This reminds me that back in the day, AT&T Security was supposedly a bunch of bmf's.
In about 1980, when I was in high school, I discovered an unused phone extension line in my bedroom closet and started experimenting with it. I quickly figured out the basics and built a little homemade phone. Later, I got the idea of using a thirty-foot spool of wire and a couple of alligator clips to quickly tap into someone's line outside of their house to steal long distance phone calls from the safety of my car. This is really trivial stuff, I know, but I thought I was clever.
But not clever enough. I called my cousin long-distance by connecting to what turned out to be the phone line of a little old lady who'd never made a long-distance phone call in her life. Her church was helping her pay her bills and noticed the phone call immediately. They called AT&T, and AT&T merely checked to see who else in my small New Mexico town had ever called that California number. Then they called my mom.
Once AT&T security found out that I hadn't actually done anything sophisticated or interesting, they just made my parents pay for the call and dropped the matter.
None of this, of course, shows that AT&T security was especially astute. But a few years later I was working as a radio disc-jockey, and I told this story to the station's chief broadcast engineer. He told me that he had worked for AT&T and that AT&T Security were among the best private security experts in the world. In his words: "Don't fuck with AT&T Security". That made an impression on me.
Later on, when I first read about the phone phreaking era, I felt lucky that a) I wasn't ingenious enough to get myself in any real trouble, and b) I didn't know anyone who was.
"Heisenberg unknowableness principle doesn't have the same ring to it though, does it?"
No, but it would perhaps be a bit more accurate. Many people believe, incorrectly, the HUP to be a problem of measurement, rather than a problem of epistemology.
The word "Apocalypse" comes from Greek meaning "unveiling".
Yes, but not quite exactly that. [1]
Apocalypse is the English transliteration of hapokalu[psi]is[3]. hApo[3] is a very common preposition meaning something like away from or off; and kalu[psi]is[3] is formed from the root, kaluptw[6][3], meaning cover, or veil. So, "unveiling" is acceptable. The only problem is that "unveil" is a term not used very generically in English, it has strong literal connotations. That makes it good for poetic imagery; but the original Koine Greek was probably not intended to specifically evoke that particular image.
Herodotus used hapokaluptw as uncover; Plato as disclose or reveal; and Plutarch as reveal one's whole mind, which I think is closer to what we're looking for here.
As to your two points....
1. The "Revelation" was made to Jesus, not just to the world. Evidence is found in Mar 13:32:
The first is a bit strange. It's trivially true, as Revelations says as much in the first verse. As to Mark 13:32, it only indicates that Christ was unaware of the exact date, not the events. Indeed, Christ seems to be aware of some of the events in Mark 13:32. Also, of course, it's contestable that what Christ is describing in Mark 13 is the same thing as what is being described in Revelations. At any rate, yes, the revelation is to Christ, who then passes it along to John; and so in that sense it's quite correct to say that the revelation is to John. Your insistence that the revelations is "really" to Christ is either niggling or important. If it's important, then it's only important in your contestable interpretation of Revelations and the NT as a whole.
2. "shortly come to pass" is a poor translation, it would be better to say "come to pass rapidly" [in a short period of time].
Similarly, your second point is also quite dubious, as hev taxei[3] is in most cases in the NT unambiguously translated as "soon" rather than "fast". Here are all the incidents of hev taxei in the New Testament. It's also quite a stretch to claim that the writer of this book is keen on specifically mentioning that the events described take place quickly right there in the first sentence. It's far more sensible to interpret this as "revelation of events to come soon"; and, furthermore, the narrative that follows describes a sequence of events that are not particularly brief. In other words, pushing the "rapidly" translation is a function of an interpretation which attempts to reconcile this with the historical fact that these events did not seem to happen "soon". Which brings me to another point.
As someone who studied Homeric and Attic Greek before Koine Greek[5], I have a greater familiarity with Greek than the average layperson or non-academic who has studied the NT in Greek as a part of their bible studies. My sister, for example, is an evangelical minister and missionary; and although her understanding of Greek has improved over time, it is still enormously tainted by doctrinal teachings of Koine Greek biased towards a particular interpretation of the NT. As a result, I am very skeptical of many fundamentalist's "translations" of the NT.
Finally, I am not at all offended by your post or the fact that you're Christian - as I noted, Larry Wall is a devout Christian and your post is marginally on-topic. But please don't assume that the sets of people that are offended by Christians and atheists are mutually exhaustive. They are not. I am an atheist. I also know a great deal more about Christianity than most Christians I've met. Stereotypes are not helpful.
[2] There used to be a second footnote. Now it's gone. It involved how to display the Greek characters I originally tried to use that I eventually discovered that/.'s preview would accept, but that the submit would not. Sigh. (Addendum: it was a different problem.)
[3] These are Latin character transliteration. We don't have a psi. Upsilon typically changes to "y" in English rather than "u" because the vowel "y" is actually closer to the presumed pronunciation. Kappa often becomes "c". Nu ["v" in my transliterations] becomes the English "n". The "w" is an omega, a long "o".)
[4] Wish I could force Unicode in this post, somehow. Ironic since one of the biggest justifications for Wall's mods to regex is that 8-bit character classes are now archaic.
[5] Koine Greek is much simpler than Homeric or Attic Greek. The New Testament was actually written in Koine Greek because it was the common language of the region. The original words, however, were probably mostly spoken in Hebrew and Aramaic. I should make it clear that I studied Greek more than ten years ago and sadly now all of my facility with it has completely evaporated. This post was constructed with frequent consultations with my Liddell & Scott. YMMV. It's also worth noting that Wall is a trained linguist as well as a studious Christian, so he means exactly what he's saying when he uses the word apocalypse.
"I guess the Christians are going to be waving banners over the naming of this one. Are they going to sue for false advertising when they realize this isn't the Apocalypse the Bible spoke of?"
Wall is a very devout Christian. Visit his website.
"The fundamental problem, as I've come to see it, with this area is the lack of a formal model that describes the *FRAMEWORK* of knowledge representation, the operations and transformations that can be applied to that knowledge, and the mathematics to back it all up."
I have't looked at your citations, but I want to make it clear that whatever I say, I am not trying to disparage the insight or utility of this work. But I think it's not going to achieve the results that you'd like.
It sounds to me like this work is trying to recapitulate epistomoligical philosophy and, essentially, mathematics itself. Math itself is the mathematics of knowledge representation and manipulation. This attempt for a fully descriptive top-down conceptual model makes many assumptions about the nature of "knowledge" and "thought" that are extremely suspect.
Let me ask a question: what is "life"? Sure, we can make some distinctions between inorganic and organic chemistry, and/or processes; but the truth is that any scientific definition of life is, upon examination, only partial and not really satisfying relative to how we perceive "life" to be a platonic ideal, a thing, something that can be well defined and understood since we think about it as if it could be. But, I think, most scientists these days have abandoned the idea of this platonic "life". Would you try to look for a complete mathematical structure which can fully describe "life"? Isn't that what biology, chemistry, and physics is doing?
Read my other post on "appropriate levels of description" if you haven't already. I'm probably overestimating how ambitious of an epistomology you really want. And I would agree that at some level of description, there's a theory and mathematical model that adequately describes the behavior of a system whose context is consciousness. But I don't think that we're in the position to discover these mathemtics. We no more understand the workings or nature of consciousness than the Greeks did the natural world. Western science only began to make progress in understanding the natural world when it scaled back its ambitions to almost nothing -- namely, to merely observe the natural world rather than formulate teleogical theories about how the natural world must work based upon assumed first principles. Trying to formulate theories of knowledge representation (in this context) and consciouness from first principles, at this point, is like reasoning about human anatomy from first principles like Aristotle did. It's both fairly hubristic and absurdly detached from experience.
