There's a Paradox

What about Paradoxes?  Paradoxes have been a central part of philosophical thinking for centuries.  For example, Achilles and the Tortoise comes from the 5th century BC.  I went to Mentalfloss.com and browsed through the best-known paradoxes.  I chose the discussion on the Raven Paradox for today.  At buzzfeed.com  they're presented with cartoons, so are quite fun. THE RAVEN PARADOX Also known as Hempel’s Paradox, for the German logician who proposed it in the mid-1940s, the Raven Paradox begins with the apparently straightforward and entirely true statement that “all ravens are black.” This is matched by a “logically contrapositive” (i.e. negative and contradictory) statement that “everything that is not black is not a raven”—which, despite seeming like a fairly unnecessary point to make, is also true given that we know “all ravens are black.” Hempel argues that whenever we see a black raven, this provides evidence to support the first statement. But by extension, whenever we see anything that is not black, like an apple, this too must be taken as evidence supporting the second statement—after all, an apple is not black, and nor is it a raven. The paradox here is that Hempel has apparently proved that seeing an apple provides us with evidence, no matter how unrelated it may seem, that ravens are black. It’s the equivalent of saying that you live in New York is evidence that you don’t live in L.A., or that saying you are 30 years old is evidence that you are not 29. Just how much information can one statement actually imply anyway? Here is Wikipedia's summary of the Raven paradox:  Raven paradox: (or Hempel's Ravens): Observing a green apple increases the likelihood of all ravens being black. I was in Toronto yesterday, and found this reflective material on the side of the umbra store near Queen Street West.  I had thought I might find some pink water reflections, but this material created prism/rainbow reflections. The first picture shows the photo without adjustments, and the next few show the finished result, with the pavement turned to black.

The Birthday Paradox

The Birthday Paradox by Alan Bellows It was my birthday yesterday.  Here are a few highlights from the parade.  In the last picture, you can see the parade is over, and the clowns are resting in the tent. So to entertain you with Birthday trivia, Alan Bellows explains how the 'same birthday' stats work... Alan Bellows writes:  "I have never had a very good relationship with Mathematics. I used to think it was me... I thought that perhaps I was just a bit put off by Math's confident demeanor and superior attitude, and by its tendency to micromanage every tiny detail of my universe. But over time I have come to the realization that I'm not the source of the problem. Math, as it turns out, is out of its bloody mind. Consider the following example: Assuming for a moment that birthdays are evenly distributed throughout the year, if you're sitting in a room with forty people in it, what are the chances that two of those people have the same birthday? For simplicity's sake, we'll ignore leap years. A reasonable, intelligent person might point out that the odds don't reach 100% until there are 366 people in the room (the number of days in a year + 1)... and forty is about 11% of 366... so such a person might conclude that the odds of two people in forty sharing a birthday are about 11%. In reality, due to Math's convoluted reasoning, the odds are about 90%. This phenomenon is known as the Birthday Paradox. If the set of people is increased to sixty, the odds climb to above 99%. This means that with only sixty people in a room, even though there are 365 possible birthdays, it is almost certain that two people have a birthday on the same day. After making these preposterous assertions, Math then goes on to rationalize its claims by recruiting its bastard offspring: numbers and formulas. It's tricky to explain the phenomenon in a way that feels intuitive. You can consider the fact that forty people can be paired up in 780 unique ways, and it follows that there would be a good chance that at least one of those pairs would share a birthday. But that doesn't really satisfy the question for me, it just feels marginally less screwy. So I did something quite out of character: I crunched the numbers. The values rapidly become unmanageable, but the trend is clear: to see the charts - go to http://www.damninteresting.com/the-birthday-paradox/ Only calculating up to eight people, we see that of the three hundred fifteen quintillion possible combinations of birthdays the group has, 7.4% of cases-- or about one in thirteen -- result in two of them having the same birthday. As each person is added, the odds do not increase linearly, but rather they curve upwards rapidly. This trend continues up to around twenty-three people, where the curve hits 50% odds, and the rate of increase starts going down. It practically flattens out when fifty-seven people are considered, and the odds rest at about 99%. Though it may not be intuitive, the numbers follow the pattern quite faithfully." And there's more… So go to the site to see the charts and read the rest.  I liked the section about how the stats of the birthday paradox are used to hack into computers! From:  http://www.damninteresting.com/the-birthday-paradox/