All of mankind recognized G-d's existence after the Great Flood. They understood that He created the world from nothing, fashioned every facet of reality from the land to the sea, the sun and moon, etc. They also understood that He rewards those who do good and, as the Flood demonstrated, will punish those who do bad. They began to fear that if they sinned, they too would suffer the same fate. In an attempt to break free from Hashem's control, the leaders of the nations devised a sophisticated plan: let us build a very tall tower that will reach Heaven and utilize it to...wage war against G-d Himself.
What were these people thinking?? If the nations recognized that G-d is real and almighty, how could they even fantasize about the idea of fighting Him - and by building a tall tower no less! Were they out of their minds?! There must have been more to their strategy than meets the eye.
R' Moshe Shapiro explains that the nations were a lot wiser than we think. They understood that Hashem invested a tremendous amount of potential and power in man. There was one specific power that they intended on tapping into that held the power to truly grant them their desire: Unity *. They knew that when all of mankind decides in unity on a single idea, Hashem will grant them their wishes, as the pasuk states "Behold, they are one people with one common language...and now, it will not be withheld from them" (11:6). This is often referred to as "Ayin Hara", for if everyone wishes bad on an individual their wishes can carry tremendous potential to be fulfilled.
The building of the tower was simply meant to be a catalyst to bring everyone to focus on one united desire - no longer wanting Hashem's involvement in man's affairs. R' Moshe Shapiro explains that indeed, had all of mankind succeeded in focusing on this one thought, Hashem would have consented to their wishes! It is for this reason that Hashem decided then and there to disperse the nations, so that they could never achieve a level of absolute unity ever again. Not only did He change their languages, He also changed their cultures and interests; from then on, each nationality has seen things in their own unique way, making universal harmony virtually impossible.
Living Inspired
The power that Hashem invested in mankind when they focus on one common goal is still as effective today as ever before! This is demonstrated when someone becomes sick and we all daven for their speedy recovery. An underlying reason why those prayers are so exceptionally powerful and can heal someone who wasn't "meant" to be healed, is because we have so many Jews focused in unity on one common goal. Never underestimate the power of an Asifa (a large gathering) of Jews who daven for a specific outcome!
If we all learn to master the practice of genuinely caring and being concerned for the needs of our nation as a whole and wishing that everyone has blessing, Hashem will fulfill our every united desire and you can be sure that Mashiach will shortly follow (as The Chofetz Chaim writes explicitly **).
Gut Shabbos
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* - As the pasuk states "The whole world was of one language and of one [common] plan" (11:1. See Rashi there).
** - See there, where he states that just as Sinas Chinam (baseless hatred) caused the long exile we are currently in, the moment we attain harmony within the Jewish nation Mashiach will come immediately.