For this reason, things like neural networks and the like are valid areas of research because they take an observation about some tiny portion of knowledge representation and attempt to abstract it. It's useful and explanatory only in this very small, limited sense. But that's something.
"So, to the extent our knowledge of physics is incomplete, I submit that so too will our understanding of conciousness and intelligence be similarly incomplete, as an upper limit on potential understanding."
I believe this is true of everything.
Here we get to an idea that I articulate as often as possible. I don't want to go into it deeply now; but I'll give you my current distilled formulation:
A "complete" description of anything is impossible. Instead, there are an innumerable number of "partial" descriptions. An individual "partial" description is the description most appropriate for some given purpose.
Humans think teleologically and they think idealistically. These two things are deeply related. Teleological thinking is thinking that is goal-oriented. We ask "Why did he do that? What is that thing for?" Idealistic thinking is thinking that abstracts our experiece of reality into idealistic, self-contained, irreducible "things". These things are like Plato's "Forms". Plato's Forms are sort of the atomic particles of his abstract universe.
Because of this, the way we try to understand the universe is from a combined top-down (teleological) and bottom-up (idealistic) analysis that, when complete, is presumed to create "understanding". This is natural; and, once we started doing this rigorously (and lightened up on the teleology), we started having great success. But this success has misled us. The culmination of this was the reductionist, determinist conceit of the nineteenth century that the universe could be fully explained in a deductive fashion, at least in principle.
But we know that this is pretty much impossible in practice, and we now know that it's not possible in principle.
The property that we are calling "intelligence" is a set of behaviors from which we intuit a gestalt. There is an appropriate level of description of a system at which this behavior resides. The other levels are superfluous for this purpose.
Your desire to "fully" understand consciousness by "fully" understanding the brain and, if necessary, physics and the state of the entire universe is this deterministic, reductionist shiboleth. It can't be done, probably not even in principle.
We can't fully solve the four-body problem in "simple" Newtonian physics. But we manage successful interplanetary probes amazingly well. This is because a sufficiently detailed approximation, aimed at accounting for the behaviors that are relevant, is both achievable and sufficient. This is true of everything.
We're not going to ever understand consciousness in the "complete" sense that we might like. But we can't do that with anything, and we seem to be doing quite well.
This kind of stuff drives me crazy. And I already have a mood disorder.
It occurs to me that people take faux-AI stuff like this seriously because, actually, they don't take AI seriously at all. This magazine writer seems to think that the sufficient characteristic of "strong" AI is some form of learning. Presumably, then, "AI" without learning is "weak" AI? Where, exactly, is the "I" part of the whole AI thing?
Don't get me wrong. I'm not an essentialist. Searle and other anti-AI people are basically asserting the tautology that something's not intelligent because it's not intelligent. And they get to decide what it means to be intelligent. But the main idea of Turing with his test was that if it is indistinguishable from intelligence, it's intelligence.
The problem here is that ALICE is easily determined to be non-intelligent by the average person. ALICE can only pass for an intelligence under conditions so severely constrained that what ALICE is emulating is merely a narrow and relatively trivial part of intelligent behavior. Humans cry out when they are injured -- I don't see anyone claiming that an animal, a rabbit for example, that screams when it's injured is intelligent.
Nobody in their right mind could think that anything we've seen even significantly approaches intelligence.
Wallace is quoted as saying that he went into the field favoring "robot minimalism", and the article writer explains this as the idea that complex behavior can arise from simple instructions. (Oops, someone better contact Stephen Wolfram and tell him he didn't invent this idea.) Wallace is clearly influenced by some important ideas of this nature that came out of, I believe, the MIT robotics lab. (Not the AI lab -- Minsky is hostile to this sort of thing, he's really is an advocate of "strong" AI; and what that really means is something like an explicitly designed AI predicated upon an understanding of consciousness that allows for a top-down description of it. I think that's, er, wrong-headed.)
Lots of folks think that this idea of complexity is the correct way to approach AI. But a really, really big problem is that I don't think that a 30,000 explicitly coded set of responses can really be described as "minimalist". Effectively, Wallace's approach has a seperate instruction for every behavior -- something quite contrary to the minimalism he seems to advocate.
For the sake of argument, let's assume that the central idea of the Turing Test is correct -- a fake indistinguishable from the original is the same kind of thing as the original. I happen to actually believe that assumption. But Wallace is also assuming that a canned set of stock responses is reasonably possible to achieve such a thing. But it clearly isn't.
A little bit of thought and math will reveal that the total number of correctly-formed English sentences is a very, very, very large number. It's effectively infinite for practical purposes. But Wallace claims that almost all of what we actually say in practice is such a tiny subset of that, that compiling a list of them is possible. So? Almost everything interesting lies in the less frequently uttered sentences; and almost everything that makes intelligence what it is is in the connections between all these sentences. Something that really could pass for intelligence would have to be able to reach, at the very least, even the least often uttered sentences; and, frankly, it'd need to be able to reach heretofore unuttered sentences, as well. More to the point, it would have to be able to do this in the same manner that a human does -- a "train of thought" would have to be apparent to an observer. Given this, we already have that practically infinite number of possible, coherent English sentences; and if you then require that sequences of sentences be constrained by an appearance of intelligence, then you've taken an enormous, practically infinite number and increased it many orders of magnitude.
I submit that such a list of possible query/response sets would be larger than the number of atoms in the galaxy (or the universe! it's not hard to get to these huge numbers quickly), or some such ridiculously large magnitude. It's just not possible to actually do it this way. If you managed it, I'd actually accept a judgment of "intelligence", since I think that the list itself would necessarily encapsulate "intelligence", though in a very brute force fashion. But so what? As in the case of Searle's Chinese Room, all the "intelligence" would implicitly be contained in the list. But this list would need to be, in physical terms, impossible large -- just to do something that the nicely (relatively) compact human brain does quite well.
So, hey, if someone wants to pursue this type of project, I can't say that as a matter of pure theory, it's "not possible". I can say that it's probably not physically possible.
The sense in which Wallace's ALICE chatbot is like trying to describe complexity arising from simplicitly is the same sense in which the Greeks (and others) tried to describe all of nature as the products of Earth, Wind, Fire, and Air. The "simple" things he's starting with
aren't really simple; they're not "atomic".
Another example from AI is the problem of computer vision -- people once thought it'd be trivial for a computer to recognize basic shapes from a camera image. Boy, were they wrong.
We'll "solve" the problem of AI. Not like this. And nothing we've seen so far, anywhere, is anything even remotely like legitimate AI.
"If they continue to do this, surely they'll be blowing big holes in any future court cases."
Hey, fantastic point. You're a smart guy. But I fear that some of the folks behind this might also be pretty sharp -- ask yourself: why are they looping portions of the real songs? Perhaps because that's enough to still be protected by copyright and be accurately referenced by the title. They could have just used a warning message, noise, what have you. This way, they may have anticipated your argument....one I don't think was wishful thinking. It would probably be one of the first defenses in a Napster-like court case.
I'm encouraged by the evidence of the posts in this thread that many slashdotters are taking the anti-piracy position on this matter. Communities such as this one are fighting the RIAA et al tooth-and-nail not because we are pirates, but because their efforts to combat pirates are extremely hostile to law-abiding consumers. For this reason, we're very suspicious of their protestations that all they're doing is trying to fight piracy.
Oh, okay. Although I think your judgment that Asimov was the connecting context something of a stretch. I would expect that if you were familiar with only one of the terms, it'd be limbo, not Valhalla. Although American Gods might explain it.
