoting another scholar here and the issue, believe it or not is, I'll just start it's precisely with regard to this point that the work of Albert Schwartz is important noting the conundrum that the routine seminal discharge is less problematized than general discharges of blood everybody is 15 or flux also 15 Albert Schwartz introduces the criteria of controllability in the discussion of spiritual purity but not denying the importance of the notions of sex and death Albert Schwartz suggests that controllability may play a role in determining which substance is defiled and how severely defiled briefly stated he argues that the less a process for an event can be controlled the more likely it is to defile and he uses the example of the expert who there are actually laws about in the Torah and they can't really be accounted for in the other two the life versus death thing although some would argue that what you excrete is the opposite of life because your body doesn't use it so you might be able to get some of that in there but tears and saliva why is that okay? because their excretion is subject to human control and these substances were not considered defiling in ancient Israel on the other hand the most defiled substances since they are the least subject to human control so again when this third category we're trying to pick up like what's the logic with tears and saliva and poop why is this thing that I discharged from my body that makes me appear that this other one doesn't they're trying to figure out a holistic approach to this so third option is controllable versus uncontrollable but here's the one we often think of food or dietary laws and this you can go part of the way with some of the things we've already discussed let's just take the life and death thing it's very clear that animals that eat dead animals carry in the use those will render you unclean to your infills so there's this life-death connection you can't really get this big god-like connection or control and it doesn't really work and then there's some oddities then we're not going to go through the whole list but we've got flying animals eating unclean insects linen animals water animals there's no unifying principle that explains why everything is on either side and so this one is kind of problematic I'm going to park up which animals here traditional explanations have been well this is all about hygiene again there are problems there are some clear instances where yeah it might be about hygiene because this animal is kind of pardon my pigish you know like the pigs always the easy target here but there are other animals well they're not white pigs and they're not unusually dirty you know why are they excluded so there's a bit of a disconnect there some say well the ones that are forbidden have some association with cults like religious cults in Israel at the time that other genitals are doing there are some clear ones for that but then there are other ones that well hey we offer that animal too and sort of the canids what's up with that they're clean for us since they're killing all the goats all the time and why aren't goats unclean because they do it there just doesn't seem to be a unifying logic moral and ethical again some people try to sort of interject or retroject you know like environmentalism certain animals are off limits because you have to control their populations and it's good for the ecosystem they're always exceptions to every approach and so that's why it's a problem Mary Douglas is a famous ritual scholar if you actually do any academic you know if she will run into her name she's pretty important not everybody agrees with her but she's still pretty important she writes the pattern she thinks it's about patterns of opposition especially inclusion and exclusion seeing the social structures of a society that's a cult that's its ritual system it's view of the body and its classification of animals may be expected to correspond to and conform one another so she's saying we might just have these oppositions because they might derive kind of unconsciously or just unconscious is not a good word they might just have arisen intuitively in Israelite or ancient patriarchal culture because there are certain laws in the Torah that are also found in other law codes like Hammurabi's law code you're going to get laws that show up there that show up in Torah and that's just one law code there's hundreds of law codes there's overlap between the Israelite law code the Torah and a lot of those and so a lot of people and I would be in this camp would say a lot of what goes on in Torah is that when God prompted when he started working with the Israelites when he prompted the writing of this and that book there's already a culture in place God doesn't have to like invent one and say well this is the Israelite culture now no he just takes people where they are and then once certain laws don't control crime and punishment and whatnot maybe be more tolerant than the Hittites are or something like that that God works with a culture already in place and so Douglas argues that maybe there's these binary oppositions this clean and unclean maybe it's just sort of intuitive maybe again it's not a perfect system either I'll give you one example of there are a lot of people who beat on the Bible because of the Zalofa had it incident everybody know who Zalofa had it she's the woman that has to go to Moses after her husband dies she doesn't have sons and ask if she's allowed to have her husband's property there are no laws permitting female inheritance