Johnicholas

Karma: 2,008

Johnicholas 11 Nov 2012 20:48 UTC
0 points
in reply to: Will_Sawin’s comment on: Logical Pinpointing
I understand your point—it’s akin to the Box quote “all models are wrong but some are useful”—when choosing among (false) models, choose the most useful one. However, it is not the case that stronger assumptions are more useful—of course stronger assumptions make the task of proving easier, but the task as a whole includes both proving and also building a system based on the theorems proven.

My primary point is that EY is implying that second-order logic is necessary to work with the integers. People work with the integers without using second-order logic all the time. If he said that he is only introducing second-order logic for convenience in proving and there are certainly other ways of doing it, and that some people (intuitionists and finitists) think that introducing second-order logic is a dubious move, I’d be happy.

My other claim that second-order logic is unphysical and that its unphysicality probably does ripple out to make practical tasks more difficult, is a secondary one. I’m happy to agree that this secondary claim is not mainstream.

Johnicholas 11 Nov 2012 15:37 UTC
1 point
in reply to: Will_Sawin’s comment on: Logical Pinpointing
If you were writing software for something intended to traverse the Interplanetary transfer network then you would probably use charts and atlases and transition functions, and you would study (symplectic) manifolds and homeomorphisms in order to understand those more-applied concepts.

If an otherwise useful theorem assumes that the manifold is orientable, then you need to show that your practical manifold is orientable before you can use it—and if it turns out not to be orientable, then you can’t use it at all. If instead you had an analogous theorem that applied to all manifolds, then you could use it immediately.

Johnicholas 10 Nov 2012 17:37 UTC
3 points
in reply to: Will_Sawin’s comment on: Logical Pinpointing
If you assume A and derive B you have not proven B but rather A implies B. If you can instead assume a weaker axiom Aprime, and still derive B, then you have proven Aprime implies B, which is stronger because it will be applicable in more circumstances.

Johnicholas 10 Nov 2012 15:56 UTC
0 points
in reply to: Eliezer Yudkowsky’s comment on: Logical Pinpointing
I agree with this statement—and yet you did not contradict my statement that second order logic is also not part of mainstream mathematics.

A topologist might care about manifolds or homeomorphisms—they do not care about foundations of mathematics—and it is not the case that only one foundation of mathematics can support topology. The weaker foundation is preferable.

Johnicholas 9 Nov 2012 21:06 UTC
0 points
in reply to: [deleted]’s comment on: Logical Pinpointing
Second-order logic is not part of standard, mainstream mathematics. It is part of a field that you might call mathematical logic or “foundations of mathematics”. Foundations of a building are relevant to the strength of a building, so the name implies that foundations of mathematics are relevant to the strength of mainstream mathematics. A more accurate analogy would be the relationship between physics and philosophy of physics—discoveries in epistemology and philosophy of science are more often driven by physics than the other way around, and the field “philosophy of physics” is a backwater by comparison.

As is probably evident, I think the good, solid mathematical logic is intuitionist and constructive and higher-order and based on proof theory first and model theory only second. It is easy to analogize from their names to a straight line between first-order, second-order, and higher-order logic, but in fact they’re not in a straight line at all. First-order logic is mainstream mathematics, second-order logic is mathematical logic flavored with faith in the reality of infinite models and set theory, and higher-order logic is mathematical logic that is (usually) constructive and proof-theoretic and built with an awareness of computer science.

Johnicholas 7 Nov 2012 16:54 UTC
−5 points
on: Logical Pinpointing
EY is talking from a position of faith that infinite model theory and second-order logic are good and reasonable things.

It is possible to instead start from a position of doubt that the infinite model theory and second order logic are good and reasonable things (based on my memory of having studied in college whether model theory and second order logic can be formalized within Zermelo-Frankel set theory, and what the first-order-ness of Zermelo-Frankel has to do with it.).

We might be fine with a proof-theoretic approach, which starts with the same ideas “zero is a number”, “the successor of a number is a different number”, but then goes to a proof-theoretic rule of induction something like “I’d be happy to say ‘All numbers have such-and-such property’ if there were a proof that zero has that property and another also proof that if a number has that property, then its successor also has that property.”

