The Ironic Paradox of Normalcy [Is my argument sound?]
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The Ironic Paradox of Normalcy [Is my argument sound?]
Note that I originally wrote this for a different siteone that's very uneducated overall. That's why I explain the "baby" topics that everyone already knows on here. But something occurred to me recentlythat normalcy is ironically and paradoxically (depending on how you look at it) abnormal. So I decided to do what I always do when such a thought pops up in my head... write an essay about it!
What I'm wondering is if my argument and logic was soundand more importantly, my factual information. I'm bad at explaining things in pure text (I can't embed images the way I want to on that other site) form, and I forgot lots of what I learned in AP Stats, so I'm not 100% sure if I'm accurate or not. But I am soon going to compile all my academic walls of text and will post them to a blag of sorts, in order to have a compilation of my works to show to colleges, especially MIT. So I don't want my recent posts especially to be factually wrong, or have a weak argument. I also assume peer review would actually make this look better to them.
But this isn't specifically for that reason. I just enjoy writing these types of things, and usually, on the other site, unsurprisingly only a few individuals actually contribute useful feedback to what I wrote. Here, not only will I get feedback, I can be corrected. And since on that site, because they don't understand math, they more or less view me as their one stop source for math knowledge and will believe anything mathematical I tell them. So I don't want to be factually incorrect, and pass on incorrect material, should they try and learn from it.
Thanks!
The text starts here:
Paradoxically, being normal is abnormal, and I can effectively prove this. We are going to assume an objective standardized measurement to make my point clearer essentially.
If we assume a normal distribution curve, unless someone is *exactly* 50% (i.e. average), then they aren't normal. They'd fall somewhere to the left or to the right (i.e. the deviationshence the term deviant i.e. abnormal) Now we aren't going to assume something with definitive values, such as SAT scores (mean 500, s.d. 100). We're going to assume a Gaussian distributionsuch that the function is continuous and not stepped (as is with the SAT, since you can only score multiples of 10 between 200 and 800).
Let's say that we have a forest of redwood trees. The heights fit a normal Gaussian distribution (as the sample size of trees tends off to infinity, and not considering the Planck length). Now when the statistical information is compiled, we will get a mean (average) and standard deviation. Now this does not mean that there will always be a tree with that exact mean height in a given finite sample size. All the mean means (heh heh) is that from that given sample size, if you were to pick a tree at random, compared to any other exact probability, you are most likely to pick something that is exactly 50%.
One should not confuse percentile with chance. The percentile ranking is where a given data point stands in comparison to all other data points in a set. It is not the chance of that particular data point occurring. For instance, a score of 600 on the SAT puts you at the 84th percentile. This does not give you an 84% chance that you will get a 600. If that were so, then over 99% of people would be getting perfect scores. It means that you scored better than 84% of people. The percentage chance of something happening is a bit more complicated. One would have to go into zscores and standard deviations to fully explain that.
Now I must bring in a bit of calculus to make my point clearer. Let's imagine a boy Herman that is assumed to have an IQ (mean=100, s.d. 15) between 100 and 115. Now IQ isn't actually Gaussian, but for the purpose of trying to get my point across, it's good enough. We don't know his exact IQ, but we know it falls in that range. We know that 100 is the defined "average" i.e. the normal IQ, so that's the 50th percentile. We also know that 115 is one standard deviation (one zscore) off from the bell curve, which is the 84th percentile.
We want to know how many other people are like Herman. The only thing is, we don't know Herman's exact IQ, so we will consider everyone that has an IQ between 100 and 115. Easily enough, we know that 34% of people (by definition the area underneath a curve in between points a and b on the normal distribution curve... i.e. the integral from a to b of the normal distribution curve; in this instance, from .50 to .84) will have IQs in between 100 and 115, since that's the distance in between our a and b.
We then say that Herman has an IQ range that includes 34% of the population's IQs. As omniscient hypothetical hypothesizers, we know that Herman's IQ is *exactly* 100. Now we're going to assume the IQ test dramatically improved, and could provide you with your exact IQ value that could fall on any of the real number system; however, this wasn't before a numerous number of past improvements. Herman was able to take IQ tests as they were improving in accuracy. , and was able to get a more and more accurate result of his score.
Herman's next test gave him a range of scores between 100 and 110. This doesn't have a nice way of figuring the value out without doing the math, so for now, take my word in this calculationif you want to know how it's done, take a Statistics course or google it. It turns out that the percentile interval accounts for about 24.75% of the population.
We continue to take smaller and smaller intervals, i.e. we let b approach a. Here's the calculus bit. Integration (the integralyou know, the long s with a number above and below; specifically, this is the definite integral, which lies between a specified interval) is a process whereby we find the area underneath a curve. The normal distribution curve works using principles of an integral. Imagine you have a curve. Now imagine that you're approximating it with a bunch of little rectangles in your interval, of which the bases are attached to the x axis, and part of the rectangle intersects the function, and all rectangles are adjacent.
Think about Minecraft or pixel art. The more pixellated an image is, the less it resembles the object, and vice versa. The smaller and smaller the pixels get, the better it approximates the true image (a crucial fact for fractal geometry and the box counting dimension, but that's irrelevant to this particular case). So essentially, you can estimate the area underneath a curve by adding more and more rectangles (i.e. raising its "resolution" in ways, by allowing more rectangles to fit in a space due to making the rectangles thinner and thinner, much like making the pixels smaller per unit area makes the picture clearer and more accurate. In fact, the pixel description is essentially a process of integration that's more often used on areas, instead of lines, and the Minecraft voxel example for 3D spaces.
Now back to 3rd grade mathwhat's the formula for the area of a rectangle? Obviously, base (or length) times height (or width). We know the height at a given intersection with the function will always be a fixed value. But we're making the width smaller and smaller and smaller i.e. it approaches zero i.e. making the base length approach zero. Now using limit properties (or common sense), we know that a base length of zero will make the area equal zero. But by making the width smaller, we also get more samples of the function's height (i.e. it's y value for a given x value, in this case, the x value that corresponds to the intersection location), and that tooat more accurate x values. Adding all these areas together in an interval [a,b] (a being the number below the integral symbol and b being the number above it) can tell us the area underneath a curve (more specifically, its distance from the x axis) As the number of rectangles underneath a curve approaches infinity, the value gets more and more accurate. After a given point, the value will be accurate enough for one's purposes. This is basically how to find the integral using the Riemann Sum method.
Now what do you think will happen as b approaches a? The area underneath the curve will clearly get smaller and smaller. You don't even need to understand integrals to see that. Now using some limits, as b approaches a, we say that the limit as b approaches a in the integral will approach zerothat is to say, if b equaled a, then the integral would be zero, since there is less and less area underneath the curve involved.
Now let's take this back to Herman. Herman's next score was in between 100 and 105. The area underneath the curve between 100 and 105 percentilerankwise is 13.05%. Only 13.05% of people would fall into Herman's range. To 104, it's 10.51%. To 103 it's 7.93%. To 102 it's 5.30%. To 101 it's 2.66%. Now the real IQ scores can't be noninteger values, but let's assume it can. To 100.5, it's 1.33%. To 100.25, it's 0.65%.
