Yes, surfboards need fins. The fins cut easily through the water when going forward, but create a lot of drag when turned sideways. This makes the board wand to keep going forward and not slip sideways (the same way a keel works for a sailboat or sailboard). Without the fins the surfer would have no way of making the board go forward, and no way of turning the boadr (it would slip sideways every time he/she tried to dig the rail into the wave).
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