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All of SteveLandsburg's Comments + Replies

Rationality Quotes: February 2010

Splat:

Thanks again for bringing insight and sanity to this discussion. A few points:

1) Your description of the structure N presupposes some knowledge of the structure N; the program that prints out the structure needs a first statement, a second statement, etc. This is, of course, unavoidable, and it's therefore not a complaint; I doubt that there's any way to give a formal description of the natural numbers without presupposing some informal understanding of the natural numbers. But what it does mean, I think, is that K-complexity (in the sense that ... (read more)

Rationality Quotes: February 2010

SteveLandsburg15y40

If you're going to write a book hundreds of pages long in which you crucially rely on the concept of complexity, you need to explicitly to define it. That's just how it works. If you know what concept of complexity is "the" right one here, you need to spell it out yourself.

Well, Silas, what I actually did was write a book 255 pages long of which this whole Dawkins/complexity thing occupies about five pages (29-34) and where complexity is touched on exactly once more, in a brief passage on pages 7-8. From the discrepancy between your descripti... (read more)

3SilasBarta15y

I have not read the entire book. I have read many long portions of it, mostly the philosophical ones and those dealing with physics. I was drawn to on the assumption that, surely you would have defined complexity in your exposition! It's misleading to say that your usage of complexity only takes 8 pages, so it's insignificant. Rather, the point you make about complexity is your grounding for broader claims about the role mathematics plays in the universe, which you come back to frequently. The explicit mention of the term "complexity" is thus a poor measure of how much you rely on it. But even if it were just 8 pages, you should still have defined it, and you should still not expect to have achieved insights on the topic, given that you haven't defined it. (I certainly wouldn't want to buy it -- why should I subsidize such confused thinking? I don't even like your defenses of libertarianism, despite being libertarian.) Ah, another suddenly-crucial distinction to make, so you can wiggle out of being wrong! ---------------------------------------- I should probably use this opportunity to both show I did read many portions, and show why Landsburg doesn't get what it means to really explain something. His explanation of the Heisenberg Uncertainty Principle (which gets widely praised as a good explanation for some reason) is this: think of an electron as moving in a circle within a square. If you measure its vertical position, its closeness to the top determines the chance of getting a "top" or "bottom" reading. Likewise the horizontal direction: if you measure the horizontal position of the electron, your chances of getting a "left" or "right" reading depends on how far it is from that side. And for the important part: why can't you measure both at the same time? Landsburg's brilliant explanation: um, because you can't. But that's what the explanation was supposed to demystify in the first place! You can't demystify by feeding that very mystery a blackbox fact

Rationality Quotes: February 2010

SteveLandsburg15y30

Splat:

The problem you encounter here is that these substructures and near-substructures, once they reach a certain size, actually require more information to specify than N itself.

This depends on what you mean by "specify". To distinguish N from other mathematical structures requires either an infinite (indeed non-recursive) amount of information or a second order specification including some phrase like "all predicates". Are you referring to the latter? Or to something else I don't know about?

2) I do not know Chaitin's de... (read more)

4Splat15y

Replying out of order: 2) A quick search of Google Scholar didn't net me a Chaitin definition of K-complexity for a structure. This doesn't surprise me much, as his uses of AIT in logic are much more oriented toward proof theory than model theory. Over here you can see some of the basic definitions. If you read page 7-10 and then my explanation to Silas here you can figure out what the K-complexity of a structure means. There's also a definition of algorithmic complexity of a theory in section 3 of the Chaitin. According to these definitions, the complexity of N is about a few hundred bits for reasonable choices of machine, and the complexity of T(N) is &infty;. 1) It actually is pretty hard to characterize N extrinsically/intensionally; to characterize it with first-order statements takes infinite information (as above). The second-order characterization. by contrast, is a little hard to interpret. It takes a finite amount of information to pin down the model[*][PA2], but the second-order theory PA2 still has infinite K-complexity because of its lack of complete rules of inference. Intrinsic/extensional characterizations, on the other hand, are simple to do, as referenced above. Really, Gödel Incompleteness wouldn't be all that shocking in the first place if we couldn't specify N any other way than its first-order theory! Interesting, yes, shocking, no. The real scandal of incompleteness is that you can so simply come up with a procedure for listing all the ground (quantifier-free) truths of arithmetic and yet passing either to or from the kind of generalizations that mathematicians would like to make is fraught with literally infinite peril. 3&4) Actually I don't think that Dawkins is talking about K-complexity, exactly. If that's all you're talking about, after all, an equal-weight puddle of boiling water has more K-complexity than a squirrel does. I think there's a more involved, composite notion at work that builds on K-complexity and which has so far resi

