Galatians 4:23
23. after the flesh—born according to the usual course of nature: in contrast to Isaac, who was born "by virtue of the promise" (so the Greek), as the efficient cause of Sarah's becoming pregnant out of the course of nature (Ro 4:19). Abraham was to lay aside all confidence in the flesh (after which Ishmael was born), and to live by faith alone in the promise (according to which Isaac was miraculously born, contrary to all calculations of flesh and blood).
Galatians 4:29
29. persecuted—Ishmael "mocked" Isaac, which contained in it the germ and spirit of persecution (Ge 21:9). His mocking was probably directed against Isaac's piety and faith in God's promises. Being the older by natural birth, he haughtily prided himself above him that was born by promise: as Cain hated Abel's piety.
him . . . born after the Spirit—The language, though referring primarily to Isaac, born in a spiritual way (namely, by the promise or word of God, rendered by His Spirit efficient out of the course of nature, in making Sarah fruitful in old age), is so framed as especially to refer to believers justified by Gospel grace through faith, as opposed to carnal men, Judaizers, and legalists. even so it is now— (Ga 5:11; 6:12, 17; Ac 9:29; 13:45, 49, 50; 14:1, 2, 19; 17:5, 13; 18:5, 6). The Jews persecuted Paul, not for preaching Christianity in opposition to heathenism, but for preaching it as distinct from Judaism. Except in the two cases of Philippi and Ephesus (where the persons beginning the assault were pecuniarily interested in his expulsion), he was nowhere set upon by the Gentiles, unless they were first stirred up by the Jews. The coincidence between Paul's Epistles and Luke's history (the Acts) in this respect, is plainly undesigned, and so a proof of genuineness (see PALEY, Horæ Paulinæ). Galatians 4:31
31. So then—The oldest manuscripts read, "Wherefore." This is the conclusion inferred from what precedes. In Ga 3:29 and Ga 4:7, it was established that we, New Testament believers, are "heirs." If, then, we are heirs, "we are not children of the bond woman (whose son, according to Scripture, was 'not to be heir,' Ga 4:30), but of the free woman (whose son was, according to Scripture, to be heir). For we are not "cast out" as Ishmael, but accepted as sons and heirs.