"And for those of you who don't recognize it, Valhalla and Limbo are both locations on the Spacer planet Inferno."
[rolling eyes]
It's things like this that give us technogeeks a bad name. I know you mean well, but you need to perform a sanity check before you jump in with little tidbits of enlightenment. For example, try doing a web search first. That way, you'll usually learn something you weren't aware that you didn't know, you may avoid making a fool of yourself as you did here, and you could even provide a nice link, too. Everyone wins, but especially you.
"I would like to comment that while Ruby is big in Japan, more western minded people might do better investigating Python, which gives all the perks of ruby, and more..."
Ruby was written specifically to answer perceived deficiencies in Python, not to mention Perl. It's more than a little misleading to represent Python as "Ruby with more". It's debatable whether Ruby is better than Python, but someone interested should do some research and decide for themselves. It's thought of very highly by a lot of people, and I suspect that the mention of it here by a US developer is an indication of its burgeoning popularity in the US.
It's also not a very good basis for understanding the theory behind calculus. The theoretical background for calculus came much later than Newton or Leibniz: think instead of Cauchy and Riemann. If you've studied only Newton and Leibniz, you've studied a small part of the history and origin of calculus---not its theory, not its practical use and not even its full history.
The concept of a limit was fuzzy at best and Leibniz worked with infinimetesimals. They didn't really understand "what the calculs really is." This was something that took a couple of centuries to figure out. The idea that these later technical refinements are not relevant to "a deep comprehension of the subject matter in general" is nonsense IMO.
Which is not to say that the larger point of studying generalities first is bad. But ultimately math is about details. Dismissing these details as being irrelevant to a deep understanding is misleading.
I don't recall saying not to read later writers like Riemann, Cauchy, or Weierstrass.
You are giving short shrift to Newton and Leibniz. The "incorrect" or "incomplete" ideas of the past are what informed the "correct" and "complete" ideas of the present. My personal experience has been that I always have a deeper, greater comprehension of the subject matter when approached in this manner; and the contemporary pedagogical method of a sort of "revelatory vision of the complete truth" is both false and misleading. There is more symmetry to mathematical and scientific discoveries in terms of precedence than you think -- they inform each other. If you only have the conventional revelatory, hubristic education, you'll think you know a subject better than you really do. As I said elsewhere, there's a reason that the very very best people go back and reexamine foundational and historical ideas, doing so relieves the myopia of the present.
I have said repeatedly that a historical or general approach to studying mathematics is not the equivalent of the type of study you and others prefer. I have repeatedly warned that this should be taken into account. I have never said that this more generalized comprehension is "better", I've said several times that the ideal is both. What you and others are reflexively attempting to say to me in reponse is that your method of study of mathematics is the only valid method, and the approach I am recommending is clearly inferior to yours. Given that I am not making an apparently chauvinistic argument about my own preference, and you are, I suspect that the bias lies with you.
Yes, you and others bristle at the connotations of my phrase "deeper comprehension", and I understand why you do. But you do so because you equate "deeper comprehension" with "greater comprehension", which is incorrect. I didn't mean it that way. Math, and science, is in the details, and a facility with those details is essential. But so is conceptual comprehension. No one can productively study these subjects without including both. Ideally, the study of both would be exhaustive. In practice, this is never true, and nowadays could never be true. Given limited resources, adjusting the relative mix of the two allows for adjusting for a desired outcome.
It's a fascinating thing to watch terms evolve. To pretty much repeat what I already wrote, I get almost breathless when I consider the increasing generalization that eventually contradicts the original usage's common sense coining. Obviously not just terms, but concepts.
It goes without saying just how badly the Greeks would go apeshit if they were presented with mathematics as it is now. And its arguable that Euclid with his strictness about not mixing different "kinds" in a ratio, the secret of incommensurability by the Pythagoreans, all kinds of stuff, that they had already glimpsed the abyss and refused to attempt to cross it. But their intellectual descendents did, and for damn good reasons. Furthermore, we could probably show them how it so often repeated that a more generalized mathematical concept that they would find abhorrent ended up being validated by physics. We'll put aside their antipathy for empiricism. (Although, is that the essential problem? If you're a mathematician, though, I think you can probably show lots of examples where, over and over, this sort of thing became compellingly necessary completely within the context of mathematics.)
At St. John's, a very interesting thing happens. Since it's a set curriculum of the "Great Books", it draws students with a fairly wide variety of intellectual predispositions. Of course, even if someone thinks of themselves as a literature person, they understand that they'll have to understand Lobachevsky, so they're not your typical student in any event. Even so, people that are very humanities oriented or even describe themselves as being mathephobes, will commonly become deeply enamored of math at the college, and leave to major in math elsewhere, or go to graduate school in math. I think that's a wonderful thing, and it indicates to me that math at the secondary school level is being mistaught. All the beauty is being leeched out of it.
Well, the "technique" you are learning is not necessarily the technique that a similar student at your school learned twenty years ago. Strangely, they were nevertheless able to understand mathematics.
The Ivy League schools are not exactly the same with regards to the approach to these matters of pedagogy. That's why, in fact, you are referring to your school as a "liberal arts" school, and you are not attending MIT. Yours may be a steller mathematics department. Certainly MIT's is. I doubt that they take the exact same approach to the subject, nor do they teach all the same "techniques".
Generally, the better the school, the more it will require that you learn deep concepts along with technique. But all scientific fields and mathematics, too, have become fragmented and specialized enough, that there simply isn't time to provide both deep comprehension and sufficient practical preperation and skill. This is just simply true, and I can't imagine that you would claim otherwise.
I suspect that you are reflexively responding to what you figured I said, rather than what I actually said. You'll notice that I never claimed that you could learn mathematics without doing mathematics, and it's also obvious that doing mathematics requires technical expertise. The question is what is useful for deep comprehension, and what is useful for the ability to accomplish another purpose? I imagine that a mathematics education today is still pretty deep in terms of general comprehension. Theoretical physics, as well. It's interesting that you chose that example, as most physicists are not theoreticians. My experience among grad students in the sciences, mostly physics, is that their comprehension of fundamentals is sometimes frighteningly uneven.
Another problem is that highly trained people like yourself (or who you will be) like to think that the only significant comprehension possible of their specialty is via their specific training. This is self-serving, and a simple function of human tendency toward chauvinism.
I am not in any way endorsing autodidactical cranks. (I am neutral with regards to autodidacticism. I just don't want to give those "I have a better theory that General Relativity!" nuts any encouragement.)
St. John's College of both Annapolis and Santa Fe. There's a required math class six of the eight semesters. Here's a general page for the reading list, unfortunately they don't provide a reading list of what appears in the math "tutorial".
I wonder if you're a johnnie like me. In any event, I heartily concur with your recomendations.
But, again, as I've said elsewhere, this type of comprehension does not prepare one sufficiently to do the type of work that people actually do now. But if you learn what they know, you'll understand the subject much better.
A footnote. As is the case with physics, I do think that eventually one needs to have at least a general understanding of what has happened in 20th century mathematics. To my mind, everything that came before is the (mostly) comfortable beginning to a story that takes a very surprising and discomfitting turn. I believe that there's something very important going on here; and, in fact, these 20th developments essentially reexamine foundational ideas and reinterpret them. Some might say undermining them. Which is pretty darn weird since these developments are the culmination of what they seem to repudiate. This is incredibly fascinating and provocative to me. So, not hitting the 20th might leave the student with a false idea of where we at present.
"What would you consider the "canon" of math to be?"
Well, with a minor in math you probably already have experience with most of the math dealt with by the authors of original texts I would recommend to you -- that's what "canon" would mean to me. So if you want something completely new, those probably wouldn't fit the bill.
