It is my opinion that Occam's Razor is a dangerous two edged sword that is more likely to decapitate the user than do any good. Occam's Razor is an assertion that the simplest explanation for a thing is the most likely explanation to be correct. Wikipedia defines it specifically as, "Among competing hypothesis, the one with the fewest assumptions should be selected." There are several problems with this, but the most important thing to understand about Occam's Razor is that is it nothing more than a maxim, or an idea that people like to live by because it sounds good.
The first problem with Occam's Razor is the people that like to cite it. They are almost always fairly well educated people, often with a background in science. It is especially popular when comparing scientific explanations with religious ones, and it also sees a lot of action in the comparison of more esoteric scientific theories (especially ones that border on pseudoscience) with accepted theories. The problem is, Occam's Razor is not science! Occam's Razor does not employ the scientific process at all. In fact, if you asked any real scientist about the Wikipedia definition ("Among competing hypothesis, the one with the fewest assumptions should be selected."), the response would be that only the correct hypothesis should be selected, and if one has not been proven correct, none should be accepted. A sufficiently disproved theory should obviously be discarded, but a theory that has not been disproved should not be eliminated merely because another one seems to be simpler. Unfortunately, the most common people to cite Occam's Razor are people who should know better, and the result is that it is treated by large parts of the scientific community as some kind of natural law. The fact, however, is that it is not. If Occam's Razor was correct, quantum theory could not be.
The second problem with Occam's Razor is that it is wrong. Quantum theory is a great example of this. Before the late 1800s and early 1900s, classical physics reigned supreme. There were a few unanswered questions, but there were several very simple explanations that only made a few small assumptions. When a particular group of scientists (including people like Albert Einstein, Neils Bohr, and Erwin Schrödinger) started digging deeper though (instead of just accepting the much simpler explanations), they found that around the level of atoms things start working differently from classical physics. Quantum physics is extremely complex. It has to rely on tons of assumptions at the lowest levels. It also can be proven with evidence, despite the reliance on more assumptions and greater complexity. Occam's Razor completely missed the mark on that one, and it has missed the mark very consistently across many different fields of science. Occam's Razor sounds good, but it is absolutely useless when put to practical use. Occam's Razor suggests that we should accept theories based on the probability, relative to other theories, that they are correct. It measures that probability purely based on complexity. It ignores things like observation and experimental evidence. Honestly, I cannot respect a scientist that treats Occam's Razor as anything other than an amusement.
What it comes down to is that complexity is not a good metric for determining if a theory is correct or not. Unfortunately, Occam's Razor has been applied to far too many good ideas in the past. There are plenty of old, discarded theories, like the luminiferous aether, as well as theories that are routinely discarded without any attempt at disproving them, commonly seen in alternative medicine and sometimes in religion, that have not been disproved to the same degree of rigor as has been expected for more popular theories. Many of these are thrown out merely because they violate Occam's Razor. The problem is that discarding theories purely on the doctrine that complexity is an indication of low quality may be cutting our own throats.
Consider this: The luminiferous aether was an 1800s theory about the propagation of electromagnetic waves in void (the vacuum of space, for example). It was easily observed that kinetic waves required some kind of medium to propagate through. Waves in water is the simplest example, but sound waves in air is also a good one. It was reasoned, since proof of visible light's wave qualities had been demonstrated, that light and other forms of EM waves must have some kind of "aether" that they propagate through, like kinetic waves in water or air. This theory was widely accepted, and this "luminiferous aether" was assigned a list of invented properties and attributes. Near the end of the 1800s, many of the apparently most important properties of the aether were disproved. The theory had lost nearly all of its followers by 1900. The thing is, none of the disproved properties were essential to central idea. The single essential requirement was that the aether was the medium through which light and other wavelengths of EM traveled. This requirement was never disproved. All that was disproved was that the aether does not have the same properties as water or air. The theory lost popularity largely because it did not fit the mold that scientists had designed for it, and it was more complex than a competing theory that EM could propagate itself, without the need for a medium to propagate through. The fact is, the theory of the luminiferous aether has never been disproved to anywhere near the degree of evidence that would be required to disprove string theory, despite the fact that there are other observed wave phenomenon that suggest the aether should exist, while there is no evidence whatsoever supporting string theory, aside from the fact that it seems to be internally consistent with itself.
Where is the problem with this though? Why should we be worried about theories coming and going? What differences does it make if we believe in the aether or in self propagating waves, so long as the math is consistent? The answer is: It makes a huge difference. What if the luminiferous aether does exist? What if the main reason science has not advanced significantly (compared to the late 1800s to early 1900s) over the last almost 100 years now is that we are following a dead end, because we discarded the correct theory (or at least a more correct one)? What if there are some totally awesome new discoveries waiting right around the corner, but that corner is back in the late 1800s, because we went down the wrong road over 100 years ago? What if our liberal use of Occam's Razor is constantly cutting off lines of inquiry that could lead us to things like extremely cheap power, teleportation, cheap and safe space travel, dramatically better medicine, more efficient and less destructive food production, and all sorts of other things that could dramatically improve us and everything around us? Modern science is a mad rush forward, using Occam's Razor as a machete to clear the path, but maybe we should slow down and take a good look at what we are about to mow down. Maybe it is time to look behind, to see what gems we might have missed in our haste. It is time to consider that maybe the reason we are having such a hard time solving our problems is that the answers have been trampled and left in the clutter behind us.
It is time to throw away Occam's Razor and wield a more precise tool. Maybe a sickle would be more appropriate, since that is what is used for harvesting, and when harvesting, the accepted process includes close scrutiny of what has been cut down, only throwing out those things that have been carefully weighed and proven to have no value. Instead of haphazardly throwing to the side anything that is hard to think about or less novel than the rest, we should be carefully reaping and collecting, and then methodically sifting through the harvest, searching for the valuable gems. By taking things slower and considering even the ideas we don't like, we might be able to advance faster. In fact, many parts of quantum theory were hated by those who discovered them, and at least two great scientists of the time spent many years of their lives trying in vain to disprove their own discoveries. Maybe the reason science advanced so quickly during that period was that scientists then had less fear of complexity and were more willing to follow leads that they did not like. Real science is not about simplicity, convenience, or proving your own beliefs to be correct. Real science deals with complexity when it arises, presses on even when it is not convenient, and changes its beliefs when it discovers them to be wrong. Occam's Razor diverts science down paths that promise easy success but ultimately lead to dead ends.
10 March 2016
24 February 2016
The Problem with Fairness
The concept of fairness is a major problem for humans. It causes massive amounts of inefficiency, it is economically harmful, it causes a great deal of unnecessary poverty, and it is inherently hypocritical.
Fairness causes a great deal of problems for queuing theory. Queuing, in the simplest terms, is a group of things lining up for something. In some places, a line of people waiting for some kind of service (a line at the bus stop or a checkout line at the store, for example). In computer science, a queue is a container that stores things and gives them to the program on a first-come-first-served basis. A major pet project in queuing theory is how to best arrange lines of people waiting for a service, when there are multiple providers of that service operating at the same time. Bank tellers and grocery store checkouts are a perfect example of this. The question is, what is the most efficient queue for this kind of situation? It turns out the answer is the kind of queue found in grocery stores. Each service provider (cashier, in this case) has a queue. When a person comes to checkout, that person goes to the shortest queue (that being the one with the least amount of work, so one person with 100 items adds more length than several people with only a few items each). This queuing strategy minimizes the amount of time spent in the queue, and it is currently considered the most efficient kind of queue, however it is not perfect. The problem is that it is not fair. With this queuing strategy, people are not always served in the order they arrived. Maybe one cashier has an item that is not scanning correctly, so the next customer has to wait an extra 30 seconds. While he is waiting, the guy that got there after him, who is in a different queue, gets served first. Yes, it seems trivial, because it is trivial. It also really bothers people. It bothers people so much, in fact, that they will spend several minutes trying to figure out which line is likely to get them to the cashier first. Some people are so bothered by this that they will pick a line and then peak around to see if another line is going faster, sometimes switching lines several times before they end up checking out. People would prefer a single serial (just one line) queue over a parallel (multiple lines) queue, even though the second one will almost always save everyone time, because the serial queue is never unfair.
Fairness is economically harmful for several reasons. The first one is that people tend to define it in whatever way is most beneficial to themselves. It is economically harmful, because it encourages people to thing about things like ownership emotionally instead of logically.
The most shallow definition is contractual fairness. This is the idea that anything you can get someone to agree to is fair. The claim that something is only worth what people will pay for it is an example of contractual fairness. A more nefarious example is the extension that something is always worth the most people will pay for it. Contractual fairness would assert that if I was the only person in a desert with access to a well, and you had exactly $100, it would be fair for me to charge you your entire $100 for a pint of water. It does not take into account need or ability to afford. In a work environment, contractual fairness asserts that any wage you pay me is fair, regardless of my productivity, because I agreed to that wage. It does not care whether I had other options, and it does not care if I only took the job because the only other option was starving, Contractual fairness even allows for slavery and coercion, because even if someone is threatening your life, you have the choice to give it up instead of agreeing to the contract.
Implied fairness, also defined as natural fairness, is fairness as we observe it in nature. "The early bird gets the worm," because it has already eaten it by the time the other birds have arrived. This is where "first come, first served" comes from. "Possession is nine tenths of the law," because it is hard to prove ownership, but it is easy to see who currently possesses something. The person who makes something implicitly owns it, for the same reason. When your parents die, you get what they owned, because you know more about what and where it is than anyone else. Unfortunately, the strong also rule over the weak. It is better to let the weak die, because they are a liability to everyone else. It is better to kill your enemies, because it makes you and your offspring a larger relative portion of the gene pool. Implied fairness is actually worse than contractual fairness, because it is based on nature, and nature does not actually care about fairness at all.
