Math Is Fun Forum
  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

#1 2015-12-06 22:24:33

Solvitur ambulando
Member
Registered: 2015-12-06
Posts: 28

The Achilles diary paradox

I would like to offer the following ‘semi-paradox’ for discussion.

“The gods tell Achilles that they have some good news and some bad news. The good news is that he will live an infinite number of days. They have even prepared a diary (with infinite pages) foretelling the fortunes that will befall him on each day. Each day when he wakes up Achilles must tear out a page of the diary, and the contents of the page will determine his fate for the day. Every page that carries the date of his birthday states, 'Today you will enjoy divine powers'. The bad news is that due to a printing error, all the remaining pages state, 'Sorry, today is basically just another day at the office'.

As compensation for the error the gods allow him to tear out the pages in any order he chooses, providing (1) he states in advance his strategy for tearing out the pages, and sticks to it, (2) the strategy involves tearing out one page each day, and (3) every page is assigned a specific day on which it will be torn out. For the sake of clarity let us say that his birthday is June 1st and that the first page of the diary is Jan 1st 0001 AD. The diary covers the years 0001, 0002 and so on indefinitely.

Achilles chooses as follows: For the first 364 days he tears out the first 364 June 1st pages; then he takes the first non-birthday page (Jan 1st of the year 0001); then he takes the next 364 June 1st pages; then the second non-birthday page (Jan 2); and so on endlessly, treating himself to an extra birthday on leap years. Thus Achilles enjoys divine powers and celebrates his birthday on all but one day of each year, and will do forever.”

Achilles’ fate seems paradoxical because it is tempting to think that his future birthdays are far less numerous than his non-birthdays and that his selection regime somehow contravenes this. The temptation is reinforced by the fact that in any finite diary his future birthdays certainly will be far less numerous than his non-birthdays; indeed the ratio of non-birthdays to birthdays converges to 364.25 (allowing for leap years). But in the case of infinite time his birthdays are neither less nor more numerous than his non-birthdays; both are countably infinite (there are infinitely many June 1sts) and the ratio of his non-birthdays to his birthdays is undefined.

If Achilles’ lifespan were finite he would eventually run out of ‘birthday’ pages and be obliged to use up a huge backlog of non-birthday pages. For example, if he lived to 100 and if the diary covered 100 years, he would run out of June 1st pages after 100 days and would have to live the remaining 99 years and 265 days without divine powers and without a birthday.

For me the appeal of the paradox lies in the nagging sense that his strategy ought to suffer a similar failure if he is immortal, even though the mathematics demonstrates otherwise.

Last edited by Solvitur ambulando (2016-01-08 20:37:40)

Offline

#2 2015-12-06 23:44:00

bob bundy
Administrator
Registered: 2010-06-20
Posts: 8,168

Re: The Achilles diary paradox

hi Zenophobia

Like your pyramid scheme that seems to work.  That's the trouble with infinity.  It can be a slippery concept to pin down.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

Offline

#3 2015-12-16 05:50:26

Relentless
Member
Registered: 2015-12-15
Posts: 624

Re: The Achilles diary paradox

I feel really slow, but I have to ask: Why can't he just take only birthday pages? I know it's stated that he "must eventually take all of the pages", but that's nonsense when it comes to infinity. He could just reply "Well I'm taking one non-birthday page every googol (or much more) years, so eventually I will". I suppose the fact that I immediately think that is due to me having little trouble in grasping this paradox; he is able to defer having non-birthdays, effectively, for an infinite amount of time as a legitimate ratio.

FUN FACTS: The Julian calendar (Old Style) had years consisting of an average 365.25 days. According to this calendar, every century year is a leap year (they are all divisible by four). But in the Gregorian (New Style) calendar, only those century years that are divisible by 400 are counted as leap years. This means that in a 400-year cycle, there will be 97 leap years instead of 100.
This makes the average year 365.2425 days. In addition, it is 52.1775 weeks (26.08875 fortnights), and the average month is 30.436875 days.

