[1] Cf. _La Logique de Leibnitz_, ch. vii., by L. Couturat (Paris,
1901).
[2] Cf. _The Principles of Mathematics_, by Bertrand Russell
(Cambridge, 1903).
[3] Cf. _Formulaire mathématique_ (Turin, ed. of 1903); earlier
formulations of the bases of arithmetic are given by him in the
editions of 1898 and of 1901. The variations are only trivial.
[4] Cf. Russell, _loc. cit._, pp. 199-256.
[5] The first unqualified explicit statement of _part_ of this
definition seems to be by B. Peirce, "Mathematics is the science
which draws necessary conclusions" (_Linear Associative Algebra_, §
i. (1870), republished in the _Amer. Journ. of Math._, vol. iv.
(1881)). But it will be noticed that the second half of the
definition in the text--"from the general premisses of all
reasoning"--is left unexpressed. The full expression of the idea and
its development into a philosophy of mathematics is due to Russell,
_loc. cit._
[6] "Una questione sui numeri transfiniti," _Rend. del circolo mat.
di Palermo_, vol. xi. (1897); and Russell, _loc. cit._, ch. xxxviii.
[7] Cf. Russell, _loc. cit._, ch. x.
[8] Cf. _Pragmatism: a New Name for some Old Ways of Thinking_
(1907).
[9] Due to Bertrand Russell, cf. "Mathematical Logic as based on the
Theory of Types," _Amer. Journ. of Math._ vol. xxx. (1908). It is
more fully explained by him, with later simplifications, in
_Principia mathematica_ (Cambridge).
[10] Cf. Stanley's _Eastern Church_, Lecture v.
[11] Cf. _A Short History of Mathematics_, by W. W. R. Ball.
MATHER, COTTON (1663-1728), American Congregational clergyman and
author, was born in Boston, Massachusetts, on the 12th of February 1663.
He was the grandson of Richard Mather, and the eldest child of Increase
Mather (q.v.), and Maria, daughter of John Cotton. After studying under
the famous Ezekiel Cheever (1614-1708), he entered Harvard College at
twelve, and graduated in 1678. While teaching (1678-1685), he began the
study of theology, but soon, on account of an impediment in his speech,
discontinued it and took up medicine. Later, however, he conquered the
difficulty and finished his preparation for the ministry. He was elected
assistant pastor in his father's church, the North, or Second, Church of
Boston, in 1681 and was ordained as his father's colleague in 1685. In
1688, when his father went to England as agent for the colony, he was
left at twenty-five in charge of the largest congregation in New
England, and he ministered to it for the rest of his life. He soon
became one of the most influential men in the colonies. He had much to
do with the witchcraft persecution of his day; in 1692 when the
magistrates appealed to the Boston clergy for advice in regard to the
witchcraft cases in Salem he drafted their reply, upon which the
prosecutions were based; in 1689 he had written _Memorable Providences
Relating to Witchcraft and Possessions_, and even his earlier diaries
have many entries showing his belief in diabolical possession and his
fear and hatred of it. Thinking as he did that the New World had been
the undisturbed realm of Satan before the settlements were made in
Massachusetts, he considered it natural that the Devil should make a
peculiar effort to bring moral destruction on these godly invaders. He
used prayer and fasting to deliver himself from evil enchantment; and
when he saw ecstatic and mystical visions promising him the Lord's help
and great usefulness in the Lord's work, he feared that these
revelations might be of diabolic origin. He used his great influence to
bring the suspected persons to trial and punishment. He attended the
trials, investigated many of the cases himself, and wrote sermons on
witchcraft, the _Memorable Providences_ and _The Wonders of the
Invisible World_ (1693), which increased the excitement of the people.
Accordingly, when the persecutions ceased and the reaction set in, much
of the blame was laid upon him; the influence of Judge Samuel Sewall,
after he had come to think his part in the Salem delusion a great
mistake, was turned against the Mathers; and the liberal leaders of
Congregationalism in Boston, notably the Brattles, found this a
vulnerable point in Cotton Mather's armour and used their knowledge to
much effect, notably by assisting Robert Calef (d. c. 1723) in the
preparation of _More Wonders of the Invisible World_ (1700) a powerful
criticism of Cotton Mather's part in the delusion at Salem.