We don’t need to talk about models at all—in particular we don’t need to talk about infinite models.

Second-order arithmetic is sufficient to get what EY wants (a nice pretty model universe) but I have two objections. First it is too strong—often the first sufficient hammer that you find in mathematics is rarely the one you should end up using. Second, the goal of a nice pretty model universe presumes a stance of faith in (infinite) model theory, but the infinite model theory is not formalized. If you do formalize it then your formalization will have alternative “undesired” interpretations (by Lowenheim-Skolem).

Johnicholas 29 Oct 2012 2:22 UTC
−19 points
on: Proofs, Implications, and Models
Uncountability is an idea that skeptics/rationalists/less-wrongers should treat with a grain of salt.

Inside of a particular system, you might add an additional symbol, “i”, with the assumed definitional property that “i times i plus one equals zero”. This is how we make complex numbers. We might similarly add “infinity”, with an assumed definitional property that “infinity plus one equals infinity”. Depending on the laws that we’ve already assumed, this might form a contradiction or cause everything to collapse, but in some situations it’s a reasonable thing to do. This new symbol, “infinity” can be definitionally large in magnitude without being large in information content—bits or bytes required to communicate it.

A specific generic member of an uncountable set is infinitely large in information content—that means it would take infinite time to communicate at finite bandwidth, infinite volume to store at finite information density. It is utterly impractical.

It is reasonable (because it is true) to believe that every mathematical object that a mathematician has ever manipulated was of finite information content. The set of all objects of finite information content is merely countable, not uncountable.

What I’m trying to say is: Don’t worry about uncountability. It isn’t physical and it isn’t practical. Mathematicians may be in love with it (Hilbert’s famous quote: “No one shall expel us from the Paradise that Cantor has created.”), but the concept is a bit like that “infinite” symbol—infinite in magnitude, not in information content.

Johnicholas 23 Oct 2012 4:51 UTC
0 points
in reply to: Bugmaster’s comment on: Babies and Bunnies: A Caution About Evo-Psych
My understanding is that the essay’s effect is via the horror a reader feels at the alternate-world presented in the essay. It opens the reader’s eyes somewhat to the degree that sexism is embedded in everyday grammar and idiom. My understanding is that it is not a persuasive essay in the usual sense.

Please elaborate.

Johnicholas 18 Oct 2012 15:27 UTC
0 points
in reply to: MaoShan’s comment on: Stuff That Makes Stuff Happen
I agree that if you don’t look at the numbers, but at the surrounding text, you get the sense that the numbers could be paraphrased in that way.

So does h, labeled “I hear universe” mean “I hear the universe tell me something at all”, or “I hear the universe tell me that they love me” or “I hear the universe tell me what it knows, which (tacitly according to the meaning of knows) is accurate”?

I thought it meant “I have a sensation as if the universe were telling me that they love me”, but the highest probability scenarios are p&u&h, and -p&u&h, which would suggest that regardless of their love, I’m likely to experience a sensation as if the universe were telling me that they love me. That seems reasonable from a skeptical viewpoint, but not from a believer’s viewpoint.

Johnicholas 17 Oct 2012 19:39 UTC
7 points
on: Stuff That Makes Stuff Happen
Twice in this article, there are tables of numbers. They’re clearly made-up, not measured from experiment, but I don’t really understand exactly how made-up they are—are they carefully or casually cooked?

Could people instead use letters (variables), with relations like ‘a > b’, ‘a >> b’, ‘a/b > c’ and so on in the future? Then I could understand better what properties of the table are intentional.

Johnicholas 12 Oct 2012 23:13 UTC
5 points
on: Causal Diagrams and Causal Models
I think it would be valuable if someone pointed out that a third party watching, without controlling, a scientist’s controlled study is in pretty much the same situation as the three-column exercise/weight/internet use situation—they have instead exercise/weight/control group.

This “observe the results of a scientist’s controlled study” thought experiment motivates and provides hope that one can sometimes derive causation from observation, where the current story arc makes a sortof magical leap.

