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Relativity, Uncertainty, Incompleteness and Undecidability

By chato in Science
Wed Aug 31, 2005 at 08:45:49 PM EST
Tags: Science (all tags)
Science

In this article four fundamental principles are presented: relativity, uncertainty, incompleteness and undecidability. They were studied by, respectively, Albert Einstein, Werner Heisenberg, Kurt Gödel and Alan Turing. This is a very simple explanation without the technical details, but which tries to show at least the general idea behind each principle.


Relativity

Relativity says that there is no privileged, "objective" viewpoint for certain observations. Let's imagine that two ships are in the sea in a very dark night, and that each of them only has a positioning light at the tip of the mast. The night is so dark that neither points at the horizon nor the waves in the water can be seen to check on which direction each ship is moving. Under these conditions, any of the captains could say that his ship is the one that is "still", while the other ship is moving, or vice versa.

Similarly, for hundreds of years we have known that the earth is not "still" in the universe, but revolves around the sun. But, at what speed does the sun move? We need a reference view point to measure speed. We can measure the speed of the sun related to a distant object, like another star, the center of the Milky Way, the center of our star cluster, but there is nothing we could call "the absolute speed of the sun". Surely, we could measure the speed of the sun relative to the space surrounding it, which is not exactly empty but has some particles, but those particles are also moving. Any attempt to determine an absolute speed is useless, as there is always an alternative view point that is equally valid and that produces a speed that is completely different. Speed is relative.

Now, if things move relative to each other, then obviously their positions at a given time are also measured relative to each other. The universe, as we know it, has no "center" with special characteristics, and even if it had it, clearly it would be completely arbitrary to say that something is "up", "down", "to the left" or "to the right". In space, these things have no meaning. Position and direction are relative. Any measurement of them requires a frame of reference.

So far, so good, but things get a bit more complicated. Let's think for a moment, how do we measure time? We measure it with changes of position and speed: the time a certain thing takes to fall to the ground, the time a ray of light takes to move between two points, etc. A direct consequence of the fact that position and speed are relative, is that time is also relative. There is no "now" that is simultaneous to the whole universe. There is no big clock ticking for the entire universe at the same time. The following experiment has been repeatedly made: if we synchronize two very precise clocks, and then we move them, for instance, by taking one of them in a plane and making a trip around the world, while the other remains in the ground, when they are together again we can see that one of the clocks has measured a longer time than the other. If we let more time pass and we increase the speed, we obtain the typical image of an astronaut on a space trip that takes a few years for him, but that returns to earth and notices that hundreds of years have passed. While relativity of motion itself is a very old concept, Albert Einstein first showed that even time is not absolute.

Do not confuse this with subjectivity: in a movie I once heard a dialog like "for Einstein, everything was relative, for instance, half an hour with a person you love or half an hour waiting for the metro are not the same". This is not relativity, it is called subjectivity and it's a different story. Relativity says that, even with the most precise instruments, there are no "absolute" measurements, and different view points can produce measurements that are "objectively" different. This does not mean there is no objective truth, but rather that there are no privileged view points.

Uncertainty

Baseball could be played with a smaller ball, the size of a ping-pong ball, or with a larger ball, the size of a volley ball; it would look weird but the game would still be playable. The rules of the game at our scale, are the same for a small, medium or large ball. We could even figure out how to play a table version of a mini-baseball with small balls of 1 millimeter of diameter. The dynamics of the game would be the same: the ball is thrown, the ball describes a curve, the ball is hit.

However, on the scale of the very small, if we continue reducing the size of the ball until it is the size of an atom or the size of an electron, the scenario is radically changed. At atomic scale, all sort of things that would look very weird at our scale happen. An example of how different things are, when we see them at atomic scale, is the following: if we have in our living room a lamp with a control to make the light brighter or darker, we can fine-tune how much light there is in the room. If the lamp has a certain luminosity, we have the impression that we can always lower a bit the luminosity without turning off the lamp completely. This is just an illusion, because if we look closely and take a very precise source of light, then we could see that there is a point in which if we lower a little more the light, we switch it off; there is a limit under which there are no more intermediary intensity levels. Light has a minimal unit, indivisible, it is called a "photon". All subatomic particles have an indivisible unit called a "quantum", and a "photon" is a quantum of light. Because of this phenomenon, the theory that tries to explain how everything works at a very small scale is called quantum mechanics.

Now, let's get back to the baseball game. Now we no longer throw a rubber and leather ball but a very small particle, let's say an electron. This electron goes on its way to a miniaturized baseball bat with which we want to hit it. We need to know where is the electron, but to see it, we need light. The only problem is that light is made of photons, and the electron is so small that a photon hitting it will move it from its trajectory. We aim a miniature lamp to the electron to see it and we receive the photons back, but once we have received the photons we have altered the trajectory of the electron. We could try to use just a single photon but that would be enough to move the electron. We can try to make this photon less capable of moving the electron, by having less momentum, but the problem is that for doing that we should have to generate light with longer waves and that would not allow us to see the electron precisely.

