Today we go on a journey to ancient Greece, and before us stands the hero of the Trojan war Achilles. Taking a break from his usual activities of slaying Trojans, Achilles has been enticed into a running race by a rather loud mouthed tortoise. Now clearly the tortoise is at a distinct disadvantage, so has managed to negotiate a small head start. Runners are of course allowed to run at their own maximum speed, but they must run at a constant speed once they cross their respective start lines. Trumpets sound and the race is on. Achilles reaches a frightening pace, the tortoise not so much. After a chunk of time Achilles has reached the position where the tortoise began. But since time has elapsed and the tortoise is moving forwards in the race it must be at a new position beyond the original. No bother the race continues; and Achilles reaches the new position of the tortoise. But hang on a second – more time has passed so the tortoise isn’t in position 2 – it is now in position 3. You see the issue here? Every time we allow Achilles to reach the tortoises old position, the tortoise has moved forwards leading to a logical conclusion Achilles can never overtake the tortoise. Yet this is not what everyday experience tells us. This is one of Zeno’s paradoxes; a paradox much older than Christ which the ancients loved to muse on. Intuition and logic at odds; but how do you resolve this mystery? Snared in the logic trap.

It isn’t hard to know that the outcome is wrong – if one object is moving faster than the other then eventually at some distance it will overtake; otherwise you could try and shoot me and I would just run away from the bullet; not an experiment me or Zeno would perform. But how do you rebuff the logic? We are inquisitive minds after all and it is not good enough to just dismiss what is at face value a very logical argument. In truth the answer isn’t so hard and we have talked about it before; infinity.

What you need to appreciate is we have a sequence. Don’t worry about the nuts and bolts of it; but for every step in the above diagram, you can appreciate it takes Achilles less time (he is travelling a smaller distance at the same speed). So what we end up with is a sum of infinitely many distances – imagine for illustrations sake, from the start to position 1 is 10 seconds, then from position 1 to position 2 is 1 second, then position 2 to position 3 is 0.1 seconds etc (no attempt to make the mathematics accurate here). In order for Zeno’s paradox to be correct, when we add up the infinite number of terms (the limit of the sequence as n tends to infinity) we must have infinite time (i.e. it never happens). This is wrong; in fact when we sum this series over infinity we have a finite time which happens to be a method for calculating the exact point where Achilles takes the lead. Phew – order is restored. The heart of this paradox is understanding the nature of infinity; without this you can quite easily convince yourself of something that is so clearly not the case, or at least land yourself in a situation where you cannot rebuff it.

That is what today’s post is all about; when you are embarking on a voyage of scientific and mathematical discovery you have to keep your wits about you. It is only by concentrated thought and a good understanding of all the components of your argument that the correct result can be arrived at. Otherwise you end up writing a very compelling case for a falsity. In the case of Achilles this was okay; because the result was so clearly wrong that I am sure nobody decided to go and race a car before they read on. But this will not always be the case; particularly if you are right on the frontier of modern science (somewhere I hope to visit in the future).

Paradoxical logic need not always take the form of the result above. Sometimes the logic isn’t actually all that tricky; it’s just the result is not the first thing that springs to mind. Try this:

*You have 100kg of good quality healthy potatoes, which unlike your average potato are comprised of 99% water and 1% “potato matter”. Unsurprisingly, because you had the ludicrous idea of buying 100kg of potato they don’t get eaten. In fact in the first week none get eaten and the potatoes loose water – now they are 98% water. You decide to move them, but need to know the weight using only the above information. How much do they weigh? Have a think before you peak! *

Well commiserations if you said anything that starts with a 9! Snared by the logic trap again. So let’s work this out with some elementary algebra. Appreciate that the potato matter is unchanged – I had 1kg before (1% of 100kg) and none of this has changed. Now if we let x represent the total weight of the spuds, the total weight must be 1kg, the unchanged potato matter, and the 98% which is water – so we can say the total weight x is expressed by the below formula.

What we see here is if we multiply through by 100, and collect our x terms on the right hand side we have 100 = 2x; i.e the potatoes now weight 50kg! I love a good mind puzzle like that; if you said 99kg then don’t feel bad; you just forgot about concentration. If the original ratio of water to potato matter was 99:1, when the water decreases by a percent the potato matter now accounts for two percent of the weight; 49:1 ratio, so 50kg in total. These types of problems are designed to confuse.

You can look more of these up if you are interested; there are thousands from critical thinking problems, through to mathematics and science that really are most excellent to engage with. They will test your brain in a very worthwhile way and get you thinking logically. It is surprising how different somethings look through a logical lens. I will leave you with Galileo’s paradox, it involves sets.