As compared to the Julian calendar: 365.25 days, 52 and 5/28 weeks (52.17[857142857142]...), 26 and 5/56 fortnights (26.089[285714285714]...), and an average month of 30.4375 days.

However, the average year is still a Julian one for all sets of years from xx01 to the next xx96. And in particular, more practically, the set of years from 1901 to 2096 still has an average 365.25 days.

Last edited by Relentless (2015-12-16 20:57:04)

Offline

#4 2015-12-19 18:19:59

Solvitur ambulando
Member
Registered: 2015-12-06
Posts: 28

Re: The Achilles diary paradox

Good point Relentless. You’ve stumbled upon Part 2 of the paradox! Whenever the gods say ‘Achilles, isn’t it time you took your first non-birthday page?’, Achilles can always reply ‘Not yet!’ without violating the conditions of his immortality. But if he *always* replies ‘Not yet!’ he will violate those terms!

How to resolve this apparent contradiction?

The answer (I think) is that the two statements relate to two different perspectives on infinity. From the perspective of the here-and-now the future is indeterminate so the first non-birthday can be delayed to some unspecified future. From a mathematical perspective, if the conditions are to be satisfied there must be some numbers n1, n2 etc such that on Days n1, n2... Achilles takes non-birthday pages. If the gods were smart they would insist that Achilles specify his page-tearing strategy in advance: that is, specify the days on which he would choose non-birthdays.

I think Zeno was driving at a similar problem with his paradox of Achilles getting from A to B. From the perspective of the halfway mark , three-quarters mark etc, Achilles always has further to go; but from the vantage of point B he has arrived. The difference is that in the spatial case the infinite ‘steps’ can be completed, whereas in the case of endlessly succeeding days they can’t – not even if one is immortal!

Offline

#5 2015-12-19 18:38:10

Relentless
Member
Registered: 2015-12-15
Posts: 624

Re: The Achilles diary paradox

Infinity is fun, isn't it? I have always thought so haha!

Once again, I don't think that it is much of a problem. If it isn't agreed otherwise, he can in fact always reply "Not yet!", because there will never come a day when it is declared that he cannot take all of the non-birthday pages with the pattern he is obeying.
I'm also not sure that Achilles would mind much if he was ordered to specify his page-tearing routine. He would just have to come up with a way of describing the largest possible finite number of days he can after which he is obligated to take a non-birthday page. And even if he only had a day to do so, he would be able to come up with a truly immense number. But then again, he would still have to suffer an infinite number of non-birthdays, so it's possible this is little compensation after all!

To be honest, I think that being immortal is the most frighteningly horrifying prospect imaginable, no matter what special powers you are given. To live until everything that could possibly happen in a finite space has happened, and then to do it again, and again, and again...

Zeno's spatial paradoxes once troubled me too, but eventually I got an explanation that satisfied me. Basically, very small distances are traversed in very small amounts of time, so time makes velocity possible. Zeno probably thought of time and motion as the same.

Last edited by Relentless (2015-12-19 18:45:26)

Offline

#6 2015-12-19 22:14:13

Solvitur ambulando
Member
Registered: 2015-12-06
Posts: 28

Re: The Achilles diary paradox

Yes, infinity is fun and also intriguing because as Zeno pointed out it seems to be unavoidable – you can’t walk across a room without stepping over it!

Achilles can delay the non-birthdays by specifying a mind-boggling large number, but from the perspective of infinity any finite span of time is but the blink of an eye. As you say, he has to pay his dues eventually.

Perhaps the real horror of hell, if there is such a place, is not that it is a place of suffering but that residence therein is endless. One could say the same of heaven! It is strange, is it not, that the prospect of dying is less frightening than the prospect of living forever? These musing probably belong in the ‘Dark discussions’ section.