On the other hand, working through those texts might give you much deeper insight into the math you already know. Is that what you want? Or do you just want to go further with what you know or to fill in the gaps? Again, do you want to do this for the pure intellectual satisfaction of comprehending something in general, or do you want to do specific stuff with what you learn?
For the life of me, I can't remember which one, but it was one of the preeminent mathematicians (but it could have been a physicist) of the last few generations, I think, that said he wanted to spend his twilight years in deep study of Newton's Principia Mathematica (obviously read at my school, re: calculus) Clearly, he thought there was something of value there to learn.
One thing about math is that some subfields can be pretty independent of all the others. I think you could start with basic set theory and go a long way without needing to (deeply) refer to other stuff. I keep wondering if I want to try to teach myself differential geometry (modern). That's because I want to understand general relativity, really. (You may notice that I agreed with the comment above that you can't understand many mathematical or physical ideas without doing the math.) I am not in a position to really evaluate how feasible this is. Yet.
You could probably find some good stuff on Amazon. Look for real mathematicians trying to write about a specific subfield in a more generalized manner. (I don't ever read popularizations of science or math by people who are not scientists or mathematicians. I think it's good advice.)
I think you're a little confused. You were the one who insulted my education. My education is useful, so is yours. For different things. I'm not saying one is better than the other. Yeah, I responded with something that has an insulting subtext, but that was only to counter yours. Again, I don't think my type of education is for everyone, nor do I think yours is, either. But it is absolutely wrong to think of eduation as being only vocationally oriented -- which is what you implied with your post.
In truth, almost all American higher eduation is now vocational education. Your attitude and comment demonstrate this. It's the only thing most people can imagine that an education could be for.
The problem is that since what they want is a vocational education, and what the economy needs is a vocational education, it's interesting that we're not doing a very good job providing one. This is because of the supposed continued commitment to a "liberal education" by most American undergraduate schools. The result is the worst of both worlds: watered down liberal arts classes that teach little and make the students resentful that they are required to take them; and too few vocationally relevant classes, often with a poor degree of contemporary technical relevancy. This is why there's been a junior/community college revolution going on in this country for about twenty years -- they're meeting the demand that the universities aren't.
Obviously, since I went to an extreme liberal arts school I believe in the ideal of a liberal education. But as a practical matter, vocational education is essential. Ideally, it'd probably make me happy if everyone did what I did, and then do a year or so of undergraduate preparatory work in a particular field, then continue on to a graduate school in that field. For the people that wouldn't have gotten an advanced degree, or don't want that much schooling, you could still do what I did but put vocational schooling and experience beginning in parallel like they do in Europe. But I don't really expect everyone to do what I did, and I'm certain it's not appropriate for everyone. What degree of a sort of liberal education is for "everyone"? Well, we started down this road before and where we're arrived is not satisfactory. I think I'd prefer to find a way to get as much as possible of this done in primary and secondary school, extending schooling to year-around and adding another year; then sending people on to vocational, liberal, or professional educations.
It's actually a pretty modern thing to think of "education" as being a vocational education. What you needed to know to work in a vocation, you learned in apprenticeship or some other such institution. America has a particular problem with all this, though, since we have a very egalitarian ideal that wants to give all citizens some sort of a liberal education, while our relentless practicality also demands that we teach people to do their jobs. The two things are in many ways disharmonious.
I am not saying that you can learn math without doing it. My liberal arts education specifically doesn't subsitute reading about something with actually learning and doing it.
But the math you should do is dependent upon what you want to do with it later. To take a trivial example supporting my point, I was really pissed off at the education I'd gotten previously when I worked my way through Book I of Euclid's Elements and came to the Pythogorean Theorem. Suddenly, I understood it in a much deeper way. Did it matter that much in regards to that algebra I had done earlier in high school? Nope, not really.
Or take irrational numbers. They are presented to students in the most prosaic fashion, and many students (not math majors or mathematicians, of course -- remember, I'm using rudimentary examples) would simply say "uh, they're numbers whose decimals go on forever? Oh, wait, they're numbers whose decimals go on forever without anything repeating?" That's literally true, and means nothing. When you stumble upon the incommensurability of the diagonal of a square to its side in the context of Euclidean geometry, such a thing is dumbfoundingly counter-intuitive.
This type of thing repeats itself as you work your way deeper into any discipline. The top people tend to better acquaint themselves with deep, fundamental ideas as necessary. It's hard to do truly original work without doing so. But today's scientists are not trained, really, for doing truly original work, and they shouldn't be. Those that want to and have the aptitude will achieve that deeper level of comprehension on their own. Everyone else will do their much more technical, incremental work. And that is, in fact, the overwhelming majority of the progress made in science and mathematics. The big stuff gets all the glory, but its the little stuff that accounts for most of the work and enables the big stuff to be discovered. This is why although I greatly personally prefer deep comprehension over facility with technique, I don't advocate that this is the proper pedagogical approach for all students.
The poster that asked the question needs to ask what he's looking for in his approach to mathematics. You know as well as I do that introductory calculus texts are more an attempt to manage to acquaint the student with calculus and then teach a variety of techniques that are likely to be of use in particular fields. If you're not working in those fields, if you're never going to use calculus either for technical purposes or as a working mathematician, you probably don't need most of those techniques. Much of this comes and goes as different technical approaches are fashionable. It just simply isn't the case that all the techniques that a student is taught in college calculus courses are essential to their understanding of the subject matter. That can't be true, as which techniques are taught change over time.
Obviously, there's a core facility with both concepts and technique that is necessary for any resonable level of comprehension. I was not disputing that. That's why, in fact, I went to a liberal arts college very unlike yours (which is every one other than mine), where actually doing the mathematical work, of say, Lobechevsky, is considered essential and where a gloss in a math survey course is rightly considered for the most part a waste of the liberal art student's time. You're right: you don't learn a subject like math by reading about it.
I use my education everyday. What you are talking about is a vocational education. You know, like shop class.
Yeah, "a lot" is two words. I conflate them to one quite often, since I think of it as a single word. I'm not the only one. It'll probably eventually appear in the OED. I'm a language pragmatist, not a proscriptivist.
"Do you want this for information's sake, or do you want to plan a career out of it?"
Yes, I second the importance of asking yourself this question.
I have an intensive classic liberal arts education. Calculus directly from Newton and Leibniz, for example. This is great for understanding what the calculus really is, but very poor for doing the kind of calculus that people do as a practical matter.
The thing to understand in science and, yes, even math today, is that these have become almost completely technical fields -- that is "technical" in the sense of "technique". To be functional at all working in any of these fields requires the acquisition of a great amount of particular knowledge and technique that is not at all about a deep comprehension of the subject matter in general. A lot of my fellow alums find this out the hard way if they continue on to graduate school in a science, even though they tend to be accepted to the best schools. They have a lot of catch-up to do about the nitty-gritty stuff. On the other hand, their deeper comprehension serves them well as students and working scientists not infrequently.
The point is that if you want to just really get into math because you want to know more about it, then you should not try to duplicate what someone does who is studying it for professional purposes. You should approach it from another angle; then, if you choose, supplement your general knowledge by beginning to acquire proficiency in the specific. You'll also have a better idea of what interests you before you go the distance by learning much of the minutae necessary to even have a decent comprehension of actual contemporay work done in these fields.
The people doing this stuff for a living (or are students until they discover that they can't find a job and do this stuff for a living) will snobbishly dismiss a liberal arts approach to these subjects as being a waste of time or as some sort of pretense of learning that's not really there. Ignore them. They can't see the forest for the trees, and they shouldn't. That's not their job. For you, it's probably more fun to first examine and think about the forest before you start getting intimate with the trees.
