Oh actually you’re right. I didn’t interpret the op correctly. I thought it was just some weird extension of Knuth’s up arrow notation but now I see what’s going on.
In that sense, (2) isn’t a real number, as infinity isn’t a real number, it’s an extended real number.
And I think you’re right again, ((2)) isn’t well defined I don’t think.
Fair enough, I was simply trying to appeal to what is probably his most familiar intuition regarding the real number line.
Most people that are going to have confusion about a big number being close to infinity probably aren’t going to know what a metric space is.
The laws of nature that physicists actually use, however, are written in the language of Calculus. And understanding the laws in Calculus form is ridiculously useful.
Hell, everyone shouldn’t just have to take Calculus, they should have to take Differential Equations as well. You cannot understand reality without calculus knowledge, or without an understanding of differential equations.
http://en.wikipedia.org/wiki/Iron_peak has a pretty straightforward explanation for why the usual channels for element production stop at Iron.
Probably the most damning criticism you’ll find, curi, is that fallibilism isn’t useful to the Bayesian.
The fundamental disagreement here is somewhere in the following statement:
“There exist true things, and we have a means of determining how likely it is for any given statement to be true. Furthermore, a statement that has a high likelihood of being true should be believed over a similar statement with a lower likelihood of being true.”
I suspect your disagreement is in one of several places.
1) You disagree that there even exist epistemically “true” facts. 2) That we can determine how likely something is to be true. or 3) That likelihood of being true (as defined by us) is reason to believe the truth of something.
I can actually flesh out your objections to all of these things.
For 1, you could probably successfully argue that we aren’t capable of determining if we’ve ever actually arrived at a true epistemic statement because real certainty doesn’t exist, thus the existence or nonexistence of true epistemic statements is on the same epistemological footing as the existence of God—i.e. shaky to the point of not concerning oneself with them all together.
2 basically ties in with the above directly.
3 is a whole ’nother ball game, and I don’t think it’s really been broached yet by anyone, but it’s certainly a valid point of contention. I’ll leave it out unless you’d like to pursue it.
The Bayesian counter to all of these is simply, “That doesn’t really do anything for me.”
Declaring we have certainty, and quantifying it as best we can is incredibly useful. I can pick up an apple and let go. It will fall to the ground. I have an incredibly huge amount of certainty in my ability to repeat that experiment.
That I cannot foresee the philosophical paradigm that will uproot my hypothesis that dropped apples fall to the ground is not a very good reason to reject my relative certainty in the soundness of my hypothesis. Such a apples-aren’t-falling-when-dropped paradigm would literally (and necessarily) uproot everything else we know about the world.
Basically, what I’m trying to say is that all you’re ever going to get out of a Bayesian is, “No, I disagree. I think we can have certainty.” And the only way you could disprove conclusions made by Bayesians are through means the Bayesian would have already seen, and thus the Bayesian would have already rejected said conclusion.
You’ve already outlined that the fallibilist will just keep tweaking explanations until an explanation with no criticism is reached. I think you might find Bayesianism more palatable if you just pretend that we aren’t trying to find certainty, just say we’re trying to minimize criticism.
This probably hasn’t been a very satisfying answer. I certainly agree it’s useful to have an understanding of the biases to our certainties. I also think Bayesianism happens to build that into itself quite well. Personally, I don’t think there’s anything I’m absolutely certain about, because to claim so would be silly.
Where don’t you like it? I don’t think anyone actually argues for your first definition, because, like I said, it’s silly. I think curi’s point is that fallibilism is predicated on your second definition not (ever?) being a valid claim.
My point is that the things we are “certain” about (as per your second definition) probably coincide almost exactly with “statements without criticism” as per curi’s definition(s).
When applying to a college, if your academic transcript contains, say, twelve passed AP Exams, it will look twice as impressive as that of a high-school valedictorian.
Minor nitpick here. At least at my High School, it was impossible to even be valedictorian without taking more AP classes than everyone else.
Otherwise I agree with the general message. When I was doing my college shopping several years ago, I actually found that colleges were far more impressed if you were taking straight up college courses during High School than if you were taking AP classes. Of course, not everyone lives close enough to a University or Community College to pull it off, but it’s definitely something worth doing.
