Our Gemara on Amud Beis discussed the opinion of Rabbi Eliezer:

In a case where limbs of burnt offerings fit for sacrifice were intermingled with limbs of blemished burnt offerings, Rabbi Eliezer says: Although all the limbs are unfit for sacrifice, if the head of one of them was sacrificed, all the heads shall be sacrificed, as the head that was sacrificed is assumed to have been that of the unfit animal in the mixture.

Yet, as some kind of insurance policy, the Gemara requires that the remaining heads, when they are sacrificed, are done two at a time. The stated rationale is that at least one of each pair is certainly from a permitted sacrifice.

Commentaries have struggled to explain the logic of this particular stringency. If we are relying on majority, so be it. In what sense does it make it any better to bring two at a time? Technically, whether you bring them one at a time or two at a time, the odds are exactly the same of bringing an improper offering. Somehow, having at least one permitted portion of the sacrifice being brought along with the other is helpful. The logic for this is not clear.

The Steipler (Kehillas Yaakov Zevachim 6:8, hearos) offers a creative suggestion. There is a concept known as ikva issura — that the prohibited object is established. The principle of ikva issura is used in regard to the obligation of an asham taluy, which is a special sacrifice that is brought when a person is unsure whether he unknowingly violated a transgression that would incur kares and require a chattas if violated due to lack of knowledge. However, the obligation for the asham taluy is only if ikva issura, such as when there are two pieces of meat in front of him — one from forbidden fat and the other regular kosher meat — and he is not sure which one he ate. This is the situation where he has to bring an asham taluy to provide temporary atonement in case he did transgress. Yet, if there was only one piece of meat in front of him, and he was unsure whether that particular piece of meat was forbidden fat or kosher meat, he does not incur an obligation to bring an asham taluy. Here, there was no established presence of forbidden meat; it was only a statistical question. (Kerisos 18a.)

This is where it gets interesting, because in terms of odds and probabilities, depending on other variables, it could go in either direction. There is nothing intrinsically different about the odds between the two cases. This is why the Steipler notes that there is something phenomenological about having the presence of an established forbidden item — and somehow the Torah relates to it differently. In a similar manner, the Steipler suggests that there also could be a concept called ikva heteira — that the permitted substance is established. Offering one head on the altar along with another, where for sure there is one permitted head in the offering, establishes heter — permissibility.

While the Steipler makes this interesting comparison, tali tanya b’delo tanya! (He is supporting one unknown concept with another unknown concept.) How do we understand this trend in which the optics of the matter influence the halacha when mathematically the odds are the same?

I have an important insight that might help us understand the lomdus. Part of our challenge is that we are inserting Western scientific sensibilities into the thought processes of the ancients. The ancients — our sages, and also the Greek mathematicians and astronomers — were no fools. They drew conclusions about the world based on their observations, and while at this time we might think differently, there are aspects of mathematics and physics that are based on postulates and a priori assumptions. If you make different postulates and different assumptions, certain things will be interpreted differently.

The first thing that is important for us to understand is that the whole science of mathematical probabilities was not something that the Rabbis or the Greek mathematicians considered a discipline. Calculating odds, while it might have mathematical significance and be useful for predicting trends, is completely irrelevant from a different perspective. Odds do not tell you whether something will or will not happen, nor do they tell you what something truly is. Furthermore, the ancients were less arrogant, and they understood that much of reality was based on the way in which God presented it for humans to see and perceive.

This is a key principle: If something looks a certain way and it is interpreted to be similar to other things that look like it, that perceptual category matters. Much of ancient medicine — which has something to offer, by the way — is based on similarities. If somebody is feeling depressed, dark, and down, there is an assumption that there is an abundance of bitter bile in the body, and therefore they need to take foods that are lighter. If a person is suffering from a fever or inflammation, then foods that create heat are to be avoided, while foods that reduce inner heat or give off less heat, or are less spicy, are the better choice. The mandrakes that Rachel and Leah desired in order to enhance fertility are not accidental in that the roots look like homonunculi, literally the “man” in “mandrakes” (Bereishis 30:14; see Ibn Ezra’s reasoning regarding body humours as well as their appearance).

Consider the following thought experiment. Imagine a person blind from birth. This person could easily identify a ball or a square by feeling whether it was round or had edges. If his sight was miraculously restored, and he saw a ball or a square, would he recognize what it was without touching it? Remember, his reality of a ball had to do with feeling curves or sharp edges, not the mental image of a ball or square that most of us have. So what is a ball in reality? What is a square in reality? Here we begin to see how much of reality is based on perception, not an objective reality.

Aristotle was a brilliant scientist and philosopher. If you asked him why a rock falls to the earth while smoke and fire rise toward the sky, he would tell you the answer is obvious. Smoke comes from fire; fire is more related to air and heavenly material; therefore it naturally “desires” to go upward and join the rest of its material, while stone is made from earth, which is on the ground, and it naturally “desires” to be with like objects. To him, this is no different than family members being attached to each other and wanting to be together. Everything is designed by the same intellect and guiding force of God, and if it looks logical and sounds logical, that is probably what it is. So if we know that things that are alike are drawn to each other, why would it make a difference whether it was a stone or a human being? He knew nothing of gravity or kinetic energy and felt very satisfied with his observation of the world, his conclusions, and what they meant. (See Book Two of Physics.)

While we can prove that gravity exists mathematically and that Newtonian physics, to a degree, allows for a more accurate predictable presentation of how objects behave, ultimately we still have no idea how gravity works. When it comes to other forces of nature — kinetic energy, light waves — we at least have some sense of the vehicles and manifestations and processes. Science has not yet discovered what a gravity wave is — if it exists — and how it works. We modern scientists know the “what” much better than the “why,” while the ancients looked at the “why” first, and that gave them an idea about the “what.” They certainly were not always right, but they were not fools.

When you look at it from this perspective, it is easier to understand why the sages believed that the Torah’s rulings were affected by perception. If something looked a certain way, tasted a certain way, or even was paired with something that was permitted this itself might create a status of permissibility and have an influential force.