Thursday, September 04, 2008

The Four Horsemen of Digital Apocalypse

by Len Hart, The Existentialist Cowboy

I have just survived the worst and most insidious 'virus/trojan' attack since I got my first PC. Despite the fact that I was loaded to the gills with anti-malware, this attacker ran and survived the gauntlet, set up shop and had the nerve to threaten to shut me down. The ensuing epic struggle was a 'virtual' Armageddon, a final, desperate battle between me and the Four Horsemen of Digital Apocalypse.

The offending malware pretended to be a Windows alert. It informed me that my computer was 'unprotected'. It demanded that I 'download' their latest whatever. To do so, I was required to click on an "I AGREE' button. I could not click that button in good conscience. I rarely agree with anything --let alone a 'button', let alone a button that was daring to 'threaten' me.

Never, ever click on a button unless you know where it comes from and trust it. Likewise, suspect any prompt that does not give you a 'I DECLINE' option.

Given the deceptive nature of these programs, I am suspicious of 'any' label given a button. Many are indistinguishable from legitimate 'windows' buttons. In this case, no alternatives were given me. No 'X' in a corner; no 'decline' option; no 'maybe later'; no way to get out! Even CTRL-ALT-DELETE had apparently been disabled. There seemed to be no way to get this piece of crap off my screen and out of my computer.

In the conditional logic of this rudimentary program, you are simply not given an exit from the loop. It is this fact that gives the game away. Only a crooked or incompetent (or both) programmer would write such a conundrum. It was deliberate. It was crooked. It was intended to wrest from me control over my computer.

I found in this experience a parable and a rule of thumb. Any proposition (save life itself) is an evil bargain if there is no escape save death. From a programmer's standpoint, the 'logic' of this code was a very simple matter of omitting a critical 'else' or 'else/if' statement here and there. If you click 'agree', you allow this 'trojan' free reign over your computer. But --there is no 'decline' or 'later' button of any kind. Doing nothing at all is, likewise, not an option. Your computer is effectively lost to you.

This 'spammer' --whom I suspect resides in either China or Florida --has much in common with the mob. It's an offer, you cannot refuse. It might have been worse. At least, I didn't awaken in bed with a dead horse. "Offers that cannot be refused" are commonly used not only by spammers but politicians and, most insidiously, religions. The Christian religion, primarily, puts a 'box' on the computer screen of your mind (or soul) with a choice: hit the 'accept' button or lose the use of your 'computer', in this case, your 'soul'.

Elsewhere in Christianity, however, it is said that such a choice must be made freely. But a coerced choice is anything BUT free! Thus --Christian theology --by enforcing a decision through blackmail --has 'spammed' your mind, denied you free will. By definition, there is no free will without choices. Christianity, by violating its own principles, nullifies itself as valid religion, philosophy, or moral guide. As a theology, as a philosophy, as a 'program', it is, therefore, fallacious and, perhaps, deliberately misleading. Any 'formal system' in violation of its own premises is false! I daresay most religions are of this form. Most, if not all, theology is false.

Decisions made under threat of death are not morally valid. Even 'confessions' made under the threat of death are not admissible in court. In the case of the Christian religion, the threat is not merely one of death, but after death an everlasting hell fire the existence of which no one can prove. No wonder much of the history of the human race is a record of bad decisions and equally bad consequences. Though I am no expert on the fine points of every religion on earth, I will venture this: of the world's major religions, only Buddhism seems free of indefensible and unsupportable dogma. I have yet to hear anyone threaten me with eternal hell fire should I decide not to follow the 'path' of the Buddha.

Of a slightly different logical structure was the 'indulgence' scam perpetrated by the Catholic Church, nothing more than Pope Leo's scheme to raise enough money to pay off his debts, build the new St. Peter's, finance his orgies, underwrite the art. Life is always a mixed bag. Amid the waste, evil, and debauchery, the monies raised by this criminal fraud paid for the lasting works of Michelangelo and Raphael.

Historically, 'religion' purports to explain everything. Those observed phenomena not explained by science or common sense are 'explained' theologically, in terms of the 'supernatural'. But a 'supernatural' explanation is, in fact, no explanation at all. Something is unexplainable if it cannot be explained in terms of 'natural' phenomena. There are, therefore, by definition, no 'super natural' explanations, only natural ones. Thus religion is tautological in its inception. Scientific explanations are only 'natural', by definition, not 'super'! Put yet another way: 'supernatural' is an oxymoron.

