This article may be confusing to those who are not familiar with different number bases, so I will give a brief explanation. A number base is generally defined by the number of digits used in that base. The most commonly used base is base 10 or decimal, which has the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Notice that there are 10 digits. The base of a number system also defines what the difference places mean. In all bases, the first place is always 1s. The next place is the the base itself. For base ten, the second place is the 10s place. The third place is the base squared (100 in base 10), the fourth is the based cubed (1000, in base 10) and so on. Another common base is base 2 or binary. Base 2 has 2 digits, 0 and 1, and the places are powers of 2. The first place is 1s, the second is 2s, the third is 4s, and so on. Some bases have more than 10 digits. Base 16 or hexadecimal has 16 digits. It uses 0 through 9, like base 10, but it runs out of traditional digits there, so it adds A, B, C, D, E, and F for the 10, 11, 12, 13, 14, and 15 digits. Other systems (like base 12) have been invented that reuse other symbols or have unique symbols for those past 9. Math in different bases works very much like base 10, except that you carry and borrow differently, based on the number of digits available.
There have been numerous discussions about numerical bases. Base 10 is rather inconvenient in many aspects, and bases like 12 have some significant benefits. Base 12 is especially useful, because it makes common fractional math extremely easy. Base 12 makes division by 2, 3, 4, 6, and 12 trivial. In base 10, 1/3 = 0.333... This makes any math involving division by 3 complicated and difficult. Even division by 4 is a bit hairy in base 10, where 1/4 = 0.25. The problem with this is that the most frequent division we do is division by small numbers. We might divide a restaurant bill between 2, 3, or 4 people. If there are more people than that, they are generally in groups, where 2, 3, or 4 people end up paying. A great example of why base 12 works so well is time. Seconds and minutes are counted in base 60, but hours are counted in base 24 (in this case, base 24 is just double base 12, so it works the same way). If we need to break up the time for some task between several people (work shifts, for example), base 24 makes it easy. An 8 hour shift is 1/3rd of a day. If you have 4 people, you have 6 hour shifts. It does not work so well with 5 people, but with 6 people, you have 4 hour shifts. With 8 people, it is 3 hour shifts. The fractions involved here, 1/2, 1/3, 1/4, and 1/6, are not just found in time math though. They are found all over the place. The argument for switching to base 12 is quite strong, but there are some problems with it.
The biggest problem with base 12 is that it is not a natural base for humans. We don't have enough fingers to count in base 12 on our hands. This is actually the only logical argument against using base 12. The other problems all stem from the fact that nearly all historical math uses base 10, and while teaching base 12 to children should not really be any more difficult than teaching base 10, it would not be sufficient. Children would have to learn both base 10 and base 12, because there is so much work done in base 10. They would also have to learn to convert between them. Besides that, a lot of practical stuff would have to be transitioned, at great cost. This would include speed limit signs, prices, a huge number of computer programs, measurement systems (though some, like the Imperial System, are already partway there), and the entire world would have to change at the same time to avoid international confusion. It is just not practical. There are certainly major benefits, but they are not worth the cost.
So, why do we use base 10 in the first place? Supposedly it is because it is the most natural way to count for a species with 10 fingers. If humans had 6 fingers and each hand, we would probably count in base 12 instead of base 10. The problem is that this is not actually true. Base 10 is not the most natural way to count for a species with 10 fingers. We actually use base 10 because of a mistake. That mistake is caused by the fact that humanity learned to count before the number 0 was discovered.
The real natural way to count with 10 fingers is base 11. Because humans had not discovered 0 though, only the 10 positive digits that can be counted on hands were considered. In fact, looking at ancient writing systems, we will find that many don't use base 10. Many also have a single glyph (or digit) for the number 10 (instead of combining 1 and 0). Many civilizations that learned to count from fingers had difficulty counting higher than 10, because the lack of a 0 made the place value system currently used almost impossible to discover. A few civilizations did create a placeholder digit for this, but it had no meaning by itself. At least one just left a blank space for empty places, which often made the numbers difficult to interpret. There was clearly a lot of confusion over this, but ultimately, base 10 came to the forefront. Our current base 10 system seems to come from Arabic, which retained the 10 digits idea, but shifted it down one to include 0 and exclude 10. The fact, however, is that on 10 fingers, you can represent digits from 0 to 10, which is actually 11 digits, not 10. Counting on 10 fingers, if you realize that 0 is a number, naturally leads to a base 11 system, not a base 10 system. (Be glad we did not get stuck with base 11 though. As a prime number, 11 is not divisible by anything but 1 and itself, which puts it in the category of the worst possible bases for practical use. In base 11, even division by 2 is difficult!) If you dig a bit deeper though, you might find something else that is interesting.
