Monthly Archives: February 2011

An Open Letter to Glen Beck from a non-Orthodox Jew

Hey, Glen.

Look, I know we don’t get along. We don’t agree on much of anything. But still, we really need to talk.

The other day, you said some really stupid, really offensive, and really ignorant things about Jews. I know you’re insulted – after all, four hundred Rabbis from across the spectrum came together to call you out for being an antisemitic asshole, and that’s gotta hurt.

But that’s no excuse for being a pig-ignorant jackass.

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Another Crank comes to visit: The Cognitive Theoretic Model of the Universe

When an author of one of the pieces that I mock shows up, I try to bump them up to the top of the queue. No matter how crackpotty they are, I think that if they’ve gone to the trouble to come and defend their theories, they deserve a modicum of respect, and giving them a fair chance to get people to see their defense is the least I can do.

A couple of years ago, I wrote about the Cognitive Theoretic Model of the Universe. Yesterday, the author of that piece showed up in the comments. It’s a two-year-old post, which was originally written back at ScienceBlogs – so a discussion in the comments there isn’t going to get noticed by anyone. So I’m reposting it here, with some revisions.

Stripped down to its basics, the CTMU is just yet another postmodern “perception defines the universe” idea. Nothing unusual about it on that level. What makes it interesting is that it tries to take a set-theoretic approach to doing it. (Although, to be a tiny bit fair, he claims that he’s not taking a set theoretic approach, but rather demonstrating why a set theoretic approach won’t work. Either way, I’d argue that it’s more of a word-game than a real theory, but whatever…)

The real universe has always been theoretically treated as an object, and specifically as the composite type of object known as a set. But an object or set exists in space and time, and reality does not. Because the real universe by definition contains all that is real, there is no “external reality” (or space, or time) in which it can exist or have been “created”. We can talk about lesser regions of the real universe in such a light, but not about the real universe as a whole. Nor, for identical reasons, can we think of the universe as the sum of its parts, for these parts exist solely within a spacetime manifold identified with the whole and cannot explain the manifold itself. This rules out pluralistic explanations of reality, forcing us to seek an explanation at once monic (because nonpluralistic) and holistic (because the basic conditions for existence are embodied in the manifold, which equals the whole). Obviously, the first step towards such an explanation is to bring monism and holism into coincidence.

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E. E. Escultura and the Field Axioms

As you may have noticed, E. E. Escultura has shown up in the comments to this blog. In one comment, he made an interesting (but unsupported) claim, and I thought it was worth promoting up to a proper discussion of its own, rather than letting it rage in the comments of an unrelated post.

What he said was:

You really have no choice friends. The real number system is ill-defined, does not exist, because its field axioms are inconsistent!!!

This is a really bizarre claim. The field axioms are inconsistent?

I’ll run through a quick review, because I know that many/most people don’t have the field axioms memorized. But the field axioms are, basically, an extremely simple set of rules describing the behavior of an algebraic structure. The real numbers are the canonical example of a field, but you can define other fields; for example, the rational numbers form a field; if you allow the values to be a class rather than a set, the surreal numbers form a field.

So: a field is a collection of values F with two operations, “+” and “*”, such that:

  1. Closure: ∀ a, b ∈ F: a + b in F ∧ a * b ∈ f
  2. Associativity: ∀ a, b, c ∈ F: a + (b + c) = (a + b) + c ∧ a * (b * c) = (a * b) * c
  3. Commutativity: ∀ a, b ∈ F: a + b = b + a ∧ a * b = b * a
  4. Identity: there exist distinct elements 0 and 1 in F such that ∀ a ∈ F: a + 0 = a, ∀ b ∈ F: b*1=b
  5. Additive inverses: ∀ a ∈ F, there exists an additive inverse -a ∈ F such that a + -a = 0.
  6. Multiplicative Inverse: For all a ∈ F where a != 0, there a multiplicative inverse a-1 such that a * a-1 = 1.
  7. Distributivity: ∀ a, b, c ∈ F: a * (b+c) = (a*b) + (a*c)

So, our friend Professor Escultura claims that this set of axioms is inconsistent, and that therefore the real numbers are ill-defined. One of the things that makes the field axioms so beautiful is how simple they are. They’re a nice, minimal illustration of how we expect numbers to behave.