Werner Heisenberg showed that if we built a machine to tell us with high precision were an electron is, this machine could not also tell us the speed of the electron. If we want to measure its speed without altering it we can use a different light but then we wouldn't know where it is. At atomic scale, no instrument can tell us at the same time exactly where a particle is and exactly at what speed it is moving. Clearly, we could try to stop the electron with a wall and in that way we would know exactly where the electron is and we would know that it is still in relation to the wall, but that has no predictive value and doesn't help us in our baseball game. When measuring, we create a distortion and we are always bound by a trade-off in the measurement of certain quantities.

Sometimes a related, but not equivalent example is given: to measure the pressure of the tires of a car, we have to let some air out. So, when the indicator reads 30.000 psi, actually the pressure is 29.999 psi or less. Measure implies interacting, and interacting implies a certain alteration. At our scale, that alteration does not matters, but when we go to the very small, this alteration is a very important part of the rules.

Incompleteness

In our normal life, there are many situations that have only two states. If you have to take a train at 8 o'clock in the morning, then there are two possibilities: either you catch the train, or you miss it; if you switch a light, it will either go on or off; the accused of a trial is declared guilty or not guilty, and so on. This logic of true or false is central in mathematics. One plus one equals two? True. Two plus two equals five? False. True or false - there are no other options.

There are many natural phenomena that have intermediate states, but logic only cares about phenomena with two possibilities: true or false, black or white, zero or one, on or off, etc. This is not a weakness of logic, but rather a definition of its scope, as for instance, botanics works with plants or geology with rocks, mathematical logic works with anything that has two states.

Logic is an old discipline whose fundamental tool is deduction or logical deduction. Let A=Mary was murdered with a knife, B=John was at Mary's place at 23:00, C=Mary died at 23:01, D=John knew Mary, E=a hair from John was found in Mary's hands, F=a knife was found with Mary's blood and John's fingerprint. A and B and C and D and E and F could imply: John is guilty. If we take G=John is left-handed and the stabs are right-handed, then A and B and C and D and E and F and G could imply: John is not guilty. Several premises are composed to reach a conclusion.

This deductive, "Sherlock Holmes"-type logic, provides the rules to compose certain facts and reach certain conclusions. For instance, if for something to be true it is necessary that other two things are true, and one of the latter is false, then the former has to be false. If to get snow we need rain and low temperatures, and the weather is warm, then there will not be snow. It's logic. There are several rules in mathematical logic that build a deductive system.

If this system is complete, then anything that is true is provable. Similarly, anything false is provable false. Kurt Gödel got the intuition that traditional mathematical logic was not complete, and devoted several years to try to find one thing, a single thing that was inside the mathematics but outside the reach of logic. Gödel searched, inside the rigidness of the laws of number theory in mathematics, a valid expression that could not neither be proven to be true nor to be false.

Let's consider this expression: "this sentence is false". If "this sentence is false" is false, then it is twice false, hence true ... and if "this sentence is false" is true, then we only need to read it to see it's false. In number theory, Gödel found a way of writing a statement p that is equivalent to "p cannot be proved", using numbers and mathematical formulas, something that is possible to handle using number theory, but that can neither be proved nor disproved within this system. To do it, he had to express many ideas as numbers and weave step by step a very long proof, but he was finally able to prove that mathematical logic is incomplete, this is, that there are true things that we could never prove that are true, and false things that we could never prove that are false.

Gödel's incompleteness means that the classical mathematical logic deductive system, and actually any logical system consistent and expressive enough, is not complete, has "holes" full of expressions that are not logically true nor false.

Undecidability

Alan Turing is considered as one of the fathers of computer science. He designed a model of how a computer works that is fundamental to computer science and showed that, for instance, a pocket calculator or a sophisticated video game console work in the same way: they read some data, check a table of rules and a memory and write some data back.

In a calculator, what is read is a key stroke, the table of rules are mathematical operations, memory is the partial result and what is written back is the final result. In a video game console what is read is the controller or joystick status, the rule table are the rules of the game, the memory is the game status and what is written back are the drawings on the screen. Memory, even when it can be very large, is never infinite and the behavior of the machine is completely determined by rules that are always obeyed, so this model is known as a "finite deterministic Turing machine".

