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This is all a lot of fun, but it's possible that the outcome of a lawsuit once actually hinged on a solution. It seems that Protagoras agreed that one of his law students would not have to pay him for his instruction until the student had won his first case. After some time had passed and the student had failed to obtain any clients (and therefore had not gone to court to argue any cases that he might win), Protagoras got tired of waiting and sued the student for the cost of the lessons.
Protagoras argued that he'd get paid either way: if the court decided in his favor, the student would have to pay the judgment; if the suit were settled in favor of the student instead, he would have won his first case and would therefore have to pay Protagoras under the terms of the original agreement.
The student argued just the opposite: if Protagoras won, the student would still not have won his first case and he wouldn't have to pay, while if the student won instead, he wouldn't have to pay because the court had issued its decision that way.
(Incidentally, in looking up the specifics of this form of the paradox, I re-discovered that the "autological/heterological" business that started this thread is known as "Grelling's paradox".)
OK, I just read up on the Grelling's paradox cited by Draney:
http://en.wikipedia.org/wiki/Grelling%E2%80%93Nelson_paradox
and see the similarity. And imho it only reinforces my assertion that the English language (or for that matter ANY language) constitutes the kind of formal system Godel was talking about. There will always be statements that are true but unprovable, false but unprovable as false, or ambiguous and undecidable.
What I find really interesting, is that our attempts to categorize certain words as autological or aptronymic or whatever, itself runs into the same epistemological problems that Godel was trying to work around. Not that he ever found a workaround.
You are correct. But there is an ambiguity surrounding that word (theorem) as well. We still refer to the Pythagorean Theorem, even though it was proved centuries ago. Likewise in science, where we still use the term "Theory of Relativity," even though it's pretty much been verified by experimental tests. Curiously, we say "Newton's Laws of Motion" but not the "Law of Relativity." It's been a sore point with many scientists, who feel we get a little sloppy with that terminology.
Case in point, many scientists suggest that by using the term "Darwin's Theory of Evolution" we leave ourselves open to criticism by those espousing "intelligent design" as an alternative to evolution. One of the first things IDers like to point out is that "it's only a theory" and thus open to question.
At some point, in science anyway, after sufficient testing and failed falsification attempts, what started out as a "theory" becomes so well accepted that it is considered, for all practical purposes, a "law" that is not likely to be falsified at any point in the future. Of course, it's the nature of science to be open to potential future falsification or revision.
In mathematics and logic, there's a bit more rigor and certainty and finality than in science. But even mathematicians are guilty of sloppy language.
It's interesting that every crackpot out there is comfortable claiming to have found a flaw in Einstein's "Theory" of Relativity, but no one attacks Newton's "Laws" of Motion.
Even if mathematicians and scientists were always precise in their language, there would still be confusion where some words have field-specific meanings.
For example, you see the same misinterpretation of theories of evolution, relativity, etc. that you see in medicine:
"There's a high correlation between poor diet and heart disease"
"Correlation? So one always causes the other!"
"Well, no. It's just a very high probability."
"I see. It's just luck."
"Argh."
In my own field, I've learned to use one set of words when talking to those in the field and another set of words when talking to those outside the field. Too often, people assign the wrong meaning to the lingo of the field - and you so seldom get a second chance to explain yourself.
"Theorem" in mathematics means something entirely different from what "theory" means in common parlance. A mathematical theorem is something that can be proven, and you either proceed to do so at that point or you take it as read that a proof exists. What non-mathematicians think of when they hear either word is actually what mathematicians would call a "conjecture" or a "hypothesis".
"Fermat's Last Theorem" was a misnomer until Andrew Wiles worked his magic on it (unless you held the opinion that Fermat himself really did have a proof that wouldn't fit in his margins, which is doubtful in light of the subsequent time and effort required before a publishable proof was actually found). Similarly, the "Four-Color Map Theorem" wasn't really a theorem until the 1970s when Haaken and Appel produced the first computer-generated proof.
They're being more careful in the literature these days. The most-discussed unproven thing at the moment is probably the "Goldbach Conjecture" (that every even number can be written as the sum of two primes). It will be interesting to see, if a proof is developed in our lifetime, whether the mathos will switch terms and call it the Goldbach Theorem from then on.
Heimgenge said:
Likewise in science, where we still use the term "Theory of Relativity," even though it's pretty much been verified by experimental tests. Curiously, we say "Newton's Laws of Motion" but not the "Law of Relativity."
Ron Draney said:
A mathematical theorem is something that can be proven, and you either proceed to do so at that point or you take it as read that a proof exists. What non-mathematicians think of when they hear either word is actually what mathematicians would call a "conjecture" or a "hypothesis".
You know, I used to know that about "theorem", but I forgot. My apologies, Heimhenge.
It's true of "theory", too, by the way. In science they call a conjecture a "hypothesis"; a theory, on the other hand, is a picture of how things work, usually a picture with wide-reaching consequences. You wouldn't speak of the Doppler "theory", because it's pretty simple and it isn't that hard to work out what would result if it's true. But relativity had to be argued out for decades, not just because some people disbelieved them (even their proponents on occasion!) but mostly because it took that long to notice some of their implications. And they're still stumbling over new results of quantum theory. Calling it a "theory" doesn't mean it isn't proven...and also it doesn't mean it is.
