Have you discussed any of these things with abstract algebra people?

the natural numbers are simple in abstract algebra, if I remember correctly, which I might not. In any case that’s not what I’m commenting about.

“A string of a million random numbers is complex, because it takes a long time to describe all of the content.”

the string of random numbers is described within the very statement you make, and so it does not take a long time to describe. One needs to differentiate between systems and projections thereof, that may be quite a bit more simple. That is, you don’t need to list the entire set to describe them.

In any case, the joke’s on us: Steve_Landsburg will continue to collect royalties on a book built from an inconsistent and haphazardly-constructed foundation. I can’t claim to be able to pull that off!

But I am uneasy that you consider thoughtful what I consider trivial, compared to my getting you to provide a reasonable* answer:
http://www.thebigquestions.com/2009/12/17/non-simple-arithmetic/#comment-1311
to the question that you yourself put to Silas:
http://www.thebigquestions.com/2009/12/17/non-simple-arithmetic/#comment-1295

Your answer is also an implicit admission of what I called your big mistake:
http://www.thebigquestions.com/2009/12/17/non-simple-arithmetic/#comment-1287
and it is also a retreat from shifting your ground:
http://www.thebigquestions.com/2009/12/17/non-simple-arithmetic/#comment-1256

Silas: Thank you for what I take as kind words, thinly disguised. Though it is strictly true that, in this issue, I am half-competent at best.

My further thoughts on this, if any, by e-mail only. I might cc to Silas. Neither of you should hold his breath.

* not quite the same answer as Silas gave today; but since the question is about a definition, there is no single correct answer.

1) your error is in being consistent across 3-4 definitions.

I’d be more inclined to attribute the problem to your unwillingness to read what I’ve actually said.

And I’d be more inclined to an excessive willingness to read what you’ve said. See, when you use (or, as is more often, imply) a definition, I keep checking back to see that that definition hasn’t subtly shifted to carry over baggage from another meaning. I already read your original definition of complexty — there wasn’t one, just a mass of contradictions if I try to apply any one of them. Your reply contains a great example:

I take complexity to be measured by the difficulty of specifying something uniquely in first-order language. … For each [computer], there is a purely arithmetical statement whose truth or falsity is undetermined by the specification. … you still haven’t answered *my* question about what you mean when you say arithmetic is “simple”.

Okay, so you use “arithmetic” in a broader sense, in which existing computers aren’t implementing or describing it. Strange, but let’s go with that. Then you go on later to talk about how effective arithmetic is at understanding the world around us. But that’s a different meaning: it’s referring to a finitely-describable, computable system. So the system can’t distinguish whether it’s the system that has all numbers as the sum of two primes? So what? It doesn’t need to, and neither do models of the universe’s physics. The universe’s physics don’t contain anything uncomputable, so where’s the uncomputable arithmetic?

And I did explain what it means for arithmetic to be simple: you can finitely describe how to perform the arithmetic operations. You *must* be able to, else, using math in physics would have no data-compressive power.

(I find it strange that you never defined complexity originally, you never before thought about how the standard definition of complexity applies to the domain you were dealing with, and yet you still feel confident your original claims reached crucial insights related to complexity.)

I’ve never seen Dawkins qualify his position with regard to domains of applicability. He makes a broad general statement that complexity, wherever it occurs, must evolve from simplicity.

He quite clearly did qualify its applicability, since he was speaking about the material, observable world. And I already explained what he would have to be saying to mean it in any other sense (note the nested quotation): “‘Biological evolution is mathematically possible’ must have evolved.” Now, has Dawkins said, *anywhere* else, anything about the varying time-history of the theoretical mathematical possibility of biological evolution? Of course not: he makes claims about biological evolution, not about the evolution of the Platonic math behind biological evolution.

It’s begging the question to count the immaterial, undescribable Platonic realm, with no influence on this world, as a kind of existence qualitatively the same as that in the observable realm It’s another fallacy of equivocation: massively explain what counts as “existing”, and act like the looser kind has the same significance as the stricter kind.