Turing wrote many programs, and he worked in deciphering codes for the Allies during the Second World War. He quickly become aware that some programs "hang" and stayed working forever without going anywhere. This happened often when there was something strange given as input. For instance, it is possible to write a program to decipher codes made of numbers that represent the coordinates where an attack will take place, but if this program is accidentally fed with letters instead of numbers, then the program fails, as it is not prepared to this type of input. Even after solving the trivial problem of accepting only numbers, a new problem appears and it is that the coordinates could be inconsistent, for instance, they could represent a place that is farther to the north than the north pole, and so on. Every new problem solved can create new problems in the program.

If we want to write a "perfect" program, that never hangs, then one way is to test it to try all different inputs, but this is often impractical as there are too many combinations; besides this, there is a deeper problem and that's that even if a long time has passed, we can never know for certain if the program is still doing something useful or if it has "hung". Turing thought that this problem could be solved with the help of another program, and he tried to write a "superprogram" that would be able of examining another program and check if it was correct, in the sense that it would terminate at some point, instead of running forever. After several attempts, he started to suspect that it might be impossible to write such a superprogram, and finally he was able to prove that in general it is not possible to check if a program will halt.

Turing's halting problem is one of the problems that fall in to the category of undecidable problems. It says that it is not possible to write a program to decide if other program is correctly written, in the sense that it will never hang. This creates a limit to the verification of all programs, as all the attempts of building actual computers, usable in practice and different from Turing machines have been proved to be equivalent in power and limitations to the basic Turing machine. It is impossible to know if a program will "hang" under some circumstance, and modern programming techniques try to minimize the probability of this occurring, as well as ensuring a smooth recovery and avoiding data loss, but for non-trivial programs, there are usually no guarantees of correctness.

A "Blind Spot"?

These fundamental principles should not be taken as limitations to science, and they do not exclude the existence of an objective reality. They are rather limitations to some operations, such as making a measurement or working with formal logic, that have to be taken into account to understand natural phenomena.

There is a certain relationship between these principles. Both relativity and uncertainty arise from physics and are related to observations, while incompleteness and undecidability arise from mathematics and are related to the limitations of formalisms. Uncertainty and undecidability govern our capacity of making predictions, while relativity and incompleteness are related to the fact that references are necessary, but prevent us from doing certain operations. I would not like to try to take these relationships too far, instead, I would like to state another analogy.

The retinal nerves of a mammal's eye converge at a single spot, the optic nerve, which transmits impulses to the brain. This design has a disadvantage and that is that right in the point where the nerves meet, there is no sensitivity to light. This produces a "blind spot", a zone of the visual field where we cannot see. At the same time, it is curious that we usually cannot see that we cannot see. First, the blind spot is rather small; second, the brain compensates the image so we don't see a black disc floating in the air, and third, we have two eyes and their blind spots do not fall in the same visual area. We could use the fact that we know about the blind spot in the design of certain things, such as in the design of the instruments board of an aircraft, but besides that in daily life and for 99.9% of the population the existence of a blind spot has little practical implications.

Something similar happens with the laws we have discussed here. They certainly restrict observations and formalisms, but they do not mean we cannot make observations or formalizations. Even with relativity, if we are caught driving at 80 miles per hour in a 65 miles per hour zone, we will have to pay, even if relative to us the car was not moving, because we have agreed in a certain frame of reference. Even with uncertainty we can still play baseball as the uncertainty in the position of an object the size of a base ball is far smaller than what we can see with our naked eyes. Even when deductive systems are incomplete, incompleteness is not a problem to most mathematicians, and every year very complex and difficult proofs are shown without problems. Even with undecidability, high quality programs control high availability systems and most of the errors do not arise from arcane halting conditions, but rather from simple and avoidable programming mistakes.

The rules of the scientific game include relativity, uncertainty, incompleteness and undecidability. From the point of view of science, understanding these laws can lead us to new discoveries on how the universe works. Through this particular "blind spot", we can see.

Acknowledgments

Pepe Flores, computing engineer, entrepreneur, and also my professor, boss, partner and friend, in that chronological order, was the person from whom I first heard that these are the four most important laws in science. Ingmar Weber and several Kuro5hin readers also provided valuable feedback and comments.

Note: there is a spanish version of this article.

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Relativity, Uncertainty, Incompleteness and Undecidability | 200 comments (129 topical, 71 editorial, 0 hidden)
-1: garbage; as a budding mathematical physicist (1.00 / 25) (#4)
by I Did It All For The Noogie on Tue Aug 30, 2005 at 04:41:34 AM EST

and current Putnam fellow, I deplore this type of nonsense. I didn't read much and couldn't find your point. My guess is you read a pop-sci book, maybe Michio Kaku's Hyperspace or Green's (?) Elegant Universe, and now you fancy yourself a philosopher of science and expert on everything. Recommendation: Go to hell and fuck off. If that's unsatisfactory, put in 8-10 hard years of hard work getting a PhD in physics.

BTW its a "quanta of light" for singular.