Mather took some part as adviser in the Revolution of 1689 in
Massachusetts. In 1690 he became a member o£ the Corporation (probably
the youngest ever chosen as Fellow) of Harvard College, and in 1707 he
was greatly disappointed at his failure to be chosen president of that
institution. He received the degree of D.D. from the University of
Glasgow in 1710, and in 1713 was made a Fellow of the Royal Society.
Like his father he was deeply grieved by the liberal theology and Church
polity of the new Brattle Street Congregation, and conscientiously
opposed its pastor Benjamin Colman, who had been irregularly ordained in
England and by a Presbyterian body; but with his father he took part in
1700 in services in Colman's church. Harvard College was now controlled
by the Liberals of the Brattle Street Church, and as it grew farther and
farther away from Calvinism, Mather looked with increasing favour upon
the college in Connecticut; before September 1701 he had drawn up a
"scheme for a college," the oldest document now in the Yale archives;
and finally (Jan. 1718) he wrote to a London merchant, Elihu Yale, and
persuaded him to make a liberal gift to the college, which was named in
his honour. During the small-pox epidemic of 1721 he attempted in vain
to have treatment by inoculation employed, for the first time in
America; and for this he was bitterly attacked on all sides, and his
life was at one time in danger; but, nevertheless, he used the treatment
on his son, who recovered, and he wrote _An Account of the Method and
further Success of Inoculating for the Small Pox in London_ (1721). In
addition he advocated temperance, missions, Bible societies, and the
education of the negro; favoured the establishing of libraries for
working men and of religious organizations for young people, and
organized societies for other branches of philanthropic work. His later
years were clouded with many sorrows and disappointments; his relations
with Governor Joseph Dudley were unfriendly; he lost much of his former
prestige in the Church--his own congregation dwindled--and in the
college; his uncle John Cotton was expelled from his charge in the
Plymouth Church; his son Increase turned out a ne'er-do-well; four of
his children and his second wife died in November 1713; his wife's
brothers and the husbands of his sisters were ungodly and violent men;
his favourite daughter Katherine, who "understood Latin and read Hebrew
fluently," died in 1716; his third wife went mad in 1719; his personal
enemies circulated incredible scandals about him; and in 1724-1725 he
saw a Liberal once more preferred to him as a new president of Harvard.
He died in Boston on the 13th of February 1728 and is buried in the
Copps Hill burial-ground, Boston. He was thrice married--to Abigail
Phillips (d. 1702) in 1686, to Mrs Elizabeth Hubbard (d. 1713) in 1703,
and in 1715 to Mrs Lydia George (d. 1734). Of his fifteen children only
two survived him.
Though self-conscious and vain, Cotton Mather had on the whole a noble
character. He believed strongly in the power of prayer and repeatedly
had assurances that his prayers were heard; and when he was disappointed
by non-fulfilment his grief and depression were terrible. His spiritual
nature was high-strung and delicate; and this condition was aggravated
by his constant study, his long fasts and his frequent vigils--in one
year, according to his diary, he kept sixty fasts and twenty vigils. In
his later years his diaries have less and less of personal detail, and
repeated entries prefaced by the letters "G.D." meaning Good Device,
embodying precepts of kindliness and practical Christianity. He was
remarkable for his godliness, his enthusiasm for knowledge, and his
prodigious memory. He became a skilled linguist, a widely read
scholar--though much of his learning was more curious than useful--a
powerful preacher, a valued citizen, and a voluminous writer, and did a
vast deal for the intellectual and spiritual quickening of New England.
He worked with might and main for the continuation of the old theocracy,
but before he died it had given way before an increasing
Liberalism--even Yale was infected with the Episcopalianism that he
hated.
Among his four hundred or more published works, many of which are
sermons, tracts and letters, the most notable is his _Magnalia Christi
Americana: or the Ecclesiastical History of New England, from Its
First Planting in the Year 1620 unto the Year of Our Lord, 1698_.