Interestingly, there is salvation of sorts to be found in the limits of logic itself. Rudy Rucker, a mathematician gifted with a redeeming sense of humor, wrote of Kurt Gödel's Incompleteness Theorem that it was '... so simple, and so sneaky, that it is almost embarrassing to relate.'
His basic procedure is as follows:
  1. Someone introduces Gödel to a UTM, a machine that is supposed to be a Universal Truth Machine, capable of correctly answering any question at all.
  2. Gödel asks for the program and the circuit design of the UTM. The program may be complicated, but it can only be finitely long. Call the program P(UTM) for Program of the Universal Truth Machine.
  3. Smiling a little, Gödel writes out the following sentence:"The machine constructed on the basis of the program P(UTM) will neversay that this sentence is true." Call this sentence G for Gödel. Note that G is equivalent to: "UTM will never say G is true."
  4. Now Gödel laughs his high laugh and asks UTM whether G is true or not.
  5. If UTM says G is true, then "UTM will never say G is true" is false. If "UTM will never say G is true" is false, then G is false (since G = "UTM will never say G is true"). So if UTM says G is true, then G is in fact false, and UTM has made a false statement. So UTM will never say that G is true, since UTM makes only true statements.
  6. We have established that UTM will never say G is true. So "UTM will never say G is true" is in fact a true statement. So G is true (since G = "UTM will never say G is true").
  7. "I know a truth that UTM can never utter," Gödel says. "I know that G is true. UTM is not truly universal."

    With his great mathematical and logical genius, Gödel was able to find a way (for any given P(UTM)) actually to write down a complicated polynomial equation that has a solution if and only if G is true. So G is not at all some vague or non-mathematical sentence. G is a specific mathematical problem that we know the answer to, even though UTM does not! So UTM does not, and cannot, embody a best and final theory of mathematics ..

    Although this theorem can be stated and proved in a rigorously mathematical way, what it seems to say is that rational thought can never penetrate to the final ultimate truth ... But, paradoxically, to understand Gödel's proof is to find a sort of liberation. For many logic students, the final breakthrough to full understanding of the Incompleteness Theorem is practically a conversion experience. This is partly a by-product of the potent mystique Gödel's name carries. But, more profoundly, to understand the essentially labyrinthine nature of the castle is, somehow, to be free of it.

    --Rudy Rucker, Infinity and the Mind
The work of Alan Turing proving that certain propositions in a 'closed logical system cannot be proved within that system' is, of course, a corollary to Kurt Gödel's famous proof. Both have had enormous consequences in academia, computing and philosophy. It is hoped that one day, the impact of this work will be felt in the field of politics.

Both Gödel and Turing were concerned with the inherent flaw in any formal system. The question raised is this: if a single trojan could very nearly take over my computer, might a much better, wider and highly co-ordinated attack seize the internet itself. Is our salvation to be found in Gödel and/or Turing? What lesson is learned by the defeat of Chess genius Gary Kasparov by IBMs 'Big Blue'.
Chess is a game of guile and strategy. Chess means putting your emotional engines out of sight and choosing moves with cold calculation. In the end, Kasparov's cool cracked. He angrily resigned -- charging, at first, that IBM had let a human call the moves. I doubt anything of the kind, just because the computer's eventual victory was predictable.

Two generations ago, Alan Turing gave us an important thought model for all this. Turing said, suppose you go into a room with a keyboard and a monitor. You type in questions and receive answers. Then you try to determine whether the answers are being given by a human or by a machine. Ever since then, we've said that a computer which can't be told from a human passes the Turing test.

Most of us have assumed that no one could ever create a Turing Machine because that veers close to creating sentient intelligence. Here the argument over Deep Blue heats up because of Kasparov's initial belief that he was dealing with humans. Deep Blue really did pass the Turning Test as far as Kasparov was concerned.

That's why I think this strange little chess game was significant -- not because the outcome was a surprise, but because Kasparov thought Deep Blue might be human.

--Dr. John Lienhard: Kasparov and Deep Blue
It does not follow, however, that because Big Blue defeated Kasparov, that computers will eventually render the human being obsolete, that eventually there may be no defense against a computer generated 'virus' or trojan by which a 'federation' of networked computers will assume complete and total dictatorial control and thus rule the world.