I have discovered a truly elegant proof of the Generalized Colour Theorem, but alas, there are not enough bytes in the sig to write it.

this is interesting (2.88 / 9) (#6)
by circletimessquare on Tue Aug 30, 2005 at 04:46:23 AM EST

it's more philosphy than anything else: what is knowable and what is unknown

just please be aware of the flaming negativity you will find here

ignore it and look to the positive feedback


The tigers of wrath are wiser than the horses of instruction.

In what way (2.50 / 6) (#13)
by forgotten on Tue Aug 30, 2005 at 05:40:49 AM EST

is relativity an example of what we cannot know?

--

No. (2.66 / 15) (#18)
by benna on Tue Aug 30, 2005 at 06:00:40 AM EST

Let's start with Einstein.  He was absolutly not saying that there could be no objective truth.  In fact, Einstein was a Platonist.  While there isn't any single reference frame which constitutes the whole truth, all of the reference frames taken togeather constitute all of reality.  The idea that reletivity says otherwise is a fantasy.

Next Heisenberg.  The Copenhagen interpretation of Quantum Mechanics does support your view somewhat, but this is by no means the only view.  Some (such as myself) view quantum information as purely statistical in nature, and therefore it has no implications regarding the objectivity of the universe itself.  Uncertainty is uncertainty in measurement, not in reality.  In fact, if one believe the multiple universe interpretation of QM, the very same reasoning that applies to reletivity applies here.

Ah, Gödel.  Another staunch platonist.  In fact, he was a good friend of Einstein for this reason.  Gödel did not see his imcompeteness theorem as a limitation on numbers, but as a limitation on formalism.  He was completly confident that the numbers were complete, in the platonic realm, just not the formal system.

Turing's undecidablity is again a failure of logic and not a failure of empirical science.  Science would check all the inputs, and form a consenous, not look for a proof.  Proofs are in the realm of logic.  I think this is fundementally your problem.  You confuse formal logic with science.

All that said I only skimmed your article, so perhaps I am completly wrong about what you meant to say.
-
"It is not how things are in the world that is mystical, but that it exists." -Ludwig Wittgenstein

+1, Author has ability to use paragraphs correctly (1.44 / 9) (#37)
by Egil Skallagrimson on Tue Aug 30, 2005 at 09:02:54 AM EST


----------------

Enterobacteria phage T2 is a virulent bacteriophage of the T4-like viruses genus, in the family Myoviridae. It infects E. coli and is the best known of the T-even phages. Its virion contains linear double-stranded DNA, terminally redundant and circularly permuted.

But... (1.26 / 15) (#56)
by Big Sexxy Joe on Tue Aug 30, 2005 at 02:02:24 PM EST

These limitations only apply to ordinary humans.  The transhumans of the future will face none of these limitations.

I'm like Jesus, only better.
Democracy Now! - your daily, uncensored, corporate-free grassroots news hour
You are wrong (1.14 / 7) (#62)
by maxsilver on Tue Aug 30, 2005 at 05:38:15 PM EST

If two clocks are moved in different reference times, they maintain the same time. Okay, before you burn me, listen to the proof.

For you to move a clock into an area and at a speed that consists of a different spatial reference, and that would have a different physical effect on the mechanics of the clock, you have got to perform an euqivalent energy transformation on this clock to get it into this position.

So in the second frame of relative reference, the clock is going quicker, but because energy transference is not instantaneous, there is no way you can compare the clocks.

When you do manage to bring back the second clock into the original frame of reference, you are going to reverse the energy effect on the clock - if it waent faster, the journey back will make it slower, and vice versa.

Don't get me wrong, you are not moving back in time, rather, which the original reference has maintained its relative speed, the other one has changed. So the other clock is ticking faster, and by the time they land at the same spot, they are the same time once again.

Of course the is the matter degradatory effect of this travel on whatever material you transport, so this experiment can never produce accurate results if actually performed.

VOTE THIS ARTICLE DOWN (1.13 / 15) (#83)
by Sigismund of Luxemburg on Wed Aug 31, 2005 at 03:51:44 AM EST

The author has been modding helpful editorial comments down in his own story!
ANONYMISED
-1, article promotes fear, uncertainty and doubt (1.08 / 12) (#85)
by United Fools on Wed Aug 31, 2005 at 05:17:26 AM EST

We need certainty and dependability. This article makes our head spin and confuses us. we have too much in this world to fear already; look at the news today, everything is negative. We don't need more.
We are united, we are fools, and we are America!
Errors of Understanding (2.85 / 7) (#91)
by Morosoph on Wed Aug 31, 2005 at 10:18:04 AM EST

Relativity
Similarly, for hundreds of years we have known that the earth is not "still" in the universe, but revolves around the sun. But, at what speed does the sun move? We need a reference view point to measure speed.
In an accelerating frame of reference, such as where the earth is travelling around the sun, you need to invoke general relativity, where acceleration and gravity are comparable. Because of the different densities of the earth an the sun, you can tell the difference; the centre of rotation is nearer the sun, for a start.

