SS: Thought experiments

Thought Experiments

William James’ squirrel:

SOME YEARS AGO, being with a camping party in the mountains, I returned from a solitary ramble to find every one engaged in a ferocious metaphysical dispute. The corpus of the dispute was a squirrel – a live squirrel supposed to be clinging to one side of a tree-trunk; while over against the tree’s opposite side a human being was imagined to stand. This human witness tries to get sight of the squirrel by moving rapidly round the tree, but no matter how fast he goes, the squirrel moves as fast in the opposite direction, and always keeps the tree between himself and the man, so that never a glimpse of him is caught. The resultant metaphysical problem now is this: Does the man go round the squirrel or not? He goes round the tree, sure enough, and the squirrel is on the tree; but does he go round the squirrel? [Stop for discussion] In the unlimited leisure of the wilderness, discussion had been worn threadbare. Every one had taken sides, and was obstinate; and the numbers on both sides were even. Each side, when I appeared therefore appealed to me to make it a majority. Mindful of the scholastic adage that whenever you meet a contradiction you must make a distinction, I immediately sought and found one, as follows: “Which party is right,” I said, “depends on what you practically mean by ‘going round’ the squirrel. If you mean passing from the north of him to the east, then to the south, then to the west, and then to the north of him again, obviously the man does go round him, for he occupies these successive positions. But if on the contrary you mean being first in front of him, then on the right of him, then behind him, then on his left, and finally in front again, it is quite as obvious that the man fails to go round him, for by the compensating movements the squirrel makes, he keeps his belly turned towards the man all the time, and his back turned away. Make the distinction, and there is no occasion for any farther dispute. You are both right and both wrong according as you conceive the verb ‘to go round’ in one practical fashion or the other.”
[“What is pragmatism?” 1904 lecture]

Descartes’ Evil Genius:

I shall then suppose, not that God who is supremely good and the fountain of truth, but some evil genius not less powerful than deceitful, has employed his whole energies in deceiving me; I shall consider that the heavens, the earth, colors, figures, sound, and all other external things are nothing but the illusions and dreams of which this genius has availed himself in order to lay traps for my credulity; I shall consider myself as having no hands, no eyes, no flesh, no blood, nor any senses, yet falsely believing myself to possess all these things; I shall remain obstinately attached to this idea, and if by this means it is not in my power to arrive at the knowledge of any truth, I may at least do what is in my power [i.e. suspend my judgment], and with firm purpose avoid giving credence to any false thing, or being imposed upon by this arch deceiver, however powerful and deceptive he may be. But this task is a laborious one, and insensibly a certain lassitude leads me into the course of my ordinary life. And just as a captive who in sleep enjoys an imaginary liberty, when he begins to suspect that his liberty is but a dream fears to awaken, and conspires with these agreeable illusions that the deception may be prolonged, so insensibly of my own accord I fall back into my former opinions, and I dread awakening from this slumber, lest the laborious wakefulness which would follow the tranquility of this repose should have to be spent not in daylight, but in the excessive darkness of the difficulties which have just been discussed.

I suppose, then, that all the things that I see are false; I persuade myself that nothing has ever existed of all that my fallacious memory represents to me. I consider that I possess no senses; I imagine that body, figure, extension, movement and place are but the fictions of my mind. What, then, can be esteemed as true? Perhaps nothing at all, unless that there is nothing in the world that is certain.

Bertrand Russell:

There is a village in which all the adult males are clean shaven. In the village is a barber. The barber shaves all and only those adult males who do not shave themselves. So, if Bob shaves himself then the barber does not shave Bob. And, if Bob does not shave himself then the barber does shave Bob.
Question: Does the barber shave himself?

Hypothesis: The world and everything in it was created five minutes ago.

Russell wants to show that the memories of something are logically independent of that something, but the hypothesis has been used to support skepticism.

English: Russell's Teapot Español: Tetera de R...

English: Russell’s Teapot Español: Tetera de Russell (Photo credit: Wikipedia)