Moral fairness is about what people "deserve." It asserts that people who work hard deserve more rewards than people who do not. People deserve to use the "fruits of their labor" however they see fit. If you take something I own, it still belongs to me. "All men are created equal." I should inherit what my parents own when they die, because I am their offspring. At the same time though, people who don't work should be allowed to starve and die. If someone is in poverty, it must be their own fault. Everyone who commits a crime, whether that crime actually harms anyone or not, should be locked away in jail. The problem with moral fairness is that it is completely objective. It does not distinguish between a lazy person and a crippled person. Both don't work much, and thus both deserve poverty. Someone who steals because they are starving to death is no better than a rich person who embezzles. Moral fairness uses social conventions to define an objective law of fairness, but it takes no effort to account for the many special circumstances.
Civilized fairness arrives from deep contemplation and critical thought. It is not based on personal feelings or emotions. This kind of fairness asserts that everyone should be given an equal chance (instead of asserting that everyone is given an equal chance, like moral fairness does). It cares very deeply about situation. Disability is not the same a laziness. Contracts are not inherently fair, and agreeing to a contract under duress (including the duress of threat of starvation) does not count as voluntary. Charging the most people are willing to pay is not inherently fair, and sometimes things are worth more than the highest bidder is offering. Acquisition does not inherently grant full rights of ownership, and owning something does not necessarily give exclusive rights of possession. The creation of a thing does not inherently grant the creator possession or control of it. Death is never a fitting punishment for laziness, and laziness is not always even the fault of the person who is lazy. Unlike the other three definitions, civilized fairness also considers more than just the individual. If first-come-first-served does not minimize average wait time, then another strategy is more fair, because it wastes less time total. Civilized fairness allows the late guy to get to the front of the line in an airport, so he won't miss his flight, even if it is his own fault that he is late, because it is better to minimize the number of people who miss flights than it is to save 30 people waiting to go through security one minute just because they got there first. Civilized fairness says it is better to have a simpler welfare system, despite the fact that it is more easily abused, because the cost of abuse is less than the cost of the extra administration required to maintain a complex welfare system. It also says that income should be reasonably well distributed, because that will ultimately enrich everyone (including the rich) and improve the economy, while contractual and moral fairness will ultimately harm the economy by starving the majority of resources. Civilized fairness takes a level of thinking that is beyond what most people are willing to do. It tends to have a higher up front cost, and it sometimes uses unintuitive strategies, but ultimately it pays of with enormous interest.
In the end, all four types of fairness have their place. Implied fairness is intuitive, because we see it constantly in nature. Contractual fairness is important to the proper function of economy and government. Moral fairness is important for keeping people motivated and useful. Without civilized fairness though, we are little more than animals who have managed to make hive-like social constructs, without hive-like anti-individualism. Civilized fairness tempers selfishness with the need to get along. It can allow us to thrive as a group, without sacrificing our individuality. It allows for the beneficial parts of the other types of fairness, without the parts that are ultimately harmful. Civilized fairness is a higher way of thinking and living, and without it, I don't think any large civilization can last long term, without eventually collapsing.
Fairness causes a great deal of problems for queuing theory. Queuing, in the simplest terms, is a group of things lining up for something. In some places, a line of people waiting for some kind of service (a line at the bus stop or a checkout line at the store, for example). In computer science, a queue is a container that stores things and gives them to the program on a first-come-first-served basis. A major pet project in queuing theory is how to best arrange lines of people waiting for a service, when there are multiple providers of that service operating at the same time. Bank tellers and grocery store checkouts are a perfect example of this. The question is, what is the most efficient queue for this kind of situation? It turns out the answer is the kind of queue found in grocery stores. Each service provider (cashier, in this case) has a queue. When a person comes to checkout, that person goes to the shortest queue (that being the one with the least amount of work, so one person with 100 items adds more length than several people with only a few items each). This queuing strategy minimizes the amount of time spent in the queue, and it is currently considered the most efficient kind of queue, however it is not perfect. The problem is that it is not fair. With this queuing strategy, people are not always served in the order they arrived. Maybe one cashier has an item that is not scanning correctly, so the next customer has to wait an extra 30 seconds. While he is waiting, the guy that got there after him, who is in a different queue, gets served first. Yes, it seems trivial, because it is trivial. It also really bothers people. It bothers people so much, in fact, that they will spend several minutes trying to figure out which line is likely to get them to the cashier first. Some people are so bothered by this that they will pick a line and then peak around to see if another line is going faster, sometimes switching lines several times before they end up checking out. People would prefer a single serial (just one line) queue over a parallel (multiple lines) queue, even though the second one will almost always save everyone time, because the serial queue is never unfair.
Fairness is economically harmful for several reasons. The first one is that people tend to define it in whatever way is most beneficial to themselves. It is economically harmful, because it encourages people to thing about things like ownership emotionally instead of logically.
The most shallow definition is contractual fairness. This is the idea that anything you can get someone to agree to is fair. The claim that something is only worth what people will pay for it is an example of contractual fairness. A more nefarious example is the extension that something is always worth the most people will pay for it. Contractual fairness would assert that if I was the only person in a desert with access to a well, and you had exactly $100, it would be fair for me to charge you your entire $100 for a pint of water. It does not take into account need or ability to afford. In a work environment, contractual fairness asserts that any wage you pay me is fair, regardless of my productivity, because I agreed to that wage. It does not care whether I had other options, and it does not care if I only took the job because the only other option was starving, Contractual fairness even allows for slavery and coercion, because even if someone is threatening your life, you have the choice to give it up instead of agreeing to the contract.
Implied fairness, also defined as natural fairness, is fairness as we observe it in nature. "The early bird gets the worm," because it has already eaten it by the time the other birds have arrived. This is where "first come, first served" comes from. "Possession is nine tenths of the law," because it is hard to prove ownership, but it is easy to see who currently possesses something. The person who makes something implicitly owns it, for the same reason. When your parents die, you get what they owned, because you know more about what and where it is than anyone else. Unfortunately, the strong also rule over the weak. It is better to let the weak die, because they are a liability to everyone else. It is better to kill your enemies, because it makes you and your offspring a larger relative portion of the gene pool. Implied fairness is actually worse than contractual fairness, because it is based on nature, and nature does not actually care about fairness at all.
Moral fairness is about what people "deserve." It asserts that people who work hard deserve more rewards than people who do not. People deserve to use the "fruits of their labor" however they see fit. If you take something I own, it still belongs to me. "All men are created equal." I should inherit what my parents own when they die, because I am their offspring. At the same time though, people who don't work should be allowed to starve and die. If someone is in poverty, it must be their own fault. Everyone who commits a crime, whether that crime actually harms anyone or not, should be locked away in jail. The problem with moral fairness is that it is completely objective. It does not distinguish between a lazy person and a crippled person. Both don't work much, and thus both deserve poverty. Someone who steals because they are starving to death is no better than a rich person who embezzles. Moral fairness uses social conventions to define an objective law of fairness, but it takes no effort to account for the many special circumstances.
Civilized fairness arrives from deep contemplation and critical thought. It is not based on personal feelings or emotions. This kind of fairness asserts that everyone should be given an equal chance (instead of asserting that everyone is given an equal chance, like moral fairness does). It cares very deeply about situation. Disability is not the same a laziness. Contracts are not inherently fair, and agreeing to a contract under duress (including the duress of threat of starvation) does not count as voluntary. Charging the most people are willing to pay is not inherently fair, and sometimes things are worth more than the highest bidder is offering. Acquisition does not inherently grant full rights of ownership, and owning something does not necessarily give exclusive rights of possession. The creation of a thing does not inherently grant the creator possession or control of it. Death is never a fitting punishment for laziness, and laziness is not always even the fault of the person who is lazy. Unlike the other three definitions, civilized fairness also considers more than just the individual. If first-come-first-served does not minimize average wait time, then another strategy is more fair, because it wastes less time total. Civilized fairness allows the late guy to get to the front of the line in an airport, so he won't miss his flight, even if it is his own fault that he is late, because it is better to minimize the number of people who miss flights than it is to save 30 people waiting to go through security one minute just because they got there first. Civilized fairness says it is better to have a simpler welfare system, despite the fact that it is more easily abused, because the cost of abuse is less than the cost of the extra administration required to maintain a complex welfare system. It also says that income should be reasonably well distributed, because that will ultimately enrich everyone (including the rich) and improve the economy, while contractual and moral fairness will ultimately harm the economy by starving the majority of resources. Civilized fairness takes a level of thinking that is beyond what most people are willing to do. It tends to have a higher up front cost, and it sometimes uses unintuitive strategies, but ultimately it pays of with enormous interest.
In the end, all four types of fairness have their place. Implied fairness is intuitive, because we see it constantly in nature. Contractual fairness is important to the proper function of economy and government. Moral fairness is important for keeping people motivated and useful. Without civilized fairness though, we are little more than animals who have managed to make hive-like social constructs, without hive-like anti-individualism. Civilized fairness tempers selfishness with the need to get along. It can allow us to thrive as a group, without sacrificing our individuality. It allows for the beneficial parts of the other types of fairness, without the parts that are ultimately harmful. Civilized fairness is a higher way of thinking and living, and without it, I don't think any large civilization can last long term, without eventually collapsing.