Begun in 1693 and finished in 1697, this work was published in London,
in 1702, in one volume, and was republished in Hartford in 1820 and in
1853-1855, in two volumes. It is in seven books and concerns itself
mainly with the settlement and religious history of New England. It is
often inaccurate, and it abounds in far-fetched conceits and odd and
pedantic features. Its style, though in the main rather unnatural and
declamatory, is at its best spontaneous, dignified and rhythmical; the
book is valuable for occasional facts and for its picture of the
times, and it did much to make Mather the most eminent American writer
of his day. His other writings include _A Poem Dedicated to the Memory
of the Reverend and Excellent Mr Urian Oakes_ (1682); _The Present
State of New England_ (1690); _The Life of the Renowned John Eliot_
(1691), later included in Book III. of the _Magnalia; The Short
History of New England_ (1694); _Bonifacius_, usually known as _Essays
To Do Good_ (Boston, 1710; Glasgow, 1825; Boston, 1845), one of his
principal books and one which had a shaping influence on the life of
Benjamin Franklin; _Psalterium Americanum_ (1718), a blank verse
translation of the Psalms from the original Hebrew; _The Christian
Philosopher: A Collection of the Best Discoveries in Nature, with
Religious Improvements_ (1721); _Parentator_ (1724), a memoir of his
father; _Ratio Disciplinae_ (1726), an account of the discipline in
New England churches; _Manuductio ad Ministerium: Directions for a
Candidate of the Ministry_ (1726), one of the most readable of his
books. He also left a number of works in manuscript, including
diaries, a medical treatise and a huge commentary on the Bible,
entitled "Biblia Americana."
See _The Life of Cotton Mather_ (Boston, 1729), by his son, Samuel
Mather; William B. O. Peabody, _The Life of Cotton Mather_ (1836) (in
Jared Sparks's "Library of American Biography," vol. vi.); Enoch Pond,
_The Mather Family_ (Boston, 1844); John L. Sibley, _Biographical
Sketches of Graduates of Harvard University_, vol. iii. (Cambridge,
1885); Barrett Wendell, _Cotton Mather, the Puritan Priest_ (New York,
1891), a remarkably sympathetic study and particularly valuable for
its insight into (and its defence of) Mather's attitude toward
witchcraft; Abijah P. Marvin, _The Life and Times of Cotton Mather_
(Boston, 1892); M. C. Tyler, _A History of American Literature during
the Colonial Period_, vol. ii. (New York, 1878); and Barrett Wendell,
_A Literary History of America_ (New York, 1900).
Cotton Mather's son, SAMUEL MATHER (1706-1785), also a clergyman,
graduated at Harvard in 1723, was pastor of the North Church, Boston,
from 1732 to 1742, when, owing to a dispute among his congregation over
revivals, he resigned to take charge of a church established for him in
North Bennett Street.
Among his works are _The Life of Cotton Mather_ (1729); _An Apology
for the Liberties of the Churches in New England_ (1738), and _America
Known to the Ancients_ (1773). (W. L. C.*)
MATHER, INCREASE (1639-1723), American Congregational minister, was born
in Dorchester, Massachusetts, on the 21st of June 1639, the youngest son
of Richard Mather.[1] He entered Harvard in 1651, and graduated in 1656.
