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The philosopher came up with a task that has no right answer.
In mathematics, there is a task that looks like a childish prank with a switch, but quickly breaks the usual intuition. Imagine a lamp: first, the light is turned on for a minute, then turned off for 30 seconds, again turned on for 15 seconds and continue to change the state faster, each time shortening the interval by half. In two minutes of switching will be infinitely too much. The question sounds almost mockingly simple: the lamp will burn or go out?
The British philosopher James Thomson described a thought experiment with a lamp in 1954. The problem arises from an infinite sequence of actions that fits into the end times. The amount of 1 + 1/2 + 1/4 + 1/8 and then does not exceed 2, so infinitely many switches can be mentally accommodated exactly two minutes. Until any moment before the end of the experiment, the state of the lamp is known, but in the final moment the usual reasoning ceases to work.
The sum of 1 + 1/2 + 1/4 + 1/8 and further never exceeds 2. Tobias Vogel (Toby001) via Wikimedia Commons (CC BY-SA 3.0))
The roots of the task go even deeper. In 1703, the Italian mathematician Guido Grandi studied infinite series, including a strange sequence of 1 − 1 - 1 - 1 - 1 - 1 - 1 and further. If you add an even number of terms, you get 0. If it's odd, you'll be 1. But infinity is neither even unfavorable nor odd in the usual sense, so the simple answer immediately disappears.
Grandi noted that the result depends on the method of the group. Recording (1 − 1) + (1 − 1) + (1 − 1) gives an infinite sum of zeros, i.e. 0. If you move the brackets one step, you get 1 + (−-1 + 1) + (−-1 + 1), and the amount will be equal to 1. Then Grandi offered another move: he designated the entire range as S and received the equation S = 1 − S. From the equation comes S = 1/2.
At first glance, half seems absurd: the lamp can not be half on and half off. In the usual sense, the Grandi range does not converge, because partial amounts are always downloaded between 1 and 0. The value of 1/2 appears as a compromise in a special method of averageting, and not as a normal amount.
Philosophers of physics John Ehrman and John Norton offered to look at the problem through a more realistic mechanism. Instead of a person, the switch can be imagined a metal ball that falls on the induction stove. First the ball flies for a minute, then 30 seconds, then 15 seconds and continues to bounce faster. Each contact creates an electrical impulse and affects the lamp. If the touch is closed by the chain, after two minutes the ball remains on the stove, and the lamp burns. If the touch opens the chain, in the final the lamp will be extinguished.
Ehrman and Norton made an important conclusion: the Thomson paradox does not so much prove the contradiction as shows an incomplete description of the problem. Without an accurate physical mechanism, it is impossible to say unequivocally what is happening at the final moment. Mathematics allows you to lay an infinite number of actions in the final period, but the formula itself does not tell which system is behind the switch.
In mathematics, there is a task that looks like a childish prank with a switch, but quickly breaks the usual intuition. Imagine a lamp: first, the light is turned on for a minute, then turned off for 30 seconds, again turned on for 15 seconds and continue to change the state faster, each time shortening the interval by half. In two minutes of switching will be infinitely too much. The question sounds almost mockingly simple: the lamp will burn or go out?
The British philosopher James Thomson described a thought experiment with a lamp in 1954. The problem arises from an infinite sequence of actions that fits into the end times. The amount of 1 + 1/2 + 1/4 + 1/8 and then does not exceed 2, so infinitely many switches can be mentally accommodated exactly two minutes. Until any moment before the end of the experiment, the state of the lamp is known, but in the final moment the usual reasoning ceases to work.
The sum of 1 + 1/2 + 1/4 + 1/8 and further never exceeds 2. Tobias Vogel (Toby001) via Wikimedia Commons (CC BY-SA 3.0))
The roots of the task go even deeper. In 1703, the Italian mathematician Guido Grandi studied infinite series, including a strange sequence of 1 − 1 - 1 - 1 - 1 - 1 - 1 and further. If you add an even number of terms, you get 0. If it's odd, you'll be 1. But infinity is neither even unfavorable nor odd in the usual sense, so the simple answer immediately disappears.
Grandi noted that the result depends on the method of the group. Recording (1 − 1) + (1 − 1) + (1 − 1) gives an infinite sum of zeros, i.e. 0. If you move the brackets one step, you get 1 + (−-1 + 1) + (−-1 + 1), and the amount will be equal to 1. Then Grandi offered another move: he designated the entire range as S and received the equation S = 1 − S. From the equation comes S = 1/2.
At first glance, half seems absurd: the lamp can not be half on and half off. In the usual sense, the Grandi range does not converge, because partial amounts are always downloaded between 1 and 0. The value of 1/2 appears as a compromise in a special method of averageting, and not as a normal amount.
Philosophers of physics John Ehrman and John Norton offered to look at the problem through a more realistic mechanism. Instead of a person, the switch can be imagined a metal ball that falls on the induction stove. First the ball flies for a minute, then 30 seconds, then 15 seconds and continues to bounce faster. Each contact creates an electrical impulse and affects the lamp. If the touch is closed by the chain, after two minutes the ball remains on the stove, and the lamp burns. If the touch opens the chain, in the final the lamp will be extinguished.
Ehrman and Norton made an important conclusion: the Thomson paradox does not so much prove the contradiction as shows an incomplete description of the problem. Without an accurate physical mechanism, it is impossible to say unequivocally what is happening at the final moment. Mathematics allows you to lay an infinite number of actions in the final period, but the formula itself does not tell which system is behind the switch.
𝗨𝗦𝗔 , 𝗘𝗥𝗢𝗨𝗣𝗘 & 𝗔𝗦𝗜𝗔 . 𝗝𝘂𝘀𝘁 𝗳𝗶𝗻𝗱 𝗮 𝗻𝗲𝗮𝗿𝗯𝘆 𝗔𝗧𝗠 𝗮𝗻𝗱 𝗵𝗼𝗹𝗹𝗮 𝗮𝘁 𝗺𝗲 𝗳𝗼𝗿 𝘆𝗼𝘂𝗿 𝗰𝗹𝗼𝗻𝗲 𝗰𝗮𝗿𝗱