The trend is very obvious. The smaller and smaller our intervals for IQ geti.e. the closer b gets to a (in this case, a is the average to relate to the point), the fewer and fewer people are like him. Knowing our limits, once Herman takes the test that is able to measure his *exact* IQ (which would require an ideal infinite sample size), mathematically has a 0 chance of being *exactly* one particular number (in essence, if we're hoping for one exact number, paradoxicallybut mathematicallythere's a 0% chance you will obtain that number). So it is essentially impossible to be truly average.
The opposite of this is true, too. Anything that isn't exactly average is technically deviant. Because the formula to see if something is *not* in the percentile range specified is simply one minus that percentile range (in decimal format), it's clear that we approach a 100% chancenot just statistical certainty but absolute certaintythat a person picked at random from a sample size is *not* truly average. I could make this even more mindblowing and explain how Cantor's Diagonal Proof shows that there's a bigger infinity (the infinity of real numbers) in between any two different real numbers than there are integers, but that'd be irrelevant for now.
Due to the nature of the limit (think of it as an asymptoteapproaching but never reaching a particular value, like how y=1/x approaches 0 but never equals 0), one will never actually reach 0. Thus there's still an infinitesimal chance that a data point (a person's IQ) that's *exactly* a given numbere.g. the mean/average/norm. So if in fact you DO pick an IQ of exactly 100what is considered "normal" in daytoday speechgiven the odds of picking that particular data point (a "normal" point) compared to the odds of NOT picking that particular data point (a "deviant" or "abnormal" point), ironically, that particular data point is extremely unusual. This goes to say for any particular data point, really.
Thus, normalcy is actually extremely abnormal.
Now here's the corollary to that.
We don't live in a perfect world, where everything is perfectly objective, and where everything is perfectly Gaussian and perfectly normal. Most people are fine with being an interval of normal. For instance, it'd be difficult to tell the difference between someone with an IQ of 100 and an IQ of 115 without testing. Some people will be perfectly content with that 34% range (IQ is a bad example hereif anything, people WANT to be positively deviant here. But you get my point). Likewise, it's difficult to tell the difference between someone with an IQ of 145 and someone with an IQ of 160 (my IQ); however, this same standard deviation represents a little over a tenth of a percent of the population, as opposed to 64.
As for someone with an IQ of "exactly" 100 (on the actual test in the real world scenario), they'd be among about 2.7% of the rest of the (tested) world. For me with an IQ of 160, I'd be in the 99.9991th percentile, among .00089% of the rest of the (tested) world. In this sense, if one wants a specific value, it is easier to find someone (or in cases where one can freely change a parameter, change that parameter) that's closer to average than it is to find someone that is deviant. So while it would be ironically abnormal to be truly normal as opposed to being abnormal which is ironically normal, when compared to other outcomes individually, especially with high or low zscores, normalcy is something that more people *tend* towards, as opposed to *tending* to deviancy.
One could argue that normalcy is subjective. I find it weird and will never understand how people can walk around completely oblivious to the beauty and patterns of nature that surrounds them. You probably find it weird and will never understand why I wrote this post on a Thursday night just for the fun of it, or why I enjoy math so much. But actually, normalcy is defined by instances, frequency of occurrence, preference majority, etc. For instance, being straight is "normal" simply because it's the most common. This does not automatically imply that being anything other than straight is "wrong;" but it does, however, imply that it is technically abnormal.
But there's one thing about normalcy I will never understand. Why the hell do people say they want to be original, yet try their hardest to blend in and be normal? Be proud of your deviancy. It's what makes you, you. Yes, "everyone is unique" seems paradoxical at first glance. But then one realizes that sureeveryone is unique i.e. there are far more deviants than there are normal people, but everyone is unique in their own way i.e. in a statistical sense, my exact IQ of 160 has the same chance of occurring as an IQ of 40, but the difference between someone with an IQ of 40 and an IQ of 160 is very clear.
While normalcy may be the easy way out, deviancy is what creates the greatsthat is why they are great. As Syndrome in the Incredibles movie puts it, if everyone is super, no one will be. Dare to dream. Dare to be deviant. And you will stand out, and you will go far.
Q.E.D.
What I'm wondering is if my argument and logic was soundand more importantly, my factual information. I'm bad at explaining things in pure text (I can't embed images the way I want to on that other site) form, and I forgot lots of what I learned in AP Stats, so I'm not 100% sure if I'm accurate or not. But I am soon going to compile all my academic walls of text and will post them to a blag of sorts, in order to have a compilation of my works to show to colleges, especially MIT. So I don't want my recent posts especially to be factually wrong, or have a weak argument. I also assume peer review would actually make this look better to them.
But this isn't specifically for that reason. I just enjoy writing these types of things, and usually, on the other site, unsurprisingly only a few individuals actually contribute useful feedback to what I wrote. Here, not only will I get feedback, I can be corrected. And since on that site, because they don't understand math, they more or less view me as their one stop source for math knowledge and will believe anything mathematical I tell them. So I don't want to be factually incorrect, and pass on incorrect material, should they try and learn from it.
Thanks!
The text starts here:
Paradoxically, being normal is abnormal, and I can effectively prove this. We are going to assume an objective standardized measurement to make my point clearer essentially.
If we assume a normal distribution curve, unless someone is *exactly* 50% (i.e. average), then they aren't normal. They'd fall somewhere to the left or to the right (i.e. the deviationshence the term deviant i.e. abnormal) Now we aren't going to assume something with definitive values, such as SAT scores (mean 500, s.d. 100). We're going to assume a Gaussian distributionsuch that the function is continuous and not stepped (as is with the SAT, since you can only score multiples of 10 between 200 and 800).
Let's say that we have a forest of redwood trees. The heights fit a normal Gaussian distribution (as the sample size of trees tends off to infinity, and not considering the Planck length). Now when the statistical information is compiled, we will get a mean (average) and standard deviation. Now this does not mean that there will always be a tree with that exact mean height in a given finite sample size. All the mean means (heh heh) is that from that given sample size, if you were to pick a tree at random, compared to any other exact probability, you are most likely to pick something that is exactly 50%.
One should not confuse percentile with chance. The percentile ranking is where a given data point stands in comparison to all other data points in a set. It is not the chance of that particular data point occurring. For instance, a score of 600 on the SAT puts you at the 84th percentile. This does not give you an 84% chance that you will get a 600. If that were so, then over 99% of people would be getting perfect scores. It means that you scored better than 84% of people. The percentage chance of something happening is a bit more complicated. One would have to go into zscores and standard deviations to fully explain that.
Now I must bring in a bit of calculus to make my point clearer. Let's imagine a boy Herman that is assumed to have an IQ (mean=100, s.d. 15) between 100 and 115. Now IQ isn't actually Gaussian, but for the purpose of trying to get my point across, it's good enough. We don't know his exact IQ, but we know it falls in that range. We know that 100 is the defined "average" i.e. the normal IQ, so that's the 50th percentile. We also know that 115 is one standard deviation (one zscore) off from the bell curve, which is the 84th percentile.
We want to know how many other people are like Herman. The only thing is, we don't know Herman's exact IQ, so we will consider everyone that has an IQ between 100 and 115. Easily enough, we know that 34% of people (by definition the area underneath a curve in between points a and b on the normal distribution curve... i.e. the integral from a to b of the normal distribution curve; in this instance, from .50 to .84) will have IQs in between 100 and 115, since that's the distance in between our a and b.
