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Archive for September, 2010

The forum meeting on Sep. 24 focused on the reading “The Implications of Quantum Physics” in When Science Meets Religion by Ian Barbour.  We began with a brief discussion of the three different categories of thought that exist when trying to understand the universe.

  1. Classical Realism: the universe is deterministic. Models are replicas of the world and allow scientists to visualize the world and its structure.
  2. Instrumentalism: theories are useful information that can be used for correlating observations, making predictions, and achieving technical control. Models can be used to construct theories but are not actual representations of the world. Lucas described this interpretation as a “pragmatic approach with no pragmatic consequences.”
  3. Critical Realism: intermediate interpretation. Theories are partial representations of the world and its interactions. Models are imperfect depictions of the world but have truth values that can increase our knowledge of the world. In this view, the goal is understanding, and not control or prediction.

The group unanimously admitted to being critical realists.

Lucas then read a passage from God’s Mechanics by Guy Consolmagno in order to address the topic of using science to defend religion. Quantum physics was initially used to explain Christianity, but when the popular beliefs of physics were upset by acquired knowledge, this in turn caused distress within the religious community. From this, it was realized that science can be used to inform belief but shouldn’t be used as an explanation for belief. Religion must have a primary belief based on religious experience, and this belief can then be informed by empiricism.

We then moved on to questions of chance, using the framework set up by our last meeting on probability. Joanna provided an insightful example to a statistical rationale for the existence of God proposed in 1710. One scientist examined the birth records of boys and girls and found that more boys were born than girls each year. If life was left completely to chance, this ratio should be 50/50.  Therefore, this deviation was proof that God does exist.

This gave way to discussion regarding the question: is intelligence, such as human intelligence, a necessary outcome of evolution? Although intelligence is hard to quantify, it has been the general trend for it to always increase. We also broached the idea of establishing confidence in the evolution of life and intelligence based on the single data point that we have. Can we say with a greater confidence that life will evolve under sufficient conditions because there was a short waiting period before the evolution of life on Earth? And likewise, can we say with a lesser confidence that eukaryotic cells will evolve considering that there was a longer time before these appeared on Earth? These questions began to raise similar inquiries, such as whether or not photosynthesis, where photons are used as an energy source, is likely to occur? If energy in a biosphere is too abundant, it is likely to result in cutthroat competition. Thus, it is more suitable for there to be a variety of abundant resources in addition to one limiting resource.

Next, we discussed interpretations of probability and how it relates to the universe. When considering electrons, we can ask whether it exists in one given place and we simply do not have the tools to perceive its location, or whether it really exists only as a probability distribution. In order to confront these questions, we turned to Schrödinger’s cat. This thought experiment, devised by Erwin Schrödinger, proposes placing a cat in a chamber with a small flask of poison. Also in the chamber is some radioactive substance which can set off a Geiger counter attached to a hammer which can break the flask and thus kill the cat. Once the experiment is set up, we would close the box and only have a probability of whether the cat is alive or dead. In a given amount of time, if the radioactive material decays, the cat will be dead. If not, then it will be alive. For the classical realist, the cat is either alive or dead and we don’t know which because we lack the tools to acquire this information. For the critical realist, the cat is 50% alive and 50% dead. Joanna brought up the point that the experiment is flawed considering that the cat is an observer and can thus know itself whether it is alive (or dead). However, since there would be no privileged observer from the outside that could know what the cat would know, we have to approach the experiment as conscious external observers. But to what extent does reality exist outside our own consciousness when all we have is a science observed by conscious beings?

When the topic of causality was raised, a quantum interference experiment was described using the setup found below:

A photon would be emitted, half would be reflected by the half mirror and the other half would pass through to the full mirror and then be reflected into one of the detectors. When there were no other components to the experiment, each of the detectors would perceive half of the photons emitted.

However, once an additional half mirror was placed into the experiment, the photons seemed to take a single path and would only be perceived by a single detector.

This experiment shows that causality is not changed by the insertion of the mirror, which is related to the property of quantum entanglement.

