I
"According to whom do we say [on Rosh Hashana] 'This is anniversary day of the start of Your handiwork, a remembrance of the first day'? According to R. Eliezer who said the world was created in Tishrei" (Rosh Hashana 27a). Elsewhere, we seem to follow R' Yehoshua, who said the world was created in Nissan. Tosafos resolves the apparent contradiction as follows: in Tishrei it rose in thought to be created, but it was not created until Nissan.
This enigmatic statement may be based on Rashi (Breishis 1:1) who says: at first (betechila) "it rose in (Hashem's) thought" to create the world with the attribute of strict judgment, but Hashem saw that the world could not last under such circumstances, so He gave precedence to the attribute of mercy and joined it to the attribute of strict judgment, as it says, "on the day Hashem Elokim's making of earth and heaven" (2:4).
Tishrei is the month of judgment, its symbol being scales of judgment. In Tishrei "it rose in (Hashem's) thought" to create the world with strict judgment. Since the world could not last, actual creation was postponed until Nissan, the month of Pesach which connotes mercy (Rashi, Shemos 12:23).
Rosh Hashana is the beginning of creation, in accordance with R' Eliezer. But actual creation occurred in Nissan, in accordance with R' Yehoshua. Thus we say on Rosh Hashana, "today is the conception [haras] of the world", not its actual birthday.
Rashi's expression "betechila - at first" implies that Hashem changed His mind. This is incorrect, as "He is not a human who changes his mind" (Shmuel I 15:29 (see Bamidbar 23:19)), and heretical (Rashi Breishis 6:6). What then, does it mean?
II
Moshe saw that people were weighing the flesh [from R' Akiva's body] in the meat market. He said "Master of the world, this is Torah and this is its reward?" Hashem said to him "Be quiet. So it rose in thought before Me" (Meanchos 29b).
The Shelah (Breishis, Torah Ohr 9) explains that Hashem did not change His mind. Rather, the original plan to create the world with strict judgment remains for the exalted few, such as R' Akiva. Betechila is not chronological, indicating a change of mind, but rather conceptual, explaining the word "Bresihis." Ideally, the world should be created with strict judgment. Practically, the world could not stand, and mercy was added except for the likes of R' Akiva.
If so, we can understand betechila as l'chatchila. Ideally it rose in Hashem's thought to create the world with strict judgment. Practically, as Hashem knew from the outset, this is impossible. Hashem answered Moshe that R' Akiva was judged strictly, as was the ideal system which rose in thought before Him. The question remains, however, why is this the ideal system?
III
The answer lies in the purpose of creation, as explained by Ramchal (Derech Hashem sec. 1 chap 2): Hashem's purpose in creation was to bestow of His good to another. His wisdom declared that for such good to be complete, the one enjoying it must have earned it himself. Therefore, He created man, the purpose of all creation, with the ability to choose good or bad. By choosing good man earns closeness to Hashem [primarily in the world to come, see Ramchal's Mesilas Yesharim, Chapter 1], deriving the greatest possible pleasure from His goodness, thereby achieving the purpose of creation.
Ideally, man should be judged strictly, earning his ultimate reward by right. However, practically, man is unable to do so and mercy must be added. Still, the choice to do good is rewarded in the world to come.
R' Akiva was judged strictly, thus enabling him to enjoy even greater reward in the world to come. The Gemara (Berachos 61b) describes the torturous death of R' Akiva. The ministering angels asked Hashem, "This is Torah and this is its reward? Is it not better to be among, 'the dead who die at Your hand Hashem of old age' (Tehilim 17:14). Hashem said to them 'Their portion is in life [of the world to come]' (ibid.) A heavenly voice proclaimed, 'Fortunate are you, R. Akiva, that you will immediately [without judgment or pain. See Tosafos Kesubos 103b d.h. mezuman] enter the world to come'."
IV
On Yom Kippur, the question of the angels is recounted (Eileh Ezkerah, Musaf) with a different answer from a heavenly voice: "If I hear another sound, I will transform he world into water, I will turn the earth into 'tohu va'vohu - desolate and empty' (Breishis 1:12)".
This enigmatic response can be explained based on the words of the Ramchal. The purpose of creation is to bestow Hashem's good onto man. In its highest form, this requires the ideal system of strict justice, reserved for the exalted few such as R' Akiva. They are punished in this world for minor errors, so that they receive incomparable reward in the world to come. If the angels insist that strict justice not take place, then the highest purpose of creation no longer exists. Therefore, Hashem said He will return the world to the pre-creation state of being desolate and empty. In fact, strict justice took place, the world continues to exist, and R' Akiva was amply rewarded in the world to come.
On Rosh Hashana and Yom Kippur we are all judged by Hashem. By focusing on the months of creation and the Divine Attributes utilized in creation, we can better understand the purpose of creation. Sincere repentance and self-improvement can tilt the balance of Hashem's judgment, granting us a good year in this world and greater reward in the world to come.