anti-inductive situations: forum for week of 17 October

Simple induction is really just  a matter of  fitting a curve to a number of points.  (Assign numbers to colours, with Black=1.  A data set of black crows is like a graph where the value for each item is 1.  So a straight line through all of them projects a prediction that all crows are black.)  This week insead of asking a question or giving a debate let me set a problem. Describe a case where the intuitively simplest line through a set of points is not what you would predict for the continuation of the data.  (Let me know if this isn't stated clearly and I'll say more. But it's good to keep these short.)  
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I'm not sure if this is a good example. When you drop a ball down a spiral tube and let it slide out at the bottom end, it may not be intuitive (especially for a child that has not taken a course in physics) that the ball does not continue to spiral in a circular motion, but follow a straight path tangent to the point where it is released from the end of the tube.

I suppose another example could be that it is not intuitive for someone to think that when dropping a lead ball from the top of a very tall building that the ball would reach a terminal velocity in which it no longer accelerates. The person may think that the ball will continue to accelerate until it reaches the ground.

Stretching a rubberband and calculating its stretch/force ratio. Intuitively at the beginning with a first couple of data, we may be tempted to think that the rubberband will stretch indefinitely at the same proportion until it break. But contrary to it, there's a point called stress limit, where material, stretched beyond this point, exhibits a totally different pattern of behavior (such as needing twice as much force to stretch the same length of material or something). To make things more interesting, some materials have been found to fit into our original intuition of simple linear correlation.

I was supposed to have contributed in Group 4 the week of October 3rd. I apologize for the delay.

My case for the problem posed above is the graphing of a nerve impulse. You can find a basic graph here:

Membrane voltage remains at resting potential until a stimulus excites it to the threshold voltage. Other stimuli may precede the ultimate stimulus of the nerve impulse; however, if they are >-55 mV (based on graph above) then the nerve impulse will nto occur. Once the threshold voltage is reached the neuron 'fires' or depolarizes and repolarizes (this is called the action potential) and is represented by the big hump on the graph. Immediately after this period the graph briefly dips below the resting potential (this stage is called hyperpolarization), and then finally returns to resting potential to await another < or = -55mV stimulus.

The simplest line through a set of data points is not what one would predict for the graphing of small incremental increases in voltage within a neuron. This is because of the function of the threshold voltage.

The example case of the crows seem to be begging for someone to mention the black swan case. Assigning the colour white as 1, with each swan as an item the graph would be a horizontal line, if 16th century Europeans were making this graph (Black swans were populated in New Zealand and Australia). The black swan was an expression as a statement of impossibility at the time, as all historical records of swans were white. The discovery of new and unexpected empirical evidence demonstrates the fragility of induction and pattern-seeking.

This example might be a little bit disturbing but it's the only one that I can come up with that's both simple and interesting. If I find a better and less disturbing example I will make another post/reply to this topic.

Most people would assume that all of the worlds animals have one head and therefore one brain. So on a graph, the data showing animals with less than one brain, one brain, and more than one brain, would show a straight line depicting that all animals have one brain. However there have been discoveries of certain animals with two heads. For example, two headed snakes, and two headed turtles. So these animals have two brains.

Well I guess this example is perfect for Halloween coming up, and perhaps an inspiration for someones Halloween costume?

Someone already used the dropping an object one so.......

The application of General Relativity moving from a macro to a micro level.

It's a shame that there is a lack of a question this week. I can easily provide an example, but rather of my own experience, I'd like to present a story which I have learned from a past "Theory of Knowledge" teacher. In his childhood, he lived in a small-town in Edmonton. It was a population of a few hundred or so at the most, and as expected, was relatively isolated. Further more, there were only Caucasians living in this town. One day, he and his family go to a much larger town (as a trip I believe), and there he notices performers (dancers I believe!) who seem to have dark skin. He had never seen anyone with a darker skin color before, and thus had trouble reconciling what he was seeing with his experiences. He then noticed that the dancer's palms were white, and much lighter than the rest of their skin. He came to the false, (though arguably, justified) conclusion that the skin must have been painted on, and the sweat of the palms must have cleaned off some of the paint.

In this example, he is taking a small sample (his hometown) and extrapolating the data to a much larger community. Thus, in a town where there seem to be no Africans, the rate of African occurrence is zero, and extrapolated would mean that no Africans exist. Of course, this is false, and just another example of how limited data, and mindless extrapolation can easily lead to error in knowledge.

JamesWu 21:39, 19 October 2011 (PDT)

I originally had a question, but its hard to formulate over text. Perhaps if I become more lucid in my text, I will try to repost the question.