Offline

#7 2015-12-19 23:31:22

Relentless
Member
Registered: 2015-12-15
Posts: 624

Re: The Achilles diary paradox

Although I think it is mathematically possible that we are in some sense crossing infinity when we walk across a room, or even situated within an infinite cosmos, philosophically I find it more probable that space is finite, and composed of units that the Ancient Greeks would have called "atoms" but perhaps what we call Planck lengths or quantums of length.

That is from the perspective of infinity, not necessarily from his own perspective (: It seems possible to me that the long spans of time between could diminish the effect of the undesired non-birthday. But then any speculation about what it is like to live forever is quite suspect. Just now I was thinking that perhaps immortality would not be so bad if you could ensure that you don't remember the same thing twice, only to lose myself in the idea that if one experiences every physical event in a finite space, they also experience every psychological one.
Even though I understand it, the fact still is quite strange, and morbid, that eventually everyone would choose to die.

Yes we could get into some trouble, since I have spent a lot of time on forums discussing philosophy and you have a strong interest in paradoxes involving the infinite xD

Offline

#8 2015-12-20 20:46:31

Solvitur ambulando
Member
Registered: 2015-12-06
Posts: 28

Re: The Achilles diary paradox

That’s the kind of trouble I’d gladly get into! I find these questions fascinating not only because they challenge (and perhaps exceed) the intellect, but also because I suspect they contain deep clues regarding our understanding, and indeed our misunderstanding, of space and time and hence of our lives generally.

For example it is fact that our powers of measurement are limited in practice, and physics suggests that they may be limited in principle; yet we persist in modelling physical space with a number system (the reals) that is not only infinite but uncountably so. It seems to me that Zeno was amongst other things attempting to draw attention to the apparent disjunction, indeed irreconcilability, between points (numbers) and extensions (intervals). Set theory has gone some way to resolving this matter mathematically, yet the continuum hypothesis remains unproven and the point/extension problem remains unresolved in terms of its application to physical reality. Indeed I suspect (I’m sticking my neck out here, being a rank amateur) that this question could be relevant to if not lie at the heart of the apparent irreconcilability of relativity and quantum theory.

I suspect we need to rethink our whole notion of points and intervals as an adequate model of our physical and temporal environment. Indeed I think the whole notion of measure could do with a thorough raking-over.

As for choosing to die, it would not necessarily be morbid if we could perceive deeply (as perhaps various ‘seers’ have over the ages) the significance of transience (time), and so in some sense be free of it. Your comments raise deep questions about the nature of self, identity, experience, and much else besides.

I’d be interested in exploring these questions further, with the proviso that I will not be able to spare much time and will often be ‘off the grid’ for extended intervals (the ‘ambulando’ part of my moniker). For starters, I’d be interested to learn why philosophically you find it more probable that space is quantised.

Would you be interested in starting a new thread in the ‘Dark discussions’ section?

Offline

#9 2015-12-21 02:59:53

Relentless
Member
Registered: 2015-12-15
Posts: 624

Re: The Achilles diary paradox

I would always be interested in having a substantive discussion smile We could start a thread about the size and/or divisibility of the universe, but I would be obliged to mention that even though I've discussed it and it is popular at least in amateur philosophy, I believe it is a question that belongs in the domain of physics rather than intuitive argument. For the latter part, though, I should at least mention that a finite universe seems more likely to me because of the irreconcilability with intervals that you pointed out, because it appears to me to be simpler by making fewer assumptions about things too small or distant to perceive, and because a continuous universe also seems to me to be unverifiable. This, the fact that a length has been identified smaller than which the current physical model of length has no meaning, and the model of the universe as expanding at a finite rate from a fixed point constitute my case in brief. But as I said, I definitely find it mathematically possible to reconcile points and intervals nonetheless, so I would not presume to know.

You will probably have to explain to me the significance of our measurements. I know that it is suggested that they are limited in principle, but compared to most I never found this fact very interesting; it says something about us and our methodology, not necessarily the structure of the world. I used to be a resolute determinist, and occasionally somebody would mention Heisenberg's uncertainty to refute it. But the fact that we do not have the power to predict something hardly seems crucial to whether it is essentially fixed or subject to probability.

