In 1870, although the census was to be as of June 1st, the census was taken on July 11th, shortly after Joseph had gotten married on June 16th. The Hart family is living next door in Cranston, RI, and Joseph & Josephine are each shown in his/her own parents' home, although Josephine is called Josephine Remington.
Joseph & Josephine's family, including their six children, is in the 1880 Federal Census at Cranston, RI, on the east side of Broad St. The youngest son has not yet been named. I believe this must be Lawrence.
Joseph F. Remington and his son E. T. Remington are living in Warwick, RI, in 1910.
From "Death Notices" at RI Historical Society Library: Joseph F. Remington, 66th yr., 5 Aug 1913, at Pawtuxet.
He may have been named for Dr. Joseph Warren Fearing, an esteemed physician for 32 years in Providence. I don't know if there was any family connection.