of land in the Torah why is that because little characters are sexist they're evil sexist pigs look you know again God didn't prompt anybody to write that down it was just part of Israelite culture and Moses says hey you know you're right we're going to let her have that life but there wasn't a Torah law they had to actually go to Moses for an opinion for a ruling because it's not part of Torah it's just part of the wider culture and Moses says give her the life but people love to beat on the Bible for stuff like that but what's the basis again it may just not again it may be just cultural maybe arbitrary maybe arbitrary but it's still intentional because no matter what the logic was that produced the clean and unclean animals once the clean and unclean columns are filled it still teaches the same thing you eat these but you don't eat those well why you know to be honest because this is the way God wants it to be even though we can't figure out exactly why everything is on the list because like this other stuff that has to do with bodily fluids and you know going this place and that place and doing this ritual and that ritual it teaches you that you have to live a certain way to be fit to occupy sacred space to be part of the sacred community which is Israel to be part of God's family you have to conform to what God wants because you're not like God God is different than you God is way better than you who gets to make these rules and he wants to see if you are willing to give up maybe eating something that you really kind of like do you like to try are you willing to give that up to be part of the community maybe it's just that simple maybe there doesn't have to be an overriding logic to it here's another question what about parts of clean animals you actually have this you have parts of clean animals that they still can't eat how does that make any sense well if you look at the parts there are certain instances where you're not allowed to consume blood which is pretty general why aren't the people allowed to consume blood because the life isn't in blood because you're assuming you're intaking something that belongs to God in sacrifice or should be spilled out on sacred space which is God's because God is the author of life it's just again it's just designed to teach an idea to teach an association doesn't have to do with hygiene doesn't have to do with calorie count it's life is life belongs connected to the life giver it is not for you it's for God so you get blood burned up on the altar you're going to be drained in sacred space even when they take the blood they'll sprinkle the tabernacle or different objects what will they do with the rest of it they'll pour it out at the base of the altar it's still on sacred space because life belongs to God doesn't belong to you belongs to God thought you're going to clean animals and probably because it retains a lot of blood too you know you're only allowed to do certain things with it the sadic nerve this is the jake of story when he wrestles with the angel you know why is that forbidden probably because of the jake of story it's associated with divine encounter so to remember the divine encounter we don't need that part it's just a simple idea carrying that has to do with life and death opposition that was pretty easy the fat and the blood again are a little odd because you have to think conceptually of what is reserved for God's consumption and what is reserved for human consumption sometimes you're allowed to if you're a priest again you're the God proxy if you're a priest you're allowed to consume certain parts of the fat and eat in the sacrificial and you know other times you're not if you're not a priest Lincoln just pretty much forget that other examples we're the famous or infamous kid boiled this mother's milk why is that some some have proposed it's a humanitarian law you know we shouldn't it's cruel to take a baby goat and kill it and then boil it in its own mother's milk let's just some will say that it's a humanitarian law some say it's associated with a cult specific cult practice some people trying to tie that into some ritual text to regard it doesn't work real well but people try it there is some evidence though in iconography which is sculpture pictures aren't holaroids of the day to call them where you have this motif a mother animal with a suckling calf that appears on cult objects so there may be something it may have something to do with gentile religious practices the real answer is nobody knows for sure why it is but these are the approaches now that's what I wanted to cover next this time we can take some questions but all this is tied to sacred space why certain areas are what they are and objects are what they are now just to prep a few things for next time this is going to take us into the concept of holiness have you ever wondered why inanimate objects are called holy this is why you only use that object to do this thing at that place otherwise you are not allowed to use it and you say well who cares why is that there see we have lost I'm not on the way we do church in the modern era but if you've ever been in a really really liturgical church they still have some sense of this you don't unless you're like ordained as a priest unless you're wearing a funny hat and lots of robes just go to the front of the church and you don't go through the little doors where the priest blesses the community and stuff you don't go there in fact the church is designed with a little gate there deliberately it's not just hey we watched an HGTV show