We’re fundamentally incapable of making statements about reality without starting on some sort of arbitrary foundation.
And I think describing it as “selectively ignoring” is doing it an injustice. We’re deductively excluding, and it there were some evidence to appear that would contradict that exclusion, those theories would no longer be excluded.
I’m actually have trouble finding a situation in which a fallibilist would accept/reject a proposition, and a Bayesian would do the opposite of the fallibilist. And I don’t mean epistemological disagreements, I mean disagreements of the form “Theory Blah is not false.”
I think in asking “would I self modify to avoid/encourage x”, you’re really just asking “Do I want good things to remain being good?” Of course you do, they’re good things.
This doesn’t at all determine what it is about things that may qualify them as good, just that you desire for things you ascribe with the property of “goodness” to keep that property. But I don’t think you’re trying to do this—correct me if I’m wrong though.
So, if you never intended to identify moral facts in the first place, and you just wanted some heuristic for determining if something is “morally significant to some individual”, you’re fine as far as I can tell. Tautologically even.
I disagree. It’s actually remarkably easy to create for all intents and purposes (barring the resolution of several outstanding problems in cryptography and computer science) impossible-to-break cryptography schemes if you know anything about RSA, lattice methods, etc.
Unless “a lot of time” means the age of the universe (precluding functional quantum computers before then).
The joke response would be to let ‘n’ represent that number.
n+1.
Somewhat related (not really at all, but your post reminded me of it) is Scott Aaronson’s digression on “who can name the biggest number”: http://www.scottaaronson.com/writings/bignumbers.html
Reality is far too interesting to allow such a limit on itself.
Well, I suppose we have to be more clear about what it even means to feel pain “as the simulation is being conducted”. If we accept that pain does occur, then that pain occurring is independent of the way we carry out the simulation. So let’s pretend our turing machine tape is a bunch of sea shells on a sea shore, and I’ve devised a way to carry out an isomorphic set of operations on the sea shells that performs the computation.
We could select the obvious point in time that corresponds to the instant before the “simulation” actually starts feeling pain, as represented by pain receptors firing, as encoded in some configuration of the sea shells. Does the “pain” begin in the next timestep? Is it only a property of completed timesteps (as there is some time it takes me to actually perform the operations on the sea shells)? What if I just stop, mid timestep?
To me, it seems much more reasonable that the “pain” is just a property of the ‘time’ evolution of the system. It seems strange to imbue the act of carrying out the computation with ‘causing pain’.
To actually know what occurred, we must carry out the computation.
Does the act of carrying out the computation change anything about the ‘time’ evolution of the system? Calling it a ‘time’ evolution perhaps puts the wrong emphasis on what I think is important, here. Like another poster has said, “2+2=4” is a good analogue. We can devise a computation that results in the answer to the query “What is 2+2?”, but I don’t think one can argue that actually performing that computation can be equivocated with the result.
When I say ‘time’ evolution, I really mean the thing floating in idea space that is the decideable answer (in a formal sense) to the question “What is the set of subsequent ‘time’ steps of this initial configuration?”
I think the process of computation matters only insofar as we do not know the result of any given computation before performing it.
So say I have performed the torture sim, and say that I have every configuration of the tape listed to a corresponding page of some really long book. Is the computation performed once again if I flip through the book? Or must I physically carry out the computation using some medium (e.g. sea shells)?
To me, it seems that the only difference between the universe before I ran the simulation and the universe after is that I know what occurred in that simulation. The simulation itself, and all of its content (that is, the sequence of states following from the initial state), was already a fact of the universe before I knew about it.
I don’t think it “creates torture” any more than saying 2+2=4 “creates” the number 4--or, at least that’s what I think a computationalist is committed to.
If I have some enumeration of the torture sim in hand, but I haven’t performed the computation myself, I have no way of trusting that this enumeration actually corresponds to the torture sim without “checking” the computation. If one thinks that now performing the torture sim on a Turing machine is equivalent to torture, one must also be committed to thinking that checking the validity of the enumeration one already has is equivalent to torture.
