komponisto2

Karma: 174

komponisto2 Apr 17, 2008, 6:54 AM
1 point
on: The Quantum Arena
a function of a real space has uncountable degrees of freedom

Right—that’s exactly the misunderstanding I was addressing in my earlier comment.

An arbitrary function does indeed have uncountable degrees of freedom, but in that context you’re notconsidering it as an element of a Hilbert space. (Those degrees of freedom do not correspond to basis vectors.)

komponisto2 Apr 17, 2008, 6:33 AM
0 points
on: The Quantum Arena
Tom McCabe: The category of Hilbert spaces includes spaces of both finite and infinite dimension, so it presumably includes both countable and uncountable infinities.

mitchell porter: But the Hilbert space of a quantum field, naively, ought to have uncountable dimension, because there are continuum-many degrees of freedom.

Given any cardinal number, there exists a Hilbert space with that orthogonal dimension. Note, however, that even if the dimension is uncountable, individual elements are still given by linear combinations with countably many terms. In other words, only countably many dimensions are used “at one time” in specifying an element.

Thus, a Hilbert space formalism cannot accommodate “uncountably many degrees of freedom” in the sense people mean here. Which is okay, because I don’t think that’s what you need anyway.

komponisto2 Apr 16, 2008, 9:41 PM
0 points
on: The Quantum Arena
Also:

I’d say that I was assuming the continuum hypothesis, except that I’m an infinite set atheist.

Not that again! Let’s not mix up the map and the territory. You may not think there are any infinite sets out there in the territory, but mathematics is about the map—or, rather, mapmaking in general. So it’s a category error to jump from the conviction that the “real world” contains only finitely many things to Kroneckerian skepticism about mathematical objects.

For what it’s worth, the cardinality of the set of reals is 2^aleph_0. The continuum hypothesis is precisely the statement that this is equal to aleph_1. So yeah, you’re definitely assuming it.

komponisto2 Apr 16, 2008, 9:31 PM
0 points
on: The Quantum Arena
I’d thought the Hilbert space was uncountably dimensional because the number of functions of a real line is uncountable

Well, the number of points in a Hilbert space of dimension 2 is uncountable, and yet the space has dimension 2!

I suspect the source of the confusion here is that you’re trying to think of the values of a function as its “coordinates”. But this is wrong: the “coordinates” are the coefficients of a Fourier series expansion of the function.

The confusion is understandable, given that the two concepts coincide in the finite-dimensional case. You should think of a point in C^2, say (3,7), as a function from the two-element set {1,2} into the complex numbers C (in this case we have f(0) = 3 and f(1)=7). You can then write every such function uniquely as the sum of two “basis” functions: one that sends 1 to 1 and 2 to 0 (call this b_1), and one that sends 1 to 0 and 2 to 1 (call this b_2). Thus f = 3b_1 + 7b_2, i.e. (3,7) = 3(1,0) + 7(0,1).

In the case of functions on the real line, however, the “basis” functions cannot be functions that send one number to 1 and the rest to 0, because in order to represent an arbitrary function, you would need to add uncountably many such things together, which is not a defined operation (or, more technically, is only defined when all but countably many of the summands are zero).

Fortunately, however, if we’re talking about the space of square-integrable functions (and that was what we wanted anyway, wasn’t it?), we do have a countable orthogonal basis available, as was discovered by Fourier.

(Technical note: Actually, the space of square-integrable functions on (an interval of) the real line doesn’t consist of well-defined functions per se, but only of equivalence classes of functions that agree everywhere except on a set of measure zero. See measure theory, Lebesgue integration, L^p space, etc.)

komponisto2 Apr 16, 2008, 3:15 PM
0 points
on: The Quantum Arena
I have the same questions for Eliezer as Jadagul and Toby Ord, namely:
- Why would the space of amplitude distributions have uncountable dimension? Unless I’ve misunderstood, it sounds like it would be something like L^2, which is separable (has countable orthogonal dimension). (Of course, maybe by “dimension” you just meant the cardinality of a Hamel basis, in which case you’re right—there’s no Hilbert space with Hamel dimension aleph_0. However, “dimension” in the context of Hilbert spaces nearly always refers to orthogonal dimension.)
- Assuming you really did mean uncountable, how do you know aleph_1 is the right cardinality, rather than, say, 2^aleph_0? Are you assuming the continuum hypothesis?

komponisto2 Apr 13, 2008, 4:09 AM
0 points
on: Where Philosophy Meets Science
suggest that, like ethics,

Link to SIAI blog here broken (404 Not Found).

komponisto2 Apr 6, 2008, 9:46 PM
3 points
on: The Generalized Anti-Zombie Principle
Michael Vassar:

The classical disproof of positivism is that it is self-contradictory. “Only the empirical can be true”, but that statement is not empirical.

I have always been mystified at how this glib dismissal has been taken as some kind of definitive refutation. To the contrary, it should be perfectly obvious that a meta-statement like () a statement is nonsense unless it describes an empirically observable phenomenon is not meant to be self-referential. What () does is to lay down a rule of discourse (not meta-discourse). Its purpose is to banish invisible dragons from the discussion.