This is the real security threat for everyone, particularly anyone with sensitive data.
Viruses and worms have been mostly merely malicious. Same with cracking. And the malice involved is not very great. But what if people get serious about stealing data?
A few years ago I had an epiphany one night, and waltzed into a network security company the next day.
"Look", sez me, "Inbound connections and activity are, in the long run, not going to be the real threat. The real threat is trojaned applications that mine for data and somehow send it offsite. You need to be monitoring outbound activity for appropriateness. For example, eventually you're going to see corporate espionage where someone writes an attractive and actually useful little app, then social engineers a targeted person within an organization to download it and compromise security. This is just an example of the general problem."
They were actually pretty impressed, but the company's strategy was deliberately to avoid concerning itself with viruses or worms (more specifically, they wanted to stay only on the servers, monitoring network activity in a sophisticated manner). But it seemed to me that this was a natural extension of their product and technology. And they thought I was a pretty bright guy, but they didn't know what to do with me. Well, anyway. The irony is that they were only a year or so later bought by one of the big antivirus firms, mostly just to acquire their technology.
In this particular case, the BitchX irc app, it looks like an outside source injected some backdoor code into the application, and hacked the ftp server to distribute it in a selective manner, presumably to help lower the risk of detection. A lot of effort for not that great of a payoff, really. Here, as is often the case, it's mostly about proving how clever you are.
But we're starting to see rudimentary examples of what I was warning about with spyware and other apps that make outbound connections that are in some sense illicit. Firewalls monitoring outbound connections can only be so successful given that they're always going to let some through. I know that some of the client based firewalling/monitoring software looks at connections on a per application basis. That's a start.
Personally, my inclination is that we need a networking monitor that operates like a virus scanner -- on the client, in the background -- that accesses a secured database of allowed application to outbound connection mapping, with secured handling of exceptions or new applications referred to a security admin (ideally) or an admin. This way we don't have to use a brute-force approach that simply locks down all allowed applications and allowed outbound connections in a non-specific, usability-destroying way.
But whatever the solution, I have little doubt that this will be a growing problem which will make a transition from script-kiddie nuisance cracking to something much more sophisticated. Although I could be wrong.
Slashdot Mirror
In about 1980, when I was in high school, I discovered an unused phone extension line in my bedroom closet and started experimenting with it. I quickly figured out the basics and built a little homemade phone. Later, I got the idea of using a thirty-foot spool of wire and a couple of alligator clips to quickly tap into someone's line outside of their house to steal long distance phone calls from the safety of my car. This is really trivial stuff, I know, but I thought I was clever.
But not clever enough. I called my cousin long-distance by connecting to what turned out to be the phone line of a little old lady who'd never made a long-distance phone call in her life. Her church was helping her pay her bills and noticed the phone call immediately. They called AT&T, and AT&T merely checked to see who else in my small New Mexico town had ever called that California number. Then they called my mom.
Once AT&T security found out that I hadn't actually done anything sophisticated or interesting, they just made my parents pay for the call and dropped the matter.
None of this, of course, shows that AT&T security was especially astute. But a few years later I was working as a radio disc-jockey, and I told this story to the station's chief broadcast engineer. He told me that he had worked for AT&T and that AT&T Security were among the best private security experts in the world. In his words: "Don't fuck with AT&T Security". That made an impression on me.
Later on, when I first read about the phone phreaking era, I felt lucky that a) I wasn't ingenious enough to get myself in any real trouble, and b) I didn't know anyone who was.
Apocalypse is the English transliteration of hapokalu[psi]is [3] . hApo [3] is a very common preposition meaning something like away from or off; and kalu[psi]is [3] is formed from the root, kaluptw [6] [3] , meaning cover, or veil. So, "unveiling" is acceptable. The only problem is that "unveil" is a term not used very generically in English, it has strong literal connotations. That makes it good for poetic imagery; but the original Koine Greek was probably not intended to specifically evoke that particular image.
Herodotus used hapokaluptw as uncover; Plato as disclose or reveal; and Plutarch as reveal one's whole mind, which I think is closer to what we're looking for here.
As to your two points....
The first is a bit strange. It's trivially true, as Revelations says as much in the first verse. As to Mark 13:32, it only indicates that Christ was unaware of the exact date, not the events. Indeed, Christ seems to be aware of some of the events in Mark 13:32. Also, of course, it's contestable that what Christ is describing in Mark 13 is the same thing as what is being described in Revelations. At any rate, yes, the revelation is to Christ, who then passes it along to John; and so in that sense it's quite correct to say that the revelation is to John. Your insistence that the revelations is "really" to Christ is either niggling or important. If it's important, then it's only important in your contestable interpretation of Revelations and the NT as a whole. Similarly, your second point is also quite dubious, as hev taxei [3] is in most cases in the NT unambiguously translated as "soon" rather than "fast". Here are all the incidents of hev taxei in the New Testament. It's also quite a stretch to claim that the writer of this book is keen on specifically mentioning that the events described take place quickly right there in the first sentence. It's far more sensible to interpret this as "revelation of events to come soon"; and, furthermore, the narrative that follows describes a sequence of events that are not particularly brief. In other words, pushing the "rapidly" translation is a function of an interpretation which attempts to reconcile this with the historical fact that these events did not seem to happen "soon". Which brings me to another point.As someone who studied Homeric and Attic Greek before Koine Greek [5] , I have a greater familiarity with Greek than the average layperson or non-academic who has studied the NT in Greek as a part of their bible studies. My sister, for example, is an evangelical minister and missionary; and although her understanding of Greek has improved over time, it is still enormously tainted by doctrinal teachings of Koine Greek biased towards a particular interpretation of the NT. As a result, I am very skeptical of many fundamentalist's "translations" of the NT.
Finally, I am not at all offended by your post or the fact that you're Christian - as I noted, Larry Wall is a devout Christian and your post is marginally on-topic. But please don't assume that the sets of people that are offended by Christians and atheists are mutually exhaustive. They are not. I am an atheist. I also know a great deal more about Christianity than most Christians I've met. Stereotypes are not helpful.
[1] Here is the beginning of Revelations in Greek.
[2] There used to be a second footnote. Now it's gone. It involved how to display the Greek characters I originally tried to use that I eventually discovered that /.'s preview would accept, but that the submit would not. Sigh. (Addendum: it was a different problem.)
[3] These are Latin character transliteration. We don't have a psi. Upsilon typically changes to "y" in English rather than "u" because the vowel "y" is actually closer to the presumed pronunciation. Kappa often becomes "c". Nu ["v" in my transliterations] becomes the English "n". The "w" is an omega, a long "o".)
[4] Wish I could force Unicode in this post, somehow. Ironic since one of the biggest justifications for Wall's mods to regex is that 8-bit character classes are now archaic.
[5] Koine Greek is much simpler than Homeric or Attic Greek. The New Testament was actually written in Koine Greek because it was the common language of the region. The original words, however, were probably mostly spoken in Hebrew and Aramaic. I should make it clear that I studied Greek more than ten years ago and sadly now all of my facility with it has completely evaporated. This post was constructed with frequent consultations with my Liddell & Scott. YMMV. It's also worth noting that Wall is a trained linguist as well as a studious Christian, so he means exactly what he's saying when he uses the word apocalypse.
[6] Thus, Homer's "Calypso", the nymph who hid.
It sounds to me like this work is trying to recapitulate epistomoligical philosophy and, essentially, mathematics itself. Math itself is the mathematics of knowledge representation and manipulation. This attempt for a fully descriptive top-down conceptual model makes many assumptions about the nature of "knowledge" and "thought" that are extremely suspect.
