Antimeta — Found this very interesting math and philosophy blog while searching for something entirely unrelated (the history and politics of the name change of Bombay). Posting it for my own record, but perhaps it may be of interest to those, if any, who read my blog? Kenny Easwaran (the author of the blog) has a very useful collection of links to other philosophy/logic/math blogs at his site.

At CounterPunch, Floyd Rudmin (who I hope to quote a lot of, from what I have seen of his writing) provides a great lesson on Bayes Theorem to demonstrate the ineffectiveness of NSA monitoring with regard to identifying terrorists. But I have some comments, which can be found after the quote below.

Floyd Rudmin: the Politics of Paranoia and Intimidation

[...]

The US Census shows that there are about 300 million people living in the USA.

Suppose that there are 1,000 terrorists there as well, which is probably a high estimate. The base-rate would be 1 terrorist per 300,000 people. In percentages, that is .00033% which is way less than 1%. Suppose that NSA surveillance has an accuracy rate of .40, which means that 40% of real terrorists in the USA will be identified by NSA's monitoring of everyone's email and phone calls. This is probably a high estimate, considering that terrorists are doing their best to avoid detection. There is no evidence thus far that NSA has been so successful at finding terrorists. And suppose NSA's misidentification rate is .0001, which means that .01% of innocent people will be misidentified as terrorists, at least until they are investigated, detained and interrogated. Note that .01% of the US population is 30,000 people. With these suppositions, then the probability that people are terrorists given that NSA's system of surveillance identifies them as terrorists is only p=0.0132, which is near zero, very far from one. Ergo, NSA's surveillance system is useless for finding terrorists.

Suppose that NSA's system is more accurate than .40, let's say, .70, which means that 70% of terrorists in the USA will be found by mass monitoring of phone calls and email messages. Then, by Bayes' Theorem, the probability that a person is a terrorist if targeted by NSA is still only p=0.0228, which is near zero, far from one, and useless.

[...]

I believe this is honest and valid reasoning. However it has to be read closely because Rudmin does not use more familiar terms such as 'false postive' and 'false negative'.

He points out that the chance is very low that a person is actually a terrorist if so identified by NSA. The if-then order here is important to note. Another way to say it is to say that (simply because of the extremely low incident rate of terrorists) there will be a lot of false positives. A lot of people who are not terrorists will be wrongly labelled so by the NSA.

What he does not say or imply, but is not clear (at least in my reading, to a layperson) is that given a high accuracy rate (of the NSA test for terrorist) the chance of a false negative is quite low. In other words, the NSA monitoring (if accurate) will not miss a real terrorist. The if-then here is reversed.

IMHO, this is a crucial difference for two reasons:

1. A high false positive rate, given a low false negative rate, is an acceptable outcome for screening tests. Further tests/filters can be applied to narrow the count and eliminate false positives. The monitoring here serves as a first, coarse, red flag.
2. To the public (to whom I assume Rudmin is addressing his argument), this is of utmost relevance. Their concern is not so much with being swept up as a false positive (for they are sure they can easily exonerate themselves in further tests), but with making sure that no terrorist gets away unnoticed (false negative).

The public has demonstrated many times over that they are willing to swallow the fear-mongering and sacrifice significant chunks of liberties (especially if they believe it to be those of others) in return for perceived security and toughness. While Rudmin makes a powerful argument in pointing out that the monitoring does a poorer job than the toss of a coin (given his assumptions on accuracy rate, etc), this argument falls on mostly deaf ears.

In a well-reasoned piece titled:

Can humans escape Goedel?:A review of "Shadows of the Mind" by Roger Penrose

Daryl McCollough provides a non-paradoxical version of the Liar's Paradox to illustrate inconsistency in human thinking. In doing so, he addresses a particular aspect of the interpretations of belief and truth with regard to debate on Gödel's incompleteness theorem (the first, for the picky). That issue is a better understanding of human fallibility (and its relationship to the phrase "there are some sentences we know to be true"). Perhaps Wittgenstein can be interpreted to also explore this in his [in]famous commentary on Gödel's Theorem but more on that later.
McCollough writes:

6. How Could Inconsistency Creep Into Human Reasoning?

6.1 As I discussed in the last section, Penrose's arguments, if taken to their logical conclusion, show us not that the human mind is noncomputable, but that either the human mind is beyond all mathematics, or else we cannot be sure that it is consistent. If we reject the "mysterian" position that mind is beyond science, we are left with the conclusion that we can't know that we are consistent. This seems very counter-intuitive. If we are very careful, and only reason in justified steps, why can't we be certain that we are being consistent?

