myriaden

Hitting a Zeno in One - Solution to the Dichotomy Paradox

2007-10-08T09:06:00.000+10:00

Zeno's Dichotomy paradox can be considered a more economical variant of his Achilles and the Tortoise paradox, and hence a tougher nut to crack.

This description of the paradox is from wikipedia.

Suppose Homer wants to catch a stationary bus.

Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on.

Because you can keep dividing the half-distances to infinity, Homer is faced with an infinite number of distances to traverse.

This is not possible according to Zeno, so he concludes that Homer is unable even to start moving.

Aristotle put it in one line:
'That which is in locomotion must arrive at the half-way stage before it arrives at the goal.' [Aristotle, Physics, v1:9 239b10]

Solution

According to Zeno and as per Aristotle's statement above, any object in motion must arrive at the half-way mark before it reaches the finish.

Since nothing can be said to 'arrive at' or 'get to' a place without having first traversed some distance to that place', to arrive at the half-way mark the object must have been able to traverse that half-distance.

(One must admit that the concept of 'arrival' necessarily implies traversal to the point of arrival, else 'motion' in general has no meaning.)

This being so, an object which is able to traverse a distance from one point to another is necessarily able to first travel half that distance.

Since the formulation of each half-distance inherently admits the ability for the object to have traversed that distance prior to considering the next half-distance, we can logically conclude that an object can indeed traverse an infinite number of smaller and smaller half-distances.

The Zeno-In-One Solution

That which is in motion must necessarily be able to arrive at the half-way stage before arriving at the goal, no matter how many successive half-stages there are.

kyy 2007-10-08 mon 0841 z+10

test

2007-02-21T17:58:00.000+11:00

'Apparently I'm turning 30' invite

2007-02-13T09:03:00.000+11:00

Here's an invite i did for a friend yesterday. Took a couple of hours.

*cheeky (vetoed) version

Benj's long lost afro-american twin*

2007-01-24T15:51:00.000+11:00

*the legendary George Foreman.

halloween '06

2006-10-23T19:40:00.000+10:00

8pm Sat 28 October 2006
9 Maria St, Petersham.

Music, nibblies, the usual.
BYO alcohol.

PS. Note the usual start time (8pm)

Miffy's Revenge (alternate)

Goodbye cookie123, hello dismembered body parts

2006-09-21T09:15:00.000+10:00

Goodbye cookie123, hello dismembered body parts

I love using my fingers and eyes as much as the next man, but with terabyte flash drives, 'brute force' is gonna get a whole lot more brutal, witness:

Says the yakuza henchman to the schleper salaryman, 'So you're saying all i need is your right eyeball?'

*Smile.

kyy 2006-09-21 0906

Petticoat Palms Housecooling Party

2006-09-19T18:06:00.000+10:00

Mrs Palmers New Ultra Spunkyy presents...

Petticoat Palms Housecooling Party
6pm Sat 23 September 2006
71 George St, Erskineville

Music, nibblies, the usual.
BYO alcohol.

PS. Note the early start time (6pm)

Alt version (sans mouse-poo writing)

Dual usefulness

2006-07-21T12:04:00.000+10:00

'Dual use' is where idealogical battlelines will increasingly be drawn in the future.

The classic example of dual use is uranium, which can be used to make nuclear weapons (a military application), as well as fuel for nuclear power plants (a civilian application). Dual use is a backdoor through which non-nuclear states can gain access to nuclear fuel because it nullifies non-proliferation arguments.

Another perhaps more mundane example of dual use is the 'business case'. An employee who would like a piece of equipment or software for their personal use can construct a business case, ie. reasons why the company should procure the equipment or software, and thereby obtain it using company funds. The company benefits (assuming the business case is not a straight-out lie) but so does the employee.

2006-07-21 11:59:00am (GMT +10) kyy

Canadian permanent residence, anyone?

2006-05-18T14:54:00.000+10:00

soft pencil - nice touch...

Golf Pro's & Tennis Ho's Party

2006-05-15T15:23:00.000+10:00

Spunkyy Ball Sports presents...