Last edited by Relentless (2015-12-21 03:35:33)

Offline

#10 2016-01-01 10:04:09

Calligar
Member
Registered: 2011-09-24
Posts: 272

Re: The Achilles diary paradox

Sorry, but I'm having a lot of difficulty understanding what this paradox is saying.  I'll give a few places that I seem to have difficulty interpreting...

As compensation for the error the gods allow him to tear out the pages of the diary in any order he chooses, each page determining his fate for the day.

     The issue I have with this is I'm not entirely sure if the pages he tears out is what occurs that day or if by tearing it out, the pages effect for that day no longer happens.

For the first 364 days he tears out the pages corresponding to his first 364 birthdays

     The biggest issue I have understanding this, specifically using the word 'corresponding' in this case really confuses me.  Is he tearing out every page that is or is not his birthday?

There are a few other minor parts I'm having difficulty with, but I think if someone would be able to explain those two parts to me, I'll be able to understand the rest.  Unfortunately, I won't be able to put my thoughts on the matter until I understand it.


There are always other variables.  -[unknown]
But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end.  -Aristotle
Everything makes sense, one only needs to figure out how.  -[unknown]

Offline

#11 2016-01-01 19:43:12

Relentless
Member
Registered: 2015-12-15
Posts: 624

Re: The Achilles diary paradox

Hello Calligar,

I think the best way to help is to restate the entire paradox in a clearer way.

A man is made immortal; he will live forever. However, he only has divine powers on his birthday. Nevertheless, he has the ability to choose which order the days of his life will take place.
The paradox is that he can prioritise his birthdays such that he experiences many more birthdays than non-birthdays, forever. (As mentioned later, he can even choose to delay having a non-birthday indefinitely.)

The book is just symbolic. The idea is that tearing a page out represents choosing that day next.
Hopefully that's a little more understandable, but if not, let me know what is unclear (:


It's really the principle that is a little paradoxical. It's like if you were told to name every number, in any order, and for every prime number you name you get $100 and every other number you lose $1,000, you will never go broke because you will only name prime numbers (even though there seem to be so few prime numbers by comparison).

Last edited by Relentless (2016-01-01 19:52:32)

Offline

#12 2016-01-02 08:18:28

Calligar
Member
Registered: 2011-09-24
Posts: 272

Re: The Achilles diary paradox

Yes, sorry, I understood most of that.  The issue I'm having is some smaller details mentioned which is causing me to get confused about the whole thing.  It is why I quoted both of them:

Solvitur ambulando wrote:

As compensation for the error the gods allow him to tear out the pages of the diary in any order he chooses, each page determining his fate for the day.

For the first 364 days he tears out the pages corresponding to his first 364 birthdays

Those are the parts I'm having the most difficulty understanding.  However, you bring up now another question, which is related to trying to understand the first part I quoted (sorry about all the confusion, I'm only trying to understand this)...

Relentless wrote:

The idea is that tearing a page out represents choosing that day next.

So does that mean that by tearing a page out, it happens the day after?


There are always other variables.  -[unknown]
But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end.  -Aristotle
Everything makes sense, one only needs to figure out how.  -[unknown]

Offline

#13 2016-01-02 16:34:04

Relentless
Member
Registered: 2015-12-15
Posts: 624

Re: The Achilles diary paradox

Calligar wrote:

So does that mean that by tearing a page out, it happens the day after?

I think that is the simplest interpretation, yes (:
There are an infinite number of pages in this book. Every day has a page that represents it. Assume they are in order. It appears, at first, that he will not get to have a birthday very often, since for every birthday page, there are 364.2425 non-birthday pages (if he isn't born on leap day; if he is, then there are 1,505 and 15/97 non-birthday pages for each birthday page!). But nevertheless, he can always flip to the next birthday page and tear that out for tomorrow, so effectively he has more birthdays.

