Our Gemara on Amud Aleph tells us about one sage that rebukes another for questioning the determinations and measures of the Sages. He says:
Rabbi Zeira said to him: Do I not always tell you that you must not take yourself out of the bounds of the halakha? All the measures of the Sages are like this; they are precise and exact. For example, one who immerses himself in a ritual bath containing forty se’a of water is rendered pure, but in forty se’a less the tiny amount of a kortov, he cannot immerse and become pure in them. Similarly, an egg-bulk of impure food can render other food ritually impure, but an egg-bulk less even the tiny amount of a sesame seed does not render food ritually impure.
The Gemara goes on to give several other examples of measures where the rabbis have stated limits that seem arbitrary, but are still binding.
Rashi explains the main thrust of Rav Zeira’s point was he should not doubt the precision of these determinations. Ritva adds to this, but along the same lines, do not assume the words of the sages in their halakhic thresholds are like secular matters and are only estimates. Tiferes Yisrael Mishna Menachos 12:4 adds that this is to be understood that the amounts stated by the sages are firmly accurate and precise.
But this does beg the question, can truth be arrived at to such an extreme degree? Is it really so that every measure is precise and not an estimate, such that they can say that this amount allows for immersion of the body, and one drop less not? In fact, we have an explicit verse in the Torah about an amount that cannot be possibly accurate. The basin built by Solomon for the Temple is described as 10 amos diameter and 30 amos circumference (I Kings 7:23). We know that the closer number is 3.14 times the radius. Tosafos (Eiruvin 14a) recognizes this mathematical fact and leaves it as a difficult question. The Rambam in his commentary on Mishna Eiruvin (1:5) offers another idea. Since Pi is anyhow an estimate and an irrational number, there is no accurate number that can be given. Thus, it is not ignorance that led to a round number, simply pragmatics.
Borrowing from this approach, we might say that of course any of the amounts set by our sages are binding in a legal sense to the most accurate point, but that is because legalities, secular or Jewish must rely on sharply drawn distinctions. That is different than capturing the objective precise truth. Can the Torah operate in this manner? Rambam in the Guide (III:26) quotes an interesting Midrash (Bereishis Rabbah 44:1) which I think speaks to this point, of technical legalities versus objective truth:
The Midrash says:
What difference does it make to God whether a beast is killed by cutting the neck in front or in the back? Surely the commandments are only intended as a means of testing man; in accordance with the verse, "The word of God is a test" (Ps. 18:31).
The Rambam says:
I will now tell you what intelligent persons ought to believe in this respect; namely, that each commandment has necessarily a cause, as far as its general character is concerned, and serves a certain object; but as regards its details we hold that it has no ulterior object. Thus killing animals for the purpose of obtaining good food is certainly useful, as we intend to show (below, ch. xlviii.); that, however, the killing should not be performed by neḥirah (poleaxing the animal), but by sheḥitah (cutting the neck), and by dividing the esophagus and the windpipe in a certain place; these regulations and the like are nothing but tests for man's obedience. In this sense you will understand the example quoted by our Sages [that there is no difference] between killing the animal by cutting its neck in front and cutting it in the back.
Others may disagree with the Rambam and hold that every single halakha, down to the edge of the letter Yud, has deep significance and wisdom. However, the Rambam was comfortable with the idea that the general rule represented a wisdom and need, while the specific may have been a wiser choice than another specific, but the point of the specific was because the law needed some specifics in order to be valid.
Let me give you an example. Say it was a mitzvah to hang a written scroll on your doorpost (oh, wait, it IS a mitzvah!) If it was left vague, over time the practice would deteriorate. You would need to specify what constitutes a scroll, what constitutes a door, what constitutes writing, etc. Now of course, some of those details might have deep meaning, but do all of them have to have deep meaning? Sometimes you just need some kind of rule to give it structure. The letter needs to be shaped this way, instead of that way. Of course, God knows what He is doing, so the detail might still have wisdom and meaning, but it still might not be essential. The Rambam might say the point of the Mezuzah is to make us mindful of God and our duties toward Him when we come home and when we leave home. The details of the law though, might be merely to preserve the law and keep everyone writing mezuzos properly and not neglecting them, but the meaning of each detail might not be so important as a truth.
Yet, what Rabbi Zeira was saying is that in the realm of halakha we must care about every detail regardless of what might appear, or possibly be, an arbitrary detail. And the reason for this is that faithfulness toward every command and every detail is absolutely necessary in order that the law and tradition preserve its integrity over time.
Translations Courtesy of Sefaria, (except when, sometimes, I disagree with the translation .)