Was the Mishnah compiled in the order we find it today? If so, this would mean "Zeraim" was written first, and finally "Tohorot"?
If the Mishnah wasn't compiled in the order we find it today, then what was the original chronology?
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The Gemara in Baba Kama 102a says that there was no specific order for the compilation of the Mishna. Let us explain what that means. The Mishna was not written at one time as a single project by one author or team of authors. There were layers and layers of compilation, with the older material being incorporated into the new layers, sometimes with modifications, sometimes with additional passages. Theses additions were put in as the latter Tanaim came to them. The final version was set by Rabbi Yehuada HaNasi. The Gemara there considers how he went through the Mishna that was already in formation before him to publicize his final version. The conclusion is that each mesechta was finalized in order internally, but the order of mesechtos was not according to any set order.
Welcome to Mi Yodeya! According to the Gemara in Bava Kama, we do not assume there is an order to which tractate of the Mishnah was originally written first. This makes a difference because within one tractate, if there is a discrepancy in the Mishnah, we follow the later one, because we assume that was the author's (Rabbi Yehuda HaNasi) final intent. But if the discrepancy comes from two tractates, we don't know the order of which came first or second, as sometimes Rabbi Yehudah HaNasi taught them out of order. (Tosafot there. To me this implies even the Rabbis of the Gemarah were not sure what the order of composition was.
But as Tosafot points out there, there is an order to the tractates in terms of progression. (Just it was not followed in the actual composition.) Sometimes the Gemara itself clarifies this, but the Rambam in his introduction to the Mishnah explains the order to all six Orders of Mishnah.