I honestly think that the role of the book and the gods is unimportant. The principle is that he can choose the order of an infinite set, such that only the least common elements appear, and that is somewhat counterintuitive.

Offline

#14 2016-01-02 22:53:37

Solvitur ambulando
Member
Registered: 2015-12-06
Posts: 28

Re: The Achilles diary paradox

Hello Calligar and Relentless,

My apologies for being ‘ambulando’ for so long, but I appreciate your discussion. Relentless, you have clearly grasped the essence and details of the paradox, and your last sentence (‘The principle is that he can choose the order of an infinite set…’) summarises the essential point concisely. The gods are of course irrelevant and added purely for the sake of spinning a yarn!

You suggested reworking the paradox so as to abolish the diary, suggesting that ‘[Achilles] has the ability to choose which order the days of his life will take place’. I considered presenting the paradox in this way, but was reluctant to do so because it introduced the additional complication of seeming to require time travel or the rearrangement of time.  I felt that this complication (adding science fiction to fantasy!) would obscure the main point of the paradox and make it harder for people, or harder for some people, to see the essential point which is concerned not with reordering time per se but with reordering an infinite set. Hence the diary.

Calligar, in response to your question: Each page that he tears out determines his fate for that day. Sorry, I can see now that ‘tearing out’ is potentially ambiguous! I envisaged tearing the pages as one might tear off the pages of an office daily planner – ‘Tuesday: Meeting with Sales Team’ etc.

I have now edited my original entry – I hope it is a little clearer.

Offline

#15 2016-01-02 23:19:51

Solvitur ambulando
Member
Registered: 2015-12-06
Posts: 28

Re: The Achilles diary paradox

Relentless - In response to your post about measurement: I do not feel qualified to discuss the size of the universe. As for its divisibility, I agree with your point that continuity (of space for example) would seem to be unverifiable. I have often wondered for example whether it makes any sense to consider (say) a space interval of 10 to the minus 100 metres, given that there is no known way even in principle of dividing 10 to the minus 99 metres into so many ‘equal intervals’.

You suggest there is a difference between whether something is essentially determined and whether it is predictable. If something were essentially determined but not predictable, how would we know? And if we could not know, on what basis could we claim that it was essentially determined? I feel that this question could potentially cast light on a deeper question concerning the relationship between ‘the world as seen through the prism of measurement’ and ‘the world as it actually is’. Put another way, is there an actuality/reality beyond the realm of measurement, and is it possible to perceive it?

Offline

#16 2016-01-02 23:55:08

Relentless
Member
Registered: 2015-12-15
Posts: 624

Re: The Achilles diary paradox

Hi Solvitur, welcome back (:

The question of whether the idea of an infinite universe makes sense is discussed by all sorts of thoughtful yet unqualified folk xD It puzzles a lot of people that, for instance, one thing could be twice as large as another, while both are equally insignificant parts of an infinitely large whole. Or, similarly, that one thing is made up of infinitesimal bits, while another seems to be made up of twice as many. It is my equally unqualified understanding that in physics, 10^-100 metres is nonsense; nothing related to length occurs at that scale to distinguish it. If my statement that the infinite divisibility hypothesis is unverifiable is valid, then it is a significant objection, because the scientific method takes experimentally unverifiable claims to be worthless.

If something were essentially determined but not predictable, how would we know? And if we could not know, on what basis could we claim that it was essentially determined?

I used to think, and some philosophers do think, that the idea of an essentially random event is logically incoherent. Your first question is rhetorical unless this can be demonstrated, and I am quite sure it cannot. As to your second question ... I suppose you have given me a very concise explanation xD The quantum theory has led us to disregard our intuitions about causality, and I think rightly so.
Incidentally, Einstein himself claimed that the universe is essentially determined despite quantum theory (which he regarded as provisional); he also admitted that this belief was entirely based on faith.