We then say that Herman has an IQ range that includes 34% of the population's IQs. As omniscient hypothetical hypothesizers, we know that Herman's IQ is *exactly* 100. Now we're going to assume the IQ test dramatically improved, and could provide you with your exact IQ value that could fall on any of the real number system; however, this wasn't before a numerous number of past improvements. Herman was able to take IQ tests as they were improving in accuracy. , and was able to get a more and more accurate result of his score.
Herman's next test gave him a range of scores between 100 and 110. This doesn't have a nice way of figuring the value out without doing the math, so for now, take my word in this calculationif you want to know how it's done, take a Statistics course or google it. It turns out that the percentile interval accounts for about 24.75% of the population.
We continue to take smaller and smaller intervals, i.e. we let b approach a. Here's the calculus bit. Integration (the integralyou know, the long s with a number above and below; specifically, this is the definite integral, which lies between a specified interval) is a process whereby we find the area underneath a curve. The normal distribution curve works using principles of an integral. Imagine you have a curve. Now imagine that you're approximating it with a bunch of little rectangles in your interval, of which the bases are attached to the x axis, and part of the rectangle intersects the function, and all rectangles are adjacent.
Think about Minecraft or pixel art. The more pixellated an image is, the less it resembles the object, and vice versa. The smaller and smaller the pixels get, the better it approximates the true image (a crucial fact for fractal geometry and the box counting dimension, but that's irrelevant to this particular case). So essentially, you can estimate the area underneath a curve by adding more and more rectangles (i.e. raising its "resolution" in ways, by allowing more rectangles to fit in a space due to making the rectangles thinner and thinner, much like making the pixels smaller per unit area makes the picture clearer and more accurate. In fact, the pixel description is essentially a process of integration that's more often used on areas, instead of lines, and the Minecraft voxel example for 3D spaces.
Now back to 3rd grade mathwhat's the formula for the area of a rectangle? Obviously, base (or length) times height (or width). We know the height at a given intersection with the function will always be a fixed value. But we're making the width smaller and smaller and smaller i.e. it approaches zero i.e. making the base length approach zero. Now using limit properties (or common sense), we know that a base length of zero will make the area equal zero. But by making the width smaller, we also get more samples of the function's height (i.e. it's y value for a given x value, in this case, the x value that corresponds to the intersection location), and that tooat more accurate x values. Adding all these areas together in an interval [a,b] (a being the number below the integral symbol and b being the number above it) can tell us the area underneath a curve (more specifically, its distance from the x axis) As the number of rectangles underneath a curve approaches infinity, the value gets more and more accurate. After a given point, the value will be accurate enough for one's purposes. This is basically how to find the integral using the Riemann Sum method.
Now what do you think will happen as b approaches a? The area underneath the curve will clearly get smaller and smaller. You don't even need to understand integrals to see that. Now using some limits, as b approaches a, we say that the limit as b approaches a in the integral will approach zerothat is to say, if b equaled a, then the integral would be zero, since there is less and less area underneath the curve involved.
Now let's take this back to Herman. Herman's next score was in between 100 and 105. The area underneath the curve between 100 and 105 percentilerankwise is 13.05%. Only 13.05% of people would fall into Herman's range. To 104, it's 10.51%. To 103 it's 7.93%. To 102 it's 5.30%. To 101 it's 2.66%. Now the real IQ scores can't be noninteger values, but let's assume it can. To 100.5, it's 1.33%. To 100.25, it's 0.65%.
The trend is very obvious. The smaller and smaller our intervals for IQ geti.e. the closer b gets to a (in this case, a is the average to relate to the point), the fewer and fewer people are like him. Knowing our limits, once Herman takes the test that is able to measure his *exact* IQ (which would require an ideal infinite sample size), mathematically has a 0 chance of being *exactly* one particular number (in essence, if we're hoping for one exact number, paradoxicallybut mathematicallythere's a 0% chance you will obtain that number). So it is essentially impossible to be truly average.
The opposite of this is true, too. Anything that isn't exactly average is technically deviant. Because the formula to see if something is *not* in the percentile range specified is simply one minus that percentile range (in decimal format), it's clear that we approach a 100% chancenot just statistical certainty but absolute certaintythat a person picked at random from a sample size is *not* truly average. I could make this even more mindblowing and explain how Cantor's Diagonal Proof shows that there's a bigger infinity (the infinity of real numbers) in between any two different real numbers than there are integers, but that'd be irrelevant for now.
Due to the nature of the limit (think of it as an asymptoteapproaching but never reaching a particular value, like how y=1/x approaches 0 but never equals 0), one will never actually reach 0. Thus there's still an infinitesimal chance that a data point (a person's IQ) that's *exactly* a given numbere.g. the mean/average/norm. So if in fact you DO pick an IQ of exactly 100what is considered "normal" in daytoday speechgiven the odds of picking that particular data point (a "normal" point) compared to the odds of NOT picking that particular data point (a "deviant" or "abnormal" point), ironically, that particular data point is extremely unusual. This goes to say for any particular data point, really.
Thus, normalcy is actually extremely abnormal.
Now here's the corollary to that.
We don't live in a perfect world, where everything is perfectly objective, and where everything is perfectly Gaussian and perfectly normal. Most people are fine with being an interval of normal. For instance, it'd be difficult to tell the difference between someone with an IQ of 100 and an IQ of 115 without testing. Some people will be perfectly content with that 34% range (IQ is a bad example hereif anything, people WANT to be positively deviant here. But you get my point). Likewise, it's difficult to tell the difference between someone with an IQ of 145 and someone with an IQ of 160 (my IQ); however, this same standard deviation represents a little over a tenth of a percent of the population, as opposed to 64.
As for someone with an IQ of "exactly" 100 (on the actual test in the real world scenario), they'd be among about 2.7% of the rest of the (tested) world. For me with an IQ of 160, I'd be in the 99.9991th percentile, among .00089% of the rest of the (tested) world. In this sense, if one wants a specific value, it is easier to find someone (or in cases where one can freely change a parameter, change that parameter) that's closer to average than it is to find someone that is deviant. So while it would be ironically abnormal to be truly normal as opposed to being abnormal which is ironically normal, when compared to other outcomes individually, especially with high or low zscores, normalcy is something that more people *tend* towards, as opposed to *tending* to deviancy.
One could argue that normalcy is subjective. I find it weird and will never understand how people can walk around completely oblivious to the beauty and patterns of nature that surrounds them. You probably find it weird and will never understand why I wrote this post on a Thursday night just for the fun of it, or why I enjoy math so much. But actually, normalcy is defined by instances, frequency of occurrence, preference majority, etc. For instance, being straight is "normal" simply because it's the most common. This does not automatically imply that being anything other than straight is "wrong;" but it does, however, imply that it is technically abnormal.
But there's one thing about normalcy I will never understand. Why the hell do people say they want to be original, yet try their hardest to blend in and be normal? Be proud of your deviancy. It's what makes you, you. Yes, "everyone is unique" seems paradoxical at first glance. But then one realizes that sureeveryone is unique i.e. there are far more deviants than there are normal people, but everyone is unique in their own way i.e. in a statistical sense, my exact IQ of 160 has the same chance of occurring as an IQ of 40, but the difference between someone with an IQ of 40 and an IQ of 160 is very clear.
While normalcy may be the easy way out, deviancy is what creates the greatsthat is why they are great. As Syndrome in the Incredibles movie puts it, if everyone is super, no one will be. Dare to dream. Dare to be deviant. And you will stand out, and you will go far.
Q.E.D.
