Gödel’s confusions— METALOGIC B
Decision processes: Metalogic B
With glosses on Turing’s approach to the Entscheidungsproblem 
|Decision processes is one in a series of documents showing how to reason clearly, and so to function more effectively in society.|
|Intentions, decision processes and acts|
|Do computers ‘decide’ and ‘act'?|
|Stopping and changing|
|Computers, decision processes and Turing|
|C O L O U R K E 'Y|
By predefining a box with a set ‘number’ of items, it is possible to count until the last of those items is reached. It is possible to count ‘all’ the ‘items’, but it must be kept firmly in mind that, as an ‘individual’ person (‘object’) counts those ‘objects’, both the person and the ‘objects’ are undergoing change.
The next person counting, or the previous person recounting, the ‘objects’ will be counting different real objects. The words they use, if they are counting aloud, will be new and different movements of the air as their ever-changing vocal chords set the air vibrating.
No, you cannot step in the same river twice.
Only by relaxing the rigour of our expressions in such a manner, when ‘same’ comes to mean “I do not at this moment care about the very real differences”, can we count the ‘same’ ‘objects’. It is possible to ‘complete’ ‘the’ count, if and only if we relax rigour.
Fortunately for us, there is sufficient practical similarity in reality that we can indeed communicate and survive. But this is an empiric pragmatic finding; it is not a statement that ‘two’ ‘objects’ can be ‘the same’.
After screwing up a game of chess, we may decide
that in future, when our king is in check, we will be particularly attentive
and cautious. Perhaps we may decide that, when our bank account falls
below some predetermined level, we will carefully revue our situation
before further actions. In the later case, we may decide that in future,
after we have written one cheque too many, we will look for a ‘job’
to redress the difficulty. We may even lay out the steps in obtaining
such a job, that is: buy appropriate journals, write applications, go
to interviews, and repeat until the medicine is effective. Such a planned
procedure is termed ‘a decision process’ or an algorithm.
All acts take time, including decisions and chosen acts.A mental act is a real world choice. Thus, the mental act of choosing to apply a decision process at a future time is a present act! That act is ‘separate’ from putting the decision process into action, or even acting on the ‘results’ of that decision process. It is vital to grasp these ‘distinctions’.
The ‘past’ is but a memory, it is encoded in our present state of mind; the ‘past’ was previous arrangement of matter now only ‘remembered’. The past has gone; to travel in ‘time’ would require resetting every atom back to a previous state, physical ‘time travel’ is meaningless unrealism.
Data is counting or measuring, but we always count and indicate our ‘measures’ in monadic (individuated) numbers. So all measuring is ‘approximate’; for space is continuous to the best of our, or my, perception. Our brains, however, seem to function monadically to a great degree. That is, either/or, on or off, yes or no etc.; but the world is just not like that. Our approximate or digital methods have often served us quite well, but they also obscure and confuse and cause us much pain and idiocy.
Apparently, Laplace had the notion that, given knowledge of the complete (information) state of the universe, it would be possible to predict all future states. The impossibility of gaining such ‘complete’ information aside, our current understanding of the quantum rules and of non-linear systems has somewhat revised such optimism. Concepts of a clockwork universe and predictions of the end of science were overrun, only to re-appear recently in such superstitious vanities as ‘theories of everything’.
The digital nature of language and food and neurones. Other animals use the equivalent of words. As an example, some monkeys have different calls to distinguish snakes, eagles and leopards. They and we distinguish ‘food’ and ‘non-food’, and much else, in pragmatic ‘categories’, the brain using digital methods for storage and for decision processes. (See also loop breaking.)
We are creatures that function with much digitality ‘in’ a world that is continuous; we widely confuse our useful digital methods with the reality. Language is monadic by nature, we respond to language in bits. Language is widely used by humans to describe a maelstrom. My purpose is to show that the methods that have grown from our pre-history into the present have not explicitly recognised this mismatch at the heart of our experience.
It is my intent to make the language function in greater accord with the real world, and to make explicit the difficulties. Only by constant awareness and attention may we avoid the worst of the problems posed by language in our strange human condition. Space and time do not arrive chopped into pieces!
|Related further reading|
|why Aristotelian logic does not work||The logic of ethics|
||the confusions of Gödel (in four parts)||Feedback and crowding|
|Decision processes||For related psycho-logical
documents, start with
Intelligence: misuse and abuse of statistics
|1.||Entscheidungsproblem - decision problem, often referred to as the halting problem. Turing’s paper, On computable numbers with an application to the Entscheidungsproblem, is available at this site.|
|2.||Note that at the end of On computable numbers..., section 8, Turing mentions expressing a formula as a Gödel number, thus allowing the application of diagonalisation in order to produce new formulae. This is rather akin to the Richardian approach mentioned in Metalogic A3, §197. Remember also that Gödelian ‘numbers’ do not include all naturals (see Metalogic A2, §150). I am, therefore, unsure whether this comment of Turing’s makes useful sense, as it is based on what I regard as insecure reasoning.|
|3.||See the start of On computable numbers..., section 9.|
“The glass is falling hour by hour,
the glass will fall for ever,
Louis MacNeice 1938 , from Bagpipe Music.
“Upon those who step into the same rivers different and ever different waters flow down.”
Heraclitus c. 540 – c. 480BC, born Ephesus, now Selcuk, Turkey.
|6.||Or series of acts, according how you choose to count.|
|7.||Yet more words meaning recursion or feedback. (Go to Feedback and crowding for much more detail on this concept.)|
|8.||MS Windows, for instance, is an operating system.|
See, for example, Marvin Minsky,
The Society of Mind.
in Why Aristotelian logic does not work,
and cause, chance and choice in The logic of ethics.
Robert Bolt, A Man for all Seasons [page no. not currently available]
Margaret: “Father, the man is bad.”
|12.||Ignoring the time signals take to travel.|
|13a||His ‘machine’ is also an idealised human calculator and, thus Turing, refers to it as ‘he’.|
|14.||Saccade: usually describes brief rapid movement of the eye between fixation points.|
|15.||See, for instance, Penrose.|
|17.||On computable numbers..., p.230.|
|18.||On computable numbers..., p.259.|
|19.||See, for instance, “ ‘discrete state machines’...strictly speaking there are no such machines” ; Computing machinery and intelligence, p.439.|
|20.||See Computing machinery and intelligence, p.452|
|21.||See The logic of ethics.|
|22.||See The confusions of Godël.|
|23.||This is strange, considering Turing’s close attention to the finitistic requirements for his definitions elsewhere in his 1936 paper. To check out Turing’s comments on the finite elements of his ‘machine’, the best way is to search for the word ‘finite’ on the paper: On computable numbers...|
|24.||One possible source of confusion seems to be to imagine that the stop detector somehow interacts with the input copy of the stop detector. But this is far from clear in the very many versions attempting to explain the stopping problem that I have seen. I am convinced, sufficiently for myself, that those attempting to describe ‘the problem’ are not thinking it through with any great clarity.|
|Robert Bolt||A Man for all Seasons||
[Play first performed in 1954]
|Marvin Minsky||The Society of Mind||
Simon & Schuster 1st ed. 1986, pbk; reprinted 1988
|Andrew Hodges||The enigma of intelligence||
HarperCollins Publishers Ltd, , 1985pbk
A workman-like biography.
Magenta highlights points of special note
All links are underlined
email abelard at abelard.org
© abelard, 2001, 31 july
the address for this document is http://www.abelard.org/metalogic/metalogicB1.htm