PS: Regarding your last question, you may find the OP here http://www.mathisfunforum.com/viewtopic.php?id=22752 to be relevant. Basically, there is almost nothing about the world we can be absolutely sure of. If you are doubtful enough, you can virtually always find a way that reality might not at all correspond to what you observe or think.
The scientific method is so successful because it comprises the set of principles (read: assumptions) that best identify consistency in what is observed. That, I think, is the basis of gaining knowledge about the world.

Last edited by Relentless (2016-01-03 00:04:34)

Offline

#17 2016-01-03 23:52:35

Solvitur ambulando
Member
Registered: 2015-12-06
Posts: 28

Re: The Achilles diary paradox

Hi Relentless, thanks for the welcome!

If you have two boxes A and B, and B fits inside A but not vice versa, it seems rather difficult not to conclude that A is larger than B, regardless of whether the universe is finite or infinite. Whether or not the boxes are infinitesimal compared to the universe has no bearing on the comparison or on the notion of comparative size. If there are folks out there who question this, they sound rather interesting and I would like to meet them! Of course if the two boxes are at opposite ends of the universe, or even a few kilometres apart, the comparison becomes a little more tenuous.

I find the notion of probability persuasive yet at the same time somehow dubious. It relies on the assumption that an event can be repeated, but one of the odd things about life is that nothing is repeated; what is is, and what was is only memory. This strikes me as one of the greatest mysteries of existence, yet for the most part we blithely disregard it. Repetition implies time and time is…?

I wholeheartedly agree that there is almost nothing about the world we can be absolutely sure of. It is by no means clear that there is such a thing as ‘the world’ other than the sum total of our beliefs, impressions and expectations, which one would be rash to assume are entirely coherent. Descartes made a brave attempt to question everything, but he foundered badly when he assumed that there was an ‘I’ who was doing the thinking – apparently it didn’t occur to him that the ‘I’ might itself be a figment of thought.

We need to consider the possibility that thinking, being an exercise in virtual reality, cannot open the window to reality (if there is such a thing), although it can investigate its own processes (such as preconceptions) that interfere with clear perception. I am inclined think the Vedic seers and their ilk were on the right track when they advocated a state of mind in which the movement of thought is in abeyance.

Offline

#18 2016-01-04 12:35:18

Solvitur ambulando
Member
Registered: 2015-12-06
Posts: 28

Re: The Achilles diary paradox

I would like to add a comment on a point that I touched on earlier. It concerns the condition that the gods place on Achilles to ‘eventually take all the pages’. This is (to use Bob Bundy’s word) slippery.

Any given page in the diary will be torn out in finite time - no page will forever remain untorn - so in that sense Achilles will fulfil the condition. Let us call this Conclusion 1.Yet a day will never arrive when Achilles has taken all the pages, so in that sense he will never fulfil the condition (Conclusion 2). This in itself is a paradox, quite independent of the order in which he tears out the pages.

How to resolve the paradox? We could argue that the gods’ condition is ambiguous, but I don’t think this is the whole story. The main point is that unlike a converging sequence of intervals of time (or space), eg ½, ¾, 7/8... seconds, an infinite sequence of equal time intervals cannot be completed – at least not as far as we know, from the perspective of daily life. Mathematically we might say that Achilles' days can be numbered by the integers, but not by the integers plus omega (the first transfinite ordinal).

Last edited by Solvitur ambulando (2016-01-07 09:55:09)

Offline

#19 2016-01-04 14:59:33

Relentless
Member
Registered: 2015-12-15
Posts: 624

Re: The Achilles diary paradox

Hi!
You will find a lot of interesting and creative (if often flawed) perspectives on forums devoted to philosophy. They seem to attract cranks at least as much as they do academics. Infinity is a staple, as well as nothingness, and infinity as it relates to space and time is often pondered. It is not that the relative sizes of objects are doubted; it is simply that it can be difficult to conceptualise how objects can have relative sizes in an infinite universe. I think it has to do with the idea that infinity cannot be divided into finite parts. I suppose that I do not find it so absolutely brain-bending, but it is interesting to think about if you get why it puzzles people.

