Our mishnah is chapter 5, mishnah 7, and we are approaching the end of the chapter. This mishnah is brief and not complicated, but in order to grasp what it is about we first need to paint the whole picture.
The case before us:
A person is harvesting his grain, and in the middle of the harvest there is a small pile lying in front of him that has reached the completion of its processing. Such a pile is called an aramah, and ordinarily that term refers to a pile that has not yet reached the completion of its processing. However, if he does not intend to smooth over the pile and level it off, or if it was already a smoothed pile - as we learned in chapter 1, mishnah 6 - then it becomes obligated in maaser already at that stage.
So let us assume, for the sake of the discussion, that this pile is already obligated in maaser: either he smoothed it over, or he never intended to smooth it at all but rather to sell it as is, and in any event it has reached the completion of its processing. But outside he still has a large field of grain that has not yet been harvested, and in practice the amount of grain scattered in the field is far greater than the amount that has reached completion of its processing and stands ready to be tithed.
One night the owner of the field sees ants near the pile of grain that is obligated in the separation of maaser, and the next morning he finds a quantity of grain - perhaps even a large quantity - inside the ant holes next to the pile. The question is what the status of this grain is: did it come from the pile nearby, in which case it is obligated in maaser, or did it perhaps come from the large quantity out in the fields, in which case it is exempt from the obligation of maaser?
Why grain that was lost is exempt from maaser:
We have already learned that the obligation of maaser takes effect at the completion of the processing, when the pile is smoothed over and the working of the grain is finished. Therefore, if part of the grain was lost to the ants before that point, the owner assumes he will never get it back and gives up on it. Even if he does not know for certain that it was lost, it is enough that he abandons any hope of recovering it once it has disappeared on him. Once he has given up on it, the grain becomes ownerless, and since it left his ownership it is exempt from the obligation of maaser even when it is later found.
By contrast, if a person declares something ownerless after it has already become obligated in maaser - for example, after the pile was smoothed over he explicitly said, "I hereby declare this pile ownerless," and walked away from the place - the declaration does not help. The pile, since it reached the completion of its processing and its working was finished, is obligated in tithing even though it was declared ownerless after that point.
Hence the question about the grain found in the ant holes:
If it came from the pile after the pile became obligated in maaser - even though it became ownerless, the laws of maaser still apply to it.
If it came from the rest of the field - then it was lost before the obligation of maaser took effect, and therefore it is ownerless and exempt from the laws of maaser.
Majority versus proximity:
The usual rule regarding something whose origin is unknown is that we follow the majority, and we say that the item came from the source that constitutes the majority. According to this principle, most of the grain present in this field is grain whose processing has not been completed, and if so, the grain in the holes should be exempt.
But there is another principle available to us: not majority but proximity. In Hebrew the two words, rov and karov, sound alike, yet they have no etymological connection at all; karov means nearness of place. According to this approach we say that the grain came from the nearest available source, and in our case the ants' holes are extremely close to the pile of grain.
The two principles therefore contradict one another: according to majority, most of the grain in the field has not yet been harvested and has never become obligated in tithes, and once it was rendered ownerless it is exempt; whereas according to proximity, the nearby source is the pile that has already become obligated in tithes, and therefore even though it became ownerless, one who takes the grain out of the ants' hole is obligated to tithe it.
The usual rule is that when majority and proximity conflict, we follow the majority, and proximity is not considered to override it. For example, in an entirely different case: a city has three butchers, two kosher and one selling non-kosher meat, and a piece of meat is found lying near the non-kosher shop. The majority is kosher and the nearest is not, and the rule is that we follow the majority and assume that the meat was lost from the kosher stock, which is the majority, and it is therefore permitted. We are not concerned that the butcher closest to the spot where the meat was found happens to be the one who is not kosher.
Seemingly the same should apply here: most of the grain in the field is unprocessed, and therefore it should have been exempt from the obligation of tithes. But the Mishnah teaches that here the principle of proximity overrides the principle of majority, and we may safely assume that the grain in the ants' holes, even a large quantity of it, came from the adjacent pile. The reason is that ants do not travel far from their holes to bring grain from a distant place when they can easily obtain it from the pile right beside them.
It emerges that this is not a question of doubt at all, but of certainty: certainly, or almost certainly, all the grain in the ants' holes came from the local and closest supply, namely the grain in the pile. And since there is no doubt here at all, the principle of majority does not push aside proximity, for it is simply not reasonable that the ants wandered far off to bring grain when it was available close at hand.
This applies even to a very large quantity of grain found in the holes, for we say that the ants labored all night long: in the evening the ants were seen, and in the morning the holes were found full of grain. It is worth noting that there is no mystery in the matter of night and day; it is simply the way of ants to come out in the evening hours, because the heat has subsided, and they do their work through the night, while in the heat of the day and in the sunshine they avoid the sun entirely so as not to dry out.
The wording of the Mishnah:
After all this introduction, the wording of the Mishnah is brief: "Chorei ha'nemalim she'lanu b'tzad ha'aremah ha'chayeves, harei elu chayavin, she'yadua she'mi'davar ha'gamur gorerin kol ha'laylah".
"Chorei ha'nemalim she'lanu" - this is not an expression of ownership, but an expression of lodging: that they remained there through the night unattended. Like the water that "lodged" overnight which is needed for baking matzos, and like the prohibition of leaving the meat of a sacrifice overnight.
"B'tzad ha'aremah ha'chayeves" - that the ants' holes are located next to a pile of grain that has already become obligated in the separation of tithes.
"Harei elu chayavin" - the grain found in the holes is obligated in the separation of tithes, as explained above.
"She'yadua she'mi'davar ha'gamur gorerin kol ha'laylah" - we know, almost with certainty, that it is from the produce whose processing has been completed, namely the adjacent pile, that the ants drag their grain all night long. They do not travel out to the field, and therefore we may safely assume that this grain in the holes came from the local pile. And one who comes to eat it must first separate tithes from it, even though it became ownerless from the moment the ants took it.
In summary: in this Mishnah we learned that grain found in ants' holes that lodged overnight next to a pile that is obligated in tithes is itself obligated in tithes. Although the usual rule is that when majority and proximity conflict we follow the majority, here we follow proximity, because this is not a question of doubt but of certainty: ants do not travel out to the field while grain is available close at hand. And consequently, the ownerlessness that came about after the completion of processing does not exempt from tithes.
Incidentally, this opens a small window onto the way of life in those days: on certain occasions people would eat wheat taken directly out of ants' holes.