*Whilst not all numbers are square numbers, there are no more numbers than square numbers.*

Unfortunately, insofar as infinity problems are concerned, it takes me an infinite amount of time to figure them out

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When I was still young and optimistic back in 1947 I tried to desperately attack my internal chaos with mathematics by taking a course in CCNY in introduction to theoretical mathematics and like Brer Rabbit got stuck on the tar baby of Geog Cantor’s multiple infinities where one form of infinity was much larger than another. Somewhat the same consternation the average fruit fly has in playing chess. My bonding to fruit flies has never been greater and I take them and oysters and coral polyps as cohort strugglers in the long battle of evolution to puzzle out the universe.

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fruit flies and chaos theory – love it!

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I’m still trying to figure out how losing 1% of water mass can make the potatoes lose half their weight… it doesn’t seem reasonable unless you use algebra. Interesting article! I applaud you for encouraging the use of logic.

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Thank you very much I am glad you enjoyed the article… Life only seems reasonable when you use algebra I often find! The trick with that is to think about the situation before and the situation after. If you just think 1% = 1kg of water so it’s just 1kg you get stuck forever!

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Reading this I’m reminded of the continuum fallacy. Cool post!

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Thank you very much! I am glad you enjoyed

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Wow! I want the Achilles and the Tortoise Graphic as a Tattoo 🙂

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A very interesting choice of tattoo!

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Common sense to the rescue but is it always trustworthy? Common sense declares we cannot create a universe from nothing yet the big bang tells us it is so.

Was mathematics discovered or invented ? Might I tentatively suggest that as it seems to explain physical phenomenon so well it was discovered. It is sort of built into nature one might be excused for saying it is the underlying foundation of creation.

Perhaps when common sense fails mathematics takes over and yet the very working blood of mathematics is common sense.

It would appear that Godel discovered it has limitations , so maybe there are things mathematics cannot investigate and a new language of thought is required to rescue us struggling humans.

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Although the common sense intuition that nothing comes of nothing seems to fetter creation to unliklihood the incessant turbulence of subatomics seems to indicate the creation out of nothing of matching negative and positive particles who’s sum is nothing nevertheless becomes creative when one of these pairs is somehow barred from mating back to oblivion, at least temporarily, near a black hole or in some other way. Perhaps the entire universe is awaiting its final total dissolution in that final total marriage when it gobbles itself back to oblivion.

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Pingback: Building digital businesses | Don’t get snared by the logic trap — Rationalising The Universe·

🙂

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I’m not quite sure I understand the point of Galileo’s paradox. It has a vague definition of “number.” Forgive me if this post is out of place, I just found this blog. But I’ve studied a bit of mathematics in the past and find the question not so much of a paradox as a mathematical statement.

The set of all square numbers and set of all natural numbers have the same cardinality. We know this because we can make a bijection between the two sets by mapping each natural number to its square. Since that natural numbers have the same cardinality as the integers and the rational numbers, we know that the set of all squares also have the same cardinality. But Cantor’s diagonal shows that the real numbers don’t have the same cardinality as these other sets.

Interestingly, this means that Galileo’s Paradox is either definitely true or definitely false depending on whether your definition of “number” includes stuff like pi or not. I’m curious if he didn’t think those were numbers, or, considering when he lived, if he assumed that they were also countable!

Sorry again if I’m preaching to the choir or totally veering into left field. Thanks for listening : )

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Hi thank you very much for your message! Not at all, your analysis shows a very deep understanding of numbers and Cantor is exactly the logic you need to follow to understand the paradox. What Galileo was saying is that there are clearly infinite numbers of integers – and also infinite numbers of square numbers. But if every single number is not a square number (which it is not, 7 for example) there must be more non-square numbers. The answer to the paradox, as you rightly point out involves the logic of Cantor, and the understanding that it is possible to have infinities of “different sizes”. It is interesting actually the majority of the early paradoxes are misunderstandings of the nature of infinity

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You read my mind, leahwasacz! I saw the paradox and thought “Don’t you mean integers, not numbers?” Then I found your great comment and was happy.

Also, every single number IS a square… just not a perfect square. Even i! http://www.murderousmaths.co.uk/books/reslabmn.htm#rooti.

– Khiyali

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Yes the paradox of course refers to the integers; otherwise it isn’t a paradox at all – and indeed complex numbers and irrational numbers can be bought into play to again make no paradox; but the best way round the paradox is as leah stated the logic of Cantor and the cardinality of set theory. This is the correct way to approach the problem.