GOLF PROs & TENNIS HOs
(Sports Theme)
Abbie & Renee's Housewarming
6pm Sat 20 May 2006
9 Maria St, Petersham

Music, nibblies, the usual. BYO alcohol.

PS. Note the early start time (6pm)

Original (naughty) version

Double prepositions reveal new 'verb+preposition' verbs

2006-04-19T15:16:00.000+10:00

'Vocal minorities are listened to by media organisations.'*

To discover new verbs that are composed of verb+preposition, make up a sentence using the verb+preposition in passive voice, and attaching a prepositional phrase (ie, starting with 'to','for','by','from' etc.). Then remove the preposition from the suspected verb and see if the two sentences give different meanings. To further confirm the discovery, a third sentence can be constructed where the prepositional phrase is removed, leaving the original verb+preposition in passive voice.

Example: 'I parked him in.'

Sentence 1: 'His car was parked in by me.'
Sentence 2: 'His car was parked by me.'
Sentence 3: 'His car was parked in the driveway.'

Sentence 1: 'His car is parked in at the shopping centre carpark.'
Sentence 2: 'His car is parked at the shopping centre carpark.'
Sentence 3: 'His car is parked in the shopping centre carpark.'

The appearance of double prepositions indicates that 'to park in (someone or their vehicle)' is verb that is distinct from the verb 'to park' + preposition 'in'.

*Stimulus: "Joe's blog has much more influence than you think", by Bobbie Johnson in London, 19 April, 2006, Sydney Morning Herald (http://www.smh.com.au/news/technology/rise-of-the-blogosphere/2006/04/18/1145344085840.html).

2006-04-19 03:15:00pm (GMT +10) kyy

'Contrariwise' - New Word Parsing

2006-04-06T13:15:00.000+10:00

'contrariwise'

Possible parsing procedure
1. Verbalise (silently or voiced)
2. Verbal word matching - sounds like 'contrary' + 'wise'
3. Contextual confirmation - rhymes with 'otherwise', 'other' and 'contrary' have overlapping meanings, and the word occurs at the beginning of a clause, like 'otherwise'.

Software imitation
1. Insert syllable breaks (imitating verbalisation): con~tra~ry~wise.
2. Try to see if any of the combinations fit existing words:
con~tra~ry~wise
contra~ry~wise
con~tra~rywise
con~trarywise ('con' means 'with' in some languages, but 'trarywise is unfamiliar)
contra~rywise
contrary~wise
3. Contextual confirmation (as per above).

Possible meaning is suggested from the most likely parsing:

'Contrary' at the beginning suggests 'contrary to popular belief', 'contrary to what has been said', 'on the contrary'

'~wise' suggests 'in all situations contrary to what has been said', since 'otherwise' means 'in all other situations'

These two meanings seem to reinforce each other, adding further support for the parsing that we first chose.

Source article:

Today's buzzword: plausible deniability
In deception problems, plausible deniability is the ability to engineer a reasonable doubt about secret behavior. An example is a backdoor that results from intentionally bad coding practice. This is plausibly deniable as an innocent mistake; contrariwise, you will have no excuse if someone finds working shellcode in some hex constants.

2006 Underhanded C Contest (http://www.brainhz.com/underhanded/).

2006-04-06 01:23:00pm (GMT +10) kyy

'Keshi' as a Sentence Breaker

2006-04-06T09:15:00.000+10:00

'Keshi' ('but') breaks a sentence into two parts that should be parsed separately.

Translation software should insert '</sentence><sentence>' immediately before 'keshi'.

<sentence> ... </sentence><sentence>keshi ... </sentence>

2006-04-06 09:16:00am (GMT +10) kyy

Disambiguating info inside the sentence

2006-04-03T17:52:00.000+10:00

Following is an example of Step 1 of the Disambiguation Procedure:

'Nèigè rén tā shì shéi?' ('那个人, 他是谁?', Lit: 'That person, he is who?)

Neige (singular): That
Neige (plural): Those
ren (singular): person
ren (plural): people/persons
ta (singular): he
shi: BE
shei: who

The first two words, 'neige' and 'ren', may be singular or plural. However, 'ta' is singular, and so provides the necessary information within the sentence to disambiguate the two options for 'neige' and 'ren'.