I love Gödel's proof and as well the related work of Alan Turing. The very language of computers is like that of human beings --'flawed' or, more precisely, incomplete. We need never fear computers taking over. But it is not only because Gödel's Proof is 'logical' that it is compelling. It is positively liberating. 'Incompleteness', itself, is liberating. 'Incompleteness' should be celebrated. In 'incompleteness' is our salvation from a dictatorship imposed upon us by computers and/or inflexible systems.

Whenever a fundie, or a Nazi, or a Republican comes peddling an all-embracing system or purports to have all the answers, a complete and unassailable ideology, system or weltanshauung, or tries to blackmail me into swallowing it, I have Gödel's proof that NO system is complete, that NO one person or organization has all the answers.

There is no holy writ!

There is no voice high or low that will replace my own conscience and my own abilities to work out the truth as best I can. There is no way that I may be blackmailed with offers I dare not refuse for fear of either hell fire or, worse, the loss of my computer forever!

There is no blackmailing me into any system or cult that, like the GOP, presumes to have all the answers but is, in fact, wrong about everything! I am a free man! I make my own choices and live with the consequences.

I am immune to coercion. No one living can make meaningful statements (let alone truthful ones) about an after life that may or may not exist. Therefore, I choose to base the only life of which I have knowledge upon matters about which meaningful knowledge is, at least in theory, obtainable. It is nonsensical to base my life upon the expectation of an after life about which no meaningful statement can be made whatseover.
"Why Richard, it profits a man nothing to lose his soul for the whole world... but for Wales?"

--Character of St. Thomas More, A Man for All Seasons, Sir Robert Bolt
I submit that it profits one nothing to compromise his/her integrity in this life in the expectation of rewards in a hypothetical afterlife the existence of which may not be known and about which nothing meaningful may be said.

For those interested in avoiding 'armegeddon' with Trojans:
As that very real war goes on in the Middle East, back here at home we continue to wage a virtual war against a different kind of spam. And of course, it's not just in the U.S. Just a few days ago, the French government announced a new project by which Internet users could alert their ISPs when they receive spam messages:
http://www.wxpnews.com/Q85JLJ/071009-Signal-spam

And we're hearing that Japanese users are getting an increasing amount of spam mail from Chinese servers, most of it advertising online dating services and adult-oriented web sites:
http://www.wxpnews.com/Q85JLJ/071009-Spam-in-Japan

Spam web sites are causing trouble for Google, as many of these sites are coming up in search results and some of them are downloading malware onto users' computers when they follow the links.

Meanwhile, the Securities and Exchange Commission (SEC) is cracking down on the recent deluge of "pump and dump" spam messages that attempt to inflate the prices of stocks issued by small companies. Recently this has become the second largest category of spam, with as many as 100 million of these messages being sent every week, many of them in the form of PDF attachments. The SEC has reported a 30 percent drop, however, since they initiated an aggressive program that includes freezing the trading on some of these companies

http://www.wxpnews.com/Q85JLJ/071009-Crackdown

Another, more malicious variety of spam that has popped up in the last few months exploits the popularity of social networking. These messages claim to be from an "old school friend" or a "childhood friend" and contain a link that's supposedly for the sender's MySpace (or other social networking) homepage. However, clicking the link takes you to a site that downloads a Trojan which can gather personal information such as account numbers and passwords and send them back to the spammer:

http://www.wxpnews.com/Q85JLJ/071009-Social-spam

Spammers and email scammers are great at taking advantage of whatever's currently in the news and trends in public opinion. Shortly after September 11, there was a spate of spam messages appealing to Americans' patriotic feelings. As the public tide turned, we now see spam messages that hook into anti-war sentiments. The recent downturn in the housing market and the subprime loan scandals have resulted in a new flood of spam messages pertaining to home financing.

You might even be a spammer yourself and not know it. Thousands of computers are infected with malicious software called 'bots that turn them into "zombies" that can be controlled by spammers and used to send spam messages (and hide the true origins of the spam).

--War Against Spammers Goes On
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17 comments:

SadButTrue said...

The Rucker paradox is easily assailed by thinking outside the artificial box in which Rucker poses the question - that box being the proposition that logic fits into a binary solution set.