Russell’s teapot, sometimes called the Celestial Teapot, was an analogy first coined by the philosopher Bertrand Russell, to refute the idea that the burden of proof lies somehow upon the sceptic to disprove the unfalsifiable claims of religion. In an article entitled Is There a God?, commissioned (but never published) by Illustrated magazine in 1952, Russell said the following:
If I were to suggest that between the Earth and Mars there is a china teapot revolving about the sun in an elliptical orbit, nobody would be able to disprove my assertion provided I were careful to add that the teapot is too small to be revealed even by our most powerful telescopes. But if I were to go on to say that, since my assertion cannot be disproved, it is intolerable presumption on the part of human reason to doubt it, I should rightly be thought to be talking nonsense. If, however, the existence of such a teapot were affirmed in ancient books, taught as the sacred truth every Sunday, and instilled into the minds of children at school, hesitation to believe in its existence would become a mark of eccentricity and entitle the doubter to the attentions of the psychiatrist in an enlightened age or of the Inquisitor in an earlier time.
In his book A Devil’s Chaplain, Richard Dawkins developed the teapot theme a little further:
The reason organized religion merits outright hostility is that, unlike belief in Russell’s teapot, religion is powerful, influential, tax-exempt and systematically passed on to children too young to defend themselves. Children are not compelled to spend their formative years memorizing loony books about teapots. Government-subsidized schools don’t exclude children whose parents prefer the wrong shape of teapot. Teapot-believers don’t stone teapot-unbelievers, teapot-apostates, teapot-heretics and teapot-blasphemers to death. Mothers don’t warn their sons off marrying teapot-shiksas whose parents believe in three teapots rather than one. People who put the milk in first don’t kneecap those who put the tea in first.
Similar concepts to Russell’s teapot are the Invisible Pink Unicorn, and Flying Spaghetti Monster.

Galileo:

compound

Compound objects thought experiment to reduce Aristotle’s theory of falling objects (heavier bodies fall faster) to absurdity.
Reduction ad Absurdum argument: Galileo argued against the common sense idea of his time that heavy objects fall faster than light objects – take object A 15 lbs and B 5 lbs; according to your claim
A will fall faster than B. But if I connect the two together then the combination will at 20 lbs fall faster than either, but wait B will act as a drag and slow the combined thing down! So, C will fall both faster and slower than itself!!

Plato’s Cave

Plato has Socrates describe a group of people who have lived chained to the wall of a cave all of their lives, facing a blank wall. The people watch shadows projected on the wall by things passing in front of a fire behind them, and begin to ascribe forms to these shadows. According to Plato’s Socrates, the shadows are as close as the prisoners get to viewing reality. He then explains how the philosopher is like a prisoner who is freed from the cave and comes to understand that the shadows on the wall do not make up reality at all, as he can perceive the true form of reality rather than the mere shadows seen by the prisoners.
The Allegory may be related to Plato’s Theory of Forms, according to which the “Forms” (or “Ideas”), and not the material world of change known to us through sensation, possess the highest and most fundamental kind of reality. Only knowledge of the Forms constitutes real knowledge.[1] In addition, the Allegory of the Cave is an attempt to explain the philosopher’s place in society: to attempt to enlighten the “prisoners.”

Judith Jarvis Thomson:

It is morally wrong to kill an innocent person unless it is done to save a life.

The Famous Violinist Argument against The Killing-the-Innocent Principle:

“You wake up in the morning and find yourself back to back in bed with an unconscious violinist. A famous unconscious violinist. He has been found to have a fatal kidney ailment, and the Society of Music Lovers has canvassed all the available medical records and found that you alone have the right blood type to help. They have therefore kidnapped you, and last night the violinist’s circulatory system was plugged into yours, so that your kidneys can be used to extract poisons from his blood as well as your own. The director of the hospital now tells you, ‘Look, we’re sorry the Society of Music Lovers did this to you—we would never have permitted it if we had known. But still, they did it, and the violinist is now plugged into you. To unplug you would be to kill him. But never mind, it’s only for nine months. By then he will have recovered from his ailment, and can safely be unplugged from you.’ Is it morally incumbent on you to accede to this situation? No doubt it would be very nice of you if you did, a great kindness. But do you have to accede to it? What if it were not nine months, but nine years? Or longer still? What if the director of the hospital says. ‘Tough luck. I agree. but now you’ve got to stay in bed, with the violinist plugged into you, for the rest of your life. Because remember this. All persons have a right to life, and violinists are persons. Granted you have a right to decide what happens in and to your body, but a person’s right to life outweighs your right to decide what happens in and to your body. So you cannot ever be unplugged from him.’ I imagine you would regard this as outrageous.” (Thomson, fourth paragraph)

  1. If the Killing-the-Innocent Principle is true, then it is morally wrong for you to unplug yourself from the famous violinist.
  2. It is not morally wrong for you to unplug yourself from the famous violinist.
  3. Therefore, the Killing-the-Innocent Principle is not true.
    Apparent Moderate Anti-Abortionism

Abortion is morally wrong except in the following circumstances:
(i) if it is necessary to save the life of the mother, or
(ii) if the mother has not granted the fetus the right to use her body.
The People Seeds

” … suppose it were like this: people-seeds drift about in the air like pollen, and if you open your windows, one may drift in and take root in your carpets or upholstery. You don’t want children, so you fix up your windows with fine mesh screens, the very best you can buy. As can happen, however, and on very, very rare occasions does happen, one of the screens is defective, and a seed drifts in and takes root. Does the person-plant who now develops have a right to the use of your house? Surely not–despite the fact that you voluntarily opened your windows, you knowingly kept carpets and upholstered furniture, and you knew that screens were sometimes defective.” (Thomson, section 4).