I'm sexually attracted to the International Space Station. You have now had your daily dose of Internet.

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Re: The Ironic Paradox of Normalcy [Is my argument sound?]
As far as the pure math is concerned, your point can be summarized as "the mean is not generally the same as the mode", regarding distributions.
Regarding the more social aspect of your point, there is some value in it. It is worthwhile to note that any specific combination of traits is extremely rare, even if the generalities are common. Uniqueness in that respect is a matter of how specific you want to be in your descriptions, and a sufficiently specific description makes anyone unique.
Regarding the more social aspect of your point, there is some value in it. It is worthwhile to note that any specific combination of traits is extremely rare, even if the generalities are common. Uniqueness in that respect is a matter of how specific you want to be in your descriptions, and a sufficiently specific description makes anyone unique.
Re: The Ironic Paradox of Normalcy [Is my argument sound?]
I don't think, at least colloquially, anyone would ever consider "normal" to mean "exactly average". I would say that in conventional usage, it's probably more correct to say that "normal" is everyone within 1 standard deviation of the mean. IMHO as soon as you define normal to mean "exactly average", then this is basically proofbydefinition.
Even with this definition, I think you can still prove everyone is abnormal in at least one respect though, using the same sort of argument. You'd just break it down by property. A person can have normal height, normal weight, but abnormal intelligence, for example. If a person is normal in every possible category, then they are abnormal because of their excessive normalcy.
Even with this definition, I think you can still prove everyone is abnormal in at least one respect though, using the same sort of argument. You'd just break it down by property. A person can have normal height, normal weight, but abnormal intelligence, for example. If a person is normal in every possible category, then they are abnormal because of their excessive normalcy.
 GirlWithAMathFetish
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Re: The Ironic Paradox of Normalcy [Is my argument sound?]
LaserGuy wrote:I don't think, at least colloquially, anyone would ever consider "normal" to mean "exactly average". I would say that in conventional usage, it's probably more correct to say that "normal" is everyone within 1 standard deviation of the mean. IMHO as soon as you define normal to mean "exactly average", then this is basically proofbydefinition.
Even with this definition, I think you can still prove everyone is abnormal in at least one respect though, using the same sort of argument. You'd just break it down by property. A person can have normal height, normal weight, but abnormal intelligence, for example. If a person is normal in every possible category, then they are abnormal because of their excessive normalcy.
That's what I mentioned in the corollary ehehehe, and why I mentioned normalcy in terms of parameters.
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 GirlWithAMathFetish
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Re: The Ironic Paradox of Normalcy [Is my argument sound?]
arbiteroftruth wrote:As far as the pure math is concerned, your point can be summarized as "the mean is not generally the same as the mode", regarding distributions.
Regarding the more social aspect of your point, there is some value in it. It is worthwhile to note that any specific combination of traits is extremely rare, even if the generalities are common. Uniqueness in that respect is a matter of how specific you want to be in your descriptions, and a sufficiently specific description makes anyone unique.
That's why I mentioned normalcy in terms of specific parameters. Hehe
I'm sexually attracted to the International Space Station. You have now had your daily dose of Internet.
Re: The Ironic Paradox of Normalcy [Is my argument sound?]
I will first say that I read that very quickly, so please forgive any details I may have missed, but here are my thoughts:
To me, it seems you are arguing about something people never actually say, a strawman of sorts. Of course, the probability of having any particular number on a continuous distribution is exactly zero, but that would defeat the purpose of having a probability distribution fumction in the first place. When someone say an "average" person, I don't think anyone means a man perfectly at the middle, but a certain range, maybe about half a standard deviation from the mean. And it is a fact that, in a normal distribution, for any finite interval, there will be less persons the further you are from the mean. There are certainly more people between 0 and 1 st. deviations from the middle than there are from 1 to 2.
Your essay also seems to have the message that it's ok to be abnormal, to accept differences etc. While I may agree with the conclusion, I think the preceding math is probably unneccessary/irrelevant. Both because, as I said, perfectly average isn't what is usually mean by normality, but also because using statistics and math to "prove" something that sounds contradictory (and, in my opinion, arguably "wrong") makes the impression that logic and statistics are spurious, with arbitrary definitons. Specially if, as you say, the intended public is not math literate and may see any kind of numerical manipulation as witchcraft. It's the sort of thing that inspire phrases like "there are three kinds of lies  lies, damned lies and statistics"
edit oh well, it seems a lot of people have replied while I was writing... I guess you can disregard a good part of my post.
To me, it seems you are arguing about something people never actually say, a strawman of sorts. Of course, the probability of having any particular number on a continuous distribution is exactly zero, but that would defeat the purpose of having a probability distribution fumction in the first place. When someone say an "average" person, I don't think anyone means a man perfectly at the middle, but a certain range, maybe about half a standard deviation from the mean. And it is a fact that, in a normal distribution, for any finite interval, there will be less persons the further you are from the mean. There are certainly more people between 0 and 1 st. deviations from the middle than there are from 1 to 2.
Your essay also seems to have the message that it's ok to be abnormal, to accept differences etc. While I may agree with the conclusion, I think the preceding math is probably unneccessary/irrelevant. Both because, as I said, perfectly average isn't what is usually mean by normality, but also because using statistics and math to "prove" something that sounds contradictory (and, in my opinion, arguably "wrong") makes the impression that logic and statistics are spurious, with arbitrary definitons. Specially if, as you say, the intended public is not math literate and may see any kind of numerical manipulation as witchcraft. It's the sort of thing that inspire phrases like "there are three kinds of lies  lies, damned lies and statistics"
edit oh well, it seems a lot of people have replied while I was writing... I guess you can disregard a good part of my post.
 GirlWithAMathFetish
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Re: The Ironic Paradox of Normalcy [Is my argument sound?]
brenok wrote:To me, it seems you are arguing about something people never actually say, a strawman of sorts.
When someone say an "average" person, I don't think anyone means a man perfectly at the middle, but a certain range, maybe about half a standard deviation from the mean.
That's why I included the corollary I grossly oversimplified my post as to not confuse the fuck out of the people on that site. xD
brenok wrote: And it is a fact that, in a normal distribution, for any finite interval, there will be less persons the further you are from the mean. There are certainly more people between 0 and 1 st. deviations from the middle than there are from 1 to 2.
I think I mentioned that in my corollary toobut if not there, it's definitely in the main body text.
brenok wrote:Your essay also seems to have the message that it's ok to be abnormal, to accept differences etc. While I may agree with the conclusion, I think the preceding math is probably unneccessary/irrelevant.
Actually, if anything, the abnormality message thing was irrelevant. My post was mainly supposed to be about the math. I'm notorious (though respected) on that site for constantly posting math stuff. xD
I'm sexually attracted to the International Space Station. You have now had your daily dose of Internet.
Re: The Ironic Paradox of Normalcy [Is my argument sound?]
You ramble on a bit, and some of the details in your explanations are a little bit sloppy, but it's still a pretty good essay, GirlWithAMathFetish. With a little bit more organization, it would be very good. OTOH, I think your conversational style makes the material a lot more accessible than a more formal essay would be.
We may have another Vi Hart or Marilyn vos Savant on our hands...
PS. Please don't doublepost, especially when you're making essentially the response to two (or more) people. Just quote both (all) the people you're replying to in a single reply. (You can easily include additional quotes once you start replying by clicking the QUOTE buttons in the message list at the bottom of the POST A REPLY page).