Sure the idea that 1= 'true' and 0= 'false' provides scriptwriters with a handy way for Spock to defeat Nomad but I don't think the number system used by today's computers is a good fit with either the real world or any theoretical philosophical construct.

Truth values are better assigned using the trinary number system, where the digits have values equal to 1= 'true', -1= 'false' and 0= 'undetermined.' (this system is called balanced ternary) But as I inferred above, we might want to stick with the binary system for electronic devices if only to provide a fail-safe. After all, it would have turned out poorly for Spock had Nomad been based on a trinary logic system.

As far as philosophy and logic are concerned I think that Godel's incompleteness theorem infers, even demands the trinary system with the demonstration that there will always be some proposition that cannot be assigned a simple value of true or false.

Unknown said...

I think trinary logic may be quite useful. In this case, however, the conundrum is premised either/or. UTM can either utter G or not.

At the heart of the problem is the fact that "UTM will never say G is true" is 'self-referential'. The example demonstrates that there is at least one 'true' theorem that UTM cannot state (without violating its 'prime directive)' Simply, if UTM says 'G' is true, then, by saying it, UTM proves it false! Thus --the theorem is proven; 'G' becomes the 'one' true theorem that cannot be uttered by UTM because to utter it is to make it false.

Thanks also for the links to ST 'Nomad' episode, which, at the time, was among my favorite episodes. Star Trek was ahead of its time. From the same era was 'Mission Impossible', like Star Trek' fodder for movie spanning generations.

Godel's Incompleteness theorem has enormous implications. Prior to Godel, Bertrand Russell had hoped to base all of mathematics upon pure logic. Until Godel is utterly disproved, basing math entirely upon logic must be considered impossible.

paul said...

Your experience sounds a little like this anatomy of a malware scam

Unknown said...

paul said...

Your experience sounds a little like this anatomy of a malware scam

Thanks for the link. Am scanning it now. This must SURELY be the one. I hope to track these bastards down. They might yet regret their bullshit.

Anonymous said...

Since you're indulging Godel, Len, perhaps I can throw in two cents worth. The mathematical contradiction is worse than it first appears, even more devastating for those who would like to believe that most if not all mathematical truths are provably true or false. An explanation may help. We can put the counting numbers in a list: 1,2,3,4,5 .... Can we list fractions in the same way? Yes we can. Here are the fractions in a table:

1/1 2/1 3/1 4/1 5/1...
1/2 2/2 3/2 4/2 5/2...
1/3 2/3 3/3 4/3 5/3...
1/4 2/4 3/4 4/4 5/4 ...
1/5 2/5 3/5 4/5 5/5 ...
... etc

Now, we can put these in a list by itemizing the diagonals:

1/1, 2/1, 1/2, 1/3, 2/2, 3/1, 4/1, 3/2, 2/3, 1/4, 1/5, 2/4, 3/3, 4/2, 5/1, 6/1...

So there are as many fractions as there are whole numbers! It is a somewhat bizarre result suggestive of a cheap trick, but it's not, as we shall see. We call such an infinite list denumerable. It's a particular type of infinity. Now let's ask the question: "Can we put all decimal numbers in such a list?"..."Are the decimal numbers denumerable?" The answer is no! There are massively more decimals than fractions or whole numbers. For our purposes we can consider decimals between 0 and 1. Let's imagine we can write a denumerable list of those all the decimals. It might look like this:

0.000000.... (=0)
0.555555.... (= fraction 5/9)
0.14159265358979323846… (= pi minus 3)
0.71288888....
0.4444444....
0.8132600042115....
...
etc

Now consider the special decimal

special = 0.130957...

I have chosen this decimal specifically because it differs from every one of the decimals in the above list in at least one decimal place (check the highlighted decimals in the list). So now we know that the number 0.130957... does not lie on the list of decimals and can never do so.

The conclusion is inescapable: the decimal numbers can never be put into a list like the fractions or the decimals. We call this new infinite amount of numbers the power of the continuum. It's not immediately obvious from the example but the number of the continuum is massively greater than that of denumerable numbers. Try thinking of a handful of sand and compare it to all the sand on all the beaches of the world and you will have some idea of the comparison of denumerable to the continuum.

Here's where we get back to Godel. Mathematicians had held out the hope that all mathematics could be proved either true or false. Godel came along and showed that "No! There are some statements that may be true that we can never prove within our system of mathematics no matter how hard we try."