What the People Seeds is supposed to show: that if a woman has taken all reasonable precautions against getting pregnant, she has not granted the use of her body to any fetus that might result from intercourse.

To deny this, Thomson claims, would commit you the claim that the homeowner in the case above has granted the use of her home to any people-seed that might result from having carpets and upholstery.

Therefore, Apparent Moderate Anti-Abortionism is only apparently anti-abortionist because it allows abortion in cases of failed contraception.
Thomson’s Position on Abortion:

Abortion is permissible when, and only when
(i) all reasonable precautions against having a child have been taken; and
(ii) not aborting would require large sacrifices (“of health, of other interests and concerns, of other duties and commitments”) for an extended period of time.
Thomson’s Theory in the Normative Ethics of Behavior:

“Minimally Decent Samaritanism” or (MDS)

MDS: an act is morally right if and only if it is what a Minimally Decent Samaritan (or better) would do.

MDS is a version of an approach to the Normative Ethics of Behavior called “Virtue Theory.” A typical version of Virtue Theory says the following:

an act is morally right if and only if it is what a virtuous person would do.

A defender of this Virtue Theory would still have to explain what a virtuous person is (just as Thomson still has to explain what a Minimally Decent Samaritan is) in order for the theory to be complete.

Thomson is a rich source. Another of her well known entries is the “Trolley Problem”. It goes like this:

Suppose you are the driver of a trolley. The trolley rounds a bend, and there come into view ahead five track workmen, who have been repairing the track. The track goes through a bit of a valley at that point, and the sides are steep, so you must stop the trolley if you are to avoid running the five men down. You step on the brakes, but alas they don’t work. Now you suddenly see a spur of track leading off to the right. You can turn the trolley onto it, and this save the five men on the straight track ahead. Unfortunately … there is one track workman on that spur of track. He can no more get off the track in time than the five can, so you will kill him if you turn the trolley onto him. Is it morally permissible for you to turn the trolley? [Yale Law Journal 94 (1985)

The Chinese Room Argument

(First published Fri Mar 19, 2004; substantive revision Tue Sep 22, 2009)

The Chinese Room argument, devised by John Searle, is an argument against the possibility of true artificial intelligence. The argument centers on a thought experiment in which someone who knows only English sits alone in a room following English instructions for manipulating strings of Chinese characters, such that to those outside the room it appears as if someone in the room understands Chinese. The argument is intended to show that while suitably programmed computers may appear to converse in natural language, they are not capable of understanding language, even in principle. Searle argues that the thought experiment underscores the fact that computers merely use syntactic rules to manipulate symbol strings, but have no understanding of meaning or semantics. Searle’s argument is a direct challenge to proponents of Artificial Intelligence, and the argument also has broad implications for functionalist and computational theories of meaning and of mind. As a result, there have been many critical replies to the argument.

John Wisdom’s “Gods”

“Once upon a time two explorers came upon a clearing in the jungle. In the clearing were growing many flowers and many weeds. One explorer says, “Some gardener must tend to this plot.” The other disagrees, “There is no gardener.”
So they pitch their tents and set a watch. No gardener is ever seen. “But perhaps he is an invisible gardener.” So they set up a barbed-wire fence. They electrify it. They patrol with bloodhounds. (For they remember how HG Wells’s The Invisible Man could be both smelt and touched though he could not be seen.)
But no shrieks ever suggest that some intruder has received a shock. No movements of the wire ever betray an invisible climber. The bloodhounds never give cry. Yet still the believer is not convinced. “But there is a gardener, invisible, intangible, insensible to electric shocks, a gardener who has no scent and makes no sound, a gardener who comes secretly to look after the garden which he loves.”
At last the skeptic despairs, “But what remains of your original assertion? Just how does what you call an invisible, intangible, eternally elusive gardener differ from an imaginary gardener or even from no gardener at all?”