20 February 2016
Modern Superstitions
A long time ago, people believed some crazy things. Acorns could bring good luck. Amber on a necklace could protect from disease. There were a whole bunch of ways to figure out who you were going to marry. Kissing a certain stone in Ireland will give you a glib tongue. There are several involving luck and black cats, depending on the culture. Clover can protect you from evil magic. A dried frog worn in a silk bag prevents epilepsy. And there are hundreds more. With the prevalence of science in our society, though, it is easy to find just as many modern superstitions, often based on misunderstandings about science.
A big enough particle accelerator could create a black hole that would swallow the Earth or at least a large portion of it. This superstition is based partially in truth. Scientists have predicted that the particle accelerator at CERN is indeed powerful enough to create microscopic black holes. If particles collide with enough energy, they can form a system with such high density that they technically form a black hole. Fortunately, black holes that small do not have enough mass to remain stable, and they evaporate almost instantly. Perhaps a bigger particle accelerator could create a stable black hole? It is incredibly unlikely, and it certainly could not happen on Earth. The problem is that black holes require enormous amounts of mass to be stable. All of the mass on Earth is not enough to create a stable black hole. The Sun may have enough mass to make a stable black hole, but it currently has far too much energy to collapse into one. There are three things that make it impossible for a particle accelerator on Earth to make a black hole that is stable and large enough to be a problem. The first is lack of sufficient matter. All of the mass of the Earth is still not enough. The second is too much energy. Even if Earth did have enough matter to make a black hole, if you tried to crush it down small enough, it would have so much energy that it would immediately explode when it was released. Third, the particle accelerator necessary to accelerate enough mass to create a stable black hole would not fit on the Earth, and it would probably not even fit on Jupiter, the largest planet in our solar system, and it is over 300 times larger than the Earth. This superstition has lost a lot momentum, since LHC scientists announced that the accelerator could not create dangerous black holes, but many people still believe it.
One of the two big ones is that "chemicals" are bad. Some people even claim that they don't believe in chemicals. This superstition is propagated by the media, which uses the term very narrowly to mean only chemicals that are dangerous. What they don't mention is that pretty much everything, even water, is a chemical or is made from chemicals. The source of this lack of understanding is school and parents that don't teach their children what the word actually means, though at this point the superstition is so entrenched that even most teachers don't know any better. What it really comes down to though, is that "chemical" means any element or compound in a homogeneous mixture. So, water is a pure mixture of the compound of one oxygen atom and two hydrogen atoms. Salt water is the homogeneous solution of salt molecules dissolve in water molecules. It turns out, using a loose definition, even mud and bread dough could be considered chemicals. In addition, most of the dangerous chemicals you hear about in the media are also extremely useful. Contrast this with the fact that the human stomach produces copious amounts of hydrochloric acid, which is an extremely dangerous chemical.
There are several other very popular modern superstitions, including ones involving "organic" foods and "GMO" foods. Both of these are based largely on a misunderstanding of science, but both are also based very loosely on true ideas. It is possible to make edible plants harmful by exposing them to certain substances while they are growing. The "organic" food movement is based on the idea that organic fertilizers are safer than industrial ones. It turns out this is not true though. Most organic fertilizers come from biological processes that are poorly understood, which makes it very difficult to predict what dangerous substances might be in them. Industrial fertilizers tend to be fairly pure chemicals that are known to contain nothing dangerous. Now, there are some environmental concerns with using more pure fertilizers, but they don't negatively impact the quality of the food grown with them. Overall though, this particular superstition is not likely to cause more harm than wasting your money paying more for the same product, because it has a special label on it.
The GMO foods superstition is likely to be very dangerous in the future. The superstition here is that using scientific methods to precisely alter the genetics of plants makes them dangerous to eat. It is true that this could happen. Certain alterations could cause a plant to produce toxins that it would not have otherwise. Doing this deliberately would be extremely expensive though, and the mandatory FDA testing would quickly reveal the problem. A more worrying concern is that alterations intended to improve crop properties (which is what most GMO is for) could accidentally have side effects that would produce toxins. The fact, however, is that this is very unlikely, and it would still be caught by the mandatory testing. It turns out that, if anything, GMO foods are probably safer than non-GMO foods! There are several reasons for this. The first is that the way plants evolve naturally (nope, even in nature, plant DNA does not just stay exactly the same forever) is far more likely to produce a toxic strain than the very focused modification of one specific gene or set of genes. The second is that, if this did happen, we would probably not know until a lot of people were being affected, because there are no mandatory tests for non-GMO foods. So, how could believing the GMO superstition actually be dangerous though? Simple: There are currently two popular purposes for genetically modifying food plants. The first is improving crop properties. This includes things like making crops more resistant to pests and diseases (as in, now we don't have to use dangerous pesticides to maintain good yields), making food plants produce more food faster, and making crops more resilient to poor growing conditions. If society as a whole rejects GMO foods, we will have go back to using seriously poisonous pesticides, and we may eventually have a hard time growing enough food to feed everyone. GMO allows us to use less land to grow the same amount of food, and it allows us to use land that would otherwise have little value for growing food. Without GMO, the Earth's population cap is dramatically smaller. The second use of GMO that is starting to get more popular (you probably won't see it in stores for a while, because it is still experimental) is making food plants healthier and more tasty. In the next few years, many GMO foods will start to be available that are significantly more nutritious and tasty than non-GMO foods. (And, Monsanto, who I very pointedly do not endorse, for ethical reasons, is using GMO to create selective breeding plans, for using the natural method to get the same genetic modifications without making the food GMO.)
Humans are naturally superstitious. Eventually old superstitions are revealed for the absurdities that they are, but at the same time as we are making fun of the superstitions of our ancestors, we are subscribing to equally absurd ones of our own. So next time you laugh at a crazy superstition from a long time ago, give a thought to the modern superstitions that are affecting your own decisions.
A big enough particle accelerator could create a black hole that would swallow the Earth or at least a large portion of it. This superstition is based partially in truth. Scientists have predicted that the particle accelerator at CERN is indeed powerful enough to create microscopic black holes. If particles collide with enough energy, they can form a system with such high density that they technically form a black hole. Fortunately, black holes that small do not have enough mass to remain stable, and they evaporate almost instantly. Perhaps a bigger particle accelerator could create a stable black hole? It is incredibly unlikely, and it certainly could not happen on Earth. The problem is that black holes require enormous amounts of mass to be stable. All of the mass on Earth is not enough to create a stable black hole. The Sun may have enough mass to make a stable black hole, but it currently has far too much energy to collapse into one. There are three things that make it impossible for a particle accelerator on Earth to make a black hole that is stable and large enough to be a problem. The first is lack of sufficient matter. All of the mass of the Earth is still not enough. The second is too much energy. Even if Earth did have enough matter to make a black hole, if you tried to crush it down small enough, it would have so much energy that it would immediately explode when it was released. Third, the particle accelerator necessary to accelerate enough mass to create a stable black hole would not fit on the Earth, and it would probably not even fit on Jupiter, the largest planet in our solar system, and it is over 300 times larger than the Earth. This superstition has lost a lot momentum, since LHC scientists announced that the accelerator could not create dangerous black holes, but many people still believe it.
One of the two big ones is that "chemicals" are bad. Some people even claim that they don't believe in chemicals. This superstition is propagated by the media, which uses the term very narrowly to mean only chemicals that are dangerous. What they don't mention is that pretty much everything, even water, is a chemical or is made from chemicals. The source of this lack of understanding is school and parents that don't teach their children what the word actually means, though at this point the superstition is so entrenched that even most teachers don't know any better. What it really comes down to though, is that "chemical" means any element or compound in a homogeneous mixture. So, water is a pure mixture of the compound of one oxygen atom and two hydrogen atoms. Salt water is the homogeneous solution of salt molecules dissolve in water molecules. It turns out, using a loose definition, even mud and bread dough could be considered chemicals. In addition, most of the dangerous chemicals you hear about in the media are also extremely useful. Contrast this with the fact that the human stomach produces copious amounts of hydrochloric acid, which is an extremely dangerous chemical.
There are several other very popular modern superstitions, including ones involving "organic" foods and "GMO" foods. Both of these are based largely on a misunderstanding of science, but both are also based very loosely on true ideas. It is possible to make edible plants harmful by exposing them to certain substances while they are growing. The "organic" food movement is based on the idea that organic fertilizers are safer than industrial ones. It turns out this is not true though. Most organic fertilizers come from biological processes that are poorly understood, which makes it very difficult to predict what dangerous substances might be in them. Industrial fertilizers tend to be fairly pure chemicals that are known to contain nothing dangerous. Now, there are some environmental concerns with using more pure fertilizers, but they don't negatively impact the quality of the food grown with them. Overall though, this particular superstition is not likely to cause more harm than wasting your money paying more for the same product, because it has a special label on it.
