I assume this is taken from books like The Revolution and the like.
The presentation of the specific cases in which the Sages were scientifically correct is simply demagoguery. Anyone who has learned even a bit of Talmud knows there are things there that are not scientifically accurate. For example, that black cumin has a poisonous smell. Of course, regarding black cumin one can specifically argue that we do not have the plant the Sages meant, and so on. But there are many more examples of scientific inaccuracies on the part of the Sages.
You can relate to this in two ways. A. To say that the Sages never intended to say anything scientific, but that within the claim that appears scientific there is really a normative claim hidden. B. To say that the Sages relied on Greek science and not on the basis of the Holy One, blessed be He. Meaning, these are not things they received from Moses our rabbi. And if so, there is no problem with their making mistakes.

To Yishai,
Please answer the question and not state your general worldview…
I asked specifically about the above examples, and about them I would be happy to get an answer (I never claimed they had to be accurate about everything, but these alone are enough for the claim).

You selectively chose what they got right (and I am not even sure they got it right; it may be a matter of retroactive interpretation) and ignored the mistakes, especially since there are many more mistakes. Hindus also have cases where they got things right. Their creation myth fits both the age of the universe and also works with modern theories. Muslims also have websites showing how the Quran is correct when one selectively chooses what to take. Sometimes this is also done through strange interpretation.

With so many claims being thrown into the air, something is bound to land on something correct. Even a person guessing on a psychometric exam will have a few correct answers.
A real examination has to check whether the general body of knowledge they gave fits science. It is not that they made only a limited number of mistakes. There are Jewish laws to this day whose source is an error.

This reminds me of demagogic arguments by outreach-to-repentance people who claim that all scientific discoveries are encoded in the Torah and bring examples.
Like “the value of pi is encoded in the first verse of the Torah” (when in fact an explicit verse claims it is 3). Or that the name of God is hidden in human DNA. By the same token, the match could just as well have been something else.
In fact, one can compare this to choosing the number of functions from a certain domain to a certain range, where the domain is everything written in the sacred books (Hebrew Bible (Tanakh), Talmud, Mishnah, Jerusalem Talmud, Zohar, from verses to sayings) and the range is the number of possible meanings that can be assigned to them in a way that fits science. Anyone who has studied combinatorics or discrete mathematics knows that the number of correspondences is enormous, even with a relatively small domain and range. With such a quantity, there are sure to be correspondences that match science. That is why it is hard for me to be taken in by such claims.

A bit of nitpicking:
A. There is no “verse” that claims it is 3, but rather a Talmudic passage (which rounded the data, something any child can check).
B. Obviously, in any text one will find codes; the question is in what quantity and under what conditions. There was a major study signed by a large group of respected scientists claiming a high number of codes, such that it was not likely to be random. Later there was a random study that claimed a “total failure” of the previous study, and the argument is still ongoing.

I Kings chapter 7:
“And he made the molten sea, ten cubits from brim to brim, round all around, and its height was five cubits, and a line of thirty cubits encompassed it around.”

Diameter — 10
Circumference 30
Pi is defined as the ratio of circumference to diameter. 30/10 = 3. In the Septuagint they give a more precise value. It is possible that the Masoretic version is corrupted in this verse, or perhaps the translator of the verse in the Septuagint had a Hellenistic education and was more precise. Even there the value is not exact (since this is an irrational number that cannot be written as a ratio of two numbers, but because of the context of the verse this can be ignored, since this is practical usage and not mathematics as a theoretical field).

B. From every verse infinitely many things can emerge that fit. It is enough to find one. After finding it, you mark the target. Since we are already talking about codes, there is a code that proves Christianity. In the verse that speaks about sacrificial worship, at intervals of 3 (representing the Trinity), there appears “the blood of Jesus woman.” I saw this recently on a Christian website, and they used the Masoretic text. Of course, they will not tell you that.
Also, the Masoretic text is not the text that was in the hands of the Sages, which itself was apparently not the original text.

And it is always enough for one thing to fit out of the infinite facts that exist in the world. If the first verse had given the base of the natural logarithm, the scaling constant, the square root of 2, or Weierstrass’s theorem by skipping letters to spell “Darboux’s theorem” in gematria, they would also claim that infinitesimal calculus is encoded in Scripture. In other words, my claim is not that this is a large quantity, but that there are many possibilities that can be connected to every verse.

The strongest refutation of this actually comes from the Sages themselves. The Sages never mentioned the code method as one of the methods of interpretation, nor did they ever mention anywhere that the Torah can be interpreted by this method.

It is worth looking at the book The Code of the Days of the World by Rabbi Trop, where he developed a unique code whose mathematical probability in other books is negligible.