Johnicholas 6 Oct 2012 15:22 UTC
3 points
on: Skill: The Map is Not the Territory
There are some aspects of maps—for example, edges, blank spots, and so on, that seem, if not necessary, extremely convenient to keep as part of the map. However, if you use these features of a map in the same way that you use most features of a map—to guide your actions—then you will not be guided well. There’s something in the sequences like “the world is not mysterious” about people falling into the error of moving from blank/cloudy spots on the map to “inherently blank/cloudy” parts of the world.

The slogan “the map is not the territory” might encourage focusing on the delicate corrections necessary to act upon SOME aspects of one’s representation of the world, but not act on other aspects which are actually intrinsic to the representation.

Johnicholas 3 Oct 2012 21:00 UTC
1 point
on: The Bayesian Agent
You might enjoy Crutchfield’s epsilon machines, and Shalizi’s CSSR algorithm for learning them:

http://masi.cscs.lsa.umich.edu/~crshalizi/notabene/computational-mechanics.html

Johnicholas 21 Sep 2012 12:28 UTC
12 points
on: New study on choice blindness in moral positions
There’s cognitive strategies that (heuristically) take advantage of the usually-persistent world. Should I be embarrassed, after working and practicing with pencil and paper to solve arithmetic problems, that I do something stupid when someone changes the properties of pencil and paper from persistent to volatile?

What I’d like to see is more aboveboard stuff. Suppose that you notify someone that you’re showing them possibly-altered versions of their responses. Can we identify which things were changed when explicitly alerted? Do we still confabulate (probably)? Are the questions that we still confabulate on questions that we’re more uncertain about—more ambiguous wording, more judgement required?

Johnicholas 17 Aug 2012 1:03 UTC
2 points
in reply to: AlanCrowe’s comment on: Who Wants To Start An Important Startup?
Yes, I (and Stross) am taking auditors, internal and external, as a model. Why do you comment specifically on internal auditors?

Johnicholas 14 Aug 2012 19:24 UTC
12 points
on: Who Wants To Start An Important Startup?
There’s a lot of similarity between the statistical tests that a scientist does and the statistical tests that auditors do. The scientist is interested in testing that the effect is real, and the auditor is testing that the company really is making that much money, that all its operations are getting aggregated up into the summary documents correctly.

Charlie Stross has a character in his ‘Rule 34’, Dorothy Straight, who is an organization-auditor, auditing organizations for signs of antisocial behavior. As I understood it, she was asking whether the organizations as a whole are likely to behave badly—though one way that the organization as a whole might behave badly is by sifting out or creating leaders who are likely to individually behave badly.

What I’m trying to say is that there will be a field of auditing an organization’s ‘safety case’ - examining why it believes that it is a Friendly organization, what its internal controls entangling it with the truth are and so on, something like GiveWell for for-profits.

Johnicholas 3 Aug 2012 22:19 UTC
1 point
in reply to: royf’s comment on: Reinforcement Learning: A Non-Standard Introduction (Part 2)
As I understand it, you’re dividing the agent from the world; once you introduce a reward signal, you’ll be able to call it reinforcement learning. However, until you introduce a reward signal, you’re not doing specifically reinforcement learning—everything applies just as well to any other kind of agent, such as a classical planner.

Johnicholas 2 Aug 2012 18:09 UTC
1 point
in reply to: duwease’s comment on: Reinforcement Learning: A Non-Standard Introduction (Part 1)
The arrows all mean the same thing, which is roughly ‘causes’.

Chess is a perfect-information game, so you could build the board entirely from the player’s memory of the board, but in general, the state of the world at time t-1, together with the player, causes the state of the world at time t.

Johnicholas 2 Aug 2012 15:25 UTC
10 points
on: Reinforcement Learning: A Non-Standard Introduction (Part 2)
It might be valuable to point out that nothing about this is reinforcement learning yet.

Johnicholas 12 Jul 2012 16:25 UTC
2 points
in reply to: lukeprog’s comment on: Reply to Holden on The Singularity Institute
Those are interesting reviews but I didn’t know they were speeches in SIAI’s voice.