we kind of like that feature no it's there so that when you see a gate it's like oh I'm not supposed to go through the gate because beyond that the priest that's where he does his thing it's to remind you that certain places in the sanctuary interesting term are off limits why? because that guy thinks he's holier than I am no it's just to enforce the idea that we come here to do God stuff and that's the guy who does the God stuff because he's a minister, he's a priest whatever the title is in Israel they're just the whole nine yards doing that you were not supposed to tread on certain places you were not supposed to touch much less use certain objects we only use that cup one day a year or every this or that ritual and only the priest can touch that cup why? because the priest thinks he's better than I am no it's to reinforce the idea that in life there are sacred things and non-sacred things God is different you want to treat him like he is different where God lives should be treated like it's significant it's not normal because God isn't normal it's not common because God isn't common these are simple ideas and in the case of Torah it's just all over the place but since we don't grow up with temple in evangelicalism we have maybe we have communion, we have Christmas and Easter but our religious calendar is abbreviated compared to a ritualistic system there's really Sabbath you've got stuff to do in Sabbath you've got collections of Sabbaths cycles of Sabbaths then you've got Jubilees you've got this festival, that festival everything is in order and some of the priests we'll talk about this in the fourth week some of them were like I want to say nutcases but that's harsh some of them were just they were willing to die for what day we hold this thing on and there's a reason for that it's a set of ideas that they attached to keeping the calendar why it was so important we've lost pretty much all sense of that and part of it is because of the New Testament where is sacred space now you're looking to look around this is why I'm not saying it's a bad thing to be liturgical in some ways when we were in Wisconsin we went to a church that had an order of service I don't remember what the terminology was for order of service it wasn't terribly liturgical or ritual but they had enough of it that it made you kind of stop and think of where you were at that was kind of nice but the reality is it's okay not to do that because the building isn't where sacred space is unless a bunch of us are there because we are the temple we are sacred space the whole system changes so after centuries of embodying sacred space who are the priests now that would be the same thing the same answer that priesthood and the believer is a concept in the New Testament every believer is a priest every believer is sacred space so theologically these things have changed and so it's really tough for us to look back in the Old Testament we didn't have any of that they had to do this so that they could draw near to the throne of grace you don't just assume as an Israeli that you're okay to go worship God well God knows my heart God knows that I touched that carcass the other day and I was supposed to bring this he'll let me off the hook because I'm doing the right thing actually he won't we assume certain liberties because of Jesus this is why the language in Hebrews mimics this but it substitutes Jesus all the time and it applies the rules to every believer everywhere it's not linked to a plot of ground anymore Hebrews 9 the blood of bulls and gums will not take away sin if you do this your sin is forgiven the Bible is a contradiction another dumb argument from atheists no it's actually true all these rituals weren't about sharing moral moral violation if you committed a moral offense on abomination there's no sacrifice for you you're either put to death or exiled from the community so the writer of Hebrews actually goes when he's talking about the blood of bulls and goats doesn't take care of these problems which is the whole point that Jesus actually does take care of these problems that's why it's better that's why the writer of Hebrews says same, if you want to go back to that do you even understand what you're talking about oh I'd rather be exiled or put to death if I happened to make one of these booties I mean you're nuts that's in the Greek somewhere you're nuts there's a point for the logic in the superior argument system so anybody have any questions we'll get into some of the tabernacle stuff official system it's just strange it's just a totally different way of looking at things first we're going to be in 7.5 talking about sex Paul says do not deprive one another except perhaps by agreement for a limited time that you may devote yourself to prayer but then come together again these sort of ideas that he's look at Paul he's a Pharisee he obviously knows the law pretty well and it's this idea that that you speak in general it's Corinthians so it's including Gentiles here too but it's the idea that you set this aside for a spiritual purpose voluntarily you have to agree to do that but the logic is the same in this case we're not going to do X, Y, or Z so that we can focus over here now the effect since we don't have temples and stuff like this in the New Testament era it's not going to make you any more of a God proxy than you already are if you have the spirit in you but the idea of getting that spiritual tunnel vision for a while is what he's shooting I kind of looked back during that time that sex is far more recreational it's what