And here's the stunner: In comparing the true yet unprovable mathematical statements uncovered by Godel with true statements that we can actually prove it turns out that we are comparing denumerable infinity with the power of the continuum! In looking at the mathematically truthful statements that we cannot prove we are looking at a clear sky ablaze with billions of stars! There is far more that we can never prove than what we can prove.

And if we have to be humble about mathematical truth, how much more hesitant should we be about the more complex truths of life, society and people. The only proper human disposition should be wonder, innocence and humility. Thanks Godel!

Anonymous said...

Sorry to hear about the attacks, Len. You can run HijackThis and SpywareWarrior (also here) can analyze the results for you and help clean up your system. There are other expert groups out there who do a similar good job.

Anonymous said...

SBT, you may be misreading Godel's theorem when you say As far as philosophy and logic are concerned I think that Godel's incompleteness theorem infers, even demands the trinary system with the demonstration that there will always be some proposition that cannot be assigned a simple value of true or false.

Godel's theorem is concerned with provability of mathematical statements that are known to be true but which cannot be proved within the framework of the axioms of arithmetic. Godel does not say they can never be proved-- they often can be by the addition of further axioms. His theorem is concerned with the limitations on provability for any given fixed set of assumptions.

Multi-valued logic system have been studied for years. The Japanese went crazy for a while over fuzzy (probabilistic) logic even building robots and washing machines that used such systems. The latest computer developments use quantum logic in which Heisenberg's principle of indeterminacy is used to define a computer chip with 64 truth value states rather than the traditional binary true/false.

It appears to me that you raise the fair question not of provability but of limits to knowingness, of what we can know. I think that's a fair point but it in no way diminishes the certainty of traditional logic systems and the conclusions they reach. The religious nuts especially would love to misread Godel's theorem and throw logic to the wind and rely on superstition. Yes, there are limits on what we can prove, but Evolution is still true and 2+2 still equals 4. Under Godel some true statements may be unprovable. For example, the Goldbach conjecture claims that every even number is the sum of two primes. Thus 10 = 7+3 and 106 = 101 +5. It's long been believed that this is true but now, following Godel, we must entertain the idea that no proof of this may be available unless we add more assumptions to our mathematics system.

And, Len, your claim needs some minor adjustments -- Prior to Godel, Bertrand Russell had hoped to base all of mathematics upon pure logic. Until Godel is utterly disproved, basing math entirely upon logic must be considered impossible.

Godel cannot be disproved. In various forms the proof has been confirmed. It's true. But I know where you are going here. Russell hoped to show that mathematics was both complete and consistent. "Complete" means that any true statement derived from the system can be proved within the system. "Consistent" means that the assumptions that form the system never give rise to contradictory statements. Godel showed that for any fixed number of mathematical assumptions that define a system that system can either be complete or consistent -- but not both. So Russel's hope was in vain. But I think he suspected that anyway. So you're right, Len, to note that "basing math entirely upon [a consistent and complete] logic must be considered impossible." Godel showed that new mathematical assumptions will always be needed to handle the underlying mathematical phenomena. It's all good fun!

Unknown said...

damien said...

It's not immediately obvious from the example but the number of the continuum is massively greater than that of denumerable numbers.

Great post, damien! And timely! Just recently, I have been reading some old stuff by Isaac Asimov as it relates to 'numbers' et al. I also dragged out and dusted off some Russell that I had not read since high school.

The mind boggles.

Your analogy to "a clear sky ablaze with billions of stars" would do Carl Sagan [R.I.P.] proud.

And if we have to be humble about mathematical truth, how much more hesitant should we be about the more complex truths of life, society and people. The only proper human disposition should be wonder, innocence and humility. Thanks Godel!

I would like to add my thanks, as well. 'Certainty' has become a political issue, unfortunately. The GOP shares many defining characteristics with the Nazi party --equally certain, equally sure of all things ideological. I may have referred to Jacob Bronouski's last episode of "The Ascent of Man", in which he urged us to resist the need to be certain.

tiago said...