The GMO foods superstition is likely to be very dangerous in the future. The superstition here is that using scientific methods to precisely alter the genetics of plants makes them dangerous to eat. It is true that this could happen. Certain alterations could cause a plant to produce toxins that it would not have otherwise. Doing this deliberately would be extremely expensive though, and the mandatory FDA testing would quickly reveal the problem. A more worrying concern is that alterations intended to improve crop properties (which is what most GMO is for) could accidentally have side effects that would produce toxins. The fact, however, is that this is very unlikely, and it would still be caught by the mandatory testing. It turns out that, if anything, GMO foods are probably safer than non-GMO foods! There are several reasons for this. The first is that the way plants evolve naturally (nope, even in nature, plant DNA does not just stay exactly the same forever) is far more likely to produce a toxic strain than the very focused modification of one specific gene or set of genes. The second is that, if this did happen, we would probably not know until a lot of people were being affected, because there are no mandatory tests for non-GMO foods. So, how could believing the GMO superstition actually be dangerous though? Simple: There are currently two popular purposes for genetically modifying food plants. The first is improving crop properties. This includes things like making crops more resistant to pests and diseases (as in, now we don't have to use dangerous pesticides to maintain good yields), making food plants produce more food faster, and making crops more resilient to poor growing conditions. If society as a whole rejects GMO foods, we will have go back to using seriously poisonous pesticides, and we may eventually have a hard time growing enough food to feed everyone. GMO allows us to use less land to grow the same amount of food, and it allows us to use land that would otherwise have little value for growing food. Without GMO, the Earth's population cap is dramatically smaller. The second use of GMO that is starting to get more popular (you probably won't see it in stores for a while, because it is still experimental) is making food plants healthier and more tasty. In the next few years, many GMO foods will start to be available that are significantly more nutritious and tasty than non-GMO foods. (And, Monsanto, who I very pointedly do not endorse, for ethical reasons, is using GMO to create selective breeding plans, for using the natural method to get the same genetic modifications without making the food GMO.)
Humans are naturally superstitious. Eventually old superstitions are revealed for the absurdities that they are, but at the same time as we are making fun of the superstitions of our ancestors, we are subscribing to equally absurd ones of our own. So next time you laugh at a crazy superstition from a long time ago, give a thought to the modern superstitions that are affecting your own decisions.
12 February 2016
Tau vs Pi
Pi (π) is an amazing number that defines the ratio of the circumference of a circle to its diameter. It has important historical significance, because many civilizations knew that it existed, but until recently, none had been able to discover what it was to any degree of precision. In the modern world, we use
π all over the place. It turns out though, that there are some places where π just does not make sense. For example, the unit circle, used in trig and other more advanced mathematics, ends up with a lot of complicated and unintuitive fractions, and a single revolution around the circle is equal to 2π in distance. This is confusing to new students and makes a lot of the math more complicated. In fact, it turns out that 2π is used all over the place, and it may even be used significantly more often than π by itself.
A group of people have started advocating the use of an alternative to π. Tau (τ) is equal to 2π. It could be defined as the ratio of the circumference of a circle to its radius. A unit circle using τ makes more sense, because a single revolution is equal to τ. A quarter revolution is τ/4 (using π, it is π/2), a half a revolution is τ/2 (π), and 3/4s of a revolution is 3τ/4 (3π/2). With τ, the fraction of the circle is the fraction of τ, but with π, it is twice the fraction of the circle, which makes it more difficult to understand and complicates the math. So, this group supporting τ says we should switch from using π to using τ, because it simplifies the math.
It turns out that this is only part of the story though, and I am writing this, because I have not seen any evidence that anyone else fully understands the issue. The problem is not that τ always makes more sense than π. The problem is that we are using π wrong. To fully understand this, we need to define both π and τ, without reference to the other. In most debates on the subject, τ is defined as 2π, which is technically true but also misleading. τ is not merely 2π. Both τ and π are ratios relating the width of a circle to its circumference. Mathematically, τ = c/r, while π = c/d, where c = circumference, r = radius, and d = diameter. If you look closely, you might see why we end up using 2π all over the place. Look at it this way: π = c/2r -> 2π = c/r. It should be obvious by now. π is a ratio of the circumference to the diameter, but we are doing all of our math using the radius. Of course we have to multiply π by 2 all the time, because we are implicitly dividing the diameter by 2 nearly everywhere we use it!
The most obvious solution is to replace 2π with τ. This is certainly a valid solution, but it is not the only solution, and it is not necessarily the best or most sensible solution either. The other solution is to keep using π, but use d instead of r. This would even fix the unit circle, as radians are defined as the distance from 0 multiplied by r. π radians is not actually π. It is the distance πr, but the r is not written, because it is implied. If we replaced r with d, a full revolution would be exactly π (the d is implied this time). The only problem is that we could no longer call them radians, because the name comes from "radius," and we would be using the diameter.
So, why do we use r instead of d? This question stumped me when I took geometry in highschool. It did not make much sense. The reason is simple: When we find the area or perimeter of a square or rectangle, we use width (w) and height (h). We don't use w/2 or h/2. So why, with circles, are we always using d/2? It just does not make much sense. Even when we are calculating the area of a triangle, we use (w * h)/2, not w/2 * h or w * h/2. It does not make sense to use half the width of the circle when finding perimeter (circumference) or area, when don't use half lengths anywhere else. I recently realized why we use r instead of d, and the answer is rather disappointing. The definition of a circle is the following equation: r^2 = x^2 + y^2. This is the only place I can find where it makes significantly more sense to use half width over whole width. So why is it that we are using this one equation to define the normal case instead of using all of the others and defining this one as a special case? Honestly, I can see no reason for doing it this way, except perhaps that this is how it was done in the past. I don't happen to subscribe to the theory that tradition trumps logic and reason. If tradition does not make sense, it is time to trade it for something that does.
It turns out that this entire argument is almost pointless though. Using 2π all over the place works, and it is pretty entrenched in our mathematics. Outside of education, math and science are not going to suddenly change because someone decides it is better to do things a bit differently. Within education is where the "almost" comes in though. I am not the only person who noticed that using r instead of d does not make sense in the context of all of the rest of geometry. I am certainly not the only person who noticed that the unit circle does not make sense using π as we do. We could just replace 2π with τ, but that would only fix the unit circle and part of the math. It does not fix the underlying problem with using only half of the distance for circles, while we use the entire distance for everything else. If we really want to make education easier, we should keep using π, but switch to using d instead of r.
A group of people have started advocating the use of an alternative to π. Tau (τ) is equal to 2π. It could be defined as the ratio of the circumference of a circle to its radius. A unit circle using τ makes more sense, because a single revolution is equal to τ. A quarter revolution is τ/4 (using π, it is π/2), a half a revolution is τ/2 (π), and 3/4s of a revolution is 3τ/4 (3π/2). With τ, the fraction of the circle is the fraction of τ, but with π, it is twice the fraction of the circle, which makes it more difficult to understand and complicates the math. So, this group supporting τ says we should switch from using π to using τ, because it simplifies the math.
It turns out that this is only part of the story though, and I am writing this, because I have not seen any evidence that anyone else fully understands the issue. The problem is not that τ always makes more sense than π. The problem is that we are using π wrong. To fully understand this, we need to define both π and τ, without reference to the other. In most debates on the subject, τ is defined as 2π, which is technically true but also misleading. τ is not merely 2π. Both τ and π are ratios relating the width of a circle to its circumference. Mathematically, τ = c/r, while π = c/d, where c = circumference, r = radius, and d = diameter. If you look closely, you might see why we end up using 2π all over the place. Look at it this way: π = c/2r -> 2π = c/r. It should be obvious by now. π is a ratio of the circumference to the diameter, but we are doing all of our math using the radius. Of course we have to multiply π by 2 all the time, because we are implicitly dividing the diameter by 2 nearly everywhere we use it!
The most obvious solution is to replace 2π with τ. This is certainly a valid solution, but it is not the only solution, and it is not necessarily the best or most sensible solution either. The other solution is to keep using π, but use d instead of r. This would even fix the unit circle, as radians are defined as the distance from 0 multiplied by r. π radians is not actually π. It is the distance πr, but the r is not written, because it is implied. If we replaced r with d, a full revolution would be exactly π (the d is implied this time). The only problem is that we could no longer call them radians, because the name comes from "radius," and we would be using the diameter.
So, why do we use r instead of d? This question stumped me when I took geometry in highschool. It did not make much sense. The reason is simple: When we find the area or perimeter of a square or rectangle, we use width (w) and height (h). We don't use w/2 or h/2. So why, with circles, are we always using d/2? It just does not make much sense. Even when we are calculating the area of a triangle, we use (w * h)/2, not w/2 * h or w * h/2. It does not make sense to use half the width of the circle when finding perimeter (circumference) or area, when don't use half lengths anywhere else. I recently realized why we use r instead of d, and the answer is rather disappointing. The definition of a circle is the following equation: r^2 = x^2 + y^2. This is the only place I can find where it makes significantly more sense to use half width over whole width. So why is it that we are using this one equation to define the normal case instead of using all of the others and defining this one as a special case? Honestly, I can see no reason for doing it this way, except perhaps that this is how it was done in the past. I don't happen to subscribe to the theory that tradition trumps logic and reason. If tradition does not make sense, it is time to trade it for something that does.
It turns out that this entire argument is almost pointless though. Using 2π all over the place works, and it is pretty entrenched in our mathematics. Outside of education, math and science are not going to suddenly change because someone decides it is better to do things a bit differently. Within education is where the "almost" comes in though. I am not the only person who noticed that using r instead of d does not make sense in the context of all of the rest of geometry. I am certainly not the only person who noticed that the unit circle does not make sense using π as we do. We could just replace 2π with τ, but that would only fix the unit circle and part of the math. It does not fix the underlying problem with using only half of the distance for circles, while we use the entire distance for everything else. If we really want to make education easier, we should keep using π, but switch to using d instead of r.
28 January 2016
Impossible Conspiracy Theories: Faked Moon Landing
I hope this is going to be a series on conspiracy theories that opponents claim to be impossible. I also want to make it clear that I do not actually believe any of these conspiracy theories. The point of this series, however, is to show how they could be true. I would like to note though, that the problem with most conspiracy theories is the number of people that would have to be involved. The more people who know, the higher the probability of a leak. Read this article to learn more about the correlation between number of people and probability of a leak.