they had to do as well they would go home and drop on Netflix or something like that because you make that same application and they have larger families too because it was just the way to cause it so would you say it's the same thing I mean should we then at times look to sacrifice those things that we might devote ourselves to wholeier things I think fasting using that as an example fasting not just not eating that gets out of it and you can fast for almost everything yeah do you think given this framework of understanding sort of the two categories if you will of violating sacred space or of moral purity but also of ritual purity how does that apply to us as believers who are sacred space like are there ways that we ritually can be unclean well what's the language Paul repeat but it's not just Paul but he repeatedly uses believers you are same but you also are the temple we're referred to as temple stones we're referred to as temples consistently I'll just fire a little shot here and again this isn't aimed at anybody's eschatology I just want you to think about it more broadly we spend so much time thinking about the temple that's going to be built someday in Israel and we kind of skip the temple talk of the New Testament the temple talk of the New Testament starts with Jesus Jesus says in three days I will raise this temple up and talk to us all these years and John says no he spoke of the temple which is his body now we can like think that's kind of a throwaway scene but the temple is Jesus's body the temple is the body of Christ we are the body of Christ post resurrection all these things loop together they're not random statements that doesn't mean we're not going to have a eschatological temple somewhere it's just we're so fixated on that language that we miss New Testament temple language all over the place now if we have time on the last week there's just some crazy stuff going on like in Ezekiel 40-48 everybody looks at that passage and thinks only of a building in the future there are almost 70 references the numbers they used in Ezekiel 40-48 there are almost 70 numerals there that are multiples of jubilee numbers what's the passage Jesus quotes when he begins his ministry in Nazareth Isaiah 61 go read the first few verses and then the language about healing the broken heart and all this kind of stuff look at where that occurs in the Torah it's a jubilee passage if you plot out the chronology of the life of Jesus the year that he begins his ministry is a jubilee year he walks into the synagogue of Nazareth in a jubilee year he quotes a jubilee passage that says the scripture is fulfilled and you're hearing he wasn't kidding he's like I've got to come up with a good line it's my first sermon he's not kidding but we just sort of fly right over is Jesus the eschatological temple in 70 jubilee references in Ezekiel 40-48 well he was the temple was his body his body was the temple of Christ and that would be us do all these things fit together isn't that kind of weird no it actually really isn't it's kind of like right there hidden in plain sight but we don't think about these things because we have a hard time thinking abstractly like this both because of who we are we're modern we're nuts and bolts we're scientific we're culture we're just not used to thinking abstractly and I'll use the dirty word here non-literally like the little people were because we're kind of taught when we see a word it's meaning is the first thing that pops into our head literally well it might be there's a lot of that in the Bible it might hook into something that is just way abstract that in Israel and I know what he's talking about there because they just this is more intuitive to them it's not intuitive to us both because of what we are kind of the way we talk about scripture we have these sort of roadblocks that get in the way of reading it through their eyes it's really hard it's hard to do one of the best things you can do I don't want this to become laws for Bible study here one of the best things you can do it's going to sound trivial look up the cross references you know when the printer printed your Bible he didn't just have like white space in the middle I don't know what to put there what do I do I'm going to put verses in there just throw verses in there I don't fill up the space no they're actually there for a reason it's very difficult when the Old Testament is being quoted to do on that that's a printing decision they do that to draw your eye to oh you're quoting the Old Testament I wonder what that is you go look in the cross references you go read that if you do that all the time then it's easier to connect dots it's a habit patterns how do the New Testament writers kind of weird abstract ways but again that's using as a teaching device for Bible study I want stuff there to make use of I don't want to buy them really well I don't need to use those cross references you don't know what that will you don't know what that will none of us do anybody else when you were talking about Sexy Five and I thought back when David went to the priest and he had this holy bread he said the only possibility that the guys have been kept from women for a certain period we shouldn't have to wait for three days so somebody okay that's his link to rationalize but it's a good illustration because the priest is sort of looking for some justification for that it's like okay we'll give you the bread you're not supposed to eat this under rules it kind of washes out that's a good illustration this might be kind of a rabbit trail