2SilasBarta15y

1) Unless they say otherwise, you should assume someone is using the standard meanings for the terms they use, which would mean Dawkins is using the intuitive definition, which closely parallels K-complexity. 2) If you're going to write a book hundreds of pages long in which you crucially rely on the concept of complexity, you need to explicitly to define it. That's just how it works. If you know what concept of complexity is "the" right one here, you need to spell it out yourself. 3) Most importantly, you have shown Dawkins's argument to be in error in the context of an immaterial realm that is not observable and does not interact with this universe. Surely, you can think of some reason why Dawkins doesn't intend to refer to such realms, can't you? (Hint: Dawkins is an atheist, materialist, and naturalist -- just like you, in other words, until it comes to the issue of math.) ETA: If any followers of this exchange think I'm somehow not getting someting, or being unfair to SteveLandsburg, please let me know, either as a reply in the thread or a PM, whether or not you use your normal handle.

1SilasBarta15y

I didn't say that. Read it again. I said that there is some finite axiom list that can describe squirrels, but it's not just the axioms that suffice to let you use arithmetic. It's those, plus biological information about squirrels. But this arithmetic is not the infinitely complex arithmetic you talk about in other contexts! You can't -- you need axioms beyond those that specify N. The fact that the biological model involving those axioms uses math, doesn't mean you've described it once you've described the structure N. So whether or not you call that "simulating it in the structure N", it's certainly more complex than just N.

Rationality Quotes: February 2010

SteveLandsburg15y20

Splat: Thanks for this; it's enlightening and useful.

The part I'm not convinced of this:

to simulate any biological creature, you need N plus a bunch of biological information

A squirrel is a finite structure; it can be specified by a sequence of A's, C's, G's and T's, plus some rules for protein synthesis and a finite number of other facts about chemistry. (Or if you think that leaves something out, it can be described by the interactions among a large but finite collection of atoms.) So I don't see where we need all of N to simulate a squirrel.

3Splat15y

Well, if you need to simulate a squirrel for just a little while, and not for unbounded lengths of time, a substructure of N (without closure under the operations) or a structure with a considerable amount of sharing with N (like 64-bit integers on a computer) could suffice for your simulation. The problem you encounter here is that these substructures and near-substructures, once they reach a certain size, actually require more information to specify than N itself. (How large this size is depends on which abstract computer you used to define your instance of K-complexity, but the asymptotic trend is unavoidable.) If this seems paradoxical, consider that after a while the shortest computer program for generating an open initial segment of N is a computer program for generating all of N plus instructions indicating when to stop. Either way, it so happens that the biological information you'd need to simulate the squirrel dwarfs N in complexity, so even if you can find a sufficient substitute for N that's "lightweight" you can't possibly save enough to make your squirrel simulation less complex than N.

-2SilasBarta15y

Okay, pretend I've given you the axioms sufficient for you to +-*/. Can simulate squirrels now? Of course not. You still have to go out and collect information about squirrels and add it to your description of the axioms of arithmetic (which suffice for all of N) to have a description of squirrels. You claim that because you can simulate squirrels with (a part of) N, then N suffices to simulate squirrels. But this is like saying that, because you know the encoding method your friend uses to send you messages, you must know the content of all future messages. That's wrong, because those are different parts of the compressed data: one part tells you how to decompress, another tells you what you're decompressing. Knowing how to decompress (i.e., the axioms of N) is different from knowing the string to be decompressed by that method (i.e. the arithmetic symbols encoding squirrels). By the way, I really hope your remark about Splat's comment being "enlightening" was just politeness, and that you didn't actually mean it. Because if you did, that would mean you're just now learning the distinction between N and T(N), the equivocation between which undermines your claims about arithmetic's relation to the universe. And much of his comment was a restatement of my point about the difference between the complex arithmetic you refer to, and the arithmetic the universe actually runs on. (I'm not holding my breath for a retraction or a mea culpa or anything, just letting people know what they're up against here.)

