Whiggishness
Whiggishness is a generic term of description for some approaches, in the fields of politics and historiography, which accept or adapt attitudes of the Whig politicians in the past of the United Kingdom. It is therefore not Whiggism, the party-political philosophy of the Whigs, but an identification in contemporary thought with an older strand. (Whiggery is ambiguous as a term: it may mean party politics or a general approach.)
Whiggishness in general terms makes much of progress, a reform agenda, constitutional government, and personal freedoms. It is the extraction of some sort of essence of Whig history, a more sharply-focussed term of art in historiography. Its antithesis can be seen in certain kinds of cultural pessimism.
Macaulay
The most commonly identified Whiggish position is that of Thomas Macaulay, a Liberal Party politician and popular historian. His attitudes, in fact, come from the historical cusp at which the Whigs became the Liberal Party; previously, one might say, they were a political faction, or grouping of various factions. Between 1688 and 1832, in British politics, to be a Whig was more a question of allegiance than ideology, for most politicians. Macaulay's History of England, coming at the end of this period, is mostly concerned with its beginning; but was placed in a setting in which the abdication of James II was a precondition of progress.
20th century takes
Preoccupation with the 'whiggish' was one aspect of the post-World War I re-evaluation of European history in general. Herbert Butterfield wrote, from one side, a celebrated critique of the so-called 'Whig theory of history'. Marxist historiography in general has emphasised the underside of history, as something blithely ignored by sweeping talk about 'progress' in the absence of discussion of its costs. In terms of British history the founding of the Whig party happens at the (somewhat ill-defined) end of the early modern period; and therefore discussion of the 'whiggish' bleeds over into discussion of 'the modern'. Self-identified whiggish historians (Butterfield was rather chary of identifying those in his sights) include J. H. Hexter.
History of science
One should, above all, be very careful about revisionist Whiggery, that is, reinterpreting the history of science during such a rich period of growth in order to construct a clear history of progress with "winners" (those who anticipated the theories which were ultimately to prevail, like evolution) and "losers" (those who advocated hypotheses which we no longer maintain, like "fixity of species").From A Handbook on the History of Modern Science
As teleology
In The Anthropic Cosmological Principle (see anthropic principle for details) John D. Barrow and Frank J. Tipler identify Whiggishness (Whiggery) with a teleological principle, of 'convergence' in history to liberal democracy.