In 1657, on his eighteenth birthday, he preached his first sermon; in
the same year he went to visit his eldest brother in Dublin, and studied
there at Trinity College, where he graduated M.A. in 1658. He was
chaplain to the English garrison at Guernsey in April-December 1659 and
again in 1661; and in the latter year, refusing valuable livings in
England offered on condition of conformity, he returned to America. In
the winter of 1661-1662 he began to preach to the Second (or North)
Church of Boston, and was ordained there on the 27th of May 1664. As a
delegate from Dorchester, his father's church, to the Synod of 1662, he
opposed the Half-Way Covenant adopted by the Synod and defended by
Richard Mather and by Jonathan Mitchell (1624-1668) of Cambridge; but
soon afterwards he "surrendered a glad captive" to "the truth so
victoriously cleared by Mr Mitchell," and like his father and his son
became one of the chief exponents of the Half-Way Covenant. He was
bitterly opposed, however, to the liberal practices that followed the
Half-Way Covenant and (after 1677) in particular to "Stoddardeanism,"
the doctrine of Solomon Stoddard (1643-1729) that all "such Persons as
have a good Conversation and a Competent Knowledge may come to the
Lord's Supper," only those of openly immoral life being excluded. In May
1679 Mather was a petitioner to the General Court for the call of a
Synod to consider the reformation in New England of "the Evils that have
Provoked the Lord to bring his Judgments,"[2] and when the "Reforming
Synod" met in September it appointed him one of a committee to draft a
creed; this committee reported in May 1680, at the Synod's second
session, of which Mather was moderator, the Savoy Declaration (slightly
modified, notably in ch. xxiv., "Of the Civil Magistrate"), which was
approved but was not made mandatory on the churches by the General
Court, and in 1708 was reaffirmed at Saybrook, Connecticut. With the
Cambridge Platform of 1646, drafted by his father, the Confession of
1680, for which Increase Mather was largely responsible, was printed as
a book of doctrine and government for the churches of Massachusetts.
After the threat of a _Quo Warranto_ writ in 1683 for the surrender of
the Massachusetts charter, Mather used all his tremendous influence to
persuade the colonists not to give up the charter; and the Boston
freemen unanimously voted against submission. The royal agents
immediately afterwards sent to London a treasonable letter, falsely
attributed to Mather; but its spuriousness seems to have been suspected
in England and Mather was not "fetch'd over and made a Sacrifice." He
became a leader in the opposition to Sir Edmund Andros, to his secretary
Edward Randolph, and to Governor Joseph Dudley. He was chosen by the
General Court to represent the colony's interests in England, eluded
officers sent to arrest him,[3] and in disguise boarded a ship on which
he reached Weymouth on the 6th of May 1688. In London he acted with Sir
Henry Ashurst, the resident agent, and had two or three fruitless
audiences with James II. His first audience with William III. was on the
9th of January 1689; he was active in influencing the Commons to vote
(1689) that the New England charters should be restored; and he
published _A Narrative of the Miseries of New-England, By Reason of an
Arbitrary Government Erected there Under Sir Edmund Andros_ (1688), _A
Brief Relation for the Confirmation of Charter Privileges_ (1691), and
other pamphlets. In 1690 he was joined by Elisha Cooke (1638-1715) and
Thomas Oakes (1644-1719), additional agents, who were uncompromisingly
for the renewal of the old charter. Mather, however, was instrumental in
securing a new charter (signed on Oct. 7, 1691), and prevented the
annexation of the Plymouth Colony to New York. The nomination of
officers left to the Crown was reserved to the agents. Mather had
expressed strong dissatisfaction with the clause giving the governor the
right of veto, and regretted the less theocratic tone of the charter
which made all freemen (and not merely church members) electors. With
Sir William Phips, the new governor, a member of Mather's church, he
arrived in Boston on the 14th of May 1692. The value of his services to
the colony at this time is not easily over-estimated. In England he won
the friendship of divines like Baxter, Tillotson and Burnet, and
effectively promoted the union in 1691 of English Presbyterians and
Congregationalists. He was at heavy expense throughout his stay, and
even greater than his financial loss was his loss of authority and
control in the church and in Harvard College because of his absence.
Mather had been acting president of Harvard College in 1681-1682, and in
June 1685 he again became acting president (or rector), but still
preached every Sunday in Boston and would not comply with an order of
the General Court that he should reside in Cambridge. In 1701 after a
short residence there he returned to Boston and wrote to the General
Court to "think of another President for the Colledge." The opposition
to him had been increasing in strength, his resignation was accepted,
and Samuel Willard took charge of the college as vice-president,
although he also refused to reside in Cambridge. That Mather's
administration of the college was excellent is admitted even by his
harsh critic, Josiah Quincy, in his _History of Harvard University_.[4]
The Liberal party, which now came into control in the college repeatedly
disappointed the hopes of Cotton Mather (q.v.) that he might be chosen
president, and by its ecclesiastical laxness and its broader views of
Church polity forced the Mathers to turn from Harvard to Yale as a truer
school of the prophets.