Rationality Quotes: February 2010

SteveLandsburg15y40

Bo102010: Thanks for the kind words. I'm not sure what the community standards are here, but I hope its not inappopriate to mention that I post to my own blog almost every weekday, and of course I'll be glad to have you visit.

0[anonymous]15y

I can second that. Though, for a lack of education, I cannot tell who's right in this debate, I don't think anybody is for that it is just pure metaphysical musing about the nature of reality. But so far I really enjoyed reading your book. I also hope you'll participate in other discussions here at lesswrong.com. It's my favorite place. Sorry for possible bad publicity, I committed the mistake to quick-share something which I've just read and found intriguing. Without the ability to portray it adequately. Especially on this forum which is rather about rationality as practical tool to attain your goals and not pure philosophy detached from evidence and prediction. I also subscribed to your blog. P.S. Send you a message, you can find it in your inbox.

Rationality Quotes: February 2010

SteveLandsburg15y10

The only error is to refuse to "cash out" the meaning of "arithmetic" into well-defined >predictions, but instead keep it boxed up into one ambiguous term,

Silas: This is really quite frustrating. I keep telling you exactly what I mean by arithmetic (the standard model of the natural numbers); I keep using the word to mean this and only this, and you keep claiming that my use of the word is either ambiguous or inconsistent. It makes it hard to imagine that you're actually reading before you're responding, and it makes it very difficult to carry on a dialogue. So for that reason, I think I'll stop here.

5Bo10201015y

When I saw this in the comment feed, I thought "Wow, Steve Landsburg on Less Wrong!" Then I saw that he was basically just arguing with one person. While I think you're not correct in this debate, I hope you'll continue to post here. Your books have been a source of much entertainment and joy for me.

-1SilasBarta15y

Are you reading my replies? Saying that arithmetic is "the standard model of the natural numbers" does not For one thing, it doesn't give me predictions (i.e. constraints on expectations) that we check to see who's right. For another, it's not well-defined -- it doesn't tell me how I would know (as is necessary for the area of dispute) if arithmetic "exists" at this or that time. (And, of course, as you found out, it requires further specification of what counts as a model...) (ETA: See Eliezer_Yudkowsky's great posts on how to dissolve a question and get beyond there being One Right Answer to e.g. the vague question about a tree falling in the forest when no one's around.) So if you don't see how that doesn't count as cashing out the term and identifying the real disagreement, then I agree further discussion is pointless. But truth be told, you're not going to "stop there". You going to continue on, promoting your "deep" insights, wherever you can, to people who don't know any better, instead of doing the real epistemic labor achieving insights on the world.

Rationality Quotes: February 2010

SteveLandsburg15y00

Eliezer: There are an infinite number of truths of euclidean geometry. How could our finite brains know them all?

This was not meant to be a profound observation; it was meant to correct Silas, who seemed to think that I was reading some deep significance into our inability to know all the truths of arithmetic. My point was that there are lots of things we can't know all the truths about, and this was therefore not the feature of arithmetic I was pointing to.

7Eliezer Yudkowsky15y

A decision procedure is a finite specification of all truths of euclidean geometry; I can use that finite fact anywhere I could use any truth of geometry. I suppose there is a difference, but even so, it's the wrong thing to say in a Godelian discussion.

5SilasBarta15y

Yes, it was. When I and several others pointed out that arithmetic isn't actually complex, you responded by saying that it is infinitely complex, because it can't be finitely described, because to do so ... you'd have to know all the truths. Am I misreading that response? If so, how do you reconcile arithmetic's infinite complexity with the fact that scientists in fact use it to compress discriptions of the world? An infinitely complex entity can't help to compress your descriptions.

Rationality Quotes: February 2010

SteveLandsburg15y00

Silas: I agree that if arithmetic is a human invention, then my counterexample goes away.

If I've read you correctly, you believe that arithmetic is a human invention, and therefore reject the counterexample.

On that reading, a key locus of our disagreement is whether arithmetic is a human invention. I think the answer is clearly no, for reasons I've written about so extensively that I'd rather not rehash them here.