Arithmetic is a counterexample to that, unless you deny either that a) arithmetic exists or b) arithmetic is complex. If you deny a), then you are within your rights, but I disagree, for reasons I’ve expounded at length in my book; in this, for what it’s worth, I am on the side of most mathematicians (and, I daresay, most non-mathematicians as well).

It “exists” in a fundamentally different sense than how you otherwise use the term, as I explained. Your only substantiation for your claim about mathetmaticians is your say-so.

You seem to be confusing a mathematical claim with an ontological one: believing in the counterfactual validity of mathematical claims, with their existence as fundamental entities. That is a subtle form of the mind projection fallacy, in that you’re equating your cognitive system’s recognition of similarity across domains (in that they all represent math somehow) with fundamental reality.

Though you’d never guess it from this thread, Snorri has proved himself, elsewhere on this blog, to be capable of thoughtful and interesting commentary.

Yes, he’s capable of thoughtful commentary, but somehow became a half-competent idiot on precisely this issue. Funny how that works out.

[C] is not a definition; it’s a theorem.

Actually I don’t like your definition of “unbound”.

Yes, I know. This is only one of many standard definitions you don’t like, and are prepared to blame me for.

I’ll believe it when you show me a refereed, published paper (not necessarily authored by you) which uses all of the above terms to refer to the same concept.

Oh for Christ’s sake. If you don’t trust my knowledge of this subject, why are you asking me to explain it to you in the first place? Pick up any textbook on model theory. I’m done here.

Is training in mathematical logic standard for economists? I’m really quite impressed.

Wrong analogy, for 2 reasons. The greatest thing in the Universe is, obviously, the Universe itself. But YOUR definition [C], not mine, of the standard model, is analogous to “the smallest thing in the Universe”.

That’s because it contains the unbound variable “Mary” and is therefore not a statement. Now you’re going to tell me that you don’t like this definition of “statement”

Actually I don’t like your definition of “unbound”. But that’s beside the point, since we are discussing a priori statements and my example was a posteriori. The point is that I reject out of hand your definition of “meaning”.

It is perfectly standard terminology, and if you don’t like it, your beef is not with me; it’s with the way language happens to have evolved.

I’ll believe it when you show me a refereed, published paper (not necessarily authored by you) which uses all of the above terms to refer to the same concept. I would settle for a set of papers, each of which uses 2 or more of the above terms to refer to the same concept; provided that the papers, taken together, establish the equivalence of the entire set of terms.

Given that “axiomatized” is a meaningless word you just invented, I suppose you’re the only person who can answer this question.

Not so fast: by defining the “standard model” in [C] as “the initial segment of all models” you have provided a unique description of it. That raises the question: a unique description in which language? in the language of set theory? but then, the parenthetical phrase is no longer superfluous.

This is only a “description” if you first describe all models of set theory. Good luck with that.

If this is counts as a “description” of the natural numbers, then “the greatest thing in the Universe” counts as a description of God.

That is a definition of “meaning” that I find extremely unpalatable. “Mary had a little lamb” makes perfect sense to me, even though, without specifying which Mary it refers to, it cannot be either true or false.

That’s because it contains the unbound variable “Mary” and is therefore not a statement. Now you’re going to tell me that you don’t like this definition of “statement” and that by invoking it I’ve committed the fallacy of this, that or the other thing.

That is saying:
* “The standard model”,
* “the standard model of the natural numbers”,
* “arithmetic”, and
* “the natural numbers”
all refer to both
- the natural numbers and
- the mapping from Peano arithmetic to the natural numbers,
though I have tried to use them consistently to mean
+ the natural numbers.

That is not just circular: it is a spaghetti junction in which you always end up at the starting place.

It is perfectly standard terminology, and if you don’t like it, your beef is not with me; it’s with the way language happens to have evolved. If you don’t understand it, I’m perfectly happy to explain it to you. If you’re going to blame me for the fact that you don’t like it, I think we should end this discussion.