Gravity and acceleration gives otherwise indistinguishable objects a frame of referrence, or else gyroscopes simply wouldn't work.

Special relativity holds for inertial frames; the above example isn't such a frame of reference.

Uncertainty

Now, let's get back to the baseball game. Now we no longer throw a rubber and leather ball but a very small particle, let's say an electron. This electron goes on its way to a miniaturized baseball bat with which we want to hit it. We need to know where is the electron, but to see it, we need light. The only problem is that light is made of photons, and the electron is so small that a photon hitting it will move it from its trajectory. We aim a miniature lamp to the electron to see it and we receive the photons back, but once we have received the photons we have altered the trajectory of the electron. We could try to use just a single photon but that would be enough to move the electron. We can try to make this photon less capable of moving the electron, by having less momentum, but the problem is that for doing that we should have to generate light with longer waves and that would not allow us to see the electron precisely.
The Heisenberg's uncertainty principle goes deeper than this. It's not just about measurablity, it's about the nature of reality itself. The particle itself has an indeterminate position or velocity (or equivalently energy/time, amoungst other pairings).

Incompleteness

Let's consider this expression: "this sentence is false". If "this sentence is false" is false, then it is twice false, hence true ... and if "this sentence is false" is true, then we only need to read it to see it's false. In number theory, Gödel found a way of writing a statement p that is equivalent to "p cannot be proved", using numbers and mathematical formulas, something that is possible to handle using number theory, but that can neither be proved nor disproved within this system. To do it, he had to express many ideas as numbers and weave step by step a very long proof, but he was finally able to prove that mathematical logic is incomplete, this is, that there are true things that we could never prove that are true, and false things that we could never prove that are false.
This is a misunderstanding of what the incompleteness theorem says. "This statement is false" is simply not well-defined. Incompleteness isn't about apparent paradoxes; rather, it means that if I wish to state a theorem, such as "the Continuum hypothesis holds true"; some of the theorems that I would wish to state could not be shown to be either true or false. If I extend my set of axioms to resolve the issue, there would be other unresolvable theorems.

Undecidability

Turing's halting problem is one of the problems that fall in to the category of undecidable problems. It says that it is not possible to write a program to decide if other program is correctly written, in the sense that it will never hang. This creates a limit to the verification of all programs, as all the attempts of building actual computers, usable in practice and different from Turing machines have been proved to be equivalent in power and limitations to the basic Turing machine. It is impossible to know if a program will "hang" under some circumstance, and modern programming techniques try to minimize the probability of this occurring, as well as ensuring a smooth recovery and avoiding data loss, but for non-trivial programs, there are usually no guarantees of correctness.
It is possible to prove some programs, and also to write software that will provably halt, or where the range of inputs over which it will not halt is known. It is simply not possible to do this for all cases. In particular, if I come to you with a program, there is no algorithmic method by which you could determine its halting status.

I voted -1 (1.66 / 6) (#94)
by bml on Wed Aug 31, 2005 at 11:07:51 AM EST

I think you make serious mistakes in each of the sections:

Relativity: The first part is not Einstein's relativity. Galileo started and Newton settled all that "frame of reference" thing. Then you speak about time not being absolute, because, you propose, "there are no absolutes". Well, Einstein actually found the one absolute: C (the speed of light). And it's precisely because C is invariant that time is not.

Uncertainty: The problem with observability is not that the measurement disturbs the momentum/position of the particle. This is a common misconception, but the urcentainty principle applies also to theoretical "ideal" measurements (those that don't affect the particle being measured). The underlying mathematical principles are rather hard to grasp, so your explanation is an acceptable simplification. But you should really indicate that it is a simplification.

Incompleteness: So wrong. You start hinting at the difference between traditional logic and fuzzy logid, but then, instead of going there, you unrelatedly mischaracterize Gödel's theorem. It has nothing to do with the "this sentence is false" thing. Not even with the "p cannot be proved" statement, which is, however, closer. The real deal is that systems (certain systems, actually) cannot be proved just with the system's own tools. You need to use things from outside the system. Your last phrase is just wrong.

Undecidibility: This one is plain wrong. I can write you a one-line program that will certainly halt for all possible inputs. Where's the undecidibility here? The "halting problem" does not apply to every single algorithm. Plus, undecidability is more than just the halting problem. Something decidable is something you can always solve using an algorithm. I don't see how undecidability can be considered a general law of physics, or the universe, or whatever.

In summary: I'm sorry, but you don't really seem to know these issues very well. Not well enough to write an article on them, at least.