You cannot appeal to the “legitimacy” of sentences like (*) in order to argue on behalf of your favorite invisible dragon. But this is exactly what is going on in exchanges like the following: A: “The concept of consciousness is meaningless because it has no empirical consequences.” B” “Silly amateur! Don’t you know that logical positivism has been refuted?”

komponisto2 Mar 31, 2008, 1:19 AM
1 point
on: Hand vs. Fingers
We can play “maybe”s all day long, but it doesn’t seem very helpful unless you can actually show that a mistake has been made.

Richard, the burden of proof is on you. You are in effect making the claim that a certain problem (“reduce consciousness to physics”) is impossible to solve. But why should we believe that? When all is said and done, you appear to be saying, “because it seems that way”. This is where Eliezer comes in.

komponisto2 Mar 29, 2008, 6:31 PM
0 points
on: Initiation Ceremony
In a traditional school environment, grades are the de facto student motivator

Motivator of what? The point is that whatever behavior it is that grades in school serve to motivate, it is not learning per se. Indeed, grades are more often than not a motivator against learning. To quote Eliezer (emphasis added):

“[S]tudents aren’t allowed to be confused; if they started saying, ‘Wait, do I really understand this? Maybe I’d better spend a few days looking up related papers, or consult another textbook,’ they’d fail all the courses they took that quarter.” (Two More Things to Unlearn from School.)

This is where the autodidact has a distinct advantage, because he/she doesn’t have to “worry” about “failing courses”, and thus doesn’t have to actively resist this kind of pressure against learning. By contrast, if a traditional school student is interested in learning, he/she has to be willing to neglect “assigned tasks” and incur the expected social consequences. It is in this sense that I claim it is (or at least can be) easier to stay motivated outside the school system than inside.

komponisto2 Mar 29, 2008, 4:28 PM
7 points
on: Initiation Ceremony
Any commited autodidacts want to share how their autodidactism makes them feel compared to traditional schooled learners? I’m beginning to suspect that maybe it takes a certain element of belief in the superiority of one’s methods to make autodidactism work

First, “traditional school learning” is itself inherently problematic. Consequently, “belief in the superiority of one’s [own] methods” is not hard to acquire.

Second, all actual learning is necessarily autodidactic anyway, because where it takes place is in your own mind, not in your interactions with others.

So yes, in order to learn, you have to be “arrogant” enough to believe that you actually can. That is, you have to have the confidence that you personally can fully appreciate the insights of “authority figures” like Issac Newton. Unfortunately, social pressures seem to discourage such “presumption” in favor of deference to authority.
What links here?
- Z_M_Davis's comment on The Dangers of Partial Knowledge of the Way: Failing in School by Gordon Seidoh Worley (Jul 6, 2009, 4:46 PM; 7 points)

komponisto2 Mar 8, 2008, 9:13 PM
7 points
on: Wrong Questions
“to what extent was underconsumption the cause of the Great Depression?”

Tabooing the word “cause”, one finds that this question is a disguise for something like “Given the economic data of the period immediately preceding the Great Depression, can we prevent an economic collapse by making sure we don’t underconsume?”

komponisto2 Feb 8, 2008, 1:33 AM
3 points
0
on: The Cluster Structure of Thingspace
But if a definition works well enough in practice to point out the intended empirical cluster, objecting to it may justly be called “nitpicking”.

You should probably put in a disclaimer excepting mathematics from this—assuming that you agree it should be excepted. (That is, assuming you agree that “Aristotelian” precision—what mathematicians call “rigor”—is appropriate in mathematics.)
What links here?
- jsalvatier's comment on The “I Already Get It” Slide by jsalvatier (Feb 10, 2017, 9:21 PM; 0 points)

komponisto2 Jan 28, 2008, 11:10 PM
4 points
on: The “Intuitions” Behind “Utilitarianism”
Larry D’anna:

You are dealing with an asymptote. There is a limit to how bad momentary eye-irritation can be, no matter how many people it happens to.

By positing that dust-speck irritation aggregates non-linearly with respect to number of persons, and thereby precisely exemplifying the scope-insensitivity that Eliezer is talking about, you are not making an argument against his thesis; instead, you are merely providing an example of what he’s warning against.

You are in effect saying that as the number of persons increases, the marginal badness of the suffering of each new victim decreases. But why is it more of an offense to put a speck in the eye of Person #1 than Person #6873?

komponisto2 Jan 23, 2008, 7:33 AM
2 points
on: Circular Altruism
Lee:

Models are supposed to hew to the facts. Your model diverges from the facts of human moral judgments, and you respond by exhorting us to live up to your model.

Be careful not to confuse “is” and “ought”. Eliezer is not proposing an empirical model of human psychology (“is”); what he is proposing is a normative theory (“ought”), according to which human intuitive judgements may turn out to be wrong.

If what you want is an empirical theory that accurately predicts the judgements people will make, see denis bider’s comment of January 22, 2008 at 06:49 PM.

komponisto2 Jan 10, 2008, 11:14 PM
2 points
on: 0 And 1 Are Not Probabilities
Eliezer:

I should mention that I’m also an infinite set atheist.

You’ve mentioned this before, and I have always wondered: what does this mean? Does it mean that you don’t believe there are any infinite sets? If so, then you have to believe that a mathematician who claims the contrary (and gives the standard proof) is making a mistake somewhere. What is it?

Frankly, even if you actually are a finitist (which I find hard to imagine), it doesn’t seem relevant to this disucssion: every argument you have presented could equally well have been given by someone who accepts standard mathematics, including the existence of infinite sets.