It's in that sense, as I understand it, that they speak of the "theory of evolution", not that it's a mere conjecture but that it's a grand, far-reaching notion of how things work whose implications are still being fought out. Some pieces of it are pretty well understood; other parts are still the merest conjecture; much of it is somewhere in between; and the news media, in an attempt to understand it themselves and then to make it understandable to others, end up swirling it all together and confidently proclaiming "science tells us that..." when in fact most scientists are still trying to work out some of the very real difficulties. But it's a misunderstanding to argue that "it's just a theory, after all".
Wait, Ron, are you saying Fermat's Conjecture has recently been proven?! I don't think I'd heard that.
Bob Bridges said:
Wait, Ron, are you saying Fermat's Conjecture has recently been proven?! I don't think I'd heard that.
It was in all the papers: http://en.wikipedia.org/wiki/Andrew_Wiles
Ron Draney said:
Bob Bridges said:
Wait, Ron, are you saying Fermat's Conjecture has recently been proven?! I don't think I'd heard that.
It was in all the papers: http://en.wikipedia.org/wiki/Andrew_Wiles
DEI SUB NUMINE VIGET.
Bob Bridges said:
You know, I used to know that about "theorem", but I forgot. My apologies, Heimhenge.
No apologies needed. I taught HS science for close to 30 years and never gave the distinction much thought. How many students did I "contaminate" with that careless language? These days I choose my words more carefully.
Funny story comes to mind … I worked on that 4-color map conjecture for close to a year back when I was in HS. Not trying to prove it, but trying to find a counter-example (a map that required >4 colors). Thought I found one. Had several friends try to color it using only 4 colors. Nobody could. Figured I was gonna "get rich" with my discovery. I was so naive. Finally took it to our geometry teacher, who promptly discovered a 4-color solution. Within a year or two after that, the 4-color conjecture was proved true using a computer (as Draney correctly points out).
These days, computer-based mathematical proofs are well accepted. But in those early days it was almost considered "cheating." In fact, computer simulation in science has spawned a whole new way to do research, as well as its own journals. Since the advent of supercomputers, introduced by Cray Research in the 70s, it's become possible to (fairly) accurately simulate such complex systems as climate, stars, aerodynamics, etc. You can ask the computer models a question, and they give scientifically reliable answers.
The power of computers today is hard to comprehend. I remember, back in the 80s, paying $200 to upgrade my PC from a 10 MB hard drive to a 20 MB hard drive, and thinking "Wow … now I'm set for life." I currently have several files on my machine that individually exceed 20 MB. And my NAS (which holds the music from what used to be my 300+ CD collection) has a 1 terabyte HD. It still has close to 90 GB free space. Gotta wonder about Toffler's "future shock" concept. Or should that be the "future shock theory?"
Heimhenge wrote:
…it's become possible to (fairly) accurately simulate such complex systems as climate, stars, aerodynamics, etc. You can ask the computer models a question, and they give scientifically reliable answers.
I'm going to take issue with this. The power of the computer allows one to test a model (ie, a "theory") even if it's very complex. It cannot tell us that the model is accurate. What I'm saying is this: If I have an hypothesis that cumulus clouds form when condition #1 and (condition #2 or condition #3) and not condition #4, with each condition affecting many others to some extent, I can create a computer program that displays the results of my theory given many different theoretical conditions, by making thousands or hundreds of thousands of calculations over time and showing the results, calculated minute by calculated minute. The computer helps to reflect my theory, by displaying what would happen if my theory is correct, but that's all.
If the actuality is hard enough to measure (as in the case of weather), a computer model can be almost impossible to test against the reality. Sure, I can look out the window and see cumulus clouds forming, and I can probably use a range-finder or something measure their altitude. But how can I find out the exact air pressure, relative humidity, temperature, wind velocity and solar candlepower at each of several frequencies in every cubit foot of the volume involved? The less I know about actual conditions as that cloud formed, the less I can correct or confirm my theory.
Don't get me wrong, I really enjoyed The Day After Tomorrow. But Dennis Quaid's computer didn't demonstrate scientifically what the weather was about to do, it only predicted what would happen if his model was correct.
All I'm saying is that if you ask the computer models a question, what they return is not a "scientifically reliable answer" in the sense of an accurate real-life prediction, just an accurate reflection of the suppositions that went into it. The better the theory (repeat "theory, theory, theory"), the better the model's outcome will reflect the real world.
Point taken. Well said. Computer models are just that ... models. Their accuracy is a function of assumptions as well as processing power. But they're getting better as time goes on. I was reading an article in Physics Today, today, that addressed that very issue. The question of whether we can accurately model reality goes right back to Godel's proof. Models will never reach certainty, but they are getting closer, and provide useful insights.
"The Day After Tomorrow" was a great flick, and supports your assertion. But that was fiction, not science. Still, it reminds us that models are just models. Can you imagine a point where models are exact? I think not. But they have advanced technology and science nonetheless.
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