Hm... I know you said that it is one of the great mysteries of existence, but unfortunately I am struggling to see what is so mysterious (with no intention of being infuriating!). It might be true that the universe as a whole is never in the same state twice, but links can be made between localised events and processes. Why should it be a mystery that we predict that a lot of coin tosses will show heads close to 50% of the time? Is all of our data only memory? :S

Your final comments about thinking remind me a lot of what I know of Immanuel Kant. He wrote a lot about how things-in-themselves are unknowable and we only experience what our concepts allow us to. In particular, he said that space, time, cause and effect are merely human constructs, without which we could not have experiences.
As for Vedic seers... they confuse my small and literal mind xD


Regarding your paradox, I honestly never thought of it like that before. I always assumed Conclusion 2 without giving Conclusion 1 a thought. This seems to me to be no semi-paradox. I will definitely think on it.

Offline

#20 2016-01-05 21:30:19

Calligar
Member
Registered: 2011-09-24
Posts: 272

Re: The Achilles diary paradox

Hmm, okay, I see you edited your original post, I got the information I needed.  Though I am curious about one thing.  Why did he have to take any non-birthday pages.  Since he lives forever, he could take birthday pages and only birthday pages for eternity.  I understand the part that he must eventually take all the pages, but with no time limit, that is actually irrelevant unless I'm mistaken.  I would argue that since he is living forever and can choose what pages (out of unlimited pages) to take out, he should be able to take arguably only birthday pages because he can theoretically always take all his non-birthday pages later (of course, that later will never happen).


There are always other variables.  -[unknown]
But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end.  -Aristotle
Everything makes sense, one only needs to figure out how.  -[unknown]

Offline

#21 2016-01-06 01:38:41

Relentless
Member
Registered: 2015-12-15
Posts: 624

Re: The Achilles diary paradox

Calligar,

Hahaha, you have basically made a duplicate of the first paragraph of post #3 smile Post #4 is a good response to this query - in short, you (and I) are right!

Offline

#22 2016-01-06 15:41:33

Solvitur ambulando
Member
Registered: 2015-12-06
Posts: 28

Re: The Achilles diary paradox

The ‘Second Achilles Diary Paradox’ that I mentioned above arises from ambiguous language.

Immortal Achilles is charged with tearing out one page each day from a diary with infinite pages: page 1 on day 1, page 2 on day 2 etc. Consider the following statements:

1. Every page will eventually be torn out.
2. All the pages will eventually be torn out.
3. All the pages will eventually have been torn out.

The first statement is true because Page n will be torn out on Day n for all n.
The third statement is false because there is no n such that on Day n, every page will have been torn out.
The second statement is potentially ambiguous: true if interpreted as meaning statement 1, false if interpreted as meaning statement 2.

Let us give Achilles a new assignment. He is required to walk a distance of 1 kilometre in a straight line from a point A to a point B. He must first cover half the distance, then half of the remaining distance, then again half of the remaining distance, and so on ad infinitum. This is one of Zeno’s paradoxes. Let us assume he walks at constant speed. Consider the following statements:

1. For every positive integer n, Achilles will eventually reach the point ½ + ¼ + 1/8 +…2^(-n) kilometres from A.
2. For all positive integers n, Achilles will eventually reach the point ½ + ¼ + 1/8 +…2^(-n) kilometres from A.
3. Achilles will eventually have reached all the points ½ + ¼ + 1/8 +…1/2^n kilometres from A.

Again statement 2 is ambiguous in that it could be taken as equivalent to statement 1 or statement 3. I assume Zeno had no difficulty accepting statement 1 but he balked at statement 3 because it implied the completion of an infinite sequence.

The funny thing is that none of the statements is true because, as noted earlier in this thread, space cannot be infinitely divided.

Discuss!

I’ll try to find time soon to respond to the recent comments above.