6.2 Let me illustrate with a thought experiment. Suppose that an experimental subject is given two buttons, marked "yes" and "no", and is asked by the experimenter to push the appropriate button in response to a series of yes-no questions. What happens if the experimenter, on a lark, asks the question "Will you push the 'no' button?". It is clear that whatever answer the subject gives will be wrong. So, if the subject is committed to answering truthfully, then he can never hit the "no" button, even though "no" would be the correct answer. There is an intrinsic incompleteness in the subject's answers, in the sense that there are questions that he cannot truthfully answer.

6.3 Now, there is no real paradox in this thought experiment. The subject knows that the answer to the experimenter's question is "no", but he cannot convey this knowledge. Thus there is a split between the public and private knowledge of the subject. But now, let's extend the thought experiment.

6.4 Someday, as science marches on, we will understand the brain well enough that we can dispense with the "yes" and "no" buttons (which are susceptible to lying on the part of the subject). Instead of these buttons, we assume that the experimenter implants probes directly into the subject's brain, and we assume that these probes are capable of directly reading the beliefs of this subject. If the probes detect that the subject's brain is in the "yes" belief state, it flashes a light labeled "yes", and if it detects a "no" belief state, it flashes a light labeled "no". Now, in this improved experiment, the subject is asked the question "Will the 'no' light flash?"

6.5 In this improved set-up, there is no possibility of the subject having knowledge that he can't convey; the probe immediately conveys any belief the subject has. If the subject believes the "no" light will flash, then the answer to the question would be "yes", and the subject's beliefs would be wrong. Therefore, if the subject's beliefs are sound then the answer to the question is "no". Therefore, since the subject cannot correctly believe the answer to be "no", he similarly cannot correctly believe that he is sound. If the subject reasons from the assumption of his own soundness, he is led into making an error.

6.6 As can be seen from this thought experiment, the inability to be certain of one's own soundness is not a deficiency of intelligence. There is no way that the subject in the experiment can correctly answer the question by just "thinking harder" about it.

And provides this conclusion:

8. Conclusion

8.1 Penrose's arguments that our reasoning can't be formalized is in some sense correct. There is no way to formalize our own reasoning and be absolutely certain that the resulting theory is sound and consistent. However, this turns out not to be a limitation on what computers or formal systems can accomplish relative to humans. Instead, it is an intrinsic limitation in our abilities to reason about our own reasoning process. To the extent that we understand our own reasoning, we can't be certain that it is sound, and to the extent that we know we are sound, we don't understand our reasoning well enough to formalize it. This limitation is not due to lack of intelligence on our part, but is inherent in any reasoning system that is capable of reasoning about itself.

I think its a refreshing angle to the old debate, one that does not get as much attention.

P.S: When talking about truth above I am hopefully not mystifying it in a way that ignores the deflationary theory of truth.

[with apologies to Jeff Foxworthy]

Among other eminent bits, Karlis Podnieks has an interesting test you can use to see if you are a Platonist. As psychologists like to say, denial is more than just a river in Egypt and the first step to recovery is accepting the problem. So, take the test… its for your health! ;-)

Foundations of Mathematics. Mathematical Logic. By K.Podnieks

Suppose, someone has proved that the twin prime conjecture is unprovable in set theory. Do you believe that, still, the twin prime conjecture possesses an “objective truth value”? Imagine, you are moving along the natural number system:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, …

And you meet twin pairs in it from time to time: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71,73), … It seems there are only two possibilities:

a) We meet the last pair and after that moving forward we do not meet any twin pairs (i.e. the twin prime conjecture is false),

b) Twin pairs appear over and again (i.e. the twin prime conjecture is true).

It seems impossible to imagine a third possibility…

If you think so, you are, in fact, a Platonist.