Logic Puzzle

Once upon a time I used the following puzzle in an elementary school class at Cilaire. I even brought some costumes and asked for volunteers.
Once upon a time a farmer went to market and purchased a fox, a goose, and a bag of corn. On his way home, the farmer came to the bank of a river and hired a boat. But in crossing the river by boat, the farmer could carry only himself and a single one of his purchases – the fox, the goose, or the bag of corn.
If left alone, the fox would eat the goose, and the goose would eat the corn.
The farmer’s challenge was to carry himself and his purchases to the far bank of the river, leaving each purchase intact. How did he do it?
The first step must be to bring the goose across the river, as any other will result in the goose or the corn being eaten. When the farmer returns to the original side, he has the choice of bringing either the fox or the corn across. If he brings the fox across, he must then return to bring the corn over, resulting in the fox eating the goose. If he brings the corn across, he will need to return to get the fox, resulting in the corn being eaten. Here he has a dilemma, solved by bringing the fox (or the corn) over and bringing the goose back. Now he can bring the corn (or the fox) over, leaving the goose, and finally return to fetch the goose.

His actions in the solution are summarised in the following steps:

  1. Bring goose over
  2. Return
  3. Bring fox or corn over
  4. Bring goose back
  5. Bring corn or fox over
  6. Return
  7. Bring goose over

Thus there are seven crossings, four forward and three back.

Science and math

Achilles and the Tortoise

Achilles and the tortoise is similar. Suppose that the swift Achilles is having a race with a tortoise. Since the tortoise is much slower, she gets a head start. When the tortoise has reached a given point a, Achilles starts. But by the time Achilles reaches a, the tortoise has already moved beyond point a, to point b. And by the time Achilles reaches b the tortoise has already moved a little bit farther along, to point c. Since this process goes on indefinitely, Achilles can never catch up with the tortoise.
Nowadays, the standard solution to these paradoxes relies on the claim that (contrary to Zeno’s assumption) an infinite series can in fact be completed. Thanks to advances in mathematics, we now know that the infinite series of fractions involved in e.g., the Racetrack, has a finite sum: (1/2 + 1/4 + 1/8 + …) = 1. Hence one will of course reach the end of the track.

While I agree that the solution must depend in some way on this fact, I’m not so sure that no problems remain. One can imagine Zeno replying to the proposed solution as follows:

“Of course half the length, plus one fourth, plus one eighth, and so on, add up to the whole length. And that’s just the point. The whole length contains an infinite number of finite parts. In order to traverse it, therefore, a runner would have to complete an infinite number of tasks. But how can such a thing to be possible?”

Some modern philosophers have argued that there are indeed serious problems with the notion of completing an infinite number of tasks. The best-known example of a current-day Zeno type paradox is the Thomson Lamp, named after James F. Thomson.

The Thomson Lamp

Suppose you have a lamp with a simple on/off switch. Press the switch when it is off and the lamp will be turned on, press it again and it will be turned off. Now suppose you run the following experiment. You turn the lamp on at the start of a minute. Thirty seconds later, you turn it off. In another fifteen seconds, you turn it back on, then 7 1/2 seconds later back off again, and so on throughout the midpoints of whatever time remains. Now the question is this. At the end of the minute, will the lamp be on or off?

Since the lamp has been turned on and off an infinite number of times, for every time it has been turned on, it has been turned off, and vice versa. At the end of the minute, therefore, it can be neither on nor off. But it must be one or the other.

Attempts to find fault in this paradox often attack some irrelevant aspect of the argument. Thus one sometimes hears the criticism that this situation is physically impossible, since no mechanism could operate indefinitely fast. The on/off switch would not be able to keep up. As a counter argument to this type of criticism, I offer the following simplified version of the Thomson Lamp:

Odd/Even Paradox

Suppose a point P is moving between points A and B (just like in the original Racetrack). And suppose also that we stipulate that P is in the state “even” for the first half of the journey, “odd” for the next 1/4, “even” for the next 1/8, and so on. That is, we simply decide to classify P based on where along the journey it is, such that it alternates between what we call an “even” and an “odd” state. We can in addition stipulate that once it is in one state it remains in that state unless it gets switched according to the above rule.

What state will P be in at B? Just as with Thomson’s lamp, it cannot be in either, yet it must be in one or the other. The only solution to this paradox, it seems, is to claim that there is something wrong with the way it is set up. The stipulated conditions simply cannot form a consistent set. But why not?