Offline

#23 2016-01-06 22:16:19

Solvitur ambulando
Member
Registered: 2015-12-06
Posts: 28

Re: The Achilles diary paradox

Calligar, yes, good point. In any finite time interval (is there any other?) a strategy of indefinite postponement is potentially compatible with the ‘take every page’ condition. If the gods were smart they would insist that Achilles declare his page-taking strategy at the outset and stick to it. To fulfil the condition he would have to specify an infinite sequence n1, n2 etc such that on each day n-sub-i he would take a non-birthday page. Achilles is free to space the terms of the sequence as widely as he wishes, eg n1 = 100, n2 = 100^100, n3 = 100^100^100 etc. He might want to take a non-birthday page every now and then just to appreciate how much better off he is. See also my response to Relentless at 2015-12-20 17:19:59.

Last edited by Solvitur ambulando (2016-01-07 16:13:18)

Offline

#24 2016-01-07 00:19:21

Relentless
Member
Registered: 2015-12-15
Posts: 624

Re: The Achilles diary paradox

Hi! I find myself in complete agreement with your post about the paradox. At first, even besides ambiguity, it seems like statements 1 and 3 are not independent and it is hard to accept #1 without accepting #3... but as soon as you attempt to formalise that thought, the tension evaporates. It is simply that the pages never run out, and the days never run out, so there will always be a definite future.

Regarding Zeno's paradox, I would disagree with all of the statements as you mention - but I should mention that I have heard explanations that satisfy me even with infinitely divisible space. Without drawing on too much of my memory at this moment, the answer involves time and limits.

Last edited by Relentless (2016-01-07 00:20:09)

Offline

#25 2016-01-08 18:51:00

Solvitur ambulando
Member
Registered: 2015-12-06
Posts: 28

Re: The Achilles diary paradox

I do not feel that we (or at least I) have fully resolved the point that Relentless and Calligar raised above, namely that Achilles can endlessly postpone the task of taking non-birthday pages unless the gods insist that he reveal his page-tearing strategy at the outset and insist that he stick to it. Let’s look at a simpler scenario.

Achilles promises to perform a task but is not specific about when he will perform it. In the case of the Diary Paradox the task could be tearing out the first non-birthday page. But it could be any task – buying his mother flowers, whatever. For simplicity, assume that the task can be accomplished in less than one day.

Recall that Achilles is immortal. Let us assume that he lives in a version of Earth where day and night continue endlessly in their present form. Consider the following statements:

1. Every day is followed by another day, indeed by infinitely many other days.
2. Achilles can postpone the task for n days without breaking his word, for any positive integer n.
3. Ditto, for all positive integers n.
4. Achilles can postpone the task forever without breaking his word.
5. If Achilles postpones the task forever he will never undertake it.
6. If he never undertakes the task, he will break his word.

Where is the flaw in these arguments? One could question whether we can meaningfully posit endless time, but let’s put that aside for the moment.

The flaw lies with Statement 6. The phrase ‘he will break his word’ implies (can only imply) that a time will come when it will be clear that he has broken his word. But no such time will ever arrive. Hence, Achilles can get away with never performing the task without breaking his word. In fact, it is impossible for him to break his word because a time will never arrive in which there is no future in which to perform it, should he so choose. Hence, his promise to perform the task ‘sometime in eternity’ has no meaning.

One could argue that Achilles has broken his word the moment he decides (secretly or otherwise) that he is never going to perform the task. But that is another issue.

If you were fooled even briefly by Statement 6, it is worth considering why. We are able to make conclusions about ‘forever’ based on logic: we have done this in Statements 1-5. We are also able to conceive the completion of infinite series, as we do for example when contemplating the series ½, ¾, 7/8 etc in Zeno’s paradox. So it is fairly easy to fall into the trap of making statements about ‘what happens at the end of infinite time’. If Achilles never performs the task it will be clear that he has broken his promise… at the end of infinite time, which never arrives.

Offline

Board footer

Powered by FluxBB