Perron’s Paradox

Let N be the largest positive integer. Then either N = 1 or N > 1.

If N > 1 then N2 > N, which breaks our definition of N as the largest integer. Therefore N = 1.

“The implications of this paradox are devastating,” writes Laurence Chisholm Young. “In seeking the solution to a problem, we can no longer assume that this solution exists. Yet this assumption has been made from time immemorial, right back in the beginnings of elementary algebra, where problems are solved by starting off with the phrase: ‘Let x be the desired quantity.'”

| Science & Math

What’s in a Name

http://commons.wikimedia.org/wiki/File:Jan_Baptist_Weenix_-_Portrait_of_Ren%C3%A9_Descartes.jpg

The disciples of Descartes made a perfect anagram upon the Latinised name of their master, ‘Renatus Cartesius,’ one which not only takes up every letter, but which also expresses their opinion of that master’s speciality–‘Tu scis res naturae’ (Thou knowest the things of nature).

— William T. Dobson, Poetical Ingenuities and Eccentricities, 1882

| Language · Science & Math

Jackpot

A guaranteed way to win at roulette, from Eugene Northrop’s Riddles in Mathematics (1945):

  1. Bet $1 on red.
  2. If you win, go to step 6. If you lose, bet $2 on red.
  3. If you win, go to step 6. If you lose, bet $4 on red.
  4. If you win, go to step 6. If you lose, bet $8 on red.
  5. (And so on.)
  6. When you win, you’ll be $1 ahead. Go back to step 1.

“Theoretically, of course, it is possible for the bank to wipe you out financially. Actually, however, runs of more than 10 or 12 successive blacks or reds are extremely rare, and your stake at the twelfth play would be only $2048. When you do win you will, as before, be $1 ahead of the bank. You can then begin all over again. Simple, isn’t it?”

| Science & Math

Benardete’s Paradox

Prometheus angers Zeus, who dispatches an army of demons with these instructions:

  • Demon 1: If Prometheus is not dead in one hour, kill him.
  • Demon 2: If Prometheus is not dead in half an hour, kill him.
  • Demon 3: If Prometheus is not dead in quarter of an hour, kill him.

And so on. When Prometheus is found dead, the council of gods is displeased, but they find it impossible to identify the guilty demon — any suspect can point to an infinity of demons who must have acted before him. Must Zeus go free?

| Science & Math

The Morning Star Paradox

http://commons.wikimedia.org/wiki/File:Venus-pacific-levelled.jpg
Image: Wikimedia Commons

As it happens, the morning star and the evening star are both Venus–but the solar system might have evolved so that Mercury, for instance, was the brightest star in the morning sky.

Thus the morning star has a property that the evening star does not have: It’s necessarily identical with the morning star.

And if the morning star and the evening star have different properties, then they’re not the same object after all.

| Science & Math

Thinking Back

Can you move an object using only your mind? Of course not. But can you move one in the past?

Since January 1997, the Retropsychokinesis Project at the University of Kent has invited Web visitors to try to influence the replay of a prerecorded bitstream. In other words, they must try to influence an event that has already happened.

The experimenters claim to be agnostic as to whether retroactive causality exists, but “the best existing database suggests that the odds are in the order of 1 in 630 thousand million that the experimental evidence is the result of chance.”

Try it for yourself here — but remember, if you have some skepticism about this, it may only be because someone in the future is influencing you.

| Oddities · Science & Math

The Paradox of Omnipresence and Timelessness

It’s an essential attribute of God that he’s omnipresent, and Thomas Aquinas held that he also stands somehow outside of time and is not bound by temporal considerations. But, Richard La Croix argues,

if God is indeed omnipresent then it would appear that he must have been in the United Nations building yesterday as well as the day before yesterday. And if God was in the United Nations building both yesterday and the day before, then it would appear that he is in time and that temporal predicates do apply to him. So, it would appear that God is not a timeless being if he is omnipresent and that two doctrines crucial to the theology of Thomas Aquinas are logically incompatible.

Omniscience poses further problems: If God knows all things, then he knows what both man and he himself will do. So how is free will possible?

| Religion · Science & Math

The Wow! Signal

http://commons.wikimedia.org/wiki/File:Wow_signal.jpg

On Aug. 15, 1977, a telescope at Ohio State University detected a strong narrowband radio signal in the constellation Sagittarius — one so unusual that astronomer Jerry Ehman marked the printout with an exclamation.

The signal’s intensity rose and then fell as the beam swept past its position in the sky. That’s consistent with an extraterrestrial origin … but in 30 years and more than 100 searches, no one has been able to relocate it.

Without a recurrence, there’s no way to know what Ehman’s telescope heard that night — it’s just a frustrating splash in a large, silent sea.

08/27/2024 UPDATE: It appears there’s a natural explanation. New research suggests that the Wow! signal occurred when an interstellar cloud of cold hydrogen was stimulated by a strong transient radiation source, causing it to brighten momentarily. This phenomenon is rare but recognized, and it matches the described characteristics of the signal. This natural explanation is much preferable to the hypothesis of an alien technology — aliens may be out there, but we don’t need them to explain what Jerry Ehman saw in 1977. Thanks to podcast listener Eugene Chang for the tip.

| Oddities · Science & Math

Hard Times

In 1820, Richard Whatley wrote a facetious elegy for Oxford geologist William Buckland:

Where shall we our great Professor inter,
That in peace he may rest his bones?
If we hew him a rocky sepulchre
He will rise and break the stones,
And examine each stratum that lies around
For he’s quite in his element underground.

Ironically, when Buckland did pass away in 1856, the gravedigger struck an outcrop of limestone just below the surface and had to use gunpowder to put Buckland to rest.

Ambrose Bierce defined geology as “The science of the earth’s crust–to which, doubtless, will be added that of its interior whenever a man shall come up garrulous out of a well.”

| Science & Math