2006-04-03 05:58:00pm (GMT +10) kyy

Observations on Plurality

2006-04-03T17:02:00.000+10:00

The general rule is that the plurality of the verb and subject must agree.

1. Plurality and modifiers

Some modifiers (adjectives) use the plural form while others use the singular form, regardless of whether the noun being modified is in singular or plural form.

For example, number words use singular form when they are functioning as adjectives (e.g. 'one cat', 'three dogs'), whereas demonstrative pronouns agree with the plurality of the noun (e.g. 'that car', 'those cars')

2. Plurality and meaning vs form

Plurality is a concept that deals with the meaning (semantic) of the word, rather than the form. (e.g. 'the fish are swimming to the sea', 'the fish is dead')

Note: Sometimes the subject of the sentence may not be the last occurring noun before the verb.

For example, in the sentence, 'Three loaves of bread are being eaten by the pigs,' the subject is 'loaves' not 'bread'.

2006-04-03 05:05:00pm (GMT +10) kyy

Disambiguation Procedure

2006-04-03T16:13:00.000+10:00

Procedure for disambiguating translation options

1. Search for info inside the sentence.
2. Search for info outside the sentence (contextual search).
3. Ask user to input the extra information. (If user is given options to choose from, also give an 'other' option, e.g. 'other', 'neither' or 'both/all')
4. Ask user to select appropriate option.
5. Select default option.

If steps 3, 4 or 5 are used, then gather data about which option was selected for later analysis, to improve steps 1 and 2.

2006-04-03 04:19:00pm (GMT +10) kyy

Split Sentences

2006-04-01T17:29:00.000+11:00

Some sentences are split on a page in such a way that a computer might have trouble guessing where it ends.

For example:
'A sentence may begin normally, then morph into a list of things, namely
1. Item 1
2. Item 2
3. Item 3
before coming to an end.'

To parse this sentence properly, you need to indicate the start and end of a sentence unequivocally. This can be done by using opening and closing tags which have no visual effect, like so:

<sentence>In this example, it is clear where the sentence starts and ends, even though it may be split into two with a list, namely
1. Item 1
2. Item 2
3. Item 3
before coming to an end.</sentence>

2006-04-01 05:42:00pm (GMT +10 -1) kyy

'Do not leave your baggage unattended'

2006-03-30T10:36:00.000+11:00

'Do not leave your baggage unattended.'
(A common sign on Sydney buses.)

Observation
Not sure how to parse this sentence; options are:

(a) Split Verb
The verb 'to leave something unattended' has been split by 'your baggage'.
ie. [Do not leave] + [your baggage] + [unattended]

(b) Reverse Noun Modifier
The modifier 'unattended' is placed after the modified 'your baggage'.
ie. [Do not leave] + [your baggage < unattended]

Both are possible, but which one is preferred for ease of parsing?

2006-03-30 10:33:00am (GMT +10 -1) kyy

'You're sooo funny.'

2006-03-30T10:19:00.000+11:00

'You're sooo funny.'

Observation
If a word cannot be found in a dictionary lookup, check to see if there is a string of 3 or more repeated letters. If so, then remove a repeated letter one at a time and look up the dictionary to see if the remaining letters form a word.

NB: The condition must be a string of 3 or more repeats because some words have double repeats, e.g. vacuum.

Other examples
1. 'Whateverrr!'
2. 'Shiiiiiit!'
3. 'Zzzzzzzzz....' (There should be a word in the dictionary 'zzz' meaning *Indication that author is sleepy.)

2006-03-30 10:16:00am (GMT +10 -1) kyy

'It's oh-so elegant.'

2006-03-30T09:56:00.000+11:00

'It's oh-so elegant.'

(Ad copy for Westfield)

Observations
1. If a hyphenated word cannot be found in a dictionary lookup, parse its constituent words separately. eg. Parse 'oh-so' as 'oh' (modifier) + 'so' (modified)
2. 'oh' is being used here as a modifier of 'so', meaning 'very', ie. the sentence above translates as 'It's so very elegant' or 'It's soooo elegant'.

