[1]Ludwig Wittgenstein. Philosophical Investigations (3rd Ed.), G.E.M. Anscombe (Trans.). MacMillan, New York, 1953, 1958, 1966, 1968.
[2]References to the numbered sections of the Investigations will be of the form [I.x]; references to sections of the Tractatus will be of the form [T.x], where x is the section number.
[3]Ludwig Wittgenstein. Tractatus Logico-Philosophicus (2nd Ed.), D.F. Pears and B.F. McGuinness (Trans.). Routledge, London, 1921, 1961, 1974. Most of my references to the Tractatus refer to Pears and McGuiness. However, I chose to use atomic fact where they translate state of affairs, following the Ramsey-Ogden translation: F.P. Ramsey and C.K. Ogden (Trans.), Routledge, London, 1922. Abridged in: J. Baillie. Contemporary Analytic Philosophy, A. Kenny (Ed.), pp. 75-95. Prentice Hall, Upper Saddle River, New Jersey, 1921, 1922, 1997.
[4]F.H. Bradley. Writings on Logic and Metaphysics. Clarendon Press, Oxford, 1883, 1893, 1897, 1914, 1922, 1935, 1994.
[5]Allan F. Randall. Truth, Coherence and Correspondence in the Metaphysics of F.H. Bradley. http://www.elea.org/Bradley/, Toronto, 1996.
[6]Peter Roosen-Runge, to whom I am indebted for many useful discussions about the Tractatus, suggests using the term "state" instead of "fact". This is a better term for those who wish to use Wittgenstein's system but drop his splitting of facts into positive and negative. See: Peter Roosen-Runge, States and Propositions, http://www.cs.yorku.ca/~peter/MH/states-props.html, 1997.
[7]I am indebted to R.E. Tully for challenging me with this example.
[8]However, as we will see later, our hypothetical idealist Wittgenstein would actually be quite close to some later idealists like F.H. Bradley, who were becoming more and more objective in their idealism and less mentalist. Bradley, in many ways, comes close to leaving mentalism behind altogether.
[9]For an introduction to the metaphysical significance of the lambda-calculus, and its use in a purely objective, nonmentalist idealism, see: Allan F. Randall. Computational Platonism. http://www.elea.org/Plato/, Toronto, 1995, 1996.
[10]Wittgenstein's description of the NOR operator [T.5.5-5.5.1] is somewhat obscure. Russell, in his introduction to the Tractatus, clarifies the role of the universal operator and identifies it clearly with NOR. However, Wittgenstein's operator may sound more like NAND on a first reading than NOR, but it is actually NOR. This is because he expresses NOR in a form that sounds more like (~a & ~b & ~c) than the equivalent ~(a v b v c). NAND, the logical inverse of NOR, would be ~(a & b & c), or equivalently (~a v ~b v ~c).
[11]Gottfried Wilhelm Leibniz. G.W. Leibniz's Monadology: an edition for students, Nicholas Rescher (Trans.). University of Pittsburgh Press, Pittsburgh, 1720, 1991. [Leibniz's views were highly computational for his time. It is worth noting that he contributed greatly to the foundations of computer science, inventing the binary number system and perfecting Pascal's calculating machine.]
[12]I have worded these in terms of Tractarian logic, although personally I prefer those formulations that do not pretend to refer to "truth" values (for instance the lambda-calculus and Turing machines).
[13]David Hume. A Treatise of Human Nature. Penguin Books, London, 1739, 1740, 1969.
[14]George Berkeley. A Treatise concerning the Principles of Human Knowledge. In: A Treatise concerning the Principles of Human Knowledge. Three Dialogues between Hylas and Philonous, G.J. Warnock (Ed.), pp. 43-146. Open Court, La Salle, Illinois, 1710, 1713, 1962.
[15]Allan F. Randall. Truth, Coherence and Correspondence in the Metaphysics of F.H. Bradley. http://www.elea.org/Bradley/, Toronto, 1996.
[16]Reducing the infinite set to a single object is natural for Bradley, since he holds a strictly logic-based metaphysics, and therefore any two indiscernible objects are automatically really just the same object. Wittgenstein echoes this principle in the Tractatus, at least when he sticks to logic [T.2.0233-2331].
[17]Wittgenstein does not actually insist that this is the only meaning of "meaning" [I.43], but it is the sense he gives the word through most of the Investigations, and I will take it as his working definition.
[18]Saul A. Kripke. Wittgenstein on Rules and Private Language: an elementary exposition. Harvard University Press, Cambridge, Massachusetts, 1982.
[19]I use Kripke's example because it is a useful catch-all for the numerous paradoxes and puzzles presented by Wittgenstein in the Investigations, not because I am advocating Kripke's particular interpretation. I will talk about Kripke's example as if it were used by Wittgenstein himself, assuming the reader understands that Wittgenstein made equivalent points with his own examples.
[20]For Bradley,4 this manipulation was necessarily mental in nature. However, he did not think this meant it was necessarily carried out by a particular mind. It was a principle of experience that did not itself constitute an actual experience in, for instance, the mind of God. Thus, Bradley's absolute idealist whole was already heavily dementalized and more analytic in nature than, for example, the absolute of Hegel [G.W.F. Hegel. Phenomenology of Spirit (5th Ed.), A.V. Miller (Trans.). Oxford University Press, Oxford, 1807, 1952, 1977].
[21]Although beyond the scope of this essay, it is worth noting that quantum theory, our current best scientific theory as to the nature of reality, seems to force us to recognize the reality of other possible worlds. Our particular world can only be understood scientifically if embedded in an ensemble of possible worlds. Quantum theory thus provides strong empirical evidence for the kind of nonmentalist idealism I advocate throughout this essay. For more details on my own views and pointers to the literature, see: Allan F. Randall. Quantum Superposition, Necessity and the Identity of Indiscernibles. http://www.elea.org/Indiscernibles/, Toronto, 1996.