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German mathematician whose influential lecture at Paris, "Mathematical Problems" (1900), outlined the development of classical mathematics as the application of Kant's notion of a regulative principle. Hilbert's Grundlagen der Geometrie (Foundations of Geometry) (1899), "Axiomatisches Denken" ("Axiomatic Thinking") (1917), "Die Grundlagen der Mathematik" ("Foundations of Mathematics") (1926), and Principles of Mathematical Logic (1931) proposed the axiomatic formalization of mathematics in order to demonstrate consistency by syntactical or metamathematical methods.
Recommended Reading: Constance Reid, Hilbert (Copernicus, 1996) and Jeremy Gray and David Rowe, The Hilbert Problems: A Perspective on Twentieth Century Mathematics (Oxford, 2000).
Presocratic philosopher and mathematician who emphasized the use of empirical methods in pursuit of knowledge. Although his efforts to trisect the angle geometrically failed, they led Hippias to the discovery of the quadratrix, a curve satisfying the modern algebraic formula y = x tan[py/2a] , whose construction would render trisection of acute angles unproblematic.
Belief that social structures, events, and texts are best to be understood in the context of their historical development. Versions of this view were defended by Dilthey, Lukacs, and Gramsci. More recently, Popper and Hayek criticized the extreme version of this view, according to which the historical outcomes are inevitably determined. In the milder form embraced by Croce, Kuhn, and Gadamer, however, historicism is simply the notion that a purely ahistorical perspective on human affairs would be misleading.
Recommended Reading: H. Aram Veeser, The New Historicism Reader (Routledge, 1993); Paul Hamilton, Historicism: The New Critical Idiom (Routledge, 1996); Charles R. Bambach, Heidegger, Dilthey, and the Crisis of Historicism: History and Metaphysics in Heidegger, Dilthey, and the Neo-Kantians (Cornell, 1995); Robert D'Amico, Historicism and Knowledge (Routledge, 1992); and Karl Popper, The Poverty of Historicism (Routledge, 1993).
American moral philosopher. In Lesbian Ethics: Toward New Value (1988) Hoagland attributes a host of individual and social evils to their origin in a patriarchal and heterosexualist culture and proposes a moral revolution based on the formation of lesbian communities purposefully separated from the society at large.
Recommended Reading: For Lesbians Only: A Separatist Anthology, ed. by Sarah Lucia Hoagland and Julia Penelope (Onlywomen, 1992).
For a discussion of his life and works, see Hobbes.
American computer scientist and philosopher. In Gödel, Escher, Bach: an Eternal Golden Braid (1979), Hofstadter offers an insightful account of major developments in the metamathematics of recursive functions, tracing its importance for artificial intelligence research and human self-understanding through metaphorical comparisons with art and music. Hofstadter is also author of Metamagical Themas (1985) and co-editor (with Dan Dennett) of The Mind's I (1981).
Recommended Reading: Douglas R. Hofstadter, Fluid Concepts & Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought (Basic, 1996) and Douglas R. Hofstadter, Le Ton Beau De Marot: In Praise of the Music of Language (Basic, 1998).
Distinction between concepts or words. A homological term applies to itself, a heterological term does not. Thus, for example:
"short" and "having fewer than ten syllables" are homological terms;
"big" and "having more than ten syllables" are heterological terms.
Although "homological" is itself a homological term, a self-referential paradox arises when we consider the word "heterological." If we suppose that it applies to itself (thus being homological), then it is not heterological and does not apply to itself. But if we suppose that it does not apply to itself (thus being heterological), then it does apply to itself and is homological.
What, then, are we to make of an expression such as, "The smallest integer not namable in fewer than twenty syllables?"
Aristotle's distinction among different uses of a term: they are said to be homonymous if the uses are entirely distinct, synonymous if they are the same, and paronymous if they are different but related. Thus, for example:
In "Colleen is a cat," and "Garfield is a cat," "cat" is used homonymously.
In "Carter was president in 1978," and "Bush was president in 1990," "president" is used synonymously.
In "Jean was brave," and "What Jean did was brave," "brave" is used paronymously.
Recommended Reading: Aristotle's Categories and De Interpretatione, tr. by J.L. Ackrill (Oxford, 1975).
American educator, essayist, and poet. In such books as Ain't I a Woman? (1981), Feminist Theory: From Margin to Center (1984) one chapter of which is available on-line as "Ending Female Sexual Oppression", Talking Back: Thinking Feminist, Thinking Black (1989), Yearning: Race, Gender, and Cultural Politics (1990), Teaching to Transgress: Education as the Practice of Freedom (1994), Killing Rage: Ending Racism (1995), and Bone Black: Memoirs of a Girlhood (1996), hooks decries the implicit racism of the women's movement and eloquently describes how a complex interaction of racism, sexism, and classism contributes to the deplorable living conditions of non-white working women.
Recommended Reading: bell hooks, Feminism Is for Everybody: Passionate Politics (South End, 2000) and bell hooks, All About Love: New Visions (Harper, 2001).
American mathematician and computer scientist. During her service in the U.S. Navy (where she rose to the rank of Admiral) Hopper pioneered the development of programming languages (including COBOL) for digital computers and introduced use of the term "bug" to denote a software flaw.
Recommended Reading: Grace Murray Hopper and Steven L. Mandell, Understanding Computers (Wadsworth, 1990) and Nancy Whitelaw, Grace Hopper: Programming Pioneer.