I'm not sure, though, that I've read you correctly, because you occasionally say things like "The Map Is Not The Territory" wh... (read more)

1XiXiDu15y

Map and territory More: Map and Territory (sequence)

-1SilasBarta15y

Then you agree that your "counterexample" amounts to an assumption. If a Platonic realm exists (in some appropriate sense), and if Dawkins was haphazardly including that sense in the universe he is talking about when he describes complexity arising, then he wrong that complexity always comes from simplicity. If you assume Dawkins is wrong, he's wrong. Was that supposed to be insightful? It's a false dispute, though. When you clarify the substance of what these terms mean, there are meanings for which we agree, and meanings for which we don't. The only error is to refuse to "cash out" the meaning of "arithmetic" into well-defined predictions, but instead keep it boxed up into one ambiguous term, which you do here, and which you did for complexity. (And it's kind of strange to speak for hundreds of pages about complexity, and then claim insights on it, without stating your definition anywhere.) One way we'd agree, for example, is if we take your statements about the Platonic realm to be counterfactual claims about phenomena isomorphic to certain mathematic formalisms (as I said at the beginning of the thread). The definitions aren't incredibly different, which is why we have the same term for both of them. If you spell out that definition more explicitly, the same problems arise, or different ones will pop up. (By the way, this doesn't surprise me. This is the fourth time you've had to define a term within a definition you gave in order to avoid being wrong. It doesn't mean you changed that "subdefinition". But genuine insights about the world don't look this contorted, where you have to keep saying, "No, I really meant this when I was saying what I meant by that.")

Rationality Quotes: February 2010

SteveLandsburg15y-10

mattnewport: This would seem to put you in the opposite corner from Silas, who thinks (if I read him correctly) that all of physical reality is computably describable, and hence far simpler than arithmetic (in the sense of being describable using only a small and relatively simple fragment of arithmetic).

Be that as it may, I've blogged quite a bit about the nature of the complexity of arithmetic (see an old post called "Non-Simple Arithmetic" on my blog). In brief: a) no set of axioms suffices to specify the standard model of arithmetic (i.e. ... (read more)

Splat15y130

Your biggest problem here, and in your blog posts, is that you equivocate between the structure of the standard natural numbers (N) and the theory of that structure (T(N), also known as True Arithmetic). The former is recursive and (a reasonable encoding of) it has pretty low Kolmogorov complexity. The latter is wildly nonrecursive and has infinite K-complexity. (See almost any of Chaitin's work on algorithmic information theory, especially the Omega papers, for definitions of the K-complexity of a formal system.)

The difference between these two structu... (read more)

2Eliezer Yudkowsky15y

Then what do you mean when you say "integers"^H^H "natural numbers", if no set of premises suffices to talk about it as opposed to something else? Anyway, no countable set of first-order axioms works. But a finite set of second-order axioms work. So to talk about the natural numbers, it suffices merely to think that when you say "Any predicate that is true of zero, and is true of the successor of every number it is true of, is true of all natural numbers" you made sense when you said "any predicate". It is this sort of minor-seeming yet important technical inaccuracy that separates "The Big Questions" from "Good and Real", I'm afraid.

4Zack_M_Davis15y

Again, this word complexity is used in many ways. Complexity in the sense of humans find this complicated is a different concept from complexity in the sense of Kolmogorov complexity.

1SilasBarta15y

No, it wouldn't -- he's saying basically the same thing I did. The laws of physics are computable. In describing observations, we use concepts from math. The reason we do so is that it allows simpler descriptions of the universe.

Rationality Quotes: February 2010

SteveLandsburg15y-10

Silas:

First---I have never shifted meanings on the definition of arithmetic. Arithmetic means the standard model of the natural numbers. I believe I've been quite consistent about this.

Second---as I've said many times, I believe that the most plausible candidates for the "fabric of the Universe" are mathematical structures like arithmetic. And as I've said many times, obviously I can't prove this. The best I can do is explain why I find it so plausible, which I've tried to do in my book. If those arguments don't move you, well, so be it.... (read more)

0SilasBarta15y

Right, I've explained before why your arguments are in error. We can talk more about that some other time. No, I accept that they're separate errors. Okay: If what you describe here is what you mean by both "the natural numbers" and "the actual standard model of the natural numbers", then I will accept this definition for the purposes of argument, but that, using it consistently, it doesn't have the properties you claim. Disagree with this. Dawkins has been referring to existing complexity in the universe and the context of every related statement confirms this. But even accepting it, the rest of your argument still doesn't follow. Disagree. Again, let's keep the same definition throughout. Recall what you said the natural numbers were: The model arose from something simpler (like basic human cognition of counting of objects). The Map Is Not The Territory. Ah, but now I know what you're going to say: you meant the sort of Platonic-space model of those natural numbers, that exists independently of whatever's in our universe, has always been complex. So, if you assume (like theists) that there's some sort of really-existing realm, outside of the universe, that always has been, and is complex, then you can prove that ... there's a complexity that has always existed. Which is circular.

