Plato’s Beard

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.

Over at K’s blog, she writes:

Bitch responds: Is Cultural Feminism Pomo Feminism?
But, anyway, I’d say that, no, cultural feminism is rather different from postmodern thought. And I will warn you: While I wouldn’t say I’m a postmodernist, I certainly didn’t spend my time studying it and in fact mostly wrote criticisms of it, I do have a big problem when I read dismissive crits of their work.

Since I posted recently about the Sokal prank and the uncharitable (and inconclusive) attack it represents, the above jogged my memory of an interesting paper by Gabriel Stolzenberg, a mathematician at BU, in response to the attacks on postmodernism by various physicists and philosophers (Sokal, Weinberg, Nagel, to name a few). The paper is Reading and Relativism (PDF) and is a wonderful read and includes this section, a quotation from Luce Irigaray by Thomal Nagel, which Nagel then goes on to criticize:

Is E = Mc2 a sexed equation? Perhaps it is. Let us make the hypothesis that it is insofar as it privileges the speed of light over other speeds that are vitally necessary tous. What seems to me to indicate the possibly sexed nature of the equation is not directly its uses by nuclear weapons, rather it is having privileged what goes thefastest….”

Stolzenberger comments on Nagel’s response:

This may send Nagel into convulsions but how does he know that it is her problem not his? How can he possibly know unless he knows what Irigaray means by “sexed” and “privileges” and that her reference to speeds is not an ironic metaphor? If he does not know these things, he is kidding himself. But if he does know, why does he not tell us, so we can join in the fun of mocking Irigaray? Instead of fulfilling his obligation as a philosopher to give us a reason to believe what he says, Nagel encourages us to trust that whatever Irigaray means is refuted by the authors’ “comically patient” observation,

Whatever one may think about the “other speeds that are vitally necessary to us,” the fact remains that the relationship E = Mc2 between energy (E) and mass (M) isexperimentally verified to a high degree of precision, and it would obviously not be valid if the speed of light (c) were replaced by another speed.

This shows especially poor judgement. If Sokal and Bricmont think that something privileged can easily be replaced, there is little reason to suppose that they have any idea of what Irigaray is talking about. And by mocking her instead of giving us an argument, Nagel makes it appear that neither does he.

As Stolzenberger points out elsewhere, a kinder reading of the text might produce other interpretations which make a lot more sense than the narrow sense in which Nagel uses it.

I am reminded of Heidegger’s famous “science does not think” essay. One reading of Irigaray’s text may yield a point similar to the one Heidegger makes.

Doron Zeilberger at Rutgers publishes a page of opinions that is a wonderful read, even if you are not a mathematician. In opinion 11 he points out better than I can exactly what was wrong with Sokal’s prank on the pomo philosophers:

Opinion 11 of Doron Zeilberger:
Great Scientists, Lousy Philosophers

The intersection of the sets of great mathematicians or scientists and great philosophers is a rapidly decreasing function of time.

[...]

Nowadays, Traditional God has been replaced, in part, by another God: `Absolute Truth’. Practicing scientists get really annoyed when they are reminded that after all they are also human, and their view of science is time- and fashion- dependent. So Alan Sokal had a good laugh at the expense of post-modern cultural-relativists. But he used the same cheap trick of Euler, intimidation by jargon. He went one step farther: making fun of the sociologists’ jargon. He had the advantage that their jargon is closer to spoken English than his, so he could master it superficially.

Making fun of other people’s language is the lowest form of humor. Like Euler, Sokal did not prove anything, except that physical scientists and mathematicians are arrogant and look down on everybody else. They are also religious fanatics, for whatever religion they may have. Social science has probably lots of rubbish, but so does regular science, and in either case it is not the content that matters so much as the act of expressing oneself’s.

For more info on the Sokal Prank see the Wikipedia.