We may have another Vi Hart or Marilyn vos Savant on our hands...
PS. Please don't doublepost, especially when you're making essentially the response to two (or more) people. Just quote both (all) the people you're replying to in a single reply. (You can easily include additional quotes once you start replying by clicking the QUOTE buttons in the message list at the bottom of the POST A REPLY page).
Re: The Ironic Paradox of Normalcy [Is my argument sound?]
As an aside, I would drop the comment about your IQ being 160. It's fine to use it as an example, but telling people your IQ, especially if it's high (which is the only time people ever mention itif you looked at the distribution of IQ scores that people say they have, it would definitely not be a normal curve) always just seems to come across as intellectual dickmeasuring to me. If you're smart, it will come across fine without needing to put a (possibly meaningless) number on it.
Re: The Ironic Paradox of Normalcy [Is my argument sound?]
I thought IQ was a gaussian distribution  like isn't it, more or less, supposed to be "okay, rank everyone from best to worst and now fix them to a gaussian distribution"?
A much more succinct version of the paradox you have is that, suppose we choose a number x at random from the distribution [0,1]. The probability that, were we to choose another y from the same distribution, x would equal y is 0. So, somehow, every time we choose a random number from the distribution, we get a result which is infinitely unlikely.
In a similar vein, it is highly unlikely that any individual would win the lottery  and yet someone does.
A much more succinct version of the paradox you have is that, suppose we choose a number x at random from the distribution [0,1]. The probability that, were we to choose another y from the same distribution, x would equal y is 0. So, somehow, every time we choose a random number from the distribution, we get a result which is infinitely unlikely.
In a similar vein, it is highly unlikely that any individual would win the lottery  and yet someone does.
Mathematical hangover (n.): The feeling one gets in the morning when they realize that that short, elementary proof of the Riemann hypothesis that they came up with at midnight the night before is, in fact, nonsense.
 GirlWithAMathFetish
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Re: The Ironic Paradox of Normalcy [Is my argument sound?]
Moole wrote:I thought IQ was a gaussian distribution  like isn't it, more or less, supposed to be "okay, rank everyone from best to worst and now fix them to a gaussian distribution"?
Yes, but the IQ test only includes integer values.
LaserGuy wrote:As an aside, I would drop the comment about your IQ being 160. It's fine to use it as an example, but telling people your IQ, especially if it's high (which is the only time people ever mention itif you looked at the distribution of IQ scores that people say they have, it would definitely not be a normal curve) always just seems to come across as intellectual dickmeasuring to me. If you're smart, it will come across fine without needing to put a (possibly meaningless) number on it.
On that site, few view me as such. xD That too, I wasn't "showing off." I legitimately used it for emphasis.
I hardly mention my IQ on that site. Normally I just spam math stuff in order to nerdfish on that site. Else, it's just a nice place to post something intellectual in a weird wayit gives a decent idea of how the public would react to such topics if I was able to announce such stuff to the public, since it holds a very typical collection of people. Usually, people actually *overstate* my capability on that site. I don't like when people do that, since I feel bad when people hold such high expectations of me that I can never meet.
PM 2ring wrote:We may have another Vi Hart or Marilyn vos Savant on our hands...
Funny you say that xD I was actually planning on making ViHartlike videos over the past summer (except I'd go deeper into the topics). Unfortunately, I ran into severe technical problems. I had a green screen and everything, but my laptop... bleh. The videos essentially would teach topics in depth, and I'd ramble quite a bit, but I'd probably often throw in some sexual innuendo, since often, that's what makes people pay attention. And you can assume by my username that I'm quite adept at doing such with math. xD Eventually I'll get around to doing the videos, though. xD
But I'm no savant. Unless I get some sort of head injury that somehow makes my incredibly terrible memory much better (I guess you could say that I have an i7 processor, an exabyte of hard disk space, but only 2 kilobits of RAM), I'll be nothing more than an extremely ADHD nerd. xD The weird thing about my memory is that in certain circumstances, it's nearly flawless. If someone's reading out a short story with lots of imagery, it's easy to turn everything into one string, i.e. a movie in my head. But when I'm working with numbers, I'm creating all sorts of strings. So while I would be able to do the mental calculation itself rapidly, I can only hold twothree at moststrings of numbers in my head. x'D Stupid memory. x'D
I'm sexually attracted to the International Space Station. You have now had your daily dose of Internet.
Re: The Ironic Paradox of Normalcy [Is my argument sound?]
There's a lot of words there....
A couple things. You can completely remove the bit about how to find a Riemann integral. You can make the argument that the integral from a to a is 0 by just noting that the integral from a to b is smaller than (ba)M where M is any value larger than the function on that interval.
The bit about 'it's not exactly 0' is incorrect. The probability of selecting a single point from a normal distribution, or indeed any continuous distribution (assuming you define continuous distributions to be those that arise from PDFs), is precisely and exactly 0. You're correct that none of the areas you are taking the limit of will be 0, but the probability is not any of the values you are taking the limit of, but rather the limit itself. There's nothing infinitesimal about it, it is precisely 0, and the chance of NOT picking a particular value is not approaching 1, it is precisely 1.
A couple things. You can completely remove the bit about how to find a Riemann integral. You can make the argument that the integral from a to a is 0 by just noting that the integral from a to b is smaller than (ba)M where M is any value larger than the function on that interval.
The bit about 'it's not exactly 0' is incorrect. The probability of selecting a single point from a normal distribution, or indeed any continuous distribution (assuming you define continuous distributions to be those that arise from PDFs), is precisely and exactly 0. You're correct that none of the areas you are taking the limit of will be 0, but the probability is not any of the values you are taking the limit of, but rather the limit itself. There's nothing infinitesimal about it, it is precisely 0, and the chance of NOT picking a particular value is not approaching 1, it is precisely 1.
addams wrote:This forum has some very well educated people typing away in loops with Sourmilk. He is a lucky Sourmilk.
Re: The Ironic Paradox of Normalcy [Is my argument sound?]
mikel wrote:The bit about 'it's not exactly 0' is incorrect. The probability of selecting a single point from a normal distribution, or indeed any continuous distribution (assuming you define continuous distributions to be those that arise from PDFs), is precisely and exactly 0. You're correct that none of the areas you are taking the limit of will be 0, but the probability is not any of the values you are taking the limit of, but rather the limit itself. There's nothing infinitesimal about it, it is precisely 0, and the chance of NOT picking a particular value is not approaching 1, it is precisely 1.
To add on to that, the distinction that should be made here is that, one would say "Event X never happens" if it's not even part of the space of events  like, having an IQ of "elephant"^{1} or picking 1 out of the uniform distribution on [0,1]. In the strictest mathematical sense, the idea of "never happens" and "always happens" is fairly useless  so instead, we have "almost surely" and "almost never"  that is, even if event X has probability 0, but could, in theory, happen, we say "Event X almost never happens" is correct  though it's not the colloquial usage of "almost", since that would imply "the probability is very low", where it really means "the probability is 0"  the difference mainly being that one never observes an event which occurs never, but in a continuous distribution, one always observes an even which occurs almost never (which ultimately the paradox here).
_{1. My IQ is "xkcd"  I think it's just some meaningless acronym.}
Mathematical hangover (n.): The feeling one gets in the morning when they realize that that short, elementary proof of the Riemann hypothesis that they came up with at midnight the night before is, in fact, nonsense.