Len;
You know some one has taken over your computer when your wife calls you at work and tells you that your Pakistani mail order bride has arrived.
Seriously, when I find myself locked into a site with no options, I have one. It is called the power button. When the computer powers back up, you are free of that site. Then, check your cookies and delete the cookie from that site.
A computer’s gates are designed to know two states, high, any thing above 4.1 volts, and low, anything below .7 volts. Any thing in between is limbo. Another way of saying it is the gates are either on or off, hence binary, (or hexadecimal as I was taught and that is ancient history).
The logic is then; true or false. There is no in between, buts or maybes.
This is the very reason I refuse to participate in surveys. The results are predicated/slanted in the logic of whoever writes the survey, (or computer program).

Unknown said...

tiago said...

You know some one has taken over your computer when your wife calls you at work and tells you that your Pakistani mail order bride has arrived.

Good one, tiago : )

when I find myself locked into a site with no options, I have one. It is called the power button.

I use it often. In this case, however, Satan had already taken his dump among the many files that make up the system.

Paul's link at anatomy of a malware scam is the very culprit that attacked me but with some updates. Whoever is doing this keeps revising it. Some of the offending files did not show up among 'cookies' at all.

Basically, these 'folk' have written a crappy anti-virus program and are blackmailing folk into buying their junky crap by CREATING a need to get 'spyware protection'. This is very much like a mob 'protection racket' except that the mob could probably be depended upon to defend you against the competition. These people will blackmail you, dump junk on your computer and leave you even more exposed than before. But in the meantime, they will have gotten your CC numbers and cleaned out your account. Needless to say, they got shit from me!

They should be castrated.

Anonymous said...

I had this same virus. It was so annoying and my Norton didn't detect it at all. I had to call Friendly Computers and pay someone to fix it. I do online banking and don't want some stranger going through my accounts. The "tech guy" was really nice and it only took him an hour to remove it. Their site is http://www.friendlycomputers.com

Anonymous said...

The article is good if you don't delve too far into the Kasparov part. The contract he played under actually did allow some human interference!

Also, don't put it past Kasparov to throw one match in hopes of getting another and winning the bigger goal of making more money. Of course, that hope was smashed when IBM decided they'd had enough and the sore winners took their computer and went home -- leaving Kasparov miffed, but not poor.

Kasparov is perhaps the best chess player to have ever lived, so it's hard to say what his strategy was and whether he succeeded or failed. He's still an intelligent interesting fellow.

Oddly, 13 is his lucky number!

Unknown said...

MarkH said...

The article is good if you don't delve too far into the Kasparov part.

Don't worry! I have no plans to write bios of either Kasparov or Fisher. Besides --it doesn't matter than Deep Blue won. I once beat a grand master whose name I can't even remember. [BTW --it was one on one, not a 'demonstration' in which he played many folk at once] As far as I'm concerned, it was a fluke. And, as shocked as he may have been at the time, he was still a grand master and I was not even ranked and never would be.

And I have played very little chess since because 1) I am not willing to devote my life to it as Fisher had done; and 2) what a way stop playing!!

paul said...

I've always been baffled by the idea of kasparov/deep blue 'meaning' anything much apart from a useful bit of PR for technocratic dreamers.
The defeat of kasparov was the work of many men, several of them grand masters, using deep blue as a brute force prosthetic, nothing more.
The idea that the machine 'triumphed' (sneaky anthropomorphism) is a little like claiming your president is a self made man.

Unknown said...

damien said...

Sorry to hear about the attacks, Len. You can run HijackThis and SpywareWarrior (also here) can analyze the results for you and help clean up your system.

Thanks Damien. I am willing to try them all and trust YOUR suggestions. A caveat may be in order. Not all 'anti-spyware' is created equally; therefore, other 'victims' can benefit from your experience. That is preferable to just downloading the first so-called 'anti-spyware' scam that pops up on a search. It must be kept in mind that the offender (in my case) has pretended to be 'anti-spyware', a seemingly perfect cover.

The Japanese went crazy for a while over fuzzy (probabilistic) logic even building robots and washing machines that used such systems. The latest computer developments use quantum logic in which Heisenberg's principle of indeterminacy is used to define a computer chip with 64 truth value states rather than the traditional binary true/false.

I recall the 'fuzzy logic' craze. One of its first applications, as I recall, were Nikon, Canon, et al still cameras, video et al. 'Fuzzy logic' was said to have been especially useful in program or semi program exposure modes.

It appears to me that you raise the fair question not of provability but of limits to knowingness, of what we can know. I think that's a fair point but it in no way diminishes the certainty of traditional logic systems and the conclusions they reach.

