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.