So, Professor Escultura: to claim that that the field axioms are inconsistent, what you’re saying is that this set of axioms leads to an inevitable contradiction. So, what exactly about the field axioms is inconsistent? Where’s the contradiction?


I just recently realized that I only wrote about computability back in the earliest days of this blog. Those posts have never been re-run, and they only exist back on the original blogger site. When I wrote them, I was very new to blogging – looking back, I think I can do a much better job now. So I’m going to re-do that topic. This isn’t just going to be a re-post of those early articles, but a complete rewrite.

The way that I’m going to cover this is loosely based on the way that it was first taught to me by a wonderful professor, Dr. Eric Allender at Rutgers, where I went to college. Dr. Allender was a really tremendous professor: he managed to take an area of computer science that could seem hopelessly abstract and abstruse, and turned it into something fun and even exciting to learn about.

Computability is the most basic and fundamental sub-field of theoretical computer science. It’s the study of what a mechanical computing device can do. Not just what a specific mechanical computing device can do, but what can any mechanical computing device do? What are the limits of what you can do mechanically? And once we know the limits, what can we discover about the nature of computation?

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Fuzzy Logic vs Probability

In the comments on my last post, a few people asked me to explain the difference between fuzzy logic and probability theory. It’s a very good question.

The two are very closely related. As we’ll see when we start looking at fuzzy logic, the basic connectives in fuzzy logic are defined in almost the same way as the corresponding operations in probability theory.

The key difference is meaning.

There are two major schools of thought in probability theory, and they each assign a very different meaning to probability. I’m going to vastly oversimplify, but the two schools are the frequentists and the Bayesians

First, there are the frequentists. To the frequentists, probability is defined by experiment. If you say that an event E has a probability of, say, 60%, what that means to the frequentists is that if you could repeat an experiment observing the occurrence or non-occurrence of E an infinite number of times, then 60% of the time, E would have occurred. That, in turn, is taken to mean that the event E has an intrinsic probability of 60%.

The other alternative are the Bayesians. To a Bayesian, the idea of an event having an intrinsic probability is ridiculous. You’re interested in a specific occurrence of the event – and it will either occur, or it will not. So there’s a flu going around; either I’ll catch it, or I won’t. Ultimately, there’s no probability about it: it’s either yes or no – I’ll catch it or I won’t. Bayesians say that probability is an assessment of our state of knowledge. To say that I have a 60% chance of catching the flu is just a way of saying that given the current state of our knowledge, I can say with 60% certainty that I will catch it.

In either case, we’re ultimately talking about events, not facts. And those events will either occur, or not occur. There is nothing fuzzy about it. We can talk about the probability of my catching the flu, and depending on whether we pick a frequentist or Bayesian interpretation, that means something different – but in either case, the ultimate truth is not fuzzy.

In fuzzy logic, we’re trying to capture the essential property of vagueness. If I say that a person whose height is 2.5 meters is tall, that’s a true statement. If I say that another person whose height is only 2 meters is tall, that’s still true – but it’s not as true as it was for the person 2.5 meters tall. I’m not saying that in a repeatable experiment, the first person would be tall more often than the second. And I’m not saying that given the current state of my knowledge, it’s more likely than the first person is tall than the second. I’m saying that both people possess the property tall – but in different degrees.

Fuzzy logic is using pretty much the same tools as probability theory. But it’s using them to trying to capture a very different idea. Fuzzy logic is all about degrees of truth – about fuzziness and partial or relative truths. Probability theory is interested in trying to make predictions about events from a state of partial knowledge. (In frequentist terms, it’s about saying that I know that if I repeated this 100 times, E would happen in 60; in Bayesian, it’s precisely a statement of partial knowledge: I’m 60% certain that E will happen.) But probability theory says nothing about how to reason about things that aren’t entirely true or false.

And, in the other direction: fuzzy logic isn’t particularly useful for talking about partial knowledge. If you allowed second-order logic, you could have fuzzy meta-predicates that described your certainty about crisp first-order predicates. But with first order logic (which is really where we want to focus our attention), fuzzy logic isn’t useful for the tasks where we use probability theory.

So probability theory doesn’t capture the essential property of meaning (partial truth) which is the goal of fuzzy logic – and fuzzy logic doesn’t capture the essential property of meaning (partial knowledge) which is the goal of probability theory.