Cases Against Intuitive Line

In section 6 Justifying Induction, of chapter 4, of Dr. Morton’s book A Guide Through the Theory of Knowledge, is a reference to philosophers on inductive inferences as follows:

Some philosophers have tried to give reasons why it is reasonable to believe the conclusions of inductive inferences, and some have argued that it is only out of confusion or misunderstanding that one could think that any such reasons were possible or necessary.

Philosopher Karl Popper and physicist Freeman Dyson are offered as example, in support of conclusions of inductive inferences as unreasonable, as philosopher, and as predictive in specific application.

One of the most influential and controversial views on the problem of induction has been that of Karl Popper, announced and argued in (Popper LSD). Popper held that induction has no place in the logic of science. Science in his view is a deductive process in which scientists formulate hypotheses and theories that they test by deriving particular observable consequences.

Stanford Encyclopedia of Philosophy on Induction, 4.2.

Thirty-one years ago [1949], Dick Feynman told me about his "sum over histories" version of quantum mechanics. "The electron does anything it likes," he said. "It just goes in any direction at any speed, forward or backward in time, however it likes, and then you add up the amplitudes and it gives you the wave-function." I said to him, "You're crazy." But he wasn't.

Freeman J. Dyson, in a statement of 1980, as quoted in Quantum Reality : Beyond the New Physics (1987) by Nick Herbert

An example for the question posed can be seen in temperature and the freezing point of water. It would seem that, with water, as the temperature decreases within the atmosphere so does the temperature of the water itself. On a chart, the decrease of temperature could be directly correlated to the decrease in water temperature. One could assume that the water will continue to decrease as the temperature decreases - however this would be incorrect. There is a point in temperature where water will cease to decrease in temperature and simply freeze and turn into solid form (instead of remaining in a liquid state). Thus, while believing in the direct correlation of temperature drop of the weather and the water seemed to be accurate - it would prove to be incorrect if simple induction was followed. It is important to note that since simple induction is formulated through the basis of observation, simple induction is never one hundred percent accurate. While some theoretical laws are based upon empirical evidence that are in support of simple induction, it is not enough evidence to provide one hundred percent accuracy. In other words, the idea of observation as being infallible is incorrect when it is based simply off of routine assumptions.

This may be a long shot, but perhaps this can be thought about economically as well. My example is related to the 4 cycles of a business year(expansion, peak, recession, and recovery. If this is looked at on a graph, it may be inferred that one stage could last longer than another. Also, similar to the post before mine, the pressure point in a vacuum gauge or a manometer could be assumed to continue to rise.

So it seems that any and all facts that do not follow indefinitely an original linear path can be an answer to this question. I'll use the relationship between stress and strain of any solid material as an example. The applied force is proportional to the displacement for a certain distance of displacement. After said distance, the material will deform plastically and basically be screwed up. So it is an error to use simple induction when confronted with a material to which you do not know what the proportionality limit is.

I found today's lecture on the shortfalls of induction to be particularly interesting. I am specifically interested in exploring in what circumstances are we allowed to induce beliefs from incomplete knowledge? Suppose it were possible to produce everything we believe induction, and that induction is simple a matter of convenience (a highly improbable position in my opinion). This simplification is still obviously necessary – if nothing else, waiting for simple, deductive knowledge takes too much time. Case in point, people knew about the symptoms of diabetes long before they came to know its underlying physiological causes. Inductive beliefs in this case can be formed by doctors who know just of the symptoms and cases, and but of the underlying framework. So, induction serves as a sort of bridge between ignorance and knowledge (sound rather like the Daimon in the Symposium…).

So in what cases are we right in drawing such conclusions from such evidence? In class, we were talking about chartists and the markets. Sure, there seems to be certain patterns which dominate in certain assets – e.g. Fibonacci retracement. We hardly know why it should be the case that 61.8 or 38.2 should be support levels in stocks, but that seems to have been the case for quite some time. The pattern has some predictive powers, though, of course, it’s not very accurate by any measure. But the fact that through all the transactions that take place in the market place every second of every day, there exist a (weak) pattern is still pretty extraordinary – it’d very strange if it were just a quirk that disappeared tomorrow. Perhaps there are certain immutable laws about trading and exchanges that lead to these patterns, and it’s simply the case that we haven’t quite figured them out yet – that’s one view. On the other hand, there’s LTCM and the spectacular failure of algorithms and pattern seeking. But where is it that we actually draw the deciding line?

My example is Benazir Bhutto. I chose her because of two reasons: she was the first women prime minister of Pakistan, and she was the first female leader of any muslim country. Throughout history countries have been led by men, especially in the muslim community. Intuitively, we would expect another male figure in such a key position; therefore, her accomplishments were unexpected, defying the worldly trend of male dominance in that sector. This of course applies to other female leaders who have gained political power; however, Bhutto brings a greater sense of unexpectedness to the society due to her accomplishments in such a conservative society.