son of the sun
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(general) the past |
Vacation - Did ok in everything at school. - Thinking about hosting the blog at my own computer. - Not in the mood to write here. [ 21/07/2004 11:21 | 11 comentários ] Spoonful I have always wondered what it meant. Here. I'll get signifying down your high yeller's stingaree 'cause of my rider's brown goo'old jelly roll. [ 16/07/2004 13:00 | 1 comentário ] "Fahrenheit 9/11" "Britney Spears was not alone". I don't think Fahrenheit 9/11 has untrue information. It is populist advertisement, and as such, it uses hyperbolic means. A bad quality screener, but worth it. [ 1/07/2004 14:44 | 3 comentários ] Philosopher: Bozo Paradox, Berry's A simplified variant of Richard's paradox first published in a paper of Russell's in 1906 and attributed to the philosopher George Berry. Consider the [1] least [2] natural [3] number [4] not [5] definable [6] in [7] fewer [8] than [9] twelve [10] words [11]; this number is itself here defined in fewer than twelve words, a contradiction. (Since the number of descriptions less than twelve words long in the English language must be finite, and the set of natural numbers is infinite, there must be at least one natural [1] number [2] which [3] cannot [4] be [5] identified [6] with [7] a [8] twelve-word [9] description [10] of [11] itself [12], so the least such natural number is well-defined.) Routledge Encyclopedia of Philosophy, Version 1.0, London: Routledge Numeration in brackets courtesy Eduardo Pinheiro. [ 1/07/2004 02:16 | 5 comentários ] More Fun From Gödel's Theorems, under Philosophical Applications: There have been many attempts to apply Gödel’s incompleteness theorems to philosophical concerns. For the most part, these have to do with issues in the philosophy of mathematics and the philosophy of mind, though there have also been attempts in other areas of philosophy, and also in areas of intellectual life not lying wholly within philosophy (see Thomas 1995). Here we consider some of the better-known and/or better-developed applications. In the philosophy of mathematics the chief application is to Hilbert’s programme which is widely believed to have been refuted by G2. This is so because G2 is seen as prohibiting the finitary consistency proof that Hilbert demanded for classical mathematics (see, for example, Gödel 1958; Kitcher 1976; Kreisel 1958; Prawitz 1981; Resnik 1974; Smorynski 1977, 1985; Simpson 1988; for a dissenting view see Detlefsen 1986, 1990). G1 has been presented as having a similar effect through its exhibition of a recognizedly true real sentence of classical mathematics (namely, G) that is not finitarily provable (see Smorynski 1977, 1985; Hilbert’s programme and formalism §4). Gödel’s theorems have also been used to argue against logicism. One example of this is in Hellman (1981), which maintains that G2 implies that no finitely axiomatizable logicist system exists. For non-finitely axiomatizable systems, the claim is weaker: such systems may exist, but G2 prohibits our being able to know of any particular system that it is one of them. A major attraction of Hellman’s argument is that, unlike other arguments against logicism, it does not require that one know where to draw the line between logic and non-logic. Gödel’s theorems have also figured in philosophical discussions concerning the nature of proof. Among the more interesting of these are Myhill (1960) and Reinhardt (1985, 1986). Myhill argues that G1 and G2 establish that for any correct system containing arithmetic there are ‘correct’ inferences - inferences to whose acceptability the user of the system is rationally committed - that cannot be captured in it. There is thus, he says, an absolute epistemic notion of provability (neither syntactic, nor semantic nor psychological in character), according to which Gödel’s undecidable sentences are provable. Hence, this notion of absolute provability is not formalizable. Reinhardt (1985, 1986) refines aspects of this argument and also argues that if ‘humanly provable’ means ‘formally provable’ there must be absolutely undecidable sentences of arithmetic. He urges the view that G2 is an epistemic phenomenon - one that arises from the epistemic properties of that type of belief that mathematical proof is supposed to sponsor. These properties, it turns out, lead to something tantamount to the derivability conditions. In a related vein, Dummett (1963) has used G1 to argue that mathematical proof (including that which lies within a well-circumscribed subject area) is an ‘indefinitely extensible concept’ and cannot therefore be identified with derivation in any formal system. Since, he believes, the meanings of mathematical statements are to be given in terms of a concept of proof, this means that the notion of proof that