Re: The Ironic Paradox of Normalcy [Is my argument sound?]
Honestly? I think it's a really long piece of writing with lots of unnecessary and distracting comments embedded throughout. You mention stuff like Planck's length and Cantor's diagonal argument. These have absolutely nothing to do with your point, and makes your writing difficult to follow, especially if the target audience isn't wellversed in math or physics. On top of that, I doubt that a typical person would agree with your definition of "normal" to mean "exactly the average" anyway. The essential mathematical content of your post is that "the probability of picking any particular value from a normal distribution is zero", and this can probably be explained with half the words you used and without any references to your IQ.
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.
 GirlWithAMathFetish
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Re: The Ironic Paradox of Normalcy [Is my argument sound?]
mikel wrote:A couple things. You can completely remove the bit about how to find a Riemann integral.
I'm known on that site to ramble, and that's what people generally like about me there x'D
mikel wrote:The bit about 'it's not exactly 0' is incorrect. The probability of selecting a single point from a normal distribution, or indeed any continuous distribution (assuming you define continuous distributions to be those that arise from PDFs), is precisely and exactly 0. You're correct that none of the areas you are taking the limit of will be 0, but the probability is not any of the values you are taking the limit of, but rather the limit itself. There's nothing infinitesimal about it, it is precisely 0, and the chance of NOT picking a particular value is not approaching 1, it is precisely 1.
I think I was also trying to explain a limit (again, that's a site filled with laypeople), but I might have stopped halfway. I already forgot 95% of what I wrote. xD
brenok wrote:moole wrote:1. My IQ is "xkcd"  I think it's just some meaningless acronym.
You IQ is pretty high.
Alternatively, if he's using base 34, his IQ in denary would then be 1320573, in base 35 it would be 1439808, and in base 36, it would be 1566013. So it IS pretty high.
z4lis wrote:Honestly? I think it's a really long piece of writing with lots of unnecessary and distracting comments embedded throughout.
I tend to ramble on that site, which is why I have 90% of the followers I do. They love how I ramble x'D It's not supposed to be formal. If you read some of my other things there, I ramble a lot in those, too. x'D If I don't ramble, then people get bored quite easily, and don't read my thing x'D
I'm sexually attracted to the International Space Station. You have now had your daily dose of Internet.
 Forest Goose
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Re: The Ironic Paradox of Normalcy [Is my argument sound?]
You can say what you like about rambling making people like you, but, seriously, that doesn't justify rambling. Bad writing is bad writing  a good excuse is never: but my audience wants me to spend 300 words on something entirely irrelevant. Honestly, I doubt they do.
As for the piece:
This makes you sound pretentious and like you fancy yourself wise and bright (I think it makes you sound all around the opposite). It is written from a smug perspective and takes a very long time to make a point that misses the point.
More than anything, though, the whole premise is flawed: what are you even trying to say? That people should be "original"? That people should care less about being "normal"? That people should embrace the wonderful snowflakes that they are? I'm sorry, but I think more people are themselves and have talents they are proud of than you might think. I know tons of highly creative individuals that act just like regular folks most of the time, I know lots of talented people, in general, who act perfectly normal day to day. And why shouldn't they?
Social conventions exist for a reason, those who don't follow them are, generally, criminals, obnoxious, childish, or intolerable to be around. They are not "greats". There have been deviant "great people", they were not great because of any deviance, they were great in spite of it  and there have been all sorts of "greats" who were perfectly reasonable individuals who happened to be extremely talented. This idea of the eccentric genius is more the subject of popular accounts and fiction than it is a reality  there are plenty of brilliant people all over the place, and in almost all contexts, they look just like everyone else. And, again, why shouldn't they?
I'm not arguing that people ought not express themselves, or that social convention should dictate who we are, but, to be honest, I don't think anyone else is either. You are rebutting a mindset that doesn't happen to exist, save as dreamt up in the minds who fancy themselves "unique".
As for the piece:
This makes you sound pretentious and like you fancy yourself wise and bright (I think it makes you sound all around the opposite). It is written from a smug perspective and takes a very long time to make a point that misses the point.
More than anything, though, the whole premise is flawed: what are you even trying to say? That people should be "original"? That people should care less about being "normal"? That people should embrace the wonderful snowflakes that they are? I'm sorry, but I think more people are themselves and have talents they are proud of than you might think. I know tons of highly creative individuals that act just like regular folks most of the time, I know lots of talented people, in general, who act perfectly normal day to day. And why shouldn't they?
Social conventions exist for a reason, those who don't follow them are, generally, criminals, obnoxious, childish, or intolerable to be around. They are not "greats". There have been deviant "great people", they were not great because of any deviance, they were great in spite of it  and there have been all sorts of "greats" who were perfectly reasonable individuals who happened to be extremely talented. This idea of the eccentric genius is more the subject of popular accounts and fiction than it is a reality  there are plenty of brilliant people all over the place, and in almost all contexts, they look just like everyone else. And, again, why shouldn't they?
I'm not arguing that people ought not express themselves, or that social convention should dictate who we are, but, to be honest, I don't think anyone else is either. You are rebutting a mindset that doesn't happen to exist, save as dreamt up in the minds who fancy themselves "unique".
Forest Goose: A rare, but wily, form of goose; best known for dropping on unsuspecting hikers, from trees, to steal sweets.
Re: The Ironic Paradox of Normalcy [Is my argument sound?]
Tangential How often is it the case that people actually know their IQ? In most of my experiences with people claiming IQ's it's been people basically pulling a number out of thin air ("I'm smarter than most people I know, so I'm probably around 130") or getting it off of some random online test (or even a chain email) which I wouldn't put much faith in.
In the one case where I did get a fairly formal test where my school brought in someone (as I was a bit of a disruptive student, but tended to ace tests), they never actually told me the result, and instead only told my parents who I understand were told they shouldn't tell me. (Presumably since low/'average' might be demotivating, and 'high' might lead to arrogance.) Perhaps in some parts formally administered IQ tests are more common and the results aren't kept as secret?
In the one case where I did get a fairly formal test where my school brought in someone (as I was a bit of a disruptive student, but tended to ace tests), they never actually told me the result, and instead only told my parents who I understand were told they shouldn't tell me. (Presumably since low/'average' might be demotivating, and 'high' might lead to arrogance.) Perhaps in some parts formally administered IQ tests are more common and the results aren't kept as secret?
 Forest Goose
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Re: The Ironic Paradox of Normalcy [Is my argument sound?]
I took two in school, one when I was very young. I was never told the results, I didn't find out till I was in my mid twenties what the one was, my mother forgot the other. I don't really see that it mattered, it wouldn't have made a difference either way; it doesn't seem that informative of a number to know.
If you press most people quoting IQ scores, I've found this to be the case. Of course, that's just my experience, I have no idea if that's the norm.
Dopefish wrote:Tangential How often is it the case that people actually know their IQ? In most of my experiences with people claiming IQ's it's been people basically pulling a number out of thin air ("I'm smarter than most people I know, so I'm probably around 130") or getting it off of some random online test (or even a chain email) which I wouldn't put much faith in.
If you press most people quoting IQ scores, I've found this to be the case. Of course, that's just my experience, I have no idea if that's the norm.
Forest Goose: A rare, but wily, form of goose; best known for dropping on unsuspecting hikers, from trees, to steal sweets.
Re: The Ironic Paradox of Normalcy [Is my argument sound?]
