In 1833, Joseph married Mary Gullan, and in 1837, their son Joseph's birth is registered with his mother "MARY GULLAN". However, the two later births record the mother as GUNN. There was no MATHER+GUNN marriage, only GULLAN.
In Q4 1838, Son Joseph's birth was registered at Bolto with mother recorded as Mary Gullan.
In 1841, Joseph and Mary (35 & 30) were at Spring Gardens, Bolton with son Joseph (3). Mary was Irish.
In 1851, Joseph (of Wigan) and Mary (45 & 40), were at 8 Supton Street, Bolton with son James, aged 9 (no daughter).
In 1861, Joseph and Mary (both 50) are at 9 Hanover Street, Wigan. Joseph is a Coal Miner born at Aspull. Mary is a Bonnin Minder from Ireland. Many of the neighbours are also Irish. Their son James and Nancy (20 & 19) are with them. Although so much fits, other records suggest this in noa a mining family.
In 1871, Joseph Mather and Mary (68 & 63) were at 9 Partridge Street, Bolton with their son James, widower (30) and his two children. Joseph was a brewer.
In 1881, Joseph Mather (77, Wid) was a lodger, recorded as a British Subject, born at Sea. Is this him?