The Liberal leaders, John Leverett (1662-1724), William Brattle
(1662-1713)--who graduated with Leverett in 1680, and with him as tutor
controlled the college during Increase Mather's absence in
England--William Brattle's eldest brother, Thomas Brattle (1658-1713),
and Ebenezer Pemberton (1671-1717), pastor of the Old South Church,
desired an "enrichment of the service," and greater liberality in the
matter of baptism. In 1697 the Second Boston Church, in which Cotton
Mather had been his father's colleague since 1685, upbraided the
Charlestown Church "for betraying the liberties of the churches in their
late putting into the hands of the whole inhabitants the choice of a
minister." In 1699 Increase Mather published _The Order of the Gospel_,
which severely (although indirectly), criticized the methods of the
"Liberals" in establishing the Brattle Street Church and especially the
ordination of their minister Benjamin Colman by a Presbyterian body in
London; the Liberals replied with _The Gospel Order Revived_, which was
printed in New York to lend colour to the (partly true) charge of its
authors that the printers of Massachusetts would print nothing hostile
to Increase Mather.[5] The autocracy of the Mathers in church, college,
colony and press, had slipped from them. The later years of Mather's
life were spent almost entirely in the work of the ministry, now
beginning to be a less varied career than when he entered on it. He died
on the 23rd of August 1723. He married in 1662 Maria, daughter of Sarah
and John Cotton. His first wife died in 1714; and in 1715 he married Ann
Lake, widow of John Cotton, of Hampton, N.H., a grandson of John Cotton
of Boston.
Increase Mather was a great preacher with a simple style and a splendid
voice, which had a "Tonitruous Cogency," to quote his son's phrase. His
style was much simpler and more vernacular than his son's. He was an
assiduous student, commonly spending sixteen hours a day among his
books; but his learning (to quote Justin Winsor's contrast between
Increase and Cotton Mather) "usually left his natural ability and his
education free from entanglements." He was not so much self-seeking and
personally ambitious as eager to advance the cause of the church in
which he so implicitly believed. That it is a mistake to consider him a
narrow churchman is shown by his assisting in 1718 at the ordination of
Elisha Callender in the First Baptist Church of Boston. Like the most
learned men of his time he was superstitious and a firm believer in
"praesagious impressions"; his _Essay for the Recording of Illustrious
Providences: Wherein an Account is Given of many Remarkable and very
Memorable Events which have Hapned in this Last Age, Especially in New
England_ (1684) shows that he believed only less thoroughly than his son
in witchcraft, though in his _Cases of Conscience Concerning Evil
Spirits_ (1693) he considered some current proofs of witchcraft
inadequate. The revulsion of feeling after the witchcraft delusion
undermined his authority greatly, and Robert's Calef's _More Wonders of
the Spiritual World_ (1700) was a personal blow to him as well as to his
son. With Jonathan Edwards, than whom he was much more of a man of
affairs, and with Benjamin Franklin, whose mission in England somewhat
resembled Mather's, he may be ranked among the greatest Americans of the
period before the War of Independence.
The first authority for the life of Increase Mather is the work of his
son Cotton Mather, _Parentator: Memoirs of Remarkables in the Life and
Death of the Ever Memorable Dr Increase Mather_ (Boston, 1724); there
are also a memoir and constant references in Cotton Mather's
_Magnalia_ (London, 1702) especially vol. iv.; there is an excellent
sketch in the first volume of J. L. Sibley's _Biographical Sketches of
Graduates of Harvard University_ (Cambridge, 1873), with an exhaustive
list of Mather's works (about 150 titles); there is much valuable
matter in Williston Walker's _Ten New England Leaders_ (New York,
1901) and in his _Creeds and Platforms of Congregationalism_ (New
York, 1893); for literary criticism of the Mathers see ch. xii. of M.
C. Tyler's _History of American Literature, 1607-1676_ (New York,
1878), and Barrett Wendell's _Cotton Mather_ (New York, 1891).
Mather's worth has been under-estimated by Josiah Quincy, Justin
Winsor and other historians out of sympathy with his ecclesiastical
spirit, who represent him as only an ambitious narrow-minded schemer.
(R. We.)