Forest Goose wrote:This makes you sound pretentious and like you fancy yourself wise and bright (I think it makes you sound all around the opposite). It is written from a smug perspective and takes a very long time to make a point that misses the point.
I agree with this.
Re: The Ironic Paradox of Normalcy [Is my argument sound?]
GirlWithAMathFetish wrote:a compilation of my works to show to colleges, especially MIT. So I don't want my recent posts especially to be factually wrong, or have a weak argument.
I would recommend that you don't show that sort of thing to colleges, because in my opinion, it'll just make you look bad. It seems that your goal was to convince the reader that you're really clever, which in my opinion is indicative of a bad attitude, which I consider to be much worse than just being factually incorrect.
Re: The Ironic Paradox of Normalcy [Is my argument sound?]
GirlWithAMathFetish wrote:I tend to ramble on that site, which is why I have 90% of the followers I do. They love how I ramble x'D It's not supposed to be formal. If you read some of my other things there, I ramble a lot in those, too. x'D If I don't ramble, then people get bored quite easily, and don't read my thing x'D
This just comes down to a question of who your audience is. If your intention is to submit this to a blog/discussion forum/whatever, and that's what your audience likes to hear from you, then I suppose that's what you should write. If your intention is to submit this as part of a university entrance application package, the same sort of writing may not be appropriate for that audience.
Re: The Ironic Paradox of Normalcy [Is my argument sound?]
mark999 wrote:GirlWithAMathFetish wrote:a compilation of my works to show to colleges, especially MIT. So I don't want my recent posts especially to be factually wrong, or have a weak argument.
I would recommend that you don't show that sort of thing to colleges, because in my opinion, it'll just make you look bad. It seems that your goal was to convince the reader that you're really clever, which in my opinion is indicative of a bad attitude, which I consider to be much worse than just being factually incorrect.
Yeah, I'd have to agree with that; I mean, the piece feels like it's trying to bash the reader over the head with the idea. To the right audience, the title would've sufficed to explain your idea  and there was certainly no doubt in what the conclusion would be by the end of the first two sentences. Anyone who's ever measured anything is going to see that the probability of something being exactly 1 as opposed to 1.01 or .99 is vanishingly small. It's kind of insulting to the reader's intellect to assume that this is not obvious. Certainly, were it not already clear, the explanation given is not going to be much help  the paradox makes perfect sense without knowing about limits and integrals and Gauss  and comes off as "hey, look at this smattering of theorems I know" (namedropping Cantor doesn't help  the relevance of a Gaussian is dubious as well), even if it was intended as, "see how beautiful the math behind this is?"  I think a less intuitive result would be necessary to demonstrate the power of math, because every step of your proof is routine, so it's not particularly astonishing that, at the end, we reach a routine result  it'd be like proving the quadratic formula by computing f((b+sqrt(b^24ac))/(2a)) where, yeah, it works, but the proof isn't worth reading, especially given that more insightful solutions, like completing the square, substituting u=x+b/2, or showing that the results yielded from the quadratic formula have the correct algebraic properties (i.e. satisfy Vieta's formulas).
I think that your post shows good thought; you've connected the relevance of integration and limits to probability, and you've noticed a confusing aspect of probability: that the probability of being exactly anything is slim, and yet that it is necessary that results which, a priori, are improbable, must occur. And it's probably good to recognize that there is no such thing as a "typical" person*  and that even people who want to fit in are still unique. Yet, the essay resulting from this is not good for all contexts.
_{*Though, if such a person existed, man, would the sociologists and psychologists have a field day on them.}
Mathematical hangover (n.): The feeling one gets in the morning when they realize that that short, elementary proof of the Riemann hypothesis that they came up with at midnight the night before is, in fact, nonsense.
Re: The Ironic Paradox of Normalcy [Is my argument sound?]
I second that it's probably not appropriate to include such posts in college applications. That post is certainly not written in scientific language, independent of the factual truth of the argument. That is not necessarily terrible as you're usually not expected to be able to use scientific methods before you enter college. So while my answer may sound discouraging your skill of writing scientific papers will certainly improve when you're in college.
If you want to write a scientific article the post has to be much more concise and rigorous and not include irrelevant facts as others said. In this case I'd also advice you to remove all references to nonmathematical concepts i.e. normality in a social sense. Otherwise if you want to write a pseudoscientific essay I'd suggest to formulate the post in a more anecdotal way and not claim that it is mathematically rigorous. In any case I strongly suggest to remove references that make you sound pretentious. I also think that the post is quite confusing to read. It is not clear what your main point or theorem is. An outline of the article at the beginning would be nice.
If you want to write a scientific article the post has to be much more concise and rigorous and not include irrelevant facts as others said. In this case I'd also advice you to remove all references to nonmathematical concepts i.e. normality in a social sense. Otherwise if you want to write a pseudoscientific essay I'd suggest to formulate the post in a more anecdotal way and not claim that it is mathematically rigorous. In any case I strongly suggest to remove references that make you sound pretentious. I also think that the post is quite confusing to read. It is not clear what your main point or theorem is. An outline of the article at the beginning would be nice.
Re: The Ironic Paradox of Normalcy [Is my argument sound?]
This article is basically nonsense. The mathematics of the main section are largely irrelevant (for the reasons others have posted above)
In the "corollary", you basically say "But no one really thinks in those terms anyway", thus explicitly rendering the entire main body as irrelevant.
Yep. Cause I'm an idiot with no ability to relate to others.
Questionable. Some claim that being straight has roots in perpetuation of the species and thus would stay "normal" even if the minority practiced it.
Only because you've forced a definition of normalcy that requires a binary state
Forrest Goose's critique is very accurate.
Also, you write:
If you don't read this as showing off, perhaps you need a little more self awareness.
In the "corollary", you basically say "But no one really thinks in those terms anyway", thus explicitly rendering the entire main body as irrelevant.
"You probably find it weird and will never understand why I wrote this post on a Thursday night just for the fun of it, or why I enjoy math so much."
Yep. Cause I'm an idiot with no ability to relate to others.
"For instance, being straight is "normal" simply because it's the most common."
Questionable. Some claim that being straight has roots in perpetuation of the species and thus would stay "normal" even if the minority practiced it.
"This does not automatically imply that being anything other than straight is "wrong;" but it does, however, imply that it is technically abnormal."
Only because you've forced a definition of normalcy that requires a binary state
Forrest Goose's critique is very accurate.
Also, you write:
... and someone with an IQ of 160 (my IQ); ...
... For me with an IQ of 160, ...
... i.e. in a statistical sense, my exact IQ of 160 ...
If you don't read this as showing off, perhaps you need a little more self awareness.
Re: The Ironic Paradox of Normalcy [Is my argument sound?]
Let me give a more positive suggestion for what you might want to put into a college application to show that you like math: Some solutions of mathematical problems, carefully written up! I would imagine that a selection of solutions to problems far more involved that a typical high school math problem and carefully, correctly solved would be much more impressive than a blog post for a general audience. Look at stuff on the Art of Problem Solving website (or even the Putnam!) for an idea of what kinds of problems you'll want to attack to convince an undergraduate math department you're the real deal.
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.
Re: The Ironic Paradox of Normalcy [Is my argument sound?]
OK, speaking as a professional mathematician, if a potential student put that under my nose I'd yell "CRANK!" and run. The reason? A wall of text posing as something mathematical and talking about nonmathematical concepts, naming Cantor, this definitely spikes my crankometer. Don't do it. Do some actual mathematics.
