Like many of the house mottoes in the series it can carry multiple meanings.
One more literal meaning could refer to the fact that the Iron Islands, the area ruled by House Greyjoy, are sparse and rocky, and the thin soil makes it hard to plow and grow any crops.
Another literal interpretation could refer to the fact that the majority of the ironborn are seafarers and survive by raiding and pillaging.
They "do not sow" - they only reap.
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Ok, forget the biblical quote, for someone to reap, someone must sow but it doesn't have to be the same someone.
The Words of house Grejoy aren't in fact missing a noun.
The Greyjoys, and all of the Iron Islands, are seafarers who've traditionally survived by plundering all along the shores of Westeros.
They take joy and pride in their strength and ruthlessness, which is why they only wear jewelry they've won in battle (they don't buy such things).
The words, "We do not sow" are at once a very literal reminder to all who hear them that the Greyjoys are not farmers (like the rest of Westeros.
Keep in mind most of Westeros' population are agriculturalists or traders) whom they disdain and they show contempt for those farmers.
Obviously the phrase could be changed to "we do not sow crops" without really changing the meaning too much, although it would change the tone.
The word "sow" itself when used alone generally implies sowing some sort of crop
so it's not really necessarily.
Probably the only thing that the Greyjoys have ever sown is discord, and as I said, I'm pretty sure that wasn't the kind of sowing they were talking about.
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House Greyjoy does not sow -- they pay the iron price and they take what they need.
They do not believe in money or trade -- just pillaging and plundering.
Hence, they do not sow.
They only reap.
They're pirates, if that makes it easier to understand.
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Theon reflects on the motto of his house and the legacy from which it came... When we still kept the Old Way, lived by the axe instead of the pick, taking what we would, be it wealth, women, or glory.
In those days, the ironborn did not work mines; that was labor for the captives brought back from the hostings, and so too the sorry business of farming and tending goats and sheep.
War was an ironman's proper trade.
The Drowned God had made them to reave and rape, to carve out kingdoms and write their names in fire and blood and song.
A Clash of Kings
- Theon I
As seen the saying is quite literal, the Iron Islanders (at least those of yore) did not bother themselves with menial labor.
They took what they wanted and more from those who were too weak to protect it.
This is backed up by the historical account of Westeros by Maester Yandel...
Whenever autumn waned and winter threatened, the longships would come raiding after food.
And so the Iron Islands ate, even in the depths of winter, whilst oft as not the men who had planted, tended, and harvested the crops starved.
"We do not sow," became the boast of the Greyjoys, whose rulers began to style themselves Lords Reaper of Pyke.
The reavers brought more than gold and grain back to the Iron Islands; they brought captives as well, who would henceforth serve their captors as thralls.
Amongst the ironborn, only reaving and fishing were considered worthy work for free men.
The endless stoop labor of farm and field was suitable only for thralls.
The same was true for mining.
The World of Ice and Fire - The Iron Islands One should note that this style of life did not apply to only "nobles" on the Iron Islands but to all "iron born".
Regardless of your status on the islands, you were still better than those weakling mainlanders.
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Another interpretation here: If you look back into history and our own middle ages, to sow had connotations of servitude, as those who would sow were the lowliest peasants, locked into servitude to every other social class.
In this regard, "We Do Not Sow" is a satement of defiance meaning "we bow to no man".
It's the words that make the Greyjoy sigil my favourite in the whole world of 'Ice and Fire' even though the family themselves are less than likeable.
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I personally believe that the House motto literally means that "they do not sow", sowing commonly referred to as the task of planting seeds in a field, this most often being a job for a slave or a commoner acting under the whim of their lord.
This being said it implies that House Greyjoy will never succumb to such arduous tasks, and were in fact born to rule and not to serve under someone else.
In essence it means that they will not serve as a commoner and will risk their very lives to rule and not serve.
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What does the motto of House Greyjoy, "We Do Not Sow", in the TV series  Game of Thrones  mean?  It seems to be missing a noun, to me.   Is it something like,  we do not sow discord ? Or  we do not sow ,  we reap the whirlwind ?
What does House Greyjoy's motto, "We Do Not Sow", mean?
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The Greyjoys, and all of the Iron Islands, are seafarers who've traditionally survived by plundering all along the shores of Westeros. They do not believe in money or trade -- just pillaging and plundering. They took what they wanted and more from those who were too weak to protect it.  Therefore, the saying is quite literal, the Iron Islanders did not bother themselves with menial labor. The motto of the house of Greyjoy emphasises that they are pirates and live by plundering and pillaging rather than farming. 
The Greyjoys are pirates. They live by pillage and plunder and prey on the weak.  "We do not sow" emphasises that they are not farmers or commoners. Moreover their land was too sparse for cultivation.