We are going to begin the series with moon landing, since the theory that it was faked seems to be one of the more popular conspiracy theories. It is also central to the Flat Earth theory, which rests on rather a lot of space exploration conspiracy theories.
I can think of only two reasons the U.S. government and/or NASA would want to fake a moon landing. The first is that they don't have the technology to do it, but they want to show someone that they can. Given the time period, the U.S. government might have wanted a faked moon landing to show that we could get there before Russia. This would have involved a lot of people though, and there is a high probability the plan would have been leaked before it was even completed. The second is that the government or NASA had discovered that going to the moon was actually impossible, regardless of technology, but they wanted people to believe that it was possible (the Flat Earth theory suggests that the government knows about an impenetrable dome around the Earth that would prevent space travel at any serious distance). Even if it was only NASA that knew, that would be a lot of people, and it would still probably have leaked before the mission was complete. For the sake of argument though, let's assume that only a few people knew.
The big problem with conspiracy theories is that the chance of success is inversely proportional to the number of people that know about it and the amount of time passed. As such, it is important to minimize the number of people who know. In the case of the moon landing, it is estimated that over 400,000 people would have had to be in on it, but I think that we could reduce that to less than 1,000, with enough resources.
So, who needs to know to pull off a fake moon landing? It depends on what you can fake. It turns out that during that time period, technology was sufficiently advanced to fake quite a bit, given enough resources. There are several groups of people involved. First, there are the planners. Only one planner actually needs to know. The planners are who provide the information on what needs to be faked, and they will do a much better job if they think the mission is real. One planner needs to be in on things, because someone needs to collect information on what needs to be faked. The official plan documents might omit or marginalize information that is not important for a real moon landing but which is essential in a fake one.
The next group of people involved are those who are going to build the equipment. The planners will have created a list of requirements (is air tight, for instance). The people designing and constructing the space craft, rockets, and everything else will use those requirements. They only need to know if the crafts are going to be faked as well. Building real space craft is expensive, but fake ones are much cheaper. Involving all of the engineers dramatically reduces chances of success though. Minimizing people who know is more important than price here, so the only engineers that will know are those who will build fake stuff. We will need a little bit of fake stuff, because we are faking a moon landing here, but without the real stuff, many more people have to get involved. The fake stuff we will need is a fake moon module and some support equipment to fake space travel and the environment of the moon.
The most obvious people involved are the astronauts. Unfortunately, they are also the most conspicuous, and the public will be continuously watching them after the event. They are the weakest links, so they absolutely cannot know. This is going to be very difficult, because they are supposed to actually experience the event. This is why we need the fake stuff. We have to fool the people right at the center of the thing.
There is going to be a launch party watching the blast off. It is going to include some high profile people who definitely should not know. This is why we need the real stuff. These people need to see a group of astronauts get into a rocket and blast off into space. This is going to be the second hardest thing to fake. We cannot let the astronauts get onto the rocket, because they are going to be boarding the fake one at the same time. So we need fake astronauts, and they may even need to look an awful lot like the real ones. If they are expendable, they could be launched in the rocket, where they will likely ultimately die. Otherwise, we could have some means for them to leave the rocket after boarding, without being seen. This is a complicated one without involving more engineers, and if we picked some lookalikes from the engineers that know, we could reduce the number of people that know by launching them to their certain doom (of course, we would have to lie to them, because they would probably object if they knew). This strategy keeps the launch party safely ignorant.
Most of mission control does not need to know. They are just communicating with the real astronauts, who think that they are really visiting the moon. We probably want a few people there who know, in case something goes wrong.
Somewhere else nearby, we will need lots of people who know. The engineers who know will have built a second moon landing module that looks nearly exactly like the real one. The real astronauts will have a separate launch party, because they need to think everything is real. They will get into the rocket, and then a bunch of things have to happen. For launch, the capsule has to be mounted on equipment to produce the expected acceleration of lift off. This does not have to be perfect, but the process needs to be smooth. Mounting the capsule on a large centrifuge would work.
Once the astronauts are in the capsule and think that they have lifted off, the big thing is keeping the pictures on the windows right, so it looks like they are going through space. Some projectors mounted to the outside, with the right projection media on the windows might be able to manage that one. Simulating zero gravity is going to be hard though. Supposedly an extremely strong alternating magnetic field can achieve this effect, though at extreme expense. To my knowledge, the only reported experiments on this effect with living things was done on frogs, but I believe they did survive. So, simulating zero G is possible, though extremely expensive. Alternatively, they could have lined the space suits with ferrous materials, which would have dramatically reduced the energy requirements.
The landing would have been pretty simple, but the moon environment would have been a huge amount of work. Enough distance would have had to be created to allow an astronaut to walk around a bit, just in case. It would have to be created either in a large building or underground, and the ceiling and walls would have to show images of what space would look like from the moon (something no one would have seen before, so it would have to be a guess). Simulating low gravity in such a large area would be much harder than simulating it in a small space capsule. Since the astronaut could be guaranteed to be wearing a space suit though, the ferrous metal lining with a large electromagnet would be more viable. Low gravity simulated this way would feel wrong, but for someone who has never experienced the real thing, it would probably not be noticeable. Faking the rest of the trip would be easy from there.
Everything else just comes down to faking pictures. Eventually most faked pictures are figured out, but NASA had the funding to hire the best, and it is not like we have anything similar to compare. Most of the usual means for detecting faked pictures or videos rely on existing things to compare against.
What it comes down to is that all of the technology to fake a moon landing existed. There were two possible motives (though one is incredibly unlikely). NASA was provided with huge amounts of funding at the time, so even cost would not have been much a problem. In other words, the fake moon landing theory is possible.
The problem with the theory is not whether it was possible or not. There are so many minor problems that make it incredibly unlikely. First, the minimum number of conspirators required to pull it off in the time given is still far too large. According to the article mentioned in the introduction, even 1,000 people would probably not be able to keep it quite for more than a decade. Just the number of engineers required for such an elaborate hoax would be at least half that, and much more if you expected it to take less than a decade or two of work. Add in probably one or two hundred people involved in construction of the larger parts, and all of the various experts on things like film manipulation, disguise (for the fake astronauts and the fake dignitaries), and other minor but essential details, and you have right around 1,000. There would have just been too many people involved to keep it quiet for this long. Unless, of course, most of them were killed afterwards to keep them quiet, but then we have to wonder why he have not heard anything about almost 1,000 people going missing who all happened to work for or near NASA...
The second problem is scale. The scale and detail of the moon landing is huge. There is far more information than is necessary to convince even experts that we actually sent someone to the moon. A fake could probably have been done at half the price, by eliminating most of the unimportant details, and people would have been equally well convinced. Anyone so out of touch with those they are trying to fool almost certainly would have screwed up somewhere, and yet no one has been able to produce conclusive evidence of such a mistake.
The third problem is that neither of the motivations really make much sense to spend so much effort and money on a hoax. If the Earth really is flat with a giant dome, you don't need to fake sending a man to the moon to convince people otherwise, especially since they already believed otherwise. Sending a machine to explore the moon would have been equally effective and much easier and cheaper to fake. For only slightly more money than faking a simple landing and a small group of additional people in on the hoax, you could fake an entire moon colony. Likewise, if the U.S. did not have the technology and was worried that Russia would get there first, a much cheaper fake could have been done, or a much more elaborate fake (the colony thing again) could have been done for only a bit more money and involvement.
In short, while there may have been sufficient resources, motivation, and technology to fake the moon landing, the odds of it are incredibly small. The motivation is questionable, the execution does not fit any of the motivations, and the chance of successfully concealing the hoax for this long is almost nothing. So, it is not impossible, but it is improbable in the extreme.
We are going to begin the series with moon landing, since the theory that it was faked seems to be one of the more popular conspiracy theories. It is also central to the Flat Earth theory, which rests on rather a lot of space exploration conspiracy theories.
I can think of only two reasons the U.S. government and/or NASA would want to fake a moon landing. The first is that they don't have the technology to do it, but they want to show someone that they can. Given the time period, the U.S. government might have wanted a faked moon landing to show that we could get there before Russia. This would have involved a lot of people though, and there is a high probability the plan would have been leaked before it was even completed. The second is that the government or NASA had discovered that going to the moon was actually impossible, regardless of technology, but they wanted people to believe that it was possible (the Flat Earth theory suggests that the government knows about an impenetrable dome around the Earth that would prevent space travel at any serious distance). Even if it was only NASA that knew, that would be a lot of people, and it would still probably have leaked before the mission was complete. For the sake of argument though, let's assume that only a few people knew.
The big problem with conspiracy theories is that the chance of success is inversely proportional to the number of people that know about it and the amount of time passed. As such, it is important to minimize the number of people who know. In the case of the moon landing, it is estimated that over 400,000 people would have had to be in on it, but I think that we could reduce that to less than 1,000, with enough resources.
So, who needs to know to pull off a fake moon landing? It depends on what you can fake. It turns out that during that time period, technology was sufficiently advanced to fake quite a bit, given enough resources. There are several groups of people involved. First, there are the planners. Only one planner actually needs to know. The planners are who provide the information on what needs to be faked, and they will do a much better job if they think the mission is real. One planner needs to be in on things, because someone needs to collect information on what needs to be faked. The official plan documents might omit or marginalize information that is not important for a real moon landing but which is essential in a fake one.