For our last topic of discussion, we considered randomness and why evolutionists seem to be satisfied with it as an explanation for certain processes (such as mutations), but physicists do not accept it. Non-deterministic biology is not considered acceptable by religion, and it was speculated that this is because it leaves less room for the existence of God. However, non-deterministic physics keeps a certain mystery which permits a religious explanation to fill in the blanks. Where science is concerned, people are expecting to find proof that their faith is the correct faith. Joanna discussed a conversation where a colleague said that theologians sent scientists out in the first place to prove the existence of God and report on what he was thinking. Today, they are still waiting on such a report. Lucas then suggested that science was used by theologians for the sake of power. Conclusively, it was claimed that both science and religion are a response to nature, while the former is a response to the physical world while the latter is more of a response to experience.

The discussion ended with the question: is religion a natural product of cultural evolution? This is the primary question for next semester and so be sure to stay tuned to find out the answer.

(Images from http://homepage.mac.com/stevepur/physics/qw/qw_session_6.pdf)

Summary provided by K.P.

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Basic Probability—The Goat Problem (AKA The Monty Hall Problem)

You are presented with three curtains.  Behind two of those curtains are goats, and behind the remaining door is a red convertible (or whatever type of expensive car that strikes your fancy).

1
2
3

You pick curtain 3, and the game show host is kind enough to reveal to you that behind curtain 1, there is a goat.  Do you switch? Or stay? What is the probability now that the car is in fact behind curtain 3?  ½ you say? Are you sure?

1
GOAT
2
3
YOUR   PICK

What most people fail to recognize is that revealing one of the goats does not affect the probability of the car being behind any of the three curtains.  At the beginning of the game, before you pick your curtain, and before a goat is revealed, there is a 1/3 chance of the car being behind any of the curtains.

1
CAR
2
3
1
2
CAR
3
1
2
3
CAR

You picked curtain 3.

1
CAR
2
3
YOUR   PICK
1
2
CAR
3
YOUR   PICK
1
2
3
CAR
YOUR   PICK

The game show host shows you a goat behind one of the curtains that is not curtain 3.

1
CAR
2
GOAT
3
YOUR   PICK
1
GOAT
2
CAR
3
YOUR   PICK
1
GOAT
2
3
CAR
YOUR   PICK

In two of these situations, you will win if you switch.  In the remaining, you will win if you stay.  So you have a 2/3 chance of winning if you switch, and a 1/3 chance of winning if you stay.  This is not the ½ you assumed at the beginning.  But don’t feel bad if you still don’t get it, sometimes even the brightest of mathematicians can’t seem to wrap their heads around it.  This only goes to show that we do not reason about probability very well, and many of our intuitions about religion and evolution probabilities are flawed.

Basic Probability—Bayes Theorem

A French male would like to obtain a green card to enter the United States.  He must first test negative for HIV according to government standards.  The French male has no history of risky behavior that would make him more likely to carry the virus.  However, he tests positive.  Knowing the following probabilities:

Probability of being HIV+ given French male with no risk factors

P(HIV+|French male w/no risk factors) = 0.01%

Therefore, probability of being HIV- given French male with no risk factors

P(HIV-|French male w/no risk factors) = 99.99%

Probability of testing positive given HIV+

P(test+|HIV+) = 99.9%

Therefore, probability of testing negative given HIV+

P(test-|HIV+) = 0.1%

Probability of testing negative given HIV-

P(test-|HIV-) = 99.99%

Therefore, probability of testing positive given HIV-

P(test+|HIV-) = 0.01%

What is the likelihood that this man does indeed carry the virus given that he is a French male with no risk factors and he tested positive?  A probability tree might make the solution more obvious.

10,000 French men w/no risk factors

↓                                                              ↓

1 HIV+                                                             9999 HIV-

↓                  ↓                                                     ↓                      ↓

1 test+             0 test –                                    1 test+             9998 test –

We know that this man tested positive, so he can fall into 1 of two options—testing positive when HIV+ and testing positive when HIV-.  Given that these options are on the same level of the probability tree, they have equal weight.  This man therefore has a ½ chance of actually carrying the virus.