2006-03-30 10:03:00am (GMT +10 -1) kyy

Generality from the Singular or the Plural

2006-03-22T12:00:00.000+11:00

A general concept may be indicated by the singular or plural form in English.

e.g. 'Gift Certificate Available' or 'Gift Certificates Available'

Stimulus: Sign in shop window, Petersham.

2006-03-22 12:04:00pm (GMT +10 -1) kyy

Solution to Zeno's Second Paradox

2006-03-22T11:43:00.000+11:00

Zeno’s Second Paradox purports to show that in a chase, the chaser can never catch the person or object being chased. In the paradox the Greek hero Achilles graciously gives his slower rival, which legend has turned into ‘The Tortoise’, a head start.

At the start of the race, Achilles is at position 0 while the tortoise is at the 1 kilometre mark. For convenience, imagine that the tortoise can only run half as fast as Achilles.

Common sense may lead you to think that Achilles would overtake the tortoise after running 2 kilometres. However, by the time Achilles reaches the tortoise’s starting point at the 1 km mark, the tortoise has travelled ½ km ahead, which is 1 + ½ km from Achilles’ starting point at position 0. When Achilles reaches the 1½ km point, the tortoise has reached 1 + ½ + ¼ km, and so on.

When, after N steps, Achilles reaches a distance 2 – (½ )N-1 from the start the tortoise is still in the lead because it is at a distance 2 – (½ )N+1 from the start. No matter how big N (the number of divisions of the journey) becomes, Achilles never overtakes the tortoise!

The Solution

The solution I propose involves the observation that the pursuer and the pursued are finite entities. (As such is substantially similar to my resolution of Zeno’s First Paradox.)

Say Achilles has a stride length of 2 metres, and the tortoise has a stride length of 1 metre – this tortoise likes to bound along rather than waddle. For convenience, let’s give the tortoise a 10 metre head start. All else being the same (i.e. Achilles and the tortoise takes the same time to make each stride), after 5 strides Achilles is at the 1 km mark, and the tortoise is at the 1.5 km mark. After another 5 strides Achilles will catch up to the tortoise at the 2 km mark.

The crucial thing here is that both Achilles and the tortoise have a finite stride length that remains constant throughout the race. In Zeno’s footrace on the other hand, Achilles and the tortoise make smaller and smaller strides, so that near the 2 km mark they are making infinitessimally small strides. In fact, Achilles never catches up to the tortoise in the Zeno’s Paradox because neither of them ever reaches the 2 km mark. (If you imagine Achilles running the race by himself, you get a situation that is identical to Zeno’s First Paradox.)

The upshot of comparing these two descriptions of the race is that for Zeno’s Second Paradox to be true, both the pursuer and the pursued must be able to reduce their ‘stride length’ an infinite number of times. The only thing that fits this condition is a point (or a line that runs perpendicular to the direction of the race track).

Finite objects such as people and tortoises can reduce their stride length, but not any smaller than the size of their bodies. So the number of reductions they can achieve is finite.

Assuming that there can be no object of a size between a point and an object with finite size, what this means is that when a finite object moves, it does so in finite increments, and there is a minimum finite amount of movement for each finite object.

Further questions

One question which springs to mind is whether there is a smallest finite amount of movement.

Another is whether there is such a thing as a smallest finite anything. While it’s true that a point, being infinitessimal, is smaller than all finite objects, this does not answer the question of whether there is a smallest among finite objects.

Stimulus: "The Infinite Book", by John D. Barrow. The description of Zero's Second Paradox is largely drawn from pages 21-22 of Barrow's book and also from Zeno's Paradoxes at Wikipedia.com.

2006-03-22 11:48:00am (GMT +10 -1) kyy

The Third Article (Beyond the Definite and Indefinite)

2006-03-21T16:54:00.000+11:00

Examples of sentences where neither the definite article 'the' nor the indefinite article 'a' is best

1. "What's the best software to use for watching dvd on computer?"

Compare: "what's the best software to use for watching dvd on (the) computer?
or "what's the best software to use for watching dvd on (a) computer?"

2. "Save it to disk."