The next group of people involved are those who are going to build the equipment. The planners will have created a list of requirements (is air tight, for instance). The people designing and constructing the space craft, rockets, and everything else will use those requirements. They only need to know if the crafts are going to be faked as well. Building real space craft is expensive, but fake ones are much cheaper. Involving all of the engineers dramatically reduces chances of success though. Minimizing people who know is more important than price here, so the only engineers that will know are those who will build fake stuff. We will need a little bit of fake stuff, because we are faking a moon landing here, but without the real stuff, many more people have to get involved. The fake stuff we will need is a fake moon module and some support equipment to fake space travel and the environment of the moon.
The most obvious people involved are the astronauts. Unfortunately, they are also the most conspicuous, and the public will be continuously watching them after the event. They are the weakest links, so they absolutely cannot know. This is going to be very difficult, because they are supposed to actually experience the event. This is why we need the fake stuff. We have to fool the people right at the center of the thing.
There is going to be a launch party watching the blast off. It is going to include some high profile people who definitely should not know. This is why we need the real stuff. These people need to see a group of astronauts get into a rocket and blast off into space. This is going to be the second hardest thing to fake. We cannot let the astronauts get onto the rocket, because they are going to be boarding the fake one at the same time. So we need fake astronauts, and they may even need to look an awful lot like the real ones. If they are expendable, they could be launched in the rocket, where they will likely ultimately die. Otherwise, we could have some means for them to leave the rocket after boarding, without being seen. This is a complicated one without involving more engineers, and if we picked some lookalikes from the engineers that know, we could reduce the number of people that know by launching them to their certain doom (of course, we would have to lie to them, because they would probably object if they knew). This strategy keeps the launch party safely ignorant.
Most of mission control does not need to know. They are just communicating with the real astronauts, who think that they are really visiting the moon. We probably want a few people there who know, in case something goes wrong.
Somewhere else nearby, we will need lots of people who know. The engineers who know will have built a second moon landing module that looks nearly exactly like the real one. The real astronauts will have a separate launch party, because they need to think everything is real. They will get into the rocket, and then a bunch of things have to happen. For launch, the capsule has to be mounted on equipment to produce the expected acceleration of lift off. This does not have to be perfect, but the process needs to be smooth. Mounting the capsule on a large centrifuge would work.
Once the astronauts are in the capsule and think that they have lifted off, the big thing is keeping the pictures on the windows right, so it looks like they are going through space. Some projectors mounted to the outside, with the right projection media on the windows might be able to manage that one. Simulating zero gravity is going to be hard though. Supposedly an extremely strong alternating magnetic field can achieve this effect, though at extreme expense. To my knowledge, the only reported experiments on this effect with living things was done on frogs, but I believe they did survive. So, simulating zero G is possible, though extremely expensive. Alternatively, they could have lined the space suits with ferrous materials, which would have dramatically reduced the energy requirements.
The landing would have been pretty simple, but the moon environment would have been a huge amount of work. Enough distance would have had to be created to allow an astronaut to walk around a bit, just in case. It would have to be created either in a large building or underground, and the ceiling and walls would have to show images of what space would look like from the moon (something no one would have seen before, so it would have to be a guess). Simulating low gravity in such a large area would be much harder than simulating it in a small space capsule. Since the astronaut could be guaranteed to be wearing a space suit though, the ferrous metal lining with a large electromagnet would be more viable. Low gravity simulated this way would feel wrong, but for someone who has never experienced the real thing, it would probably not be noticeable. Faking the rest of the trip would be easy from there.
Everything else just comes down to faking pictures. Eventually most faked pictures are figured out, but NASA had the funding to hire the best, and it is not like we have anything similar to compare. Most of the usual means for detecting faked pictures or videos rely on existing things to compare against.
What it comes down to is that all of the technology to fake a moon landing existed. There were two possible motives (though one is incredibly unlikely). NASA was provided with huge amounts of funding at the time, so even cost would not have been much a problem. In other words, the fake moon landing theory is possible.
The problem with the theory is not whether it was possible or not. There are so many minor problems that make it incredibly unlikely. First, the minimum number of conspirators required to pull it off in the time given is still far too large. According to the article mentioned in the introduction, even 1,000 people would probably not be able to keep it quite for more than a decade. Just the number of engineers required for such an elaborate hoax would be at least half that, and much more if you expected it to take less than a decade or two of work. Add in probably one or two hundred people involved in construction of the larger parts, and all of the various experts on things like film manipulation, disguise (for the fake astronauts and the fake dignitaries), and other minor but essential details, and you have right around 1,000. There would have just been too many people involved to keep it quiet for this long. Unless, of course, most of them were killed afterwards to keep them quiet, but then we have to wonder why he have not heard anything about almost 1,000 people going missing who all happened to work for or near NASA...
The second problem is scale. The scale and detail of the moon landing is huge. There is far more information than is necessary to convince even experts that we actually sent someone to the moon. A fake could probably have been done at half the price, by eliminating most of the unimportant details, and people would have been equally well convinced. Anyone so out of touch with those they are trying to fool almost certainly would have screwed up somewhere, and yet no one has been able to produce conclusive evidence of such a mistake.
The third problem is that neither of the motivations really make much sense to spend so much effort and money on a hoax. If the Earth really is flat with a giant dome, you don't need to fake sending a man to the moon to convince people otherwise, especially since they already believed otherwise. Sending a machine to explore the moon would have been equally effective and much easier and cheaper to fake. For only slightly more money than faking a simple landing and a small group of additional people in on the hoax, you could fake an entire moon colony. Likewise, if the U.S. did not have the technology and was worried that Russia would get there first, a much cheaper fake could have been done, or a much more elaborate fake (the colony thing again) could have been done for only a bit more money and involvement.
In short, while there may have been sufficient resources, motivation, and technology to fake the moon landing, the odds of it are incredibly small. The motivation is questionable, the execution does not fit any of the motivations, and the chance of successfully concealing the hoax for this long is almost nothing. So, it is not impossible, but it is improbable in the extreme.
25 January 2016
My Qualifications
Some might consider this a pet peeve, but I think it is a very serious problem in our society. Many Americans seem to think that job titles and pieces of paper make them experts. Some even think that a future qualification makes them experts now. Here are a few experiences I have had, to illustrate my point.
I worked in a hardware store around 8 years ago. I worked in the electrical department. One of the products we sold was ceiling fans. We also sold different lengths of rod for attaching them to the ceiling, since they generally came with something around a foot long. There was one brand with a few fans that did not fit the rods that we stocked. The head of the department recommended that we open the box and make sure the rod would fit right when we helped customers with this particular brand of fan. One day, another employee was helping a customer with a fan, and I overheard the customer ask if the rod would fit the fan. The employee said yes, without even checking. I noticed that the brand of fan was the one we had been warned about, so I interrupted and explained what I had been told. I then went off to do some other task. A bit later, during my break, the employee involved decided to tell me off. He opened with "I am a certified electrician," and he explained to me that he knew exactly what he was doing. He complained that my comment had caused the customers to doubt his word, and he had been required to show them that the parts would fit.
Now, let me explain my thought processes. First, his antagonistic attitude did not help the situation. Second, when he stated his credentials, it made it worse. Certified electricians can make much more money doing electrical work than they can working at a chain hardware store. I have a hard time believing that a certified electrician who is worth his salt would stoop to what amounts to a retail job. Even if he did do it entirely voluntarily though, he had not been doing electrical work for many years, so it is again hard to believe that he had any personal experience with the products he was selling. Lastly, he did not start by stating his experience. He only stated his credentials. Credentials without experience is worthless, and if he did not have sufficient experience to state them, then the credentials are meaningless to me.
My second experience was on an internet forum. These are rife with people who claim credentials they don't have. Many people will try to settle an argument by explaining why they must be right, instead of proving it or explaining why they are right. I had been doing a chemistry experiment. If you put an electric current through water, you can break the molecules into oxygen and hydrogen gasses. Water is not naturally very conductive though, so this process is very slow. To improve the situation, an electrolyte can be added to the water, and a readily available one is table salt. So, I had a device with a pair of electrodes, some tubes for capturing the gasses, and a 12 volt battery, and I had filled it with salt water. It worked quite well, but I noticed something unexpected: I could smell chlorine coming off of the experiment. The water was well water, so it was not chlorinated. There was only one possible source for the gas, and that was the salt (which is made of sodium and chlorine). I decided to do some research on this. It turns out that this is not well documented, so I ended up looking through forums where people related their personal experiences. Indeed, some others had experienced the same thing, however not everyone had. An argument was ensuing about whether or not it was even possible. One group claimed that they clearly smelled chlorine coming off of their experiments. Another claimed that they did not. A third group was trying to convince everyone else that the voltages involved were way too low to break up the salt molecules, and thus it was completely impossible that the chlorine was coming from the salt (despite the fact that no other source existed). Many in the third group claimed to be chemists, and some of the rest claimed to have asked chemist friends. The fact is, however, that every single person in the third group was wrong. I could smell chlorine coming from my experiment, and the salt was the only source. The credentials were meaningless, because the people that had them (or claimed to...) were still wrong.