Another way to approach this problem is to use Bayes theorem.  Bayes theorem evaluates the probability of a hypothesis (H) being true given data (D).  It states,

So for this situation, the likelihood of the man carrying the virus (our hypothesis) given he is a French male with no risk factors and tested positive (our data) would be calculated:

P(HIV|French with no risk factors and tests+) =

Given that this semester focuses on Chance, a basic understanding of probability is key. There are essentially two schools of thought in the realm of probability—the Bayesians, and the Frequentists.

E. Sober’s Four Approaches to Probability

Kolomogorov’s three axioms of probability state that:

P() is a probability measure precisely when the following conditions are satisfied for any propositions A and B:

0≤ P(A) ≤ 1.

P(A) = 1, if A must be true.

If A and B are incompatible, then P(A or B) = P(A) + P(B).

1. Frequentists-objective probability

The frequentist approach to probability takes the actual frequency of events to be the probability of occurrence.  For example, say you toss a coin 100 times and, in those 100 tosses, 62 land on heads.  From a frequentist perspective, the probability of getting heads would be P(H) = 0.62.  In the example of the French male above, a frequentist would disagree about the probability of being HIV+.  The probability of being HIV+ could only be 1 (if he has it), or 0 (if he does not).

2. Bayesians-subjective probability

The Bayesian approach claims that we can have a reasonable degree of belief to the outcome of an event.  This confidence is based on previous evidence or observations. A Bayesian says that we can make inferences on the probability of landing heads in a coin toss.  Based on previous observations, we know that the coin will either land heads (1) or tails (0), so the probability of landing heads should be somewhere between 0 and 1.  Also, with the HIV problem above, using observed frequencies, we were able to calculate a probability of French male being HIV+ that is not 0 or 1.  Not coincidentally, this probability was calculated using Bayes Theorem.


3. Hypothetical relative frequency-somewhere in between

Most of us fall into the category of “What is the probability of landing heads in a coin toss?”—“Why, ½ of course.”  Where does this come from?  Well we assume that hypothetically, if you toss a coin an infinite number of times, the frequency of heads in a fair coin toss should converge to 0.5.

Just as the probability of landing heads converges to ½, the probability of any specific sequence of heads and tails converges to 0 as n increases.  The problem is, however, that one of these sequences DOES happen…so we cannot equate a probability of 0 with impossibility. The Law of Large Numbers states that

if and only if

, where e is a small number.

So the frequency of landing heads does not have to converge to 0.5 only approach it, there is wiggle room—namely e.  Notice that a probability appears on both sides of the if and only if statement. Unfortunately, our “somewhere in between” interpretation of probability is circular…it defines probability in terms of probability.

4. Propensity interpretation of probability-“The Imaginary Invalid”

Propensities refer to dispositional properties can be described with if/then statements. For example:

If a compound is soluble, then it will dissolve in water.

But, dissolving in water is the definition of solubility.  Sober uses the analogy of a part from the play “The Imaginary Invalid.”  A quack doctor claims that he discovers why opium puts people to sleep.  He gives opium the property of a “dormative virtue.”  This simply means that opium has the ability to put people to sleep. Propensity is a redundant way to say probability.

We like to talk about what will probably happen because (due to chaos described below), we will probably never have a way of knowing with absolute certainty.

Chaos

Chaos theory deals with phenomena that appear chaotic, but are in truth deterministic.  These occurrences are unpredictable, the weather being a perfect example.  The most well known description of chaos theory is the butterfly effect which states that a butterfly flapping its wings in Taiwan will bring about a chain of events that ultimately cause a hurricane in Brazil.  Most people believe that, even if the universe is entirely deterministic, we can never know enough to make absolute predictions.

Summary provided by L.B.

Admin apologizes for the weirdly formatted equations. If they really bother you, let me know in the comments, and I will come back to pretty them up a bit.

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