Pondering Equivalence and Galileo

I teach a freshman-level course in which we study the ideas of Einstein’s theory of General Relativity. It seems that one of the most difficult concepts for my students to comprehend is what Einstein described as the most beautiful thought of his life: “A falling person does not feel his/her weight.” It is now called the “Equivalence Principle”, and it basically says that gravity can be treated, not as a force, but as a reaction to an acceleration (similar to your reaction to a centripetal acceleration when a car goes around a tight turn). These reactionary forces are not forces in the sense of electromagnetism or other forces of nature, but they are consequences of Newton’s 3rd law of motion: every action has an equal, but opposite, reaction.
Here are some thoughts about the Equivalence Principle, and how it was actually discovered by Galileo, even though he didn’t know it. It might be wordy, but I couldn’t help myself. :^)
Newton and Galileo had already tackled the notion of measurements within “inertial” (non-accelerating) frames of reference. Einstein, in fact, developed his “Special” Theory of Relativity by considering the behavior of electromagnetic phenomena in such frames.
General Relativity and the Equivalence Principle tell us how to deal with accelerating frames of reference. The canonical thought experiment to understand the Equivalence Principle is to consider yourself inside a sealed box, with no windows or other means to measure anything outside the box. Now consider two different scenarios and their possible realities:
1. In the first, you are standing in the box feeling your normal weight. Are you and the box (a) sitting safely on the surface of Earth, both being pulled downward by Earth’s gravity, or (b) being towed through space by a rocketship at an acceleration of 9.8 m/s2 toward a hostile alien planet? Is there any observation or measurement that you could make from inside the box to tell the difference?
2. In the second scenario, you are floating weightless inside the box. Are you (a) adrift in the depths of space, far from any source of gravity, or (b) in free-fall toward the surface of the Earth about to be smashed upon impact? Is there any way to tell the difference in this situation?
Common responses from students are that “you can tell when you’re falling”, or “the air in the box would be moving around in the falling box”, etc.
Einstein realized that in each scenario, there was no experiment you could perform inside the box that would differentiate between the two possibilities. The conclusion then is that the effects of the force of gravity are equivalent to the effects of an acceleration through space. Moreover, gravity can be treated mathematically as such an acceleration, and anything being affected by gravity can be considered to be undergoing an acceleration through space.
This might sound like a simple statement, because a falling object is obviously accelerating as it falls, but consider a person standing on the surface of the Earth, feeling their usual weight. The person is not moving, and yet one may consider that this person feels their weight because he/she is undergoing an acceleration in space (just as if inside an accelerating rocketship). But if this person is not moving through space, then how can they be accelerating through it?
This is where the notion of bent spacetime and worldlines come in. I don’t want to go into much detail here about these topics, but basically gravity can be treated as the effect of motion through the 4 dimensions of “spacetime”, and when you travel through a part of spacetime that is bent, you feel the effects of gravity. This is analogous to how you feel pushed around when a car goes around a curved road.
What I find really interesting is that Galileo was already doing the experiments that prove the Equivalence Principle to be true. Consider the mass of an object. Newton (after Galileo) described the “inertial mass” as being the ratio of the external force applied to an object to its resulting acceleration (Fext=minertial a leads to minertial = Fext/a). Your intertial mass basically defines the way you react when something pushes or pulls on you.
But there is another kind of mass called the “gravitational mass”. This mass is associated with an object’s reaction to the force of gravity (Fgrav = mgrav g, or mgrav = Fgrav/g). If gravity is a true “force” (like the electromagnetic force), there is no reason why this mass should be the same as the intertial mass (for example, the electric charge of a particle does not seem to be related to its inertial mass). However, the Equivalence Principle says that an object’s inertial mass is equal to its gravitational mass.
When Galileo showed that all objects fall at the same rate, he effectively showed this. Since in this case, Fext = Fgrav = mgrav g, then a = g (mgrav/minertial). All objects were observed by Galileo to have the same acceleration due to gravity, so mgrav = minertial. In other words, an object’s reaction to the force of gravity is identical to its reaction to an acceleration through space!
However, Galileo’s experiments predated Newton’s theory of mass and inertia by the better part of a century, so Galileo lacked the mathematical formalism of Newton and had no notion of mass in the first place. But how fascinating that this idea leads almost directly to Einstein’s General Theory of Relativity, nearly 300 years before Einstein wrote it!


Article printed from FFS!: http://fightingforscience.com/2006/08/31/pondering-equivalence-and-galileo/
posted by Perry Gerakines on Thursday, August 31st, 2006 at 2:22 pm in physics, science

3 thoughts on “SS: Thought experiments

  1. I just realized that the story Jim Ayers told me a few weeks ago about a geostationary satellite over Winnipeg that goes around with the earth is a variant of William James’ squirrel. Aha! 🙂

    During the philosophy summer camp, I covered the Chinese Room experiment with our kids and grandparents/parents when we talked about knowing other minds. We found there was a real generational gap between the youngsters and those young at heart: Almost none of the adults thought that a machine that passed the Chinese Room experiment possessed intelligence, whereas all the kids thought it did.

    Like

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