The internet is full of people who lie about their credentials to convince people that they are right. I sincerely believe, however, that at least some of those claiming to have various degrees in chemistry actually did. The fact is, all a degree proves is that you managed, one way or another, to get grades high enough to pass your required classes. Most modern colleges will graduate students with all Cs and Ds. In other words, you have to know between 60% and 70% of the material for a fairly brief period of time (4 years is the typical maximum for a Bachelors degree, but often you can forget material as soon as you complete the course without too much harm). Stated more plainly, a degree is evidence that you have been exposed somewhat to the subject. It does not imply either a working or a complete knowledge of the subject. Real world experience in a specific domain of your degree that applies to the discussion is what really matters. When someone says, "I have a degree in chemistry," what it tells me is that they are not confident enough in their knowledge or experience to cite anything meaningful, so they fall back on something that they perceive to have more value than it really does.
My last experience took place in a hardware store, but not the one I worked at. I was shopping for a shovel, so I could turn up the soil in our garden (essentially I was going to hand till using a shovel). I told the employee that I needed something suitable for digging about 2 feet deep. I had taken (and passed with a high grade) a college class on organic gardening, and we had learned about some research showing that digging that deep had proved, in experiments, to have a significant beneficial effect on crops. Some other guy shopping there overheard the conversation, and he interrupted to tell me that digging deeper than 1 foot was a waste of time. Then, he stated his credentials: He was an agriculture major at the local college.
This situation is absurd. The guy did not even have a degree. The only credential he stated was the fact that he had told the college the he intended to get a degree in agriculture. He was using a future degree as his credentials. For all I know, he could have been a freshman taking only generals his first semester. Even if he had been a senior though, the agricultural knowledge taught in most of academia applies very specifically to industrial farming, not to organic gardening or even run-of-the-mill home gardening (I was applying some organic gardening techniques to home gardening). Things like the depth you dig the dirt depend on a lot of factors, including things like what crops you are growing, the climate you are in, how frequently you water, and what you use to fertilize. Even a degree in agriculture would not have qualified this guy to tell me how I should garden. The fact is, I had one class, many hours of research, and some real life experience on the subject as my credentials. This guy had made a tentative commitment to learn about the subject.
Americans seem to place far too much value on credentials and not enough on experience. This is not just a pet peeve. It is a real problem with serious economic implications. I know people with tons of experience in a field who were laid off merely because they did not have a degree. In fact, in some fields it is becoming common practice to require older workers to go back to school for a degree that did not exist when they started working. The employers generally pay for this, but often the employees have far more experience and knowledge in the field than the people who are supposed to be teaching them. This is stupid and wasteful. A degree is supposed to be evidence that a person knows a bit about a specific subject. Experience is far better evidence of that. In my field, this is fairly well recognized. I have a BS in Computer Science. Most employers don't even look at a degree if you have significant experience. Many (but not all) will still discard a resume without looking at experience, if there is no degree, but with a few years experience, it hardly matters what the degree is in, so long is it is there. Of course, since the mid 90s, business experts have been recommending that software companies look for employees that are self taught, with or without a degree, because self taught people do far superior work.
Anyhow, next time you are tempted to spout credentials, consider these lists:
It is only appropriate to share credentials if:
I worked in a hardware store around 8 years ago. I worked in the electrical department. One of the products we sold was ceiling fans. We also sold different lengths of rod for attaching them to the ceiling, since they generally came with something around a foot long. There was one brand with a few fans that did not fit the rods that we stocked. The head of the department recommended that we open the box and make sure the rod would fit right when we helped customers with this particular brand of fan. One day, another employee was helping a customer with a fan, and I overheard the customer ask if the rod would fit the fan. The employee said yes, without even checking. I noticed that the brand of fan was the one we had been warned about, so I interrupted and explained what I had been told. I then went off to do some other task. A bit later, during my break, the employee involved decided to tell me off. He opened with "I am a certified electrician," and he explained to me that he knew exactly what he was doing. He complained that my comment had caused the customers to doubt his word, and he had been required to show them that the parts would fit.
Now, let me explain my thought processes. First, his antagonistic attitude did not help the situation. Second, when he stated his credentials, it made it worse. Certified electricians can make much more money doing electrical work than they can working at a chain hardware store. I have a hard time believing that a certified electrician who is worth his salt would stoop to what amounts to a retail job. Even if he did do it entirely voluntarily though, he had not been doing electrical work for many years, so it is again hard to believe that he had any personal experience with the products he was selling. Lastly, he did not start by stating his experience. He only stated his credentials. Credentials without experience is worthless, and if he did not have sufficient experience to state them, then the credentials are meaningless to me.
My second experience was on an internet forum. These are rife with people who claim credentials they don't have. Many people will try to settle an argument by explaining why they must be right, instead of proving it or explaining why they are right. I had been doing a chemistry experiment. If you put an electric current through water, you can break the molecules into oxygen and hydrogen gasses. Water is not naturally very conductive though, so this process is very slow. To improve the situation, an electrolyte can be added to the water, and a readily available one is table salt. So, I had a device with a pair of electrodes, some tubes for capturing the gasses, and a 12 volt battery, and I had filled it with salt water. It worked quite well, but I noticed something unexpected: I could smell chlorine coming off of the experiment. The water was well water, so it was not chlorinated. There was only one possible source for the gas, and that was the salt (which is made of sodium and chlorine). I decided to do some research on this. It turns out that this is not well documented, so I ended up looking through forums where people related their personal experiences. Indeed, some others had experienced the same thing, however not everyone had. An argument was ensuing about whether or not it was even possible. One group claimed that they clearly smelled chlorine coming off of their experiments. Another claimed that they did not. A third group was trying to convince everyone else that the voltages involved were way too low to break up the salt molecules, and thus it was completely impossible that the chlorine was coming from the salt (despite the fact that no other source existed). Many in the third group claimed to be chemists, and some of the rest claimed to have asked chemist friends. The fact is, however, that every single person in the third group was wrong. I could smell chlorine coming from my experiment, and the salt was the only source. The credentials were meaningless, because the people that had them (or claimed to...) were still wrong.
The internet is full of people who lie about their credentials to convince people that they are right. I sincerely believe, however, that at least some of those claiming to have various degrees in chemistry actually did. The fact is, all a degree proves is that you managed, one way or another, to get grades high enough to pass your required classes. Most modern colleges will graduate students with all Cs and Ds. In other words, you have to know between 60% and 70% of the material for a fairly brief period of time (4 years is the typical maximum for a Bachelors degree, but often you can forget material as soon as you complete the course without too much harm). Stated more plainly, a degree is evidence that you have been exposed somewhat to the subject. It does not imply either a working or a complete knowledge of the subject. Real world experience in a specific domain of your degree that applies to the discussion is what really matters. When someone says, "I have a degree in chemistry," what it tells me is that they are not confident enough in their knowledge or experience to cite anything meaningful, so they fall back on something that they perceive to have more value than it really does.
My last experience took place in a hardware store, but not the one I worked at. I was shopping for a shovel, so I could turn up the soil in our garden (essentially I was going to hand till using a shovel). I told the employee that I needed something suitable for digging about 2 feet deep. I had taken (and passed with a high grade) a college class on organic gardening, and we had learned about some research showing that digging that deep had proved, in experiments, to have a significant beneficial effect on crops. Some other guy shopping there overheard the conversation, and he interrupted to tell me that digging deeper than 1 foot was a waste of time. Then, he stated his credentials: He was an agriculture major at the local college.
This situation is absurd. The guy did not even have a degree. The only credential he stated was the fact that he had told the college the he intended to get a degree in agriculture. He was using a future degree as his credentials. For all I know, he could have been a freshman taking only generals his first semester. Even if he had been a senior though, the agricultural knowledge taught in most of academia applies very specifically to industrial farming, not to organic gardening or even run-of-the-mill home gardening (I was applying some organic gardening techniques to home gardening). Things like the depth you dig the dirt depend on a lot of factors, including things like what crops you are growing, the climate you are in, how frequently you water, and what you use to fertilize. Even a degree in agriculture would not have qualified this guy to tell me how I should garden. The fact is, I had one class, many hours of research, and some real life experience on the subject as my credentials. This guy had made a tentative commitment to learn about the subject.
Americans seem to place far too much value on credentials and not enough on experience. This is not just a pet peeve. It is a real problem with serious economic implications. I know people with tons of experience in a field who were laid off merely because they did not have a degree. In fact, in some fields it is becoming common practice to require older workers to go back to school for a degree that did not exist when they started working. The employers generally pay for this, but often the employees have far more experience and knowledge in the field than the people who are supposed to be teaching them. This is stupid and wasteful. A degree is supposed to be evidence that a person knows a bit about a specific subject. Experience is far better evidence of that. In my field, this is fairly well recognized. I have a BS in Computer Science. Most employers don't even look at a degree if you have significant experience. Many (but not all) will still discard a resume without looking at experience, if there is no degree, but with a few years experience, it hardly matters what the degree is in, so long is it is there. Of course, since the mid 90s, business experts have been recommending that software companies look for employees that are self taught, with or without a degree, because self taught people do far superior work.
Anyhow, next time you are tempted to spout credentials, consider these lists:
It is only appropriate to share credentials if:
- Someone asks. It is almost never inappropriate to share your credentials if someone asks, though you might consider also mentioning experience, if it is relevant.
- You are filling out a job application (or a College application). Even in companies that care more about experience, credentials can make the difference between two candidates with the same work experience.
- You have been asked to share something about yourself and either the credentials are relevant to the situation or you cannot think of anything about yourself that is more interesting. In the second case, please consider rethinking your life. Again, experience is better, if it is relevant.
- You have no other experience in the field and the credentials are relevant and meaningful. For the sake of honesty, you should admit that you have no actual experience in this case.