but the priest is not having sex is that why the Catholic Church why their priests can't marry them aren't supposed to have sex I think if you read deeply enough in Catholic thought the answer that would probably be no because the priests weren't prevented from marrying them they were prevented in Israel Israel like priests could be married and you might have part of the Catholic argument sort of hook into what we're talking about tonight to make their decision of celibacy holier but typically they're going to look at things like they're going to kind of model it after Jesus Jesus wasn't married some of it's going to be a little more in a historical condition but they're going to look for some sort of template to model it like that because the the Old Testament priests could marry but I think it's conceivable that they can say well look this is special we're going to go the end of the degree here and we're not going to marry Jesus wasn't married we're going to be more sanctified in certain times so I think you'd probably run into it but I don't think it's the major part of it Paul or he said except that first pope had a mother in law so that it doesn't mess it up but they'll deny that too Paul's status is a little ambiguous if you're a Catholic and you come down on Paul being unmarried which is a possibility then you'd have some leverage there too Paul may have been divorced he may have been deserted by his wife we just don't really know he just says when he's talking about being unmarried which is right this down the state of being spouses then he gets into that I wish that all men were like you know I was then he goes into some primary reasons for it but the unmarried term there occurs only four times in the Old Testament and so it's first contained seven so the real issue this gets into the divorce and marriage issue too if Agametto the state of being spouseless can we here's the key question for that passage as it relates to divorce and marriage and ostensibly preferably with this question on what exegetical basis do we exclude divorce from the list of conditions that make you spouseless in other words is divorce included in 1 Corinthians 7's references to being unmarried because there isn't depending on what you think there you're going to take later statements like Paul's in a certain light you're going to take right around verse 27 or so when Paul says you know if you lose from a wife or you don't seek to be bad if you're bad don't seek to be loosed nevertheless hey basically if you marry you don't sin so when he says if you marry you don't sin is he talking to divorced people too because he's talking about people who are loosed you know the ultimate answer is well who really knows for sure and the answer to that is nobody I mean how many ways can you be spouseless desertion divorce death it's not really the state of being an unmarried virgin because there's a separate word in the passage for that but you know that we could conceivably be part of the big list how do we know what Paul was thinking of and not thinking of or is he thinking of the whole list or just a couple we just don't know that's why it's tough for sure and if Corinthians was written before Matthew was written it gets even more complicated because then Paul doesn't have Jesus thinking to refer back to it which is why he never quotes Matthew in 1 Corinthians 7 I don't know what you do with that well you do with that what we've always done good and that's our reason that's pretty much where we're at you know who knows anyway well I don't want to keep everybody over so thanks for coming see you next week


We have seen when we were studying the isomorphism theorems for groups that kernels of group homomorphisms were actually equivalent to the notion of a normal subgroup. And the other direction is also true that normal subgroups are the only types of subgroups of a group which can then be, you know, modded out when you take a quotient group of some kind. That is, normal subgroups are the only things you can quotient out to get a well-defined multiplication. Essentially, normal subgroups and kernels are equivalent notions inside of group theory. What is the ring theoretic analog of this? We've now introduced the idea of a ring homomorphism and we saw previously for groups that kernels were always normal subgroups. So what do kernels of ring homomorphisms, what do they form? It is sort of natural to think that from the isomorphism theorems that the kernel is what you want to mod out to get a quotient ring. And that's exactly what you want to do for ring theory, right? You quotient out a kernel of a homomorphism. But what if I don't want to explain it using homomorphism? What if I want to do it entirely intrinsic, internal to the ring itself, without reference to some external ring that we're mapping towards? How do you do that? What is a normal subring, so to speak? Well, the short answer is it's an ideal, but let me give you, before we define what an ideal is, let us explain what conditions must be on the set i so that our mod i makes a well-defined quotient ring, right? Which we'll of course define in a future lecture here. Now, just so you're aware, r, if we forget the multiplication, is an abelian group. Therefore, every subring of r will automatically be an abelian subgroup. And since r is an abelian group itself with respect to addition, every subgroup of r is a normal subgroup. So if since r, with respect to addition, is abelian, if you mod out by anything, you'll get an abelian group. So if we want to make a quotient ring, right, we want this to be well-defined, it's only multiplication that we're really concerned about. We need that multiplication. We want multiplication here to be well-defined. If we can get that, we're as good as gold. But we don't know yet. Sorry about the wrap around there. So let's think about multiplication for a moment. But also, as we mod out by i, as we have this additive structure, we can talk about the additive cosets. Because the symbol r mod i is the collection of all additive cosets of this subring i that you see here on the screen. And so a typical element of r mod i would be something like r plus i. So I want to consider the problem of multiplication.  