Base 10 is not the only common number system used anciently. Aside from some very unusual systems, like Roman Numerals that do not have a clear base, there are some that use smaller bases very effectively. The Mayan system (which did include 0) was written in base 5, presumably originating from the fact that humans have 5 fingers. The Mayans are not the only example of a civilization that used base 5. Base 10 systems seem to be the most common, historically, but base 5 systems are actually quite common as well. Even modern tally marks use base 5. The interesting thing with base 5 with fingers though, is that when you include 0 and get base 6, you get a base that is actually quite useful!
Like base 10, base 5 is not natural, because 5 fingers offer 6 digits, if 0 is not forgotten. Base 6 is the natural base for counting with 5 fingers. Including 0 through 5, you can represent a single base 6 digit on one hand. Unlike 11 or even 10, base 6 is quite flexible. It is not as flexible as base 12, but it is, perhaps, close enough. Base 6 fixes the division by 3 problem just as well as base 12. In base 6, 1/3 = 0.2. Base 6 does not fix division by 4, but it does not make it any worse: 1/4 = 0.15 in base 6. It still manages division by 2 as gracefully as base 10 or 12: 1/2 = 0.3 in base 6. The only thing lost that base 10 provides is division by 5, which is only important because we use base 10 everywhere: 1/5 = 0.12 (though, this is no worse than 1/4 in base 10). The advantage base 6 has over base 12 is that you can count in base 6 on your hands. Base 6 has another significant advantage over base 12 and base 10 in hand counting though: Since you only need 6 digits, and these can be supplied with only one hand, the second hand can be used for a second place, which in base 6 is the 6s place. In other words, you can count to 35 on your hands in base 6! (In base 6, 35 is written 55, which is 5 fingers on each hand.)
It turns out that there are two natural ways for humans to count using their fingers. The most obvious one is base 11, and the less obvious one is base 6. Base 11 is completely unsuitable for practical math, but base 6 is surprisingly good. Not only that, but base 6 allows counting to 35 using two hands.
30 March 2016
14 March 2016
Pi Day 2016
I recently wrote an article on the subject of whether we should use π (pi) or τ (tau) in circle related math. As a refresher, π is the ratio of the circumference of a circle to its diameter. It turns out that in a majority of equations that use π, it is multiplied by 2. A growing group of mathematicians and scientists is pushing to replace π with τ, which is just 2π. They argue that this would simplify a lot of math and make geometry and trig significantly easier to learn. This is probably true, but it addresses the problem from the wrong side. I argue that π is not the problem. The problem is that we have this unreasonable attachment to radius. The reason for having to apply a multiple of 2 everywhere is not that π is the wrong value. The reason is that π is a ratio of the diameter of a circle, but we always pair π with the radius, which requires a multiple of 2 to correct this.
There are a lot of different arguments for why we should use τ instead of π, but in the end, they are based on the idea that the radius is the natural way to measure a circle. This comes from the idea that a circle is composed of an infinite number of infinitely small triangles, but while that is a valid representation, it is not actually true. A circle is composed of a continuous, regular curve, with no straight lines anywhere. Just as we would not try to define any other shape as the distance from the center to any point along its perimeter, radius is not the natural way to measure a circle. Outside of mathematics and science, where it has become tradition to use radius, measuring merely half the width of an object is not something that is often useful.