- You don't actually have them yet. Being a "major" in a subject is not a credential. It is a commitment to obtain credentials. Undergoing current training in a subject (maybe as an apprentice) is also not credentials. If you know about the subject being discussed because you took a class on it, cite the class or the teacher. Likewise, if someone who is training you told you something relevant, cite that person as a source, don't claim that participation in the training itself makes you a qualified source until the training is completed.
- You don't have them at all. If you claim to have credentials that you don't have, you are a liar. Aside from being morally and ethically wrong, you will eventually ruin your reputation. Do this too many times, and people won't believe you even when you do have the credentials you claim to have. It is not worth perjuring yourself to convince people that you are right, whether you are or not. Even if you are right, let them be wrong if they want to.
- You are not absolutely certain your claims are true. If you make an incorrect claim and then use your credentials to back it, not only do you make yourself look stupid, you also make anyone else with similar credentials look stupid. If one chemist makes a false claim, people will start to assume that most or all chemists don't know what they are talking about.
- The credentials don't guarantee the knowledge you are claiming to have. A chemistry degree does not mean that you completely understand how electrolysis works. An electrician certification does not mean you have experience with every fixture and brand of electrical device that exists. Even PhDs don't know everything in their field (I had a professor with a PhD who did not know some of the features in a very common programming language that he was very experienced with). Credentials are evidence of general knowledge. For specific knowledge, you should cite either personal experience or a reliable reference. (PhDs tend to have more focused knowledge, but instead of citing their PhD, it is better for them to cite the research and thesis that earned them the PhD. That is the real experience.)
- You have actual experience. Experience trumps credentials. You might be able to show me math that "proves" my machine cannot produce chlorine, but I have experience proving that chlorine is actually being produced. My experimental evidence proves your math wrong (in fact, this is largely how science advances; someone figures out a theory about something, and someone else proves the theory wrong through experimentation). The only place where experience does not trump credentials is in job applications for companies that are too stupid to realize that experience is more valuable.
20 January 2016
Crashiness
I discovered a new metric for software stability. A set of programs were described by a blog writer as "crashy." While not well defined, this is a useful metric when comparing applications that do similar things. This is a bit of a tangent of what I want to discuss though.
I have found that most computer users have a poor understanding of software stability. I suppose this is to be expected, as most computer users do not even understand the technology at a basic level, but there are some things that people should know when they are using computers. Unfortunately, these things are not really taught anywhere. I am going to fix that in this article.
First, stability is a measure of how well a program handles various situations. A program that has errors or crashes when given typical expected input is very unstable. A program that has errors or crashes on unusual input is somewhat unstable. A program that has errors or crashes in only a few very rare cases is considered fairly stable. A program that never crashes is considered highly stable. In industry, programs go through phases of development, and often one factor in determining when to advance to the next phase is stability. Different companies tend to have different standards on what is stable enough to sell. This is why some companies have a reputation for very stable software while others have reputations for poor stability.
Second, instability is typically caused by programmer error. There are some errors in programs that cannot be avoided, however if an error can be predicted, the program can handle it without crashing. Stable programs do not crash when they encounter an error. Instead they either work around it, or if that is not possible, they notify the user and give the user options for how to respond. There are only a few extremely rare cases where an error cannot be predicted (hardware issues, like bad memory or cosmic radiation changing memory data). In short, instability exists because it was programmed in. Not to imply that it was deliberate, because it is almost never intentional, but when a program crashes, it is the fault of a programmer somewhere for not handling the error appropriately. In other words, when your applications crash, it is not your fault! I have to say this because I frequently hear people say things like, "I should have known it would do that, because it always crashes when I do this thing," as if the crash was their fault. The user should not have to memorize a bunch of cases that crash the program to avoid crashes. The programmer should have taken care of that in the first place.
Now, this is kind of a hard nosed approach. Something important to keep in mind is that it is nearly impossible to create large applications without any bugs. Software development went way beyond the level of human comprehension decades ago. When a group makes a large application, it is divided into parts, because no single human can completely understand the entire application all at once. Each group works on a part of the application, and they are given information on how their part should work together with all of the other parts. Sticking to best practices minimizes any clashing that could occur between parts, but since no person can fully understand all of the parts, there are always holes. These holes, where unexpected interaction (called "side effects" in some types of programming) between parts of a program occur can result in bugs that are extremely difficult to find. For large software companies, what it comes down to is, "How much money are we willing to spend on debugging before we start selling the software?" Cheaper companies will try to maximize profits by reducing debugging spending. Higher end companies will try to maximize quality by putting a bit more into debugging. This affects product price though, and in most cases, spending the time and money to eliminate all bugs would be so expensive that no one could afford to buy the software, and it would take so long it would be obsolete by the time it was ready anyway. In short, unless you are paying millions or billions of dollars on software that is years or even decades behind its time, you should not expect it to be bug free.
The question then is, what is reasonable to expect? The answer is that it depends on criticality. Start with the question, "What do I lose if the software crashes?" The higher the loss, the higher the criticality. For example, if your word processor crashes, you could lose hours of work. If your word processor has a decent autosave feature, you could lose minutes or seconds of work. The word processor without autosave has higher criticality than the one with. If your online video game client crashes, you lose a few seconds or minutes of leisure time to restart it, which is hardly critical at all. If your game is not online and it does not have autosave, it is more critical, but it is still just leisure time that you are losing, so it is less critical than the word processor. On the other hand, your operating system is extremely critical. This is because everything else depends on it. Your word processor could be 100% bug free, but an unstable operating system can cause it to crash anyway, and if your operating system itself crashes, you lose everything that has not been saved, regardless of the stability of the applications. On any system, the operating system is always the most critical piece of software, because everything else depends on it.
So, next time someone tells you that a piece of software is crashy, but it still does what you need, consider the cost of the crashes. You might find that you are ultimately more productive with a less popular product with fewer features, because you lose less work less often and spend less time waiting for to program to start back up after a crash.
I have found that most computer users have a poor understanding of software stability. I suppose this is to be expected, as most computer users do not even understand the technology at a basic level, but there are some things that people should know when they are using computers. Unfortunately, these things are not really taught anywhere. I am going to fix that in this article.
First, stability is a measure of how well a program handles various situations. A program that has errors or crashes when given typical expected input is very unstable. A program that has errors or crashes on unusual input is somewhat unstable. A program that has errors or crashes in only a few very rare cases is considered fairly stable. A program that never crashes is considered highly stable. In industry, programs go through phases of development, and often one factor in determining when to advance to the next phase is stability. Different companies tend to have different standards on what is stable enough to sell. This is why some companies have a reputation for very stable software while others have reputations for poor stability.
Second, instability is typically caused by programmer error. There are some errors in programs that cannot be avoided, however if an error can be predicted, the program can handle it without crashing. Stable programs do not crash when they encounter an error. Instead they either work around it, or if that is not possible, they notify the user and give the user options for how to respond. There are only a few extremely rare cases where an error cannot be predicted (hardware issues, like bad memory or cosmic radiation changing memory data). In short, instability exists because it was programmed in. Not to imply that it was deliberate, because it is almost never intentional, but when a program crashes, it is the fault of a programmer somewhere for not handling the error appropriately. In other words, when your applications crash, it is not your fault! I have to say this because I frequently hear people say things like, "I should have known it would do that, because it always crashes when I do this thing," as if the crash was their fault. The user should not have to memorize a bunch of cases that crash the program to avoid crashes. The programmer should have taken care of that in the first place.
Now, this is kind of a hard nosed approach. Something important to keep in mind is that it is nearly impossible to create large applications without any bugs. Software development went way beyond the level of human comprehension decades ago. When a group makes a large application, it is divided into parts, because no single human can completely understand the entire application all at once. Each group works on a part of the application, and they are given information on how their part should work together with all of the other parts. Sticking to best practices minimizes any clashing that could occur between parts, but since no person can fully understand all of the parts, there are always holes. These holes, where unexpected interaction (called "side effects" in some types of programming) between parts of a program occur can result in bugs that are extremely difficult to find. For large software companies, what it comes down to is, "How much money are we willing to spend on debugging before we start selling the software?" Cheaper companies will try to maximize profits by reducing debugging spending. Higher end companies will try to maximize quality by putting a bit more into debugging. This affects product price though, and in most cases, spending the time and money to eliminate all bugs would be so expensive that no one could afford to buy the software, and it would take so long it would be obsolete by the time it was ready anyway. In short, unless you are paying millions or billions of dollars on software that is years or even decades behind its time, you should not expect it to be bug free.
The question then is, what is reasonable to expect? The answer is that it depends on criticality. Start with the question, "What do I lose if the software crashes?" The higher the loss, the higher the criticality. For example, if your word processor crashes, you could lose hours of work. If your word processor has a decent autosave feature, you could lose minutes or seconds of work. The word processor without autosave has higher criticality than the one with. If your online video game client crashes, you lose a few seconds or minutes of leisure time to restart it, which is hardly critical at all. If your game is not online and it does not have autosave, it is more critical, but it is still just leisure time that you are losing, so it is less critical than the word processor. On the other hand, your operating system is extremely critical. This is because everything else depends on it. Your word processor could be 100% bug free, but an unstable operating system can cause it to crash anyway, and if your operating system itself crashes, you lose everything that has not been saved, regardless of the stability of the applications. On any system, the operating system is always the most critical piece of software, because everything else depends on it.
So, next time someone tells you that a piece of software is crashy, but it still does what you need, consider the cost of the crashes. You might find that you are ultimately more productive with a less popular product with fewer features, because you lose less work less often and spend less time waiting for to program to start back up after a crash.
Subscribe to:
Posts (Atom)