The Internet is vast, and contains many people. This is the way of things. -- Russell Dovey

+1, but not because there weren't errors... (3.00 / 2) (#106)
by debillitatus on Wed Aug 31, 2005 at 06:17:23 PM EST

because there were. But it's not possible to write about these topics to the layman and not make a bunch of incorrect statements. I mean, hell, it's definitely no worse than Michio Kaku, and that bitch is in bookstores.

I do agree with you that these are pretty damned important concepts, and they are grossly misunderstood by the layman. Anything on them which is not completely wrong helps, IMHO.

Damn you and your daily doubles, you brigand!

This is not right.. (1.25 / 4) (#107)
by StephenThompson on Thu Sep 01, 2005 at 01:30:10 AM EST

These descriptions are wrong. I wish this hadn't been voted up as it just perpetuates some wacky ideas. I'm not going to even try to explain whats wrong, but just say that the characterizations made here are essentially gibberish. If people want to get a grip on this stuff, go take a course on the subject; or at least read a book from a physicist or computer scientist.

READ ME FIRST: ERRATA (3.00 / 5) (#109)
by chato on Thu Sep 01, 2005 at 04:43:43 AM EST

Several readers pointed to errors and suggested corrections after the article left the editing queue. I am keeping a page with ERRATA and an updated version with these errors corrected. Perfection does not exist :-) so if you find a mistake, please try to be specific about what do you think is still wrong after the errata and how it can be corrected.

Writing and editing this article has been quite an experience, there is a lot of flamebait, and sometimes plain rudeness, but also many helpful, constructive comments from well educated people. Thanks everybody.



Theories or the concepts behind them. (2.50 / 4) (#113)
by Saggi on Thu Sep 01, 2005 at 06:10:36 AM EST

This is a very good article. The subjects are not easy to handle, and the details usually takes many hours of study to understand.

It does contain minor errors but to my knowledge it's fairly good in describing the subjects. Unfortunately this kind of article is the source of spawning trolls and flamebaits. This is bad. I have read several comments and most goes into the "religious" type. Some of these subjects are hard to understand, but experiments show us they exist. The universe doesn't always behave as everyday logic wants us to believe.

One of the comments note that one constant is known in the relative universe; the speed of light. This is the kind of comments that are useful. Stuff like "this is just wrong..." are not really interesting.

There is one note I would like to make:

A great deal of these laws and rules are known to be wrong! They are theories that describe the universe as best we can. But they are not the universe it self! A theory is a law that describes some effect we can measure in the universe. Einstein's theories are known to be working for large scale effects in the universe, while quantum mechanics describe the tiny parts.

We do know that Einstein's theories and quantum mechanics contradict each other, so one (or both) is flawed. But they are still the best to describe the micro and macro cosmoses. And all experiments within each field prove them to be true.

This is the kind of flaws we have in our theories. But they are still the best we have.

Before Einstein developed his theory we used the laws of Newton. What law is the best? Newton's laws work in our daily life, while it starts to fail on the galactic scale. So in a sense Einstein's laws are a refinement of Newton's laws in regards to usability. On a daily scale both provide the same answers.

Today new theories (like string theory) try to patch the faults. This is the work physics do today. If we knew everything there would be no more to do...

But what does this have to do with the article? The article establishes some fundamental ways and issues all these laws need to address. Some ideas and concepts we need to be aware of. It doesn't matter if the laws themselves are faulty. It doesn't matter if we can describe them better (even thou anyone who wish to purchase the subject would need some more detail, that might be found in thousand of other papers).

It does matter that we need to address the concepts of; relativity, uncertainty, incompleteness and undecidability.

For those who wish to read more about the relativity theories, quantum mechanics and super strings, I recommend "The Elegant Universe" by Brian Greene. It's a good book for laypeople. (If you want the hardcore mathematics and physics, sign up for your local university and study for 15 years...)

-:) Oh no, not again.
www.rednebula.com
Light example (none / 1) (#115)
by Eustace Cranch on Thu Sep 01, 2005 at 09:26:38 AM EST

The example of turning down the dimmer switch on a light doesn't really make sense. Let's assume a fixed wavelength and measure the intensity by the average amount of energy emitted per second. This value can in theory be made arbitrarily small without switching the light off.

What you perhaps mean is that the emission gets lumpy, and some seconds there will be a burst, and other seconds there will not, but that's not quite the same thing. It's also a big leap from there to quantum mechanics (nothing in the experiment really suggests this is something fundamental about light).

Anal retentive mathematical logician chimes in (none / 1) (#118)
by city light on Thu Sep 01, 2005 at 04:24:26 PM EST

You say:

There are many natural phenomena that have intermediate states, but logic only cares about phenomena with two possibilities: true or false, black or white, zero or one, on or off, etc. This is not a weakness of logic, but rather a definition of its scope

I wouldn't limit its scope so readily - Boolean (true/false) logic plays an important role in lots of formal systems, including those used to describe and prove things about continuous phenomena, things with intermediate states.