determines the meanings of mathematical statements must be an essentially vague and unformalizable one. In this he sounds a theme similar in certain respects to one associated with the intuitionist, Brouwer. Chaitin (1974, 1975, 1982) offers an analysis of Gödel’s theorems that links them to limits on the extent to which information in a given mathematical field can be ‘compressed’ into a formal system. They thus, according to Chaitin, point up the need for a continuing search for new axioms (see Computability and information). This view of the relationship between Gödel’s theorems and algorithmic information theory is challenged in van Lambalgem (1989). In the philosophy of mind and metaphysics, the main application of the theorems is to the question of mechanism (‘Do human-like minds have only such capabilities as are simulable by computational devices?’ - see Mind, computational theories of §5) and, more generally, to the question of materialism (‘Are all objects, events and/or forces in the world reducible to physical matter and its physical properties?’ - see Materialism; Materialism in the philosophy of mind). Gödel himself (1951) presented such an application, arguing that either the mind of the human mathematician cannot be codified by any formal system or that there exist absolutely unsolvable arithmetical problems of an elementary sort. On the former alternative, supposing Church’s thesis, mechanism clearly fails. On the latter, Gödel argued, materialism fails. For if there are mathematical problems that are absolutely unsolvable, then mathematics is not our own creation; and if this is so, its objects must exist independently of us, in which case a materialist view of reality fails. See Wang (1993) for a useful discussion of Gödel’s argument. The most well-known anti-mechanist argument is that of Lucas (1961), modified by Benacerraf (1967) and repeated in Penrose (1989, 1994). In outline, the argument is as follows. (1) If human minds were mechanizable (or, in some versions, if they could be known to be identical to some particular machine), then, by G2, they could not know that their beliefs are consistent. But (2) human minds can know that their beliefs are consistent; therefore, (3) they are not mechanizable (or, as in Benacerraf’s variant, they cannot be known to be identical to any particular machine). Lucas’ argument has been widely criticized (see, for example, Boolos 1968; Chihara 1972; Dennett 1978; Putnam 1960 (before the fact); Smart 1961; Wang 1974; Webb 1968, 1980). Its chief difficulty is perhaps one that is partially concealed by the unclarity of premise (2). It is true that humans are capable of examining various of their sets of beliefs and arriving at credible judgments regarding their consistency. It does not follow from this, however, that among the sets of beliefs they can judge to be consistent are some that contain that very consistency judgment itself. Indeed, if fixing a set of beliefs is a typical precondition for its evaluation, it would seem that consistency judgments do not typically belong to the sets to which they are applied; for at the time the beliefs are fixed such judgments would not yet have been made and so would not exist as elements of the set being evaluated. For consistency evaluations not requiring such prior fixing of beliefs, on the other hand, it is not clear that G2 applies. Either way, Lucas-type arguments face serious difficulties (see Detlefsen 1995). Routledge Encyclopedia of Philosophy, Version 1.0, London: Routledge [ 1/07/2004 02:02 ] Absolute Fun The Routledge Encyclopedia of Philosophy, a ten-volume printed work of which I downloaded the CD-ROM version on emule. Look at these excerpts: From the entry Yoruba epistemology When ìmò and ìgbàgbó are challenged and an argument or àríyàn jiyàn results, the tactics recommended for resolving the dispute are complex and interwoven. Often the disputants do not share the same (relevant) first-hand experiences. This may be corrected by a process of verification so long as it is possible to test the claim empirically and thereby enable people to see the results for themselves: counterclaims about the effectiveness of a certain medicine, for example, can be checked by testing the treatment. When such testing is impossible, there is no way in which the ìmò of one person can become the ìmò of another. In such a case, the only recourse is to a process of justification. The parties concerned should each explain (àlàyé) their own position by giving a full account of relevant first- or second-hand experiences. Any witnesses whose experiences may be relevant should also be called in the hope that their testimony will help decide which account, if any, should be favoured. Such things as the moral character, or ìwà, of each participant must be considered, as this might