6mattnewport15y

It seems to me that the definition of complexity is the root of any disagreement here. It seems obvious to me that the natural numbers are not complex in the sense that a human being is complex. I don't understand what kind of complexity you could be talking about that places natural numbers on an equivalent footing with, say, the entire ecosystem of the planet Earth.

Rationality Quotes: February 2010

SteveLandsburg15y-10

Then the universe doesn't use that arithmetic in implementing physics,

How do you know?

Like I said just above, it uses the kind of arithmetic that can be captured in a small set of axioms.

What kind of arithmetic is that? It would have to be a kind of arithmetic to which Godel's and Tarski's theorems don't apply, so it must be very different indeed from any arithmetic I've ever heard of.

3SilasBarta15y

Mainly from the computability of the laws of physics. Right -- meaning the universe doesn't use arithmetic (as you've defined it). You're getting tripped up on the symbol "arithmetic", for which you keep shifting meanings. Just focus on the substance of what you mean by arithmetic: Does the universe need that to work? No, it does not. Do computers need to completely specify that arithmetic to work? No, they do not. By the way: 1) To quote someone here, use the greater-than symbol before the quoted paragraph, as described in the help link below the entry field for a comment. 2) One should be cautious about modding down someone one is a direct argument with, as that tends to compromise one's judgment. I have not voted you down, though if I were a bystander to this, I would.

Rationality Quotes: February 2010

SteveLandsburg15y20

Richard: Gotcha. Sorry if it was unclear which part the "not so" referred to.

Rationality Quotes: February 2010

SteveLandsburg15y30

Richard Kennaway:

I don't know where Landsburg gets the claim that we can know all the truths of arithmetic.

I don't know where you got the idea that I'd ever make such a silly claim.

1Richard_Kennaway15y

I misinterpreted this: "we can never know all the truths of euclidean geometry, but we can still specify euclidean geometry via a set of axioms. Not so for arithmetic."

Rationality Quotes: February 2010

SteveLandsburg15y20

RichardKennaway:

Ariithmetic is complex because it can not be captured in a small set of axioms. More precisely, it cannot be specified by any (small or large) set of axioms, because any set of (true) axioms about arithmetic applies equally well to other structures that are not arithmetic. Your favorite set of axioms fails to specify arithmetic in the same way that the statement "bricks are rectangular" fails to specify bricks; there are lots of other things that are also rectangular.

This is not true, for example, of euclidean geometry, which... (read more)

3SilasBarta15y

Here we go again. Then the universe doesn't use that arithmetic in implementing physics, and it doesn't have the significance you claim it does. Like I said just above, it uses the kind of arithmetic that can be captured in a small set of axioms. And like I said in our many exchanges, it's true that modern computers can't answer every question about the natural numbers, but they don't need to. Neither does the universe. Yes, but you only need finite space to specify bricks well enough to get the desired functionality of bricks. Your argument would imply that bricks are infinitely complex because we don't have a finite procedure for determining where an arbitrary object "really" is a brick, because of e.g. all the borderline cases. ("Do the stones in a stone wall count as bricks?")

0Richard_Kennaway15y

What is this "it"? There are some who claim that when we think about arithmetic, we are thinking about a specific model of the usual axioms for arithmetic, which appears to be your view here. Every statement of arithmetic is either true or false in that model. But what reason is there to make this claim? We cannot directly intuit the truth of arithmetical statements, or mathematicians would not have to spend so much effort on proving theorems. We may observe that we have a belief that we are indeed thinking about a definite model of the axioms, but why should we believe that belief? To say that we intuit a thing is no more than to say we believe it but do not know why.

2Paul Crowley15y

That doesn't sound right. Can you point me to for example a Wikipedia page about this?