Small Number
My professor of mathematics recently emailed out an interesting problem to his students:
Problem 2006-4. Small Number. What is the smallest number that cannot be described by fewer than thirteen words?
My first response to this question which I thought was close to being right was: "The smallest number that can not be described by fewer than thirteen words." This sounded really nice to me as i counted up the words and saw there were thirteen. However very quickly it becomes apparent that this same number is described by the sentence "The smallest number that can't be described by fewer than thirteen words." (It is also ruined when I spell "can not" correctly as "cannot") Which contains twelve words and therefore ruins my first solution.
My next train of thought was to describe as concisely as possible a number which was less than any number that can be described by 13 words or less. Results in this category had rather bad grammar and were probably too vague to describe one number. I was trying to come up with sentences like "Number less than number less than smallest number minimally described in thirteen words." And this last attempt seems pretty good because by its own definition it is smaller than any other solution that might be described minimally by thirteen words.
However at this point i started thinking about other languages. Other languages definitely still use words and the problem mentions nothing about English only solutions. Also in other languages different amounts of words can be used to describe the same thing described in English. German is one language that is somewhat known for compound words. So maybe if i find some language where the translation of "less than" only takes up one word i could describe the same number but with less than thirteen words.
So at this point the problem takes a quick turn into linguistics where the goal is to find the language which can mostly concisely describe numbers and relationships between them. This would probably lead to the language of mathematics where mathematical characters might be considered words. However mathematics is not a natural language. That is to say it isn't tied to any specific cultural group, instead it is a man made language. If we take the step to allow words from synthesized languages any sentence of words in a given language can be translated into a newly forged language in which that sentence is translated as "figomstormo" and now describes the same number is described in one word which is fewer than thirteen. This reasoning applies to any solution to the problem which uses words at all. The solution "one to the negative nine to the negative nine to the negative nine" Doesn't work as a solution because in my new language that quantity is given the name poiusltur.
At this point it might seem as though the puzzle is unsolvable. However another possibility exists. In order to avoid the translation problem the sentence itself must reference itself and its form so that if it is translated to another language it is no longer the same sentence. For example "The number described in this sentence" would not describe the same number as it would if it were translated into french because it would then be a sentence in french and would be a different sentence. So somehow the sentence must be untranslatable and it seams as though being self referential might somehow accomplish this. Of course my self-referential sentence above doesn't describe a number with a specific value, and also it has fewer than thirteen words. So since I am trying to find a sentence that meets these requirements it is untranslatable (or at least it can't lose words in correct translation), it has thirteen words and is is the smallest number that cannot be described by fewer than thirteen words?
"This sentence contains thirteen words or more and describes the smallest number that cannot be described by fewer than thirteen words" The previous sentence describes the number required in the problem. The sentence cannot be changed or translated to contain fewer than thirteen words. It could be translated but any correct translation would have to contain thirteen words or more and it would describe the same number not a smaller one. The only part of it that seems incorrect is that it doesn't describe an exact number, however if it did describe an exact number then by previous arguments it would not be a solution.
"This sentence contains thirteen words or more and describes a number smaller than the smallest number that cannot be described by fewer than thirteen words" Hmm.. of course this self referential thing could get out of hand. Perhaps an infinite loop could be used here to create a infinitely long sentence with nested "a number smaller than a number smaller than..." But really as long as these sentences contain a bit of self reference with respect to the number of words in them (which they must because other wise they could be translated down to one word) then they cannot be described by fewer than thirteen words and therefore the original sentence describes the same number.
Of course the really troubling thing about "This sentence contains thirteen words or more and describes the smallest number that cannot be described by fewer than thirteen words", is that despite the sentence's inability to lose numbers it still only describes "The smallest number that cannot be described by fewer than thirteen words." which has twelve words and therefore fails as a solution. Of course it is now on this wonderful trip that I finally see that the question itself asks for "The smallest number that cannot be described by fewer than thirteen words?" The nature of questions and answers indicate that the best solution will be the one that is described by "the smallest number that cannot be described by fewer than thirteen words?" and if it is described by this then it is described by fewer than thirteen words and therefore is not a solution. So there can be no solution.
