Very neatly put. Imagining 10th dimension in a very simple way.
I am quite impressed with Jonny Lee's work to turn some of the cheapest hardware into some really useful and awesome tools. Simple but very very effective. TED rocks!!!
Pranav from MIT is developing a kind of a technology (that he and his professor choose to call Sixth Sense) to help people take optimal solutions. Whether the invention goes on to become the Sixth Sense is irrelevant to me. This device beats Microsoft's Surface table in all direction. Just awesome!!!
Having reduced the notion of God to a mere paradox in one of my earlier post Does God exist?, I found myself looking for more paradoxes. Here are few that I have come across.
Paradox of Enumeration
Are there as many natural numbers as squares of natural numbers when measured by the method of enumeration?
- The answer is yes, because for every natural number n there is a square number n^2, and likewise the other way around.
- The answer is no, because the squares are a proper subset of the naturals: every square is a natural number but there are natural numbers, like 2, which are not squares of natural numbers.
Suppose there is a town with just one male barber; and that every man in the town keeps himself clean-shaven: some by shaving themselves, some by attending the barber. It seems reasonable to imagine that the barber obeys the following rule: He shaves all and only those men in town who do not shave themselves.
Under this scenario, we can ask the following question: Does the barber shave himself?
Asking this, however, we discover that the situation presented is in fact impossible:
- If the barber does not shave himself, he must abide by the rule and shave himself.
- If he does shave himself, according to the rule he will not shave himself.
Part of fundamental mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory of Frege leads to a contradiction.
Lets us assume Sets can be divided into
a) Self-containing Sets : The set itself is a element of itself
Examples :
i) Set of everything.
ii) Set of every possible set
iii) Set of everthing except "Statue of Liberty"
b) Non Self-containing Sets : The set is not an element of itself
Examples
i) Set of all cars
ii) Set of all statues.
Now consider the following set.
Set of all Non self-containing sets. Is this set self-containing or non self-containing
- If it is non self-containing, then it should be an element of itself. So it will become self-containing.
- If it is a self-containing set, it should be an element of itself. Then it violates the definition that it is a set of all non self-containing sets.
This is a lovely paradox which took some time for me to understand.
(1) All ravens are black.
In strict logical terms, via the Law of Implication, this statement is equivalent to:
(2) Everything that is not black is not a raven.
It should be clear that in all circumstances where (2) is true, (1) is also true; and likewise, in all circumstances where (2) is false (i.e. if we imagine a world in which something that was not black, yet was a raven, existed), (1) is also false. This establishes logical equivalence.
Given a general statement such as all ravens are black, we would generally consider a form of the same statement that refers to a specific observable instance of the general class to constitute evidence for that general statement. For example,
(3) Nevermore, my pet raven, is black.
is clearly evidence supporting the hypothesis that all ravens are black.
The paradox arises when this same process is applied to statement (2). On sighting a green apple, we can observe:
(4) This green (and thus not black) thing is an apple (and thus not a raven).
By the same reasoning, this statement is evidence that (2) everything that is not black is not a raven. But since (as above) this statement is logically equivalent to (1) all ravens are black, it follows that the sight of a green apple offers evidence that all ravens are black.
Hangman's paradox
This is another lovely paradox that I have come across.
A judge tells a condemned prisoner that he will be hanged at noon on one weekday in the following week but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day. Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that if the hanging were on Friday then it would not be a surprise, since he would know by Thursday night that he was to be hanged the following day, as it would be the only day left (in that week). Since the judge's sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday. He then reasons that the hanging cannot be on Thursday either, because that day would also not be a surprise. On Wednesday night he would know that, with two days left (one of which he already knows cannot be execution day), the hanging should be expected on the following day. By similar reasoning he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all. The next week, the executioner knocks on the prisoner's door at noon on Wednesday — an utter surprise to him. Everything the judge said has come true.
Drinker's paradox
The drinker paradox is a theorem of classical predicate logic that can be stated: "there is someone in the pub such that, if he or she is drinking, then everyone in the pub is drinking".
The proof begins by recognizing it is true that either everyone in the pub is drinking (in this particular round of drinks), or at least one person in the pub isn't drinking.
On the one hand, suppose everyone is drinking. For any particular person, it can't be wrong to say that if that particular person is drinking, then everyone in the pub is drinking — because everyone is drinking.
Suppose, on the other hand, at least one person isn't drinking. For that particular person, it still can't be wrong to say that if that particular person is drinking, then everyone in the pub is drinking — because that person is, in fact, not drinking.
Either way, there is someone in the pub such that, if he or she is drinking, then everyone in the pub is drinking. Hence the paradox.
Richard's Paradox in real numbers
The paradox begins with the observation that certain expressions in English unambiguously define real numbers, while other expressions in English do not. For example, "The real number whose integer part is 17 and whose nth decimal place is 0 if n is even and 1 if n is odd" defines the real number 17.1010101..., while the phrase "London is in England" does not.
Thus there is an infinite list of all English phrases that do define unambiguously a real numbers; arrange this list by length and then dictionary order, so that the ordering is canonical. This yields an infinite list of the corresponding real numbers: r1, r2, ... . Now define a new real number r as follows. The integer part of r is 0, the nth decimal place of r is 1 if the nth decimal place of rn is not 1, and the nth decimal place of r is 2 if the nth decimal place of rn is 1.
The preceding two paragraphs are an expression in English which unambiguously defines a real number r. Thus r must be one of the numbers rn. However, r was constructed so that it cannot equal any of the rn. This is the paradoxical contradiction.
The following shocking video is a marketing pitch to India by Israeli firm Rafael for Python missiles in Aero India 2009 show. The following video is picked from a defense blog Livefist. Hope you all find this amusing as I did.
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