What would happen with their product? Now, for the moment being, I'm going to consider the Frobenius product. That is, I'm going to take the element-wise product of sets. So recall, we did this with groups, of course. If you have two subsets, c and d of your group, not necessarily subgroups, but there's just two subsets, we can define the product of the subsets, often referred to as the Frobenius product. This is going to be the collection of all possible products, c and d, where c belongs to c and d belongs to d. Like so. This is not just true in groups. If we have any binary operation, if we have any set with a binary operation, we can take subsets of that set and then take the set-wise product, if it's multiplication or sum, if it's addition. We can combine the sets using the binary operation that's in play here. So that's what one means when you talk about r plus s, or r plus i, excuse me. You can treat r like a single ten, and so we're just taking all of the possible sums between something with i and something from r, which of course is just r in that situation. So that makes sense. We can also talk about the product here. Now, when it comes to rings, since the ring multiplication does not form a group, we don't get all the nice cute little properties we had when we talked about Frobenius products with group subsets. So when that's a little bit more carefully, quality is not always available to us. Now, because products of sets are defined element-wise, and elements have, of course, additive associativity, additive commutativity, multiplicative associativity, distributed property, those properties will be inherited on the set products as well, because again, it just works element-wise. So if you take r plus i times s plus i as sets, we have the FOIL method, because the distributed property is going to work here, in which case we'll get r times s, we'll get r times i, we'll get i times s, and we'll get i times i. Now, we have to be careful here. We don't know that our ring multiplication is commutative, so the set r times i is not necessarily the same thing as i times r, and same thing with s i versus i s, right? We're okay when it comes to addition, right? So r plus i is the same thing as i plus r. So with regard to addition, right, we have a normal subgroup that the left and right cosets are the same, but when it comes to multiplication, we can't necessarily commute things around. But with looking at this right here, what we can't say is the following. Since we know that i is a subgroup, the product of any two things from i will belong to i. So we can actually say that i... Let me write that again. I'll just write that as i times i, which we can actually define that to be i squared. This is a subset of i, right? Because if we take two things in i, its product will be an i. So i squared is a subset of i. Now, as we're talking about multiplication, which only has a semi-group structure in general, we can't actually guarantee equality in this situation. For groups, if you take a subgroup h and you square it, you'll get back all of h. That has to do with the inversion axiom, but we don't have that for general semi-groups. So therefore we only get a best in inequality here, that i times i does belong inside of i, but there actually do exist counter examples where i squared is strictly less than i. You can take a look in the lecture notes in the footnote if you want to see a counter example of such a thing. I'm not going to provide it in this video though. So we do get that this belongs to that. Now let's consider the case where s equals zero. If s equals zero, that means s plus i equals just i. Now i is the additive identity. So if I'm making a ring using r mod i, that means i has to be the zero element. We have to identify this with zero, so to speak. So if I take r plus i and I times it by i, if this is a ring, this should equal i. And therefore we need this to belong to i if this is going to be well-defined. All right, so backing up a little bit.  Zero times anything will just be zero and adding zero doesn't make much of a difference. So this does simplify to be looking like this. r plus i times i will just be r i plus i which that should be part of i. If in fact this is a well-defined multiplication, that's what we need right there. And in the other direction, if you set r to be zero, then again you take off these parts and you need that i s plus i is less than i right there. So this is sort of an interesting thing and this is going to then lead to the definition that we're going for. If r i itself was a subset of i, then i plus i would then equal i and we would be good. And if i s was less than i, that is if i times s actually was inside of i, then we also get what we're looking for here. And so that's what leads to the definition of an ideal. That's why we call this symbol i. If r is a ring and i is a subring of r, we say that i is an