Regarding the biblical code:
https://www.osimhistoria.com/osim-tanach/ep130-otiyot

A. Interesting. In any case, it is very easy to find that it is slightly more than 3. It is more plausible that the result was simply rounded, like the count of the children of Israel in the censuses.
B. Again, obviously codes will also be found in Harry Potter. The only question is in what quantity. There was a study on this that claimed it tested the Hebrew Bible (Tanakh) against The Lord of the Rings (if I remember correctly) and found a near-zero probability that this was a random matter (Israel Aumann mentioned a number of 1 in 100,000, and afterward retracted it). On the other hand, there were those who questioned that study, and the argument is still ongoing. You can read here https://he.wikipedia.org/wiki/%D7%93%D7%99%D7%9C%D7%95%D7%92%D7%99_%D7%90%D7%95%D7%AA%D7%99%D7%95%D7%AA_%D7%91%D7%AA%D7%95%D7%A8%D7%94
As for the Christian code, you yourself answered: one code is proof of nothing.
The Sages did not necessarily know everything, and certainly did not write everything.

A. It is explicitly written that pi is 3. Direct calculation without rounding leads to that. Or in another language, 30 mod 10 = 0. Meaning, it divides exactly by 10 and the result is 3. They did not round downward, because they did not bring the explicit value; they only stated the diameter and circumference. They could have brought a more precise value, like circumference 32 and diameter 10, circumference 33 and diameter 10.5 (as in the Septuagint).
It also cannot be rounded in this way, because then it would not come out a circle. It would come out as an elliptical shape, and then one cannot speak of a diameter.

B. Again, it is very easy to create codes from nothing. In other words, the number of functions between the set of verses, skip intervals, and gematrias and the set of facts in the world with some loose connection is almost infinite. Anyone who has taken a course like discrete mathematics should understand this. “A code is proof of nothing” — נכון. Here we agree. The problem is that you are holding the stick at both ends.

If it fits your faith, you accept it, and if it does not fit (like if it proves Christianity), then it proves nothing.
It has no connection to whether the Sages knew everything or not. If someone were to bring proof from codes that Jewish law is nullified, or even proof in a legal case such as that one person paid back another even though according to the law he did not, or that the husband of an agunah is not alive even though there is no way according to Jewish law to determine this — would you accept it as evidence?
I have already seen crackpots who claimed that “God” in gematria is “nature,” and with some skips they proved that a person should behave according to his nature even if that means violating Jewish law. You can prove anything this way.

B. Be precise: “One code is not proof.” The test has to be constructing an objective and sufficiently large list of facts, and running a search by the same method in the Hebrew Bible (Tanakh) and in another book of similar size. If you show that in the Hebrew Bible (Tanakh) a particularly high number of codes is found, enough to rule out randomness, that would be good proof for the existence of codes. Such a test was indeed done; see the link I brought.

Of course, codes are proof of nothing except their very existence, and accordingly they also cannot prove any particular point, because on the level of a particular point it really could be random. One has to see the full picture.

I will note that in my view there is a completely different reason to reject the codes: the Sages themselves say that today (that is, already more than 1,500 years ago) we are not expert in defective and plene spellings, meaning we do not possess an exact version at the level of individual letters, and therefore the whole thing lacks any foundation.

But it is written in the Talmudic text about amoraim and tannaim that they had divine inspiration. Does the Rabbi not believe them?

What are the Rabbi’s criteria?

The question is what is meant by divine inspiration. Sometimes it is a rhetorical description. Maybe it comes to speak of their greatness and the like. The Raavad also writes that divine inspiration appeared in his study hall, and it has already been explained that he did not mean something transcendent, but rather a kind of certainty that he felt.

There is a sacrifice called the bull offering for an erroneous ruling of the community, which applies in a case where the Sanhedrin sitting in the Chamber of Hewn Stone made a mistake in a halakhic ruling and all the Jewish people sinned because of them!!! How could it be that they erred if they have divine inspiration???
From here we see that even the Sanhedrin are human beings, and because of that they can make mistakes.

In the Talmudic text it says that Rava went to study the laws of ritual slaughter for a very long time in order to know all the small details. He did not learn this through divine inspiration, but through toil, checking, and investigation.

And how did Maimonides and the Geonim disagree with the Talmudic text on issues like necromancy, sorcery, demons, the zodiac,??? Does the Talmudic text have divine inspiration???

And how did amoraim disagree with amoraim, and likewise tannaim with tannaim???
Did they argue with divine inspiration???!!!!

Obviously they all understood that prophecy had ceased, and therefore no one has divine inspiration, and no one has authority over the truth!!! And whoever makes a good argument, his words are accepted, and whoever does not, then not!!!!