Some Family Search records give birth place as Low Riggend, Airdrie, Lanark, Scotland in 1790; Low Riggend = Laigh Riggend would agree with birthplace of son William in 1810 and first son James in 1808, both in Scotlands People (official web site for Scotland). I can not find a birth record in Scotlands People for Thomas, although there is one for his wife, Elizabeth Shaw, in Airdrie in 1787. No marriage record seems to exist, although Family Search gives marriage on the same date in 1808 as the birth of James. According to family tradition, Thomas & Elizabeth eloped; perhaps their marriage was sworn to each other with a witness, legal in Scotland until 20th century. James would then have been born when Thomas was 18 years old.
Alternatively, there is a Scotlands People birth record for a Thomas Irvine in 1785 in Kilmaurs, Scotland, son of William Irvine and Janet Smith. This would make Thomas 23 when 1st son born, but librarian in Airdrie thinks that birth in Laigh Riggend is more likely than in Kilmaurs.