The answer is not to switch from π to τ. The answer is to use the natural measure of the size of a circle: diameter. Some "tauists" claim that the decision of ancient civilizations to use diameter instead of radius was arbitrary. If that is true though, then why did every civilization that set out to measure the ratio of the circumference of a circle to its size choose to use diameter instead of radius? The answer is not chance. The answer is that the choice was not arbitrary. Diameter is the natural way to measure a circle. To someone that has not been taught to measure a circle by radius, diameter is the obvious measure of size.
The fact is, replacing radius with diameter in circle equations simplifies them just as much as replacing π with τ, with the added benefit that new students will not be confused as to why we suddenly only care about half of the size of the circle. Besides that, how does the average person measure the radius of a circle? They measure the diameter and divide by two, because half a width is not a natural measurement to try to take!
In celebration of Pi Day, I bring you the following:
This is the real unit circle. You will notice that one full turn is not equal to 2π. This is because we are measuring in diameters instead of radians. There is no need to redefine the circle constant when using diameter, because diameter is what it was made from! Unfortunately, "diameterians" does not sound as good as "radians," so I propose just calling them "diameters." That works just fine, since the circumference of a circle is, by definition, π diameters. Besides that, I think "diameters" would be far less confusing to new students, as it does not sound like some kind of new unit like "radians" does. (Be honest, how many of you struggled with radians, because it was not initially clear that radians are literally just the distance around the circle measured in radii? Now, think about π diameters. Without the fancy sounding name, it is much more clear what it means.)
Now, with my nice new diameter based unit circle in mind, here are some basic circle and sphere equations using diameter instead of radius:
Circumference: πd
Area: 1⁄4πd2
Surface Area: πd2
Volume: 1⁄6πd3
I considered adding some trig to this, but there is so much, and most of it is already so complex, that it would have taken a lot of algebra to reduce things down. In short, I gave up. This is a good taste though.
Here are some things I notice with the above. First, in the circle equations, you can replace π with diameters (see the unit circle above) to find the values for partial circles. For example, π⁄2d will give you the circumference of half a circle (π⁄2 diameters is halfway around the circle). You can do something similar using τ with radius, but I just wanted to point out τ does not have an advantage here. Now take a look at the circle circumference equation and the sphere surface area equation. Notice the logical step from the first to the second? Merely squaring the diameter promotes the equation to the analogous equation of the equivalent shape one dimension higher.Area to volume is not so pretty, but sphere volume is simpler than its radius based version, and the additional factor of 4 on the circle area equation adds trivial complexity. (I made an algebra mistake in circle area. It is fixed now.) I also notice a similar progression with circle area to sphere volume, where the multiplier is 1/(2 * dimensions), and like circumference to surface area, the exponent is the number of dimensions of what we are measuring. The sphere volume is simpler with diameter, and the circle area is only trivially more complex. In fact, the only places that complexity is increased noticeably are the two standard form equations that are typically used as the definitions for circles and spheres, and outside of education, more general forms are typically used for these, which are so much more complex that the extra factor of 4 would not make any difference.
The big advantage τ has is that replacing a bunch of "2π"s in textbooks is much easier than doing the algebra required to simplify the equations when you swap r for d⁄2. If you aren't going to do it right though, what is the point of doing it at all? It is true that teaching τ with radius will be easier than what we are doing now, but many students will still start off confused that we are measuring only half of the circle (and isn't one of the big "tauist" arguments that it is absurd that 1 π only gets us halfway around the traditional unit circle?). Sticking with π, but using diameters instead of radians, means that we don't have to overthrow a constant that has been ingrained in mathematics over the course of many millenia. We don't have to deal with teaching two constants just so students will be able to understand even recently written papers, not to mention all of the classical mathematics treatises. It is a lot easier to teach students that r = d/2 than it is to get them to memorize two circle constants out to n digits just so they can work with math from different eras. We also don't have to try to explain to students why we only care about half the width of a circle. The only advantage τ has is convenience in fixing textbooks. Using π with diameters just makes sense, even to those without a heavy mathematical background.
I hope you like my Pi Day celebration! Give a few minutes to celebrate diameter today as well.