Let me ask a question: what is "life"? Sure, we can make some distinctions between inorganic and organic chemistry, and/or processes; but the truth is that any scientific definition of life is, upon examination, only partial and not really satisfying relative to how we perceive "life" to be a platonic ideal, a thing, something that can be well defined and understood since we think about it as if it could be. But, I think, most scientists these days have abandoned the idea of this platonic "life". Would you try to look for a complete mathematical structure which can fully describe "life"? Isn't that what biology, chemistry, and physics is doing?
Read my other post on "appropriate levels of description" if you haven't already. I'm probably overestimating how ambitious of an epistomology you really want. And I would agree that at some level of description, there's a theory and mathematical model that adequately describes the behavior of a system whose context is consciousness. But I don't think that we're in the position to discover these mathemtics. We no more understand the workings or nature of consciousness than the Greeks did the natural world. Western science only began to make progress in understanding the natural world when it scaled back its ambitions to almost nothing -- namely, to merely observe the natural world rather than formulate teleogical theories about how the natural world must work based upon assumed first principles. Trying to formulate theories of knowledge representation (in this context) and consciouness from first principles, at this point, is like reasoning about human anatomy from first principles like Aristotle did. It's both fairly hubristic and absurdly detached from experience.
For this reason, things like neural networks and the like are valid areas of research because they take an observation about some tiny portion of knowledge representation and attempt to abstract it. It's useful and explanatory only in this very small, limited sense. But that's something.
Here we get to an idea that I articulate as often as possible. I don't want to go into it deeply now; but I'll give you my current distilled formulation:
A "complete" description of anything is impossible. Instead, there are an innumerable number of "partial" descriptions. An individual "partial" description is the description most appropriate for some given purpose.
Humans think teleologically and they think idealistically. These two things are deeply related. Teleological thinking is thinking that is goal-oriented. We ask "Why did he do that? What is that thing for?" Idealistic thinking is thinking that abstracts our experiece of reality into idealistic, self-contained, irreducible "things". These things are like Plato's "Forms". Plato's Forms are sort of the atomic particles of his abstract universe.
Because of this, the way we try to understand the universe is from a combined top-down (teleological) and bottom-up (idealistic) analysis that, when complete, is presumed to create "understanding". This is natural; and, once we started doing this rigorously (and lightened up on the teleology), we started having great success. But this success has misled us. The culmination of this was the reductionist, determinist conceit of the nineteenth century that the universe could be fully explained in a deductive fashion, at least in principle.
But we know that this is pretty much impossible in practice, and we now know that it's not possible in principle.
The property that we are calling "intelligence" is a set of behaviors from which we intuit a gestalt. There is an appropriate level of description of a system at which this behavior resides. The other levels are superfluous for this purpose.
Your desire to "fully" understand consciousness by "fully" understanding the brain and, if necessary, physics and the state of the entire universe is this deterministic, reductionist shiboleth. It can't be done, probably not even in principle.
We can't fully solve the four-body problem in "simple" Newtonian physics. But we manage successful interplanetary probes amazingly well. This is because a sufficiently detailed approximation, aimed at accounting for the behaviors that are relevant, is both achievable and sufficient. This is true of everything.
We're not going to ever understand consciousness in the "complete" sense that we might like. But we can't do that with anything, and we seem to be doing quite well.
It occurs to me that people take faux-AI stuff like this seriously because, actually, they don't take AI seriously at all. This magazine writer seems to think that the sufficient characteristic of "strong" AI is some form of learning. Presumably, then, "AI" without learning is "weak" AI? Where, exactly, is the "I" part of the whole AI thing?
Don't get me wrong. I'm not an essentialist. Searle and other anti-AI people are basically asserting the tautology that something's not intelligent because it's not intelligent. And they get to decide what it means to be intelligent. But the main idea of Turing with his test was that if it is indistinguishable from intelligence, it's intelligence.
The problem here is that ALICE is easily determined to be non-intelligent by the average person. ALICE can only pass for an intelligence under conditions so severely constrained that what ALICE is emulating is merely a narrow and relatively trivial part of intelligent behavior. Humans cry out when they are injured -- I don't see anyone claiming that an animal, a rabbit for example, that screams when it's injured is intelligent.
Nobody in their right mind could think that anything we've seen even significantly approaches intelligence.
Wallace is quoted as saying that he went into the field favoring "robot minimalism", and the article writer explains this as the idea that complex behavior can arise from simple instructions. (Oops, someone better contact Stephen Wolfram and tell him he didn't invent this idea.) Wallace is clearly influenced by some important ideas of this nature that came out of, I believe, the MIT robotics lab. (Not the AI lab -- Minsky is hostile to this sort of thing, he's really is an advocate of "strong" AI; and what that really means is something like an explicitly designed AI predicated upon an understanding of consciousness that allows for a top-down description of it. I think that's, er, wrong-headed.)
Lots of folks think that this idea of complexity is the correct way to approach AI. But a really, really big problem is that I don't think that a 30,000 explicitly coded set of responses can really be described as "minimalist". Effectively, Wallace's approach has a seperate instruction for every behavior -- something quite contrary to the minimalism he seems to advocate.
For the sake of argument, let's assume that the central idea of the Turing Test is correct -- a fake indistinguishable from the original is the same kind of thing as the original. I happen to actually believe that assumption. But Wallace is also assuming that a canned set of stock responses is reasonably possible to achieve such a thing. But it clearly isn't.
A little bit of thought and math will reveal that the total number of correctly-formed English sentences is a very, very, very large number. It's effectively infinite for practical purposes. But Wallace claims that almost all of what we actually say in practice is such a tiny subset of that, that compiling a list of them is possible. So? Almost everything interesting lies in the less frequently uttered sentences; and almost everything that makes intelligence what it is is in the connections between all these sentences. Something that really could pass for intelligence would have to be able to reach, at the very least, even the least often uttered sentences; and, frankly, it'd need to be able to reach heretofore unuttered sentences, as well. More to the point, it would have to be able to do this in the same manner that a human does -- a "train of thought" would have to be apparent to an observer. Given this, we already have that practically infinite number of possible, coherent English sentences; and if you then require that sequences of sentences be constrained by an appearance of intelligence, then you've taken an enormous, practically infinite number and increased it many orders of magnitude.
I submit that such a list of possible query/response sets would be larger than the number of atoms in the galaxy (or the universe! it's not hard to get to these huge numbers quickly), or some such ridiculously large magnitude. It's just not possible to actually do it this way. If you managed it, I'd actually accept a judgment of "intelligence", since I think that the list itself would necessarily encapsulate "intelligence", though in a very brute force fashion. But so what? As in the case of Searle's Chinese Room, all the "intelligence" would implicitly be contained in the list. But this list would need to be, in physical terms, impossible large -- just to do something that the nicely (relatively) compact human brain does quite well.
So, hey, if someone wants to pursue this type of project, I can't say that as a matter of pure theory, it's "not possible". I can say that it's probably not physically possible.
The sense in which Wallace's ALICE chatbot is like trying to describe complexity arising from simplicitly is the same sense in which the Greeks (and others) tried to describe all of nature as the products of Earth, Wind, Fire, and Air. The "simple" things he's starting with aren't really simple; they're not "atomic".
Another example from AI is the problem of computer vision -- people once thought it'd be trivial for a computer to recognize basic shapes from a camera image. Boy, were they wrong.
We'll "solve" the problem of AI. Not like this. And nothing we've seen so far, anywhere, is anything even remotely like legitimate AI.