There are many different logics, not just one 'mathematical logic', and some, for example intuitionistic and multi-valued logics, allow for various kinds of uncertainty in their propositions.

The logic you describe is classical propositional logic, and it is  complete! Various complete deductive systems exist for classical propositional logic. Incompleteness doesn't come into action until your formal system gains the power to model arithmetic - something like first-order predicate logic with axioms for Peano arithmetic, is required. And even then, the incompleteness arises from the attempt to describe arithmetic with a recursive set of axioms, and not from the system of logical deduction itself.

measurement (2.50 / 2) (#119)
by squirlhntr on Thu Sep 01, 2005 at 06:09:45 PM EST

your article is pretty incorrect, as others have pointed out.

however, i would like to mention that those 4 things are all alike in a way: they all have to do with measurement. you measure, you get a result. mathematicans use the same thing, except they call it theory and proof.

somehow, those 2 actions are linked into everything and all the theories are in agreement: you can boil some (recursively enumerable) set down to 2 things, construct your axioms from that, and you're set. but you can't reduce everything to 1 thing (i.e., true, because then you have to assume there is a not-true, and you're back to where you started), because it makes no sense.

'no thing' is still a thing. this is the point of Godel/Turing's work and there is no way around it, or to be fair, it cannot be known whether there is a way around it or not.

this is also why in math not all things are true or false. some are both, or neither, or either. but chances are you didn't work on those problems in school because the fascists wanted to be sure they could grade you on some scale, and if they gave you an unsolvable problem, or even better, a true problem that is unprovably true, or unprovably unprovable in any sense, it would fuck up their master plan.

it is such a great folly, to think there is One Answer to Everything...

don't your forget your thermodynamics! something cannot come from nothing.

you are on the right track, and i'm not going to say you need a PhD in this bullshit to understand it, but i do suggest you read Godel's and Turing's original papers, several times, and work through it all. lots of other papers all hit on the same thing, as well: Cantor's work. the "No Free Lunch Theorem", P/NP (yet to be proven... hehe), Godel's relativity paradox, etc. etc. The problem exists in a billion forms, of which your 4 theories are only one explanation of them. and yet, in some way, they are all the same. the original papers that discuss these things are not as difficult as they might seem and then you really will understand the "fundamentals of the universe" as much as any professional dickheaded scientist.

it's not complicated. just read the original papers.


Nonsense (2.33 / 3) (#124)
by trhurler on Thu Sep 01, 2005 at 10:37:40 PM EST

First of all, these are four single points taken from three separate fields, and they are by far not 'the most important' things in those fields.

Second, your definition of undecidability is completely wrong. Computers don't have letters. They only have numbers. A letter is simply a representation on your screen of what to the computer is in fact a number. Undecidability has nothing to do with input validation, and everything to do with the fact that some algorithms' exit conditions cannot be guaranteed to occur for given inputs unless you actually run the algorithm or another algorithm of equivalent complexity on those inputs. That said, it is not a significant problem anymore, and hasn't been for a long time, because algorithms are no longer the core problem of computing.

Third, your description of Godel's work is both wrong and hyperbolic. What he actually proved is that in any system with finite axioms, there are statements which cannot be proven or disproven. It sounds a lot more impressive than it is. In reality, again, it has basically no impact on the real world whatsoever. Moreover, it is relatively trivial to simply establish an additional axiom that will enable you to work around the problem, and there is no shortage of axioms to be adopted or ways of verifying their accuracy without the use of logic(empirical data, usually) in any case that actually applies to the real world and not simply to some meaningless abstraction.

Fourth, uncertainty does not mean the particle doesn't have a position or a velocity(or both.) It means we cannot measure one without disturbing the other, thus rendering said other unmeasurable at the point of the first measurement. It isn't really an interesting statement, because it quite likely says more about the likely somewhat wrong nature of the math we're using to describe the subatomic world than about reality.

Fifth, relativity is not actually certain. It is easy to say "there is no privileged point of view" if you've never seen one, but that doesn't mean you've proven there isn't one. Yes, it is true that either here or there as we know it within our universe, there is no priviliged point of view. Human beings have an amusing capacity for seeing a limited distance and making grand pronouncements about all of existence, known and unknown alike.

What are the most important principles in mathematics and science(they are not the same,) for real? We don't know for sure, but it is quite likely that a slightly revised thermodynamics is one, that the real situation(presently unknown) regarding P->NP is another, and that we don't know enough to say much else.

