\nProjective geometry is usually carried out in a real or complex space. If a theoretical perspex machine
\nwere defined to operate in such a space it would be able to do more than a Turing machine. For example, it
\nwould be able to decide if any given irrational number is equal to zero. Whereas a Turing machine faced
\nwith the decimal expansion of a small number with indefinitely many leading zeros after the decimal point
\nand before the significant figures is not be able to decide this case. It is interesting to note that
\nsuch a simple condition as implementing a perspex machine in a real space would falsify the Church-Turing
\nthesis. However, all contemporary, practical perspex machines are defined in a rational space.\n<\/p><\/blockquote>\n
So, theoretically, a Perspex machine can project its state to real numbers – not just rationals,
\nbut reals. Since the state of a Turing machine can’t represent an irrational number, but a real-valued
\nPerspex machine can<\/em>, the Perspex machine is more powerful than the Turing machine.<\/p>\n Of course, we can’t really build<\/em> a real-valued Perspex machine, because the values of
\nthe irrational numbers can’t be represented with finite state by an real machine. But hey, practical reality is no concern of Anderson. The Perspex machine uses real numbers, by golly, and therefore, it’s super-Turing.<\/p>\n This is truly pathetic. I can devise all sorts of computational devices that do all sorts of
\nthings that don’t exist in the real world. This is no big surprise – there are literally decades
\nof study of the theoretical property of machines that can’t be built. (They’re studied as part
\nof an effort to understand the nature of computation.) For example, there’s a class of problems studied by computer scientists that are based on using a Turing machine with a halting oracle<\/em>. A halting oracle is a theoretical component which can solve the halting problem: in a halting-oracle augmented Turing machine, I can<\/em> solve problems that can’t be solved on a real computing device. <\/p>\n All Anderson has done is introduced a new variant of this kind of theoretical but impossible machine. That’s it. He’s introduced an interesting model of computation – programming by projective geometry. He’s created a fancy name and foolish semantics for a “halt” instruction in this projective computation model. And he’s shown that this model, like every<\/em> other computation model that’s ever been devised,
\ncan be augmented by impossible extensions that allow it, in theory, to perform super-Turing computation. But in the real world, it can’t, because any real incarnation of the machine is limited to Turing-level
\ncomputation, because at any point in time, it can only have finite state – and there are no
\nfinite state representations of arbitrary irrational numbers. Eliminate the arbitrary irrationals,
\nand reduce to some countable set of numbers in the projective space, and you’re Turing equivalent.<\/p>\n And that’s it. That’s his glorious result.<\/p>\n
What he does with that introduces whole new levels of inanity. I’ll give you one taste: Anderson’s
\nexplanation of original sin:<\/p>\n
\nThe walnut cake theorem guarantees that any robot we make, that does not have
\naccess to a time machine, will be prone to error. No matter how well it is made, no
\nmatter how good it tries to be, it will almost certainly do some evil. This is the
\nnature of original sin in a perspex robot. <\/p>\n
A perspex robot will suffer harm. It will run the constant risk of senility, mad-
\nness, and death. These are all consequences of free will. If we give robots free will,
\nwe subject them to these harms. We should only do this if we are convinced that the
\nharm is outweighed by good. I believe that free will is the ultimate good, so I am
\nprepared to construct robots with free will as described in this book. I will, how-
\never, take as much care as I can to ensure that no evil is done by this research. Part
\nof that care is to publish this book alerting others to the risks of the scientific
\nresearch, before doing the experiments aimed at creating a robot with free will. <\/p>\n
The reader should rest assured that progress in science is so painfully slow that
\nthere will be no substantial danger from this research for a very long time – perhaps
\nmillennia. <\/p>\n<\/blockquote>\n","protected":false},"excerpt":{"rendered":"
If you remember, a while back, I wrote about a British computer scientist named James Anderson, who claimed to have solved the “problem” of “0\/0” by creating a new number that he called nullity. The creation of nullity was actually part of a larger project of his – he claims to have designed a computing […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[79],"tags":[],"class_list":["post-506","post","type-post","status-publish","format-standard","hentry","category-computation"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-8a","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/506","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=506"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/506\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=506"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=506"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=506"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
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<\/p>\n