I'm encouraged by the evidence of the posts in this thread that many slashdotters are taking the anti-piracy position on this matter. Communities such as this one are fighting the RIAA et al tooth-and-nail not because we are pirates, but because their efforts to combat pirates are extremely hostile to law-abiding consumers. For this reason, we're very suspicious of their protestations that all they're doing is trying to fight piracy.
Oh, okay. Although I think your judgment that Asimov was the connecting context something of a stretch. I would expect that if you were familiar with only one of the terms, it'd be limbo, not Valhalla. Although American Gods might explain it.
I don't recall saying not to read later writers like Riemann, Cauchy, or Weierstrass.
You are giving short shrift to Newton and Leibniz. The "incorrect" or "incomplete" ideas of the past are what informed the "correct" and "complete" ideas of the present. My personal experience has been that I always have a deeper, greater comprehension of the subject matter when approached in this manner; and the contemporary pedagogical method of a sort of "revelatory vision of the complete truth" is both false and misleading. There is more symmetry to mathematical and scientific discoveries in terms of precedence than you think -- they inform each other. If you only have the conventional revelatory, hubristic education, you'll think you know a subject better than you really do. As I said elsewhere, there's a reason that the very very best people go back and reexamine foundational and historical ideas, doing so relieves the myopia of the present.
I have said repeatedly that a historical or general approach to studying mathematics is not the equivalent of the type of study you and others prefer. I have repeatedly warned that this should be taken into account. I have never said that this more generalized comprehension is "better", I've said several times that the ideal is both. What you and others are reflexively attempting to say to me in reponse is that your method of study of mathematics is the only valid method, and the approach I am recommending is clearly inferior to yours. Given that I am not making an apparently chauvinistic argument about my own preference, and you are, I suspect that the bias lies with you.
Yes, you and others bristle at the connotations of my phrase "deeper comprehension", and I understand why you do. But you do so because you equate "deeper comprehension" with "greater comprehension", which is incorrect. I didn't mean it that way. Math, and science, is in the details, and a facility with those details is essential. But so is conceptual comprehension. No one can productively study these subjects without including both. Ideally, the study of both would be exhaustive. In practice, this is never true, and nowadays could never be true. Given limited resources, adjusting the relative mix of the two allows for adjusting for a desired outcome.
It's a fascinating thing to watch terms evolve. To pretty much repeat what I already wrote, I get almost breathless when I consider the increasing generalization that eventually contradicts the original usage's common sense coining. Obviously not just terms, but concepts.
It goes without saying just how badly the Greeks would go apeshit if they were presented with mathematics as it is now. And its arguable that Euclid with his strictness about not mixing different "kinds" in a ratio, the secret of incommensurability by the Pythagoreans, all kinds of stuff, that they had already glimpsed the abyss and refused to attempt to cross it. But their intellectual descendents did, and for damn good reasons. Furthermore, we could probably show them how it so often repeated that a more generalized mathematical concept that they would find abhorrent ended up being validated by physics. We'll put aside their antipathy for empiricism. (Although, is that the essential problem? If you're a mathematician, though, I think you can probably show lots of examples where, over and over, this sort of thing became compellingly necessary completely within the context of mathematics.)
At St. John's, a very interesting thing happens. Since it's a set curriculum of the "Great Books", it draws students with a fairly wide variety of intellectual predispositions. Of course, even if someone thinks of themselves as a literature person, they understand that they'll have to understand Lobachevsky, so they're not your typical student in any event. Even so, people that are very humanities oriented or even describe themselves as being mathephobes, will commonly become deeply enamored of math at the college, and leave to major in math elsewhere, or go to graduate school in math. I think that's a wonderful thing, and it indicates to me that math at the secondary school level is being mistaught. All the beauty is being leeched out of it.
I was trolling with the experts on afu as long ago as 1994. I don't care about your trolling. Perhaps I meta-trolled you?
The Ivy League schools are not exactly the same with regards to the approach to these matters of pedagogy. That's why, in fact, you are referring to your school as a "liberal arts" school, and you are not attending MIT. Yours may be a steller mathematics department. Certainly MIT's is. I doubt that they take the exact same approach to the subject, nor do they teach all the same "techniques".
Generally, the better the school, the more it will require that you learn deep concepts along with technique. But all scientific fields and mathematics, too, have become fragmented and specialized enough, that there simply isn't time to provide both deep comprehension and sufficient practical preperation and skill. This is just simply true, and I can't imagine that you would claim otherwise.
I suspect that you are reflexively responding to what you figured I said, rather than what I actually said. You'll notice that I never claimed that you could learn mathematics without doing mathematics, and it's also obvious that doing mathematics requires technical expertise. The question is what is useful for deep comprehension, and what is useful for the ability to accomplish another purpose? I imagine that a mathematics education today is still pretty deep in terms of general comprehension. Theoretical physics, as well. It's interesting that you chose that example, as most physicists are not theoreticians. My experience among grad students in the sciences, mostly physics, is that their comprehension of fundamentals is sometimes frighteningly uneven.
Another problem is that highly trained people like yourself (or who you will be) like to think that the only significant comprehension possible of their specialty is via their specific training. This is self-serving, and a simple function of human tendency toward chauvinism.
I am not in any way endorsing autodidactical cranks. (I am neutral with regards to autodidacticism. I just don't want to give those "I have a better theory that General Relativity!" nuts any encouragement.)
St. John's College of both Annapolis and Santa Fe. There's a required math class six of the eight semesters. Here's a general page for the reading list, unfortunately they don't provide a reading list of what appears in the math "tutorial".
But, again, as I've said elsewhere, this type of comprehension does not prepare one sufficiently to do the type of work that people actually do now. But if you learn what they know, you'll understand the subject much better.
A footnote. As is the case with physics, I do think that eventually one needs to have at least a general understanding of what has happened in 20th century mathematics. To my mind, everything that came before is the (mostly) comfortable beginning to a story that takes a very surprising and discomfitting turn. I believe that there's something very important going on here; and, in fact, these 20th developments essentially reexamine foundational ideas and reinterpret them. Some might say undermining them. Which is pretty darn weird since these developments are the culmination of what they seem to repudiate. This is incredibly fascinating and provocative to me. So, not hitting the 20th might leave the student with a false idea of where we at present.
On the other hand, working through those texts might give you much deeper insight into the math you already know. Is that what you want? Or do you just want to go further with what you know or to fill in the gaps? Again, do you want to do this for the pure intellectual satisfaction of comprehending something in general, or do you want to do specific stuff with what you learn?
For the life of me, I can't remember which one, but it was one of the preeminent mathematicians (but it could have been a physicist) of the last few generations, I think, that said he wanted to spend his twilight years in deep study of Newton's Principia Mathematica (obviously read at my school, re: calculus) Clearly, he thought there was something of value there to learn.
One thing about math is that some subfields can be pretty independent of all the others. I think you could start with basic set theory and go a long way without needing to (deeply) refer to other stuff. I keep wondering if I want to try to teach myself differential geometry (modern). That's because I want to understand general relativity, really. (You may notice that I agreed with the comment above that you can't understand many mathematical or physical ideas without doing the math.) I am not in a position to really evaluate how feasible this is. Yet.
You could probably find some good stuff on Amazon. Look for real mathematicians trying to write about a specific subfield in a more generalized manner. (I don't ever read popularizations of science or math by people who are not scientists or mathematicians. I think it's good advice.)
In truth, almost all American higher eduation is now vocational education. Your attitude and comment demonstrate this. It's the only thing most people can imagine that an education could be for.
