This d'var Torah is to mark my siyum- completion- of the fifth order of the mishna, Seder Kodshim, in memory of my grandfather Iser ben Binyamin HaCohen. Kodshim is paricularly appropriate, as it describes the intricacies of the temple service, from the details of an offering to the measurements of the temple mount.
The Mishna is broken up into six sedarim, or orders. Each order is divided into a number of masechtot, or tractates. The last tractate is called keinim, which literally means nests. It refers to the bird offerings in the temple. I would like to touch on one topic in this tractate.
The most common case for bringing a bird offering was a birth, and thus the entire tractate refers to the owner of the birds in the feminine. For each birth, a woman was required to bring a pair of birds, one as an olah (burnt offering) and one as a chatat (sin offering).
I stress pair here because these birds must go together; it is not sufficient to have two separate bird offerings. They must be brought to the temple as a pair, and if the first is designated an olah, the other is automatically a chatat, and vice versa.
Two women: Rachel has two pairs of birds, and Leah has two pairs. If one of Rachel's birds flies and is mixed in with Leah's birds, the bird invalidates one in its departure. If one returns (from Leah to Rachel), it invalidates one in its return.
I chose this Mishna because it has a very mathematical nature to it, something I feel connects me with my grandfather. In particular, this branch of math is known as combinatorics, or studying different possible combinations. In order to understand what's going on, let's look at a simpler case.
Let's say that Rachel has a single pair of birds, as does Leah. If none of the birds do any funny business, one of Rachel's birds will be an olah, and one will be a chatat. The same is true of Leah's birds. It would look like this:
Now let's say one of the birds decides to become Houdini (another Cohen if I'm not mistaken). One of Rachel's birds hops the fence into Leah's cage, and now we don't know whose bird is whose. Once a bird is assigned an owner, the owner cannot be changed (in the picture, color represents the owner). However, the type of offering (circle for olah and square for chatat) has not been assigned, and is thus up for grabs.
It's not possible for us to make more than two birds a chatat or more than two an olah, so the best case scenario would be that two of the three birds in Leah's possession will be an olah, the third will be a chatat, and the single bird in Rachel's cage will also be a chatat. That would look like this:
In this case, all is fine and dandy; Leah's pair (the red birds) are an olah and chatat, and Rachel's pair (the green ones) are also an olah and chatat. No problem.
However, this assumes that we guessed correctly. Since the bird in Rachel's cage is a chatat, then we need to make sure that the green bird in Leah's cage is an olah, since we need matching pairs. We don't really know which bird actually belongs to Rachel, so we could accidently do the following:
Now we've got a real problem! Both of Rachel's birds are chatats, and both of Leah's birds are olahs. We've got no matching pairs at all, so none of the offerings were good. It turns out the best we can do, mathematically, is to offer just two of the birds from Leah's cage. Here's the possibilities:
Well, eight birds is basically the exact same things as four, just more complicated. To look at this mathematically, our goal is to understand:
Proving the first is easier: we need a counter-example, or a case that could lead to mismatched pairs. Let's start off with the base case: one of Rachel's birds flew into Leah's cage:
The mishna says that this disqualifies one of Rachel's birds, meaning that we sacrifice four from Leah's cage and two from Rachel's. There are two ways to sacrifice more than the allowed number:
We could sacrifice three from Rachel's cage instead of two. However, this could lead to a problem if we had something like the following:
That is, we sacrifice two olahs from Rachel's cage, and then inadvertently sacrifice Rachel's bird in Leah's cage as an olah. This means that, in total, we would have three olahs and one chatat for Rachel, which is unmatched.
We could sacrifice five from Leah's cage instead of four. Here the problem would be:
Here, the problem is that we might accidently sacrifice three of Leah's birds as the same type of offering. This would also be mismatched.
So how do we know that the mishna's advice works? Well, let's do some deductive reasoning. Rachel has four birds all told. If she sacrifices two of them in her cage as a pair, she still has one left over to be the pair to the one in Leah's cage. If we don't sacrifice that third bird, then the fourth has the freedom to be anything it wants (within the realm of dead birds).
That means that if Leah sacrifices Rachel's bird as either an olah or chatat, or doesn't sacrifice any of Rachel's birds, we're safe. There is nothing we can do to mess up Rachel's offerings. The only concern is Leah's offerings.
Well, if we offer four of the birds in Leah's cage, there are two possibilies:
In option 1, Leah gets one matched pair and one lone offering, while Rachel gets an offering without a pair. Cool. In option 2, Leah gets two matched pairs, and Rachel gets nothing. Also good.
Still with me? Good, we're right at the end here. So here goes the story: one of Rachel's birds flies into Leah's cage, and realizes living conditions were much better in Rachel's cage. The bird starts complaining, and eventually some bird flies into Rachel's cage. Problem is, we don't know if it was Rachel's escapee, or if one of Leah's birds has now broken out of jail.
What makes this different is we have two possible base cases, as follows:
Well, if we're talking about option 1, no problem: just go 50/50 in each cage, and all is well. The problem is in option 2.
Time to bring in more math: we have something called independent events, where a result in one case does not effect the result in another. The typical example is a coin toss: if I get heads the first time, the next toss is unaffected. Thus, if I flip a coin twice, I have four possible results:
We're going to have the same thing here. In each cage, we're going to offer two birds. In Rachel's cage, we have two possible results:
We have the reverse options in Leah's cage: either both Leah's, or one of Leah's and one of Rachel's. So all told, we have four possibilities.
Instead of displaying them all, I'll simply put in the most difficult case: where we take one from each owner from each cage. Since in this case, each owner will still have two unsacrificed birds, each bird which is sacrificed has complete freedom to be what it wants. Therefore, no problem, and all is well.
Keinim is part of Kodshim, the fifth order of the Mishna. The fourt order, Nezikin (literally meaning damages, the same root as the yiddish word mazik), discusses monetary laws.
If our case of the birds occurred in Nezikin, the answer would be obvious and logical: if you can't tell the birds apart anyway, who cares who's holding onto which bird? If they each started with four birds, let them each take four birds and be done with it!
I think this dichotomy exemplifies the distinction between the two orders. Kodshim is not about logic, or at least not human logic relating to this world. It is relating to things beyond what we can perceive. In a Kodshim perspective, the birds are connected to their owners in a way that cannot be severed. We may not see this, but Kodshim tells us it's there.
The juxtaposition of these two orders tells us to not compartmentalize these two approaches. We need to take both aspects with us through our daily lives. While we are fully immersed in this world, and fully subject to the laws and logic of it, we must appreciate that there is more than meets the eye.
While Papa may not be here in a Nezikin sense, he is still with us in a Kodshim sense. Even though we cannot see him or talk to him, we all know that the connection lives on in a way we cannot understand.