But this line of thought seems to imply that the reality of the torture is entirely determined by our state of knowledge about any given step of the turing machine. Which strikes me as absurd. What if one person has checked the computation, and one hasn’t, etc. It’s essentially the same position that ‘4’ doesn’t exist unless we compute it somehow (which, admittedly, isn’t a new idea).
Is there a particular reason that this forum doesn’t use the “goes to top of page when posted on” mechanic that’s used in many other popular forums?
I understand we’re using the reddit-like architecture, but it seems (not that I have any sort of data to back it up) like it stifles discussion to let things drop off the front page as new things appear.
Or perhaps I’m just far more used to bump style forums.
“What is the probability there is microbial-like life (other than from earth) in our solar system?”
I’m having difficulty giving this a good estimate, myself, actually.
Why does IIA allow you to rescale the two utility axes using different scaling functions?
Two interesting observations: The most recent utterances of the words “temporal pressure” (similar to the titles of these last two chapters) was in chapter 86 when discussing the Halls of Prophecy:
“”The Hall of Prophecy,” Minerva whispered. She’d read about that place, said to be a great room of shelves filled with glowing orbs, one after another appearing over the years. Merlin himself had wrought it, it was said; the greatest wizard’s final slap to the face of Fate. Not all prophecies conduced to the good; and Merlin had wished for at least those spoken of in prophecy, to know what had been spoken of them. That was the respect Merlin had given to their free will, that Destiny might not control them from the outside, unwitting. Those mentioned within a prophecy would have an glowing orb float to their hand, and then hear the prophet’s true voice speaking. Others who tried to touch an orb, it was said, would be driven mad—or possibly just have their heads explode, the legends were unclear on this point. Whatever Merlin’s original intention, the Unspeakables hadn’t let anyone enter in centuries, so far as she’d heard. Works of the Ancient Wizards had stated that later Unspeakables had discovered that tipping off the subjects of prophecies could interfere with seers releasing whatever temporal pressures they released; and so the heirs of Merlin had sealed his Hall. It did occur to Minerva to wonder (now that she’d spent a few months around Mr. Potter) how anyone could possibly know that; but she also knew better than to ask Albus, in case Albus tried to tell her. Minerva firmly believed that you only ought to worry about Time if you were a clock.”
Next, near the end of chapter 89, Harry “turns away from Dumbledore” twice, almost as if there’s a kink in time after explicitly mentioning “pressure”:
Harry opened his mouth to scream out all his fury, and then closed it again. There wasn’t any point in screaming, it wouldn’t accomplish anything. The unbearable pressure rising inside him couldn’t be let out that way.
’Harry turned away from Dumbledore and looked down at where the remains of Hermione Granger were lying in a pool of blood … another part of him already knew that this event was real, part of the same flawed world that included Azkaban and the Wizengamot chamber and
No
With a fracturing feeling, as though time was still torn to pieces around him, Harry turned away from Dumbledore and looked down at the remains of Hermione Granger lying in a pool of blood with two tourniquets tied around her thigh-stumps, and decided
No.
I do not accept this.”
Hypothesis: Something very strange just happened to Time around Harry, possibly involving a time turner, and things are definitely not as they seem.
It’s almost as if someone is trying to manipulate Harry’s reaction to these events, like when H&C tricks Hermione repeatedly. Fred and George are unconscious at this point, too. It goes without saying that Harry is probably an unreliable narrator.
Edit Edit: It seems I double misinterpreted the parent. I think we’ve agreed that (2) is finite, so if I ramble about it being infinite below, ignore me.
Being “close to infinity” doesn’t really make sense in standard real analysis, if you’re talking about any given finite number, because you can always produce another number arbitrarily larger than whatever you wanted to show was “really close to infinity”.
Edit: So if you’re asking “what’s bigger, the sum of the first billion twin primes, or (2)”, this question doesn’t make sense because (2) isn’t finite, but that sum is.
What’s even more interesting, though, is that you can meaningfully compare different sized infinities. Look into Cardinality.