--
'God dammit, your posts make me hard.' --LilDebbie

math vs. meta-math (none / 0) (#143)
by HKDD on Fri Sep 02, 2005 at 02:34:26 PM EST

In your errata article, you write certain important theorems cannot be proved.

Which ones?(rhetorical question)

In the same paragraph you write it is possible to construct an arithmetical statement which, if the theory is consistent, is true but not provable or refutable in the theory.  So, the assumption is that if you have a well formed formula in the Principia Mathematica, you must have an "arithmetical statement."  But that is false.  The statement "p cannot be proved," however "represented " or "encoded" in the PM, is NOT an arithmetical statement.  Arithmetical statements have a certain characteristic:  They are statements about numbers.  This "p cannot be proved" statement is NOT a statement about numbers.

But, I hear you saying "p cannot be proved" IS a statement about numbers if you take the time to read Godel's paper(suprise, suprise, there is no actual "incompleteness theorem").  I answer, maybe it is a statement about numbers, but the only way one can interpret it as "true" is to use meta-mathematical considerations, which would NOT be part of ANY arithmetical system.

Let me put it another way:  see footnote 9 in the text

In other words, the above-described procedure provides an isomorphic image of the system PM in the domain of arithmetic, and all metamathematical arguments can equally well be conducted in this isomorphic image

Here's another quote from the paper:  So the proposition which is undecidable in the system PM yet turns out to be decided by metamathematical considerations.

This is where Wittgenstein et. al. dig into the interpretation that "mathematical knowledge may be incomplete."  Why is it the case that ALL metmathematical arguments can be conducted in the PM?(rhetorical question, believeing in Platonic ideals makes the argument seem "intuitionistically unobjectionable," however, accepting the efficacy of logical positivism makes the argument very objectionable)

PM may be isormophic with the domain of arithmetic, but why would arithmetic be isomorphic with meta-arithmetic?(yet another rhetorical question)

In any case, EVERY MATHEMATICAL THEOREM is PROVABLE.
You can't escape the danger!

Is it logic? (none / 0) (#168)
by xee on Tue Sep 06, 2005 at 01:19:35 AM EST

"If to get snow we need rain and low temperatures, and the weather is warm, then there will not be snow."

Lets look at this a little closer...

At a most general level, this is an implication, P -> Q

P = to get snow we need rain and low temperatures, and the weather is warm
Q = there will not be snow

P can be further decomposed into a conjunction, ((R AND L) -> S) AND T

R = there is rain
L = there is a low temperature
S = there will be snow = NOT Q
T = the weather is warm = NOT L

Remember, P = ((R & L) -> S) & NOT L

Refactoring these statements we can see the argument structure...

Premise 1:  (R & L) -> NOT Q
Premise 2:  NOT L
conclusion: Q

This is an invalid argument.  Q can not be concluded from these two premises.  This argument employs the fallacy I like to call Modus Foolins.  With an implication, P -> Q, negating the P does NOT imply the negation of Q.  In other words, this type of argument is completely bogus...
P -> Q
NOT P
therefore, NOT Q.

For example, allow me to state that Q = NOT P
This yields the following...
P -> NOT P
NOT P
therefore NOT NOT P,
or just P.
Here we have allowed a contradiction, P & NOT P, which is the definition of invalid!


Proud to be a member.

Goedel's theorem (none / 0) (#182)
by Shubin on Wed Sep 07, 2005 at 07:00:12 AM EST

It is funny, but the article shows the COMPLETE MISUNDERSTANDING of the great theorem.
This text says : "the classical mathematical logic deductive system, and actually any logical system consistent and expressive enough, is not complete"
What is even more funny is the fact that Kurt Goedel himself proved the fact that the "classical mathematical logic deductive system" IS complete. This was his first work on a subject. Then he tried to prove the completeness of a formal arithmetic as a next logical step. But he failed.
Few days ago I had a discussion of a Goedel theorem and its meaning with a friend. After a long ICQ chat we came to a conclusion that this really interesting and complicated theorem is usually misunderstood by non-mathematicians.

I have a question for you all : anybody knows who was the first idiot who told others that Goedel's theorem proves incompleteness of ANY axiomatic system, including plain real world logic ? Everyone else just repeated his stupidity, caring not about even reading the theorem text at all. No wonder. But who started this myth ?


And "botanics" ... (none / 0) (#183)
by kevhito on Wed Sep 07, 2005 at 10:28:46 AM EST

... is what, exactly?

Bravo Sir (none / 0) (#200)
by CookTing on Thu Nov 03, 2005 at 06:48:51 PM EST

That's the best thing I've read since Kurt Vonnegut. And, quite possibly, the most well-written piece I've ever read on the intarwebs.

Relativity, Uncertainty, Incompleteness and Undecidability | 200 comments (129 topical, 71 editorial, 0 hidden)
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