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I recall taking a math test at school, aged 14 (a very distant past age in linear time), in which I scored a big fat zero. I haven’t improved too much during the intervening periods through which our planet has ploughed it’s elliptical course around the sun. You bamboozle me. I must return for more!

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I am glad you want to make a return! There is no pursuit more rewarding than Physics and Mathematics

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Hey can i get some followers I’m doing work for college😄

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I’m afraid I don’t think they’re transferable!

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Hahaha

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Interesting … that’s whiled away some work time 🙂

Logically we know that the paradox is wrong, but when you’re looking at the diagram it’s difficult to explain why. Because both players are visually on the same line you are led to fixate on the distance between them.

I made some sense of it when I thought of the distance between them only as a product of the relative speed / distance travelled by each independently. On my screen resolution (holding a ruler to the screen) Achilles travels about 5 units for the tortoise’s 2 units – which if O level maths serves is 5/2 = 2.5 times faster. So although they start 5 units apart (on screen) the tortoise only gets to travels just over 3 units before he is caught. If I’d paid more attention in class maybe I’d be able to figure out what the formula is to get a more accurate figure! 🙂

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I am glad you enjoyed! Yes exactly; what you need to do to truly solve it is to take the formula for the nth time interval and then sum this formula up to infinity. What you find is that the formula tends to a finite number – the exact time in which Achilles overtakes and goes on to win the race. If of course the sum had tended to infinity the tortoise would win; luckily for our sanity it does not!

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I couldn’t leave it alone, and tried to find an answer. I found an online simultaneous equation solver (cheat!) and entered s=d as the speed/distance relationship for Achilles, and 2/5*s=d+5 for the speed/distance relationship for the tortoise (representing the starting speed and distance for both). The result generated was -25/3. Which is -8.33 recurring. Does that mean that they meet at 8.33r units of distance ? (it seems close to the predicted distance on the pencil drawings that I did of the race) …. or have I just created gibberish which looks like a plausible answer

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Hold on maybe I am misunderstanding – we have set speed equal to distance? I don’t think this can work since the speed is constant (per the puzzle) but the distance is not. Additionally I think we need to give ourselves some initial conditions. For example, if Achilles is a units behind and s1 is his speed in step 1 it takes him t = (a/s1). But in this time the tortoise, with s2 as his speed has moved x = (a/s1)s2 units. So step 2 takes Achilles t = ((a/s1)s2)/s1) and so on and so forth. What we get is a general formula for the nth step that we take a limit as n tends to infinity. There are some workings here you may be interested in:

http://www.decodedscience.org/zenos-paradox-of-achilles-and-the-tortoise/1945

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Well I did mention that my maths expertise stopped at O-level 🙂

Maybe the difference – beyond simply being wrong! – is because the I’ve ‘guessed’ the additional data of relative speed of the racers, which I derived by on-screen measurement of distance travelled at the first recorded time interval (where at point 1 following the start of the race the tortoise had travelled 2/5ths of the distance of Achilles).

So my ‘speed’ wasn’t a representation of time and distance (like 10mph) – it’s more a representation of distance travelled in relation to the distance travelled by the other racer.

As, in this case, they’re both representations of distance (relative and raced) – I thought that it had some basis in logic 🙂

The starting point for Achilles has (a potential but not yet active) relative distance (s) equal to actual distance (d). The starting point for the tortoise has his relative distance (2/5 * s) equal to his actual distance (d + 5).

When I plot it in excel it still seems to have some logic 🙂 Achilles: 0,0.5,1,1.5,2,2.5,etc.; tortoise: 5,5.2,5.4,5.6. Intersection of the lines happens at about 8.3, as far as I can see it on the graph.

… So it seems that I didn’t solve the problem so much as invent my own problem and then struggle with it ! 🙂

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Indeed – your approach will give you roughly the right answer because what you are doing is constructing the sequence rather than formulating it – which for problems like this is just fine! Of course formulation becomes much more useful when we have hundreds of steps, but your method provides a good point to show the point at which the overtake will happen; and from there Achilles will get further and further ahead

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I must admit I left maths behind at A level, but I really enjoyed this post! It’s a good take on critical thinking and seeing the bigger picture rather than (always) relying only on what’s in front of you… well that’s how I see it at least lol. Great stuff 🙂

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Why thank you very much! I am glad you enjoyed the post; I do believe that whatever path you take there is still time to enjoy a bit of Maths

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Wow. I enjoyed this post so much. Great Stuff. 🙂

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Thank you very much!

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Reblogged this on the POLYMATH and commented:

a Looong Read. But worth every alphabet in it, at the end. 🙂

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