Problem 2006-4. Small Number. What is the smallest number that cannot be described by fewer than thirteen words?
My first response to this question which I thought was close to being right was: "The smallest number that can not be described by fewer than thirteen words." This sounded really nice to me as i counted up the words and saw there were thirteen. However very quickly it becomes apparent that this same number is described by the sentence "The smallest number that can't be described by fewer than thirteen words." (It is also ruined when I spell "can not" correctly as "cannot") Which contains twelve words and therefore ruins my first solution.
My next train of thought was to describe as concisely as possible a number which was less than any number that can be described by 13 words or less. Results in this category had rather bad grammar and were probably too vague to describe one number. I was trying to come up with sentences like "Number less than number less than smallest number minimally described in thirteen words." And this last attempt seems pretty good because by its own definition it is smaller than any other solution that might be described minimally by thirteen words.
However at this point i started thinking about other languages. Other languages definitely still use words and the problem mentions nothing about English only solutions. Also in other languages different amounts of words can be used to describe the same thing described in English. German is one language that is somewhat known for compound words. So maybe if i find some language where the translation of "less than" only takes up one word i could describe the same number but with less than thirteen words.
So at this point the problem takes a quick turn into linguistics where the goal is to find the language which can mostly concisely describe numbers and relationships between them. This would probably lead to the language of mathematics where mathematical characters might be considered words. However mathematics is not a natural language. That is to say it isn't tied to any specific cultural group, instead it is a man made language. If we take the step to allow words from synthesized languages any sentence of words in a given language can be translated into a newly forged language in which that sentence is translated as "figomstormo" and now describes the same number is described in one word which is fewer than thirteen. This reasoning applies to any solution to the problem which uses words at all. The solution "one to the negative nine to the negative nine to the negative nine" Doesn't work as a solution because in my new language that quantity is given the name poiusltur.
At this point it might seem as though the puzzle is unsolvable. However another possibility exists. In order to avoid the translation problem the sentence itself must reference itself and its form so that if it is translated to another language it is no longer the same sentence. For example "The number described in this sentence" would not describe the same number as it would if it were translated into french because it would then be a sentence in french and would be a different sentence. So somehow the sentence must be untranslatable and it seams as though being self referential might somehow accomplish this. Of course my self-referential sentence above doesn't describe a number with a specific value, and also it has fewer than thirteen words. So since I am trying to find a sentence that meets these requirements it is untranslatable (or at least it can't lose words in correct translation), it has thirteen words and is is the smallest number that cannot be described by fewer than thirteen words?
"This sentence contains thirteen words or more and describes the smallest number that cannot be described by fewer than thirteen words" The previous sentence describes the number required in the problem. The sentence cannot be changed or translated to contain fewer than thirteen words. It could be translated but any correct translation would have to contain thirteen words or more and it would describe the same number not a smaller one. The only part of it that seems incorrect is that it doesn't describe an exact number, however if it did describe an exact number then by previous arguments it would not be a solution.
"This sentence contains thirteen words or more and describes a number smaller than the smallest number that cannot be described by fewer than thirteen words" Hmm.. of course this self referential thing could get out of hand. Perhaps an infinite loop could be used here to create a infinitely long sentence with nested "a number smaller than a number smaller than..." But really as long as these sentences contain a bit of self reference with respect to the number of words in them (which they must because other wise they could be translated down to one word) then they cannot be described by fewer than thirteen words and therefore the original sentence describes the same number.
Of course the really troubling thing about "This sentence contains thirteen words or more and describes the smallest number that cannot be described by fewer than thirteen words", is that despite the sentence's inability to lose numbers it still only describes "The smallest number that cannot be described by fewer than thirteen words." which has twelve words and therefore fails as a solution. Of course it is now on this wonderful trip that I finally see that the question itself asks for "The smallest number that cannot be described by fewer than thirteen words?" The nature of questions and answers indicate that the best solution will be the one that is described by "the smallest number that cannot be described by fewer than thirteen words?" and if it is described by this then it is described by fewer than thirteen words and therefore is not a solution. So there can be no solution.
posted by Naf at 10:13 PM
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