The problem is that since what they want is a vocational education, and what the economy needs is a vocational education, it's interesting that we're not doing a very good job providing one. This is because of the supposed continued commitment to a "liberal education" by most American undergraduate schools. The result is the worst of both worlds: watered down liberal arts classes that teach little and make the students resentful that they are required to take them; and too few vocationally relevant classes, often with a poor degree of contemporary technical relevancy. This is why there's been a junior/community college revolution going on in this country for about twenty years -- they're meeting the demand that the universities aren't.
Obviously, since I went to an extreme liberal arts school I believe in the ideal of a liberal education. But as a practical matter, vocational education is essential. Ideally, it'd probably make me happy if everyone did what I did, and then do a year or so of undergraduate preparatory work in a particular field, then continue on to a graduate school in that field. For the people that wouldn't have gotten an advanced degree, or don't want that much schooling, you could still do what I did but put vocational schooling and experience beginning in parallel like they do in Europe. But I don't really expect everyone to do what I did, and I'm certain it's not appropriate for everyone. What degree of a sort of liberal education is for "everyone"? Well, we started down this road before and where we're arrived is not satisfactory. I think I'd prefer to find a way to get as much as possible of this done in primary and secondary school, extending schooling to year-around and adding another year; then sending people on to vocational, liberal, or professional educations.
It's actually a pretty modern thing to think of "education" as being a vocational education. What you needed to know to work in a vocation, you learned in apprenticeship or some other such institution. America has a particular problem with all this, though, since we have a very egalitarian ideal that wants to give all citizens some sort of a liberal education, while our relentless practicality also demands that we teach people to do their jobs. The two things are in many ways disharmonious.
But the math you should do is dependent upon what you want to do with it later. To take a trivial example supporting my point, I was really pissed off at the education I'd gotten previously when I worked my way through Book I of Euclid's Elements and came to the Pythogorean Theorem. Suddenly, I understood it in a much deeper way. Did it matter that much in regards to that algebra I had done earlier in high school? Nope, not really.
Or take irrational numbers. They are presented to students in the most prosaic fashion, and many students (not math majors or mathematicians, of course -- remember, I'm using rudimentary examples) would simply say "uh, they're numbers whose decimals go on forever? Oh, wait, they're numbers whose decimals go on forever without anything repeating?" That's literally true, and means nothing. When you stumble upon the incommensurability of the diagonal of a square to its side in the context of Euclidean geometry, such a thing is dumbfoundingly counter-intuitive.
This type of thing repeats itself as you work your way deeper into any discipline. The top people tend to better acquaint themselves with deep, fundamental ideas as necessary. It's hard to do truly original work without doing so. But today's scientists are not trained, really, for doing truly original work, and they shouldn't be. Those that want to and have the aptitude will achieve that deeper level of comprehension on their own. Everyone else will do their much more technical, incremental work. And that is, in fact, the overwhelming majority of the progress made in science and mathematics. The big stuff gets all the glory, but its the little stuff that accounts for most of the work and enables the big stuff to be discovered. This is why although I greatly personally prefer deep comprehension over facility with technique, I don't advocate that this is the proper pedagogical approach for all students.
The poster that asked the question needs to ask what he's looking for in his approach to mathematics. You know as well as I do that introductory calculus texts are more an attempt to manage to acquaint the student with calculus and then teach a variety of techniques that are likely to be of use in particular fields. If you're not working in those fields, if you're never going to use calculus either for technical purposes or as a working mathematician, you probably don't need most of those techniques. Much of this comes and goes as different technical approaches are fashionable. It just simply isn't the case that all the techniques that a student is taught in college calculus courses are essential to their understanding of the subject matter. That can't be true, as which techniques are taught change over time.
Obviously, there's a core facility with both concepts and technique that is necessary for any resonable level of comprehension. I was not disputing that. That's why, in fact, I went to a liberal arts college very unlike yours (which is every one other than mine), where actually doing the mathematical work, of say, Lobechevsky, is considered essential and where a gloss in a math survey course is rightly considered for the most part a waste of the liberal art student's time. You're right: you don't learn a subject like math by reading about it.
Yeah, "a lot" is two words. I conflate them to one quite often, since I think of it as a single word. I'm not the only one. It'll probably eventually appear in the OED. I'm a language pragmatist, not a proscriptivist.
Yes, I second the importance of asking yourself this question.
I have an intensive classic liberal arts education. Calculus directly from Newton and Leibniz, for example. This is great for understanding what the calculus really is, but very poor for doing the kind of calculus that people do as a practical matter.
The thing to understand in science and, yes, even math today, is that these have become almost completely technical fields -- that is "technical" in the sense of "technique". To be functional at all working in any of these fields requires the acquisition of a great amount of particular knowledge and technique that is not at all about a deep comprehension of the subject matter in general. A lot of my fellow alums find this out the hard way if they continue on to graduate school in a science, even though they tend to be accepted to the best schools. They have a lot of catch-up to do about the nitty-gritty stuff. On the other hand, their deeper comprehension serves them well as students and working scientists not infrequently.
The point is that if you want to just really get into math because you want to know more about it, then you should not try to duplicate what someone does who is studying it for professional purposes. You should approach it from another angle; then, if you choose, supplement your general knowledge by beginning to acquire proficiency in the specific. You'll also have a better idea of what interests you before you go the distance by learning much of the minutae necessary to even have a decent comprehension of actual contemporay work done in these fields.
The people doing this stuff for a living (or are students until they discover that they can't find a job and do this stuff for a living) will snobbishly dismiss a liberal arts approach to these subjects as being a waste of time or as some sort of pretense of learning that's not really there. Ignore them. They can't see the forest for the trees, and they shouldn't. That's not their job. For you, it's probably more fun to first examine and think about the forest before you start getting intimate with the trees.
I must have read that three times without noticing that stupid error.
Viruses and worms have been mostly merely malicious. Same with cracking. And the malice involved is not very great. But what if people get serious about stealing data?
A few years ago I had an epiphany one night, and waltzed into a network security company the next day.
"Look", sez me, "Inbound connections and activity are, in the long run, not going to be the real threat. The real threat is trojaned applications that mine for data and somehow send it offsite. You need to be monitoring outbound activity for appropriateness. For example, eventually you're going to see corporate espionage where someone writes an attractive and actually useful little app, then social engineers a targeted person within an organization to download it and compromise security. This is just an example of the general problem."
They were actually pretty impressed, but the company's strategy was deliberately to avoid concerning itself with viruses or worms (more specifically, they wanted to stay only on the servers, monitoring network activity in a sophisticated manner). But it seemed to me that this was a natural extension of their product and technology. And they thought I was a pretty bright guy, but they didn't know what to do with me. Well, anyway. The irony is that they were only a year or so later bought by one of the big antivirus firms, mostly just to acquire their technology.
In this particular case, the BitchX irc app, it looks like an outside source injected some backdoor code into the application, and hacked the ftp server to distribute it in a selective manner, presumably to help lower the risk of detection. A lot of effort for not that great of a payoff, really. Here, as is often the case, it's mostly about proving how clever you are.
But we're starting to see rudimentary examples of what I was warning about with spyware and other apps that make outbound connections that are in some sense illicit. Firewalls monitoring outbound connections can only be so successful given that they're always going to let some through. I know that some of the client based firewalling/monitoring software looks at connections on a per application basis. That's a start.
Personally, my inclination is that we need a networking monitor that operates like a virus scanner -- on the client, in the background -- that accesses a secured database of allowed application to outbound connection mapping, with secured handling of exceptions or new applications referred to a security admin (ideally) or an admin. This way we don't have to use a brute-force approach that simply locks down all allowed applications and allowed outbound connections in a non-specific, usability-destroying way.
But whatever the solution, I have little doubt that this will be a growing problem which will make a transition from script-kiddie nuisance cracking to something much more sophisticated. Although I could be wrong.