ideal. If the subsets r i and i s are, if these are contained inside i itself, we call this an ideal. Particularly this is called a two-sided ideal in general, because again with non-commuted rings, we do have to distinguish between these sometimes. If you only assume the first part, r i is contained in i, you call this a left ideal. If you only assume the second one, the second property that i s is contained in i, we call that a right ideal. And so having both of them is called a two-sided ideal or when there's no confusion whatsoever, it's just called an ideal for short. We will see in just a second that modding out an ideal is what you need for that well-defined operation, particularly we're going to do that via ring homomorphisms here. Because as we saw on the previous slide, if these two subsets belong to i, we're going to get a well-defined ring multiplication on the additive cosets. And this definition of an ideal is exactly what gives us the kernel of a homomorphism. In particular, the main result for this video here is that if phi is a ring homomorphism between two rings r and s, then the kernel is an ideal. So we already know, let's look at the proof of this thing, we already know that if we look at just the additive structure, just the additive structure, then the kernel of phi will be an additive subgroup of r. This will be proved with group theory. Commutivity was never a concern in that situation, but we're going to get that kernel of phi as an additive subgroup of r. All right? To make it a subring, we then have to prove that it's closed under multiplication, but being an ideal has a stronger, has a stronger multiplicative closure principle. So what I want you to see is the following. If r i is less than i as a subset, right? What that means is for all x inside of i, we have that r x, which of course belongs to r i. Since this is belonging inside of i here, this says that this belongs to i. And so if you take anything in x and times it by something in the ring, you're going to be inside the ideal. So let me rewrite that one more time. What we get here is that for all x inside of the ideal, and for all r inside of the ring, we have that r x belongs to i. So to be a subring, we just need to have, if you have two elements of the subring, the products inside the subring, but with an ideal, if you take any element of the ideal times any element of the ring, and that element can be external to the ideal, then their product is inside the ideal. The ideal sort of gobbles everyone up. Times it by anyone, you're inside of the ideal. And that's important because that's what zero does, right? Zero times anything is equal to zero, right? So the ideal in a quotient ring has to mimic that property. So it has the property that anything times an element of the ideal is still in the ideal. The ideal has this absorbing multiplicative property. Now I'm not saying that r x is equal to x, but r x is some other element of the ideal. We also get this in the other direction as well, right? If you take the right condition, we're going to get that x times s belongs to the ideal. As long as x belongs to the ideal and s is any element of the ring, that's what we're given with these ideals, all right? So to prove, so we already have that the kernel is an additive subgroup of r, and as things are abelian, it's necessarily a normal subgroup, right? We need to prove just that the ideal closure property happens because if the ideal multiplicative property is true, then general ring closure also is true because if I can take r times an arbitrary element of the ideal, then, and as r could be any element of the ring, then if they're both belonging to the ideal, then we still get closure there. So we just have to prove this stronger condition here. So what we're going to do is we're going to take r to be an arbitrary element of the ring r, and x is going to be an arbitrary element of the kernel. So in particular, I know that phi of x equals zero. I don't know what phi of r is, but I don't need to because if I take phi of rx, because it's a ring homomorphism, we can factor this as phi of r times phi of x since x belongs to the kernel, phi of x equals zero, like we observed above, and anything times zero is equal to zero. And so phi of rx is zero, meaning that rx belongs to the kernel, and therefore the set r times kernel is a subset of the kernel, which admittedly all you have to do is show this, same thing there, by a similar argument, if you take multiplication on the right, so the kernel times s is going to be a subset of the kernel. And this shows us then that the kernel is always an ideal. So this idea of closure, right, that for all r in the ring and for all x in the ideal, we have that rx comma xr belong to the ideal. This ideal closure is exactly what kernels do for ring homomorphisms, and therefore this is exactly the object we need if we want to quotient out to form a so-called quotient ring. And so ideals are, in fact, the ring theoretic analog of a normal subgroup. And so that brings us to the end of lecture 39 for Math 42.20. Thanks for watching, but I also want to remind us that this is actually the end of lecture 13 for Math 42.30, abstract algebra 2. So if you want to learn more about quotient rings, please check out the next video. If you liked what you saw or learned something, give the video a like, give all these videos a like. I appreciate that one. And by all means, subscribe to the channel if you want to see some more of mathematical videos like this in the future. Thanks, everyone. I'll see you next time.