2006-03-21 04:57:00pm (GMT+10 -1) kyy

Writing affects language

2006-03-21T11:49:00.000+11:00

'blog' from 'web log' ...

... is another example of how written script affects the (spoken) language. This kind of word creation is not possible with languages written in non-alphabetic script, e.g. Chinese.

The archetypal example is the use of acronyms to abbreviate phrases and create new words in languages with alphabetic scripts, e.g. 'scuba' from 'self-contained underwater breathing apparatus'.

2006-03-20 05:52:31pm (GMT+10 -1) kyy

Solution to Zeno's Paradox

2006-03-20T17:26:00.000+11:00

*Zeno's Paradox (actually the first of four) purports to show that motion is impossible.

If you want to walk from one point to another, you must first cross half the distance, then to cross the remaining distance, you must first cross half of the remainder, and so on.

If the distance you want to walk is 1 kilometre long, you will first reach 1/2 km from your start, then 3/4 km from your start (1/2 + 1/4), then 3/4 km (1/2 + 1/4 + 1/8), then 7/8 and so on. After you have taken N steps of the journey, you will have gone a distance equal to 1-(1/2)N kilometres. No matter how big N becomes (that is, no matter how many successive half-distances you cover), this distance will always be less than 1 and you will never arrive at your destination.

(Another permutation of Zeno's First Paradox seemingly prevents you from leaving your starting point. Before you can cross a given distance you must first cross half the distance, but before you can do that you must cross half that half-distance, ad infinitum.)

The solution I propose involves observing that Zeno's First Paradox assumes the person or object doing the moving is infinitely small, eg. a point (or a line perpendicular to the direction of travel).

Say your walking stride length is 1 metre, and for convenience the distance to your destination is 10 metres.

Using these figures, it's easy to see that it will take you 10 strides to reach your destination. Working through the paradox, to reach 10 m you must first reach 5 m, which you easily can cover in 5 strides. From this point to reach half the remainder (7.5 m from the start), you only need 3 more strides. In fact, with 3 more strides you exceed that distance. To fit with the paradox, you must reduce your stride length and keep on doing it so that you don't exceed each subsequent half-distance. And you must do so an infinite number of times in order to never reach your destination.

However, this is only possible with an object of zero size. If we started with an object with finite (positive) size, like a tortoise or a human, and measured the stride lengths from one heel to the next, at some point in our stride-length-reducing exercise the heel of a foot would exceed the length that is left to travel, so that the tortoise or human can be said to have arrived at the destination.

Interestingly, by merely changing how we measure stride-lengths in this scenario, we can illustrate the non-arrival of a zero-sized object. By measuring toe to toe rather than heel to heel, we in effect have a zero-sized object which, with ever reducing stride lengths, never arrives at its intended destination.

A funny way of describing this is that in order to 'prove' that motion is impossible using Zeno's Paradox, you must purposefully try not to reach your destination (that is, by successively reducing your stride length).

(You can perform this thought experiment using a line segment of finite size. Imagine you have a 1 metre long ruler. For the first 5 m, you 'walk' or flip the ruler end to end to cover the distance.

Then for the next 2.5 m you can flip the ruler for two-lengths, then angle the ruler so that it covers the remaining 1/2 metre without exceeding it. And so on. Over time, the final angle of the rulers gets more and more oblique as the final 'stride' distance gets progressively smaller.)

Therefore an object with a finite size can traverse a finite distance, but an object that is infinitely small cannot do the same.

This solution has a couple of interesting ramifications. The first is that since a point is infinitely small, its 'stride length' is infinitely reducible and it can satisfy Zeno's First Paradox, namely a point cannot move. Points can mark positions but they cannot traverse a finite distance.

Conversely, a moving object cannot be a point. If you observe an object travelling from one point to another, the object must have a finite size, no matter how small. Motion is a property of finiteness.

Stimulus: "The Infinite Book", by John D. Barrow. The description of Zeno's paradox above is largely drawn from pages 20-21.

kyy 2007-04-02 1542 z+10

*Note: This is version 1.1 of the solution. The first two comments below refer to version 1.0 of the solution, which published by kyy on 2006-03-21 1332 z+9.