I think that's a fair assessment. While a high school freshman, one of teachers lent me a collection of 'papers' by Russell, Ayer, Wittgenstin et al in a book called "Readings in Philosophical Analysis". It was at this time that I acquired a copy of Ayer's Language, Truth and Logic.

It would seem that at about 1900 or so, the very limits of knowledge began to be pressed in two directions --logicism (the reduction of pure math to pure logic) on the one hand, and another branch (best exemplified by A.J.Ayer) seeking a 'logicial' foundation for empiricism. The problems were presaged when Russell, in his attempts to found mathmatics upon logic, derived propositions that were false only when they were true and true only when they were false, a situation not unlike that facing my computer while under attack, a situation not unlke that the 'self-referential' statement that can never be uttered by UTM lest its utterance may it make it 'false' and thus 'true', therefore 'false'. At first blush, Godel seems to have discovered the absolute limit of 'logicism', while Ayer put some brakes on empiricism by questioning the meaning of meaning. His 'verifibility criterion of meaning' is unassailable.

Wittgenstein's analogy of the map is as good as Godels' proof in that both find limits to 'knowledge' in 'self-referential' statements, however logical. Indeed, 'consciousness' itself is an infinite regress like standing between two facing mirrors. The hypothetical UTM, for exmaple, breaks down only when it is challenged to make a 'self-referential' statement. Likewise, Wittgenstein's 'map' may not --in a finite 'legend' --contain valid instructions upon its own use. Wittgenstin's statement: "A logical picture of facts is a thought" is very close to summing up my own 'philosophy', and, is, in fact, a corollary in one of my own under-graduate papers. At the risk of over-simplification, it seems to be that the history of 20th Century philosophy is the story of the pursuit of both empirical knowledge (science?) and pure logic/mathmatics.

Wittgenstein believed that logic was based upon the idea that every proposition is either true or false but, like Godel, found limits to the ability of formal systems to derive every true or provable statement. Ayer categorized 'propositions' as either 'synthetic', capable of empirical verification, or 'analytic', capable of being proven either true or false by logic alone.

Science, at first blush, has nothing meaningful to say about 'ethics', systems of value judgements often having nothing to do with hard facts. Bronowski's critique of Ayer redeems Ayer. There is in "'Language, Truth and LOgic', says Bronouski, an implied 'social injunction' that one ought to behave in such a way that what is true may be proven to be so."

The religious nuts especially would love to misread Godel's theorem and throw logic to the wind and rely on superstition. Yes, there are limits on what we can prove, but Evolution is still true and 2+2 still equals 4. Under Godel some true statements may be unprovable. For example, the Goldbach conjecture claims that every even number is the sum of two primes. Thus 10 = 7+3 and 106 = 101 +5. It's long been believed that this is true but now, following Godel, we must entertain the idea that no proof of this may be available unless we add more assumptions to our mathematics system.

Indeed! A 'fact' that Russell had encountered. Godel is no excuse to 'throw logic to the wind' but religious fanatics can always be depended upon to draw the wrong conclusions about anything. I suspect, however, that anyone attracted to 'religion' are looking for 'completeness" and find it in religion. These people need a 'system' that gives them all the answers. Only religions of one sort or another can do this. I also doubt that any one professing to be a 'fundamentalist Christian' has read Godel. In America, the radio and, later, the tele-evangelists who made a living denouncing 'Godless Communism' had ever bothered to read either Marx or Engels. I am not terribly worried about the semi-literate suddenly findingredemption in an obscure corollary to the 'incompleteness theorem'.

So Russel's hope was in vain. But I think he suspected that anyway. So you're right, Len, to note that "basing math entirely upon [a consistent and complete] logic must be considered impossible." Godel showed that new mathematical assumptions will always be needed to handle the underlying mathematical phenomena. It's all good fun!

I think you're right! Russell was one of my early heroes and still is. He spoke his conscience and often paid the price for having done so. He lead a long and productive life and I hope that he understood how influential he would remain long after his death --not just for his work in logic and philosophy but his tireless efforts on behalf of nuclear disarmanment and peace.

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Unknown said...

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For the record: I was attacked by minions of Satan calling themselve (or their 'product') XP Antivirus 2008

The good news is: I may have --at last --removed the last vestiges of this plague and, in the meantime, have erected some pretty formidable obstacles. This is a good time to browse your list.