affect the reliability of information. From the entry Tibetan Philosophy The gSang-phu positions were strongly criticized by Sa skya Pa<*ita, the Tibetan who probably came closest to faithfully representing the positions of later Indian epistemology and logic. In his Rigs gter (The Treasure of Reasoning), we find a repeated trenchant critique of ‘Tibetans’ or ‘Tibetans who pride themselves on being logicians’ - polemical shorthand for the gSang-phu philosophers. None the less, gSang-phu thought had a persistent appeal due to what must have been its extremely seductive appearance of subtlety and rigour - indeed, as D. Jackson (1987: 137-8) points out, ]#kya mchog ldan reported that in his time, in the fifteenth century, people often felt that the ‘Summaries’ were subtle and proven correct, while the Rigs gter was ‘‘extremely rough’. From the entry Derrida, Jacques Although he has seemed, to some critics, merely to duplicate the writing strategies of Nietzsche, Derrida expresses some reservations about certain of Nietzsche’s gestures. For example, while he notes that the term ‘woman’ functions in Nietzsche’s writings as a trope for non-truth - and thus as an exemplary textual ‘site’ for deconstruction - he also calls attention to the violence of such gestures. Derrida seeks a strategy that would exploit the subversive potential of marginal subject positions without reifying the logic of marginality. In this way, deconstruction becomes a way of thinking about the political. From his earliest writings Derrida has been concerned with the relationship between justice and violence. Taking off from Heidegger’s essay on the Anaximander Fragment (1946), Derrida engages Levinas and others on the question of whether or not it is possible to thematize a purely non-violent conception of justice. From the entry Kabbalah The Kabbalist communities in Palestine were also close in their practices to Sufism, which was no doubt present in the wider community of the area. They advocated meditation and breath control in quiet dark places in order to attain illumination of the soul. The system of techniques used during these retreats included ritual purity, silence, fasting, restrictions of sleep and food, deep trust in God and the constant repetition of the divine name as aspects of the path to ecstasy and personal fulfilment. Sufism may even have influenced the types of mysticism followed in the Hasidism of the eighteenth century, which, it must be remembered, first developed in what had been a Turkish province of Poland. Shabbetai Zvi (d.1675) came into close contact with the Sufis even before his conversion to Islam, and his followers in the Ottoman empire continued to follow a form of worship which incorporated many Sufi practices. The close contact between Islamic and Jewish culture, which existed over a long and especially rich period, undoubtedly played its part in defining the specific character of Jewish mysticism. From the entry Randomness The point is this: what determines the patterns that must be broken for an object to be random is not some objective feature of the world - randomness is not a natural kind. Rather, what is random depends on the patterns that are specified within a given context and that must then be broken for an object to be random. What is counterintuitive about this approach is that randomness becomes thoroughly parasitic on the patterns with respect to which it is defined. Randomness on this view does not make sense until a given collection of patterns is specified. How then does this pattern-breaking approach to randomness relate to the four preceding approaches? For the computational complexity approach to randomness, the low complexity programs specify the patterns. For the mixing approach to randomness, far-from-equilibrium-states specify the patterns. For the simulation approach to randomness, statistical tests specify the patterns. The pattern-breaking approach to randomness also makes clear why chance is so often a dependable route to randomness: in many applications the patterns specified in advance identify a set of very small probability (for example, a full complement of statistical tests used to vet an RNG will typically designate as nonrandom only a tiny proportion of possible numerical sequences). Since small probability events are rare, chance will typically deliver objects or events that break all such patterns, that is, objects that are random in the pattern-breaking sense. And finally, from the entry Aristotle In treating the divine substance as a god, and hence as a being with a soul and an intellect, Aristotle attributes some mental life to it. But since it would be imperfect if it thought of objects outside itself (because it would not be self-sufficient), it thinks only of its own thinking. [ 1/07/2004 01:45 | 3 comentários ] |