Re: The Ironic Paradox of Normalcy [Is my argument sound?]
GirlWithAMathFetish wrote: But I am soon going to compile all my academic walls of text and will post them to a blag of sorts, in order to have a compilation of my works to show to colleges, especially MIT.
I assume the purpose of this is to distinguish you from all the sheep you will be competing against  to show that you are different from the rest. The first question to answer is do they want people who are different from the rest? Of course most organisations say they do, but:
I was at a career's event once and someone asked what the panel thought of speculative applications  i.e. people who have the initiative to apply for a job without going through the normal channels. The answer was to point to the wastepaper basket. So whatever they say, organizations may well prefer the flocks of sheep which they can herd through the system.
But suppose they do want a sign of independent thought: what should you give them? Well the main thing they will consider is likely to be your exam results. If your results were much better than most of the competition then why would you need anything extra? If not then a rambling article saying how brilliant you are is the last thing you want to submit.
Maybe though, your exam results will be similar to many others, in which case it might help to have something to distinguish you from the crowd. But I would avoid rambling essays. Nor would I go for something which just repeats what you will do in exams. Rather I would take one idea  something mathematical, but not necessarily super advanced  maybe related to stuff you've already done on fractals. Then turn it into a finished project. This will show an ability to work independently and to finish things, skills which I would think will give a better impression than pages and pages of stuff which no one will read through.
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 GirlWithAMathFetish
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Re: The Ironic Paradox of Normalcy [Is my argument sound?]
Blah, y'all are missing the point with my example... forget about it.
I'm sexually attracted to the International Space Station. You have now had your daily dose of Internet.

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Re: The Ironic Paradox of Normalcy [Is my argument sound?]
If they're missing the point badly enough that their critiques are not relevant, you need to rewrite this completely.
EDIT: Wait, I recognize you. Your username relates to fractals "over there." Funny how your IQ jumped 15 points.
EDIT: Wait, I recognize you. Your username relates to fractals "over there." Funny how your IQ jumped 15 points.
Last edited by Jay Vogler on Sun Oct 05, 2014 9:30 pm UTC, edited 1 time in total.
Re: The Ironic Paradox of Normalcy [Is my argument sound?]
GirlWithAMathFetish, I hope you'll take the comments you received seriously. It's great that you're so enthusiastic about maths, but I think you really need some guidance to get you on the right track. And there's nothing wrong with that, especially considering that you're only 17.
As you implied in your opening post, there are a lot of very mathematically knowledgeable people here. I hope you won't make the mistake of thinking that they're just being mean to you for the fun of it or because they're jealous of your ability.
Good luck.
As you implied in your opening post, there are a lot of very mathematically knowledgeable people here. I hope you won't make the mistake of thinking that they're just being mean to you for the fun of it or because they're jealous of your ability.
Good luck.
 Forest Goose
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Re: The Ironic Paradox of Normalcy [Is my argument sound?]
GirlWithAMathFetish wrote:Blah, y'all are missing the point with my example... forget about it.
When you react this way to multiple sources of legitimate criticism, especially after asking for opinions, no one will care to help you next time.
In perfect blunt sincerity, the essay you posted is not good, in fact, it's bad. Rather than acting indignant, or writing all of us off, look at what you wrote, rethink it, and try to find its faults. Learning is not about being right, it's about being wrong  if you cannot accept failures simply, openly, and easily, you won't go far in mathematics, nor in science. Develop a sense of humility; when someone finds an error: smile, learn from it, and move on  you'll be a lot better for it, mathematically, and in general.
The alternative: sulk about and act pissy when someone points out your mistakes, slowly turn into a crank. If you don't believe me, interact with some math cranks, you'll notice that their go to line is "yeah, but you didn't understand". You have some potential, get a better attitude, don't go down a bad route or develop bad habits  because you, like everyone here, are going to be wrong more often than not.
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Re: The Ironic Paradox of Normalcy [Is my argument sound?]
Which point with which example? You provide a lot of different examples that don't all seem to be related to quite the same point.GirlWithAMathFetish wrote:Blah, y'all are missing the point with my example... forget about it.
If your overall point was simply that almost no one *exactly* matches the mean value for any particular parameter, then yes, that's a true fact, but not a terribly relevant one when it doesn't actually match what "normal" means in most cases. Very few people may have IQs of exactly 100, but more than 2/3 of the population should (in theory) have IQs between 85 and 115, and I think most would be happy to call that whole range quite "normal" or "average".
You're right, though, that there is seemingly something kind of paradoxical in the fact that someone could potentially be very interestingly unusual in their total averageness. If you could come up with 10 uncorrelated quantitative measurements of different people, then we'd only expect about one person on Earth to be in the middle 10% of all those measurements. You could call such a person the most average person in the world, a record which would in fact make the person quite fascinating indeed.
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Re: The Ironic Paradox of Normalcy [Is my argument sound?]
The way the post is structured, it seems to be about normality. There is a good (and wellknown) mathematical idea in there, namely that the chance of having any specific value in a continuous distribution, even the average, is extremely small (actually 0, for a true continuous distribution); I'd suggest that if that's the main point, then either don't include the stuff about normality or make it less prominent (include less of it and don't put it at the beginning). (Possible alternate title: "Being average [or "truly average" or "exactly average"] is abnormal [or "unusual"]". Maybe even just replacing "normal" with "average" (but not "abnormal" with "not average") would work.) If your main point is about normality, then it's important that the mathematical definition of normality that you're using actually matches up with what the audience understands normality to mean, and I don't think it matches up here.GirlWithAMathFetish wrote:brenok wrote:Your essay also seems to have the message that it's ok to be abnormal, to accept differences etc. While I may agree with the conclusion, I think the preceding math is probably unneccessary/irrelevant.
Actually, if anything, the abnormality message thing was irrelevant. My post was mainly supposed to be about the math. I'm notorious (though respected) on that site for constantly posting math stuff. xD
Sure it is. If a goal is for the audience to enjoy it, and the audience enjoys rambling, then one should ramble. Whether it meets some general or abstract idea of "good" is (in this case, and in most cases) irrelevant.Forest Goose wrote:You can say what you like about rambling making people like you, but, seriously, that doesn't justify rambling. Bad writing is bad writing  a good excuse is never: but my audience wants me to spend 300 words on something entirely irrelevant.
Because if they have talent and creativity, then they may be able to use that to improve the quality of their daytoday life to something better than a normal person's life. Also for some people, acting normal doesn't work (e.g., due to lack of abilities, or due to it tiring them out or not getting them what they care about).I know tons of highly creative individuals that act just like regular folks most of the time, I know lots of talented people, in general, who act perfectly normal day to day. And why shouldn't they?
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mittfh wrote:I wish this post was very quotable...
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Re: The Ironic Paradox of Normalcy [Is my argument sound?]
chridd wrote:Sure it is. If a goal is for the audience to enjoy it, and the audience enjoys rambling, then one should ramble. Whether it meets some general or abstract idea of "good" is (in this case, and in most cases) irrelevant.
Sorry, I'll be more explicit:
1.) Most audiences don't actually seem to like rambling meandering things, I have a hard time believing that the people she showed this to genuinely like it because it is rambling. Maybe I'm wrong, but I bet it's more likely they aren't faulting it or that those faults are being ignored.
2.) Enjoying something does not