There are a lot of different arguments for why we should use τ instead of π, but in the end, they are based on the idea that the radius is the natural way to measure a circle. This comes from the idea that a circle is composed of an infinite number of infinitely small triangles, but while that is a valid representation, it is not actually true. A circle is composed of a continuous, regular curve, with no straight lines anywhere. Just as we would not try to define any other shape as the distance from the center to any point along its perimeter, radius is not the natural way to measure a circle. Outside of mathematics and science, where it has become tradition to use radius, measuring merely half the width of an object is not something that is often useful.
The answer is not to switch from π to τ. The answer is to use the natural measure of the size of a circle: diameter. Some "tauists" claim that the decision of ancient civilizations to use diameter instead of radius was arbitrary. If that is true though, then why did every civilization that set out to measure the ratio of the circumference of a circle to its size choose to use diameter instead of radius? The answer is not chance. The answer is that the choice was not arbitrary. Diameter is the natural way to measure a circle. To someone that has not been taught to measure a circle by radius, diameter is the obvious measure of size.
The fact is, replacing radius with diameter in circle equations simplifies them just as much as replacing π with τ, with the added benefit that new students will not be confused as to why we suddenly only care about half of the size of the circle. Besides that, how does the average person measure the radius of a circle? They measure the diameter and divide by two, because half a width is not a natural measurement to try to take!
In celebration of Pi Day, I bring you the following:
This is the real unit circle. You will notice that one full turn is not equal to 2π. This is because we are measuring in diameters instead of radians. There is no need to redefine the circle constant when using diameter, because diameter is what it was made from! Unfortunately, "diameterians" does not sound as good as "radians," so I propose just calling them "diameters." That works just fine, since the circumference of a circle is, by definition, π diameters. Besides that, I think "diameters" would be far less confusing to new students, as it does not sound like some kind of new unit like "radians" does. (Be honest, how many of you struggled with radians, because it was not initially clear that radians are literally just the distance around the circle measured in radii? Now, think about π diameters. Without the fancy sounding name, it is much more clear what it means.)
Now, with my nice new diameter based unit circle in mind, here are some basic circle and sphere equations using diameter instead of radius:
Circle
Definition: d2 = 4(x2 + y2)Circumference: πd
Area: 1⁄4πd2
Sphere
Definition: d2 = 4(x2 + y2 + z2)Surface Area: πd2
Volume: 1⁄6πd3
I considered adding some trig to this, but there is so much, and most of it is already so complex, that it would have taken a lot of algebra to reduce things down. In short, I gave up. This is a good taste though.
Here are some things I notice with the above. First, in the circle equations, you can replace π with diameters (see the unit circle above) to find the values for partial circles. For example, π⁄2d will give you the circumference of half a circle (π⁄2 diameters is halfway around the circle). You can do something similar using τ with radius, but I just wanted to point out τ does not have an advantage here. Now take a look at the circle circumference equation and the sphere surface area equation. Notice the logical step from the first to the second? Merely squaring the diameter promotes the equation to the analogous equation of the equivalent shape one dimension higher.
The big advantage τ has is that replacing a bunch of "2π"s in textbooks is much easier than doing the algebra required to simplify the equations when you swap r for d⁄2. If you aren't going to do it right though, what is the point of doing it at all? It is true that teaching τ with radius will be easier than what we are doing now, but many students will still start off confused that we are measuring only half of the circle (and isn't one of the big "tauist" arguments that it is absurd that 1 π only gets us halfway around the traditional unit circle?). Sticking with π, but using diameters instead of radians, means that we don't have to overthrow a constant that has been ingrained in mathematics over the course of many millenia. We don't have to deal with teaching two constants just so students will be able to understand even recently written papers, not to mention all of the classical mathematics treatises. It is a lot easier to teach students that r = d/2 than it is to get them to memorize two circle constants out to n digits just so they can work with math from different eras. We also don't have to try to explain to students why we only care about half the width of a circle. The only advantage τ has is convenience in fixing textbooks. Using π with diameters just makes sense, even to those without a heavy mathematical background.
I hope you like my Pi Day celebration! Give a few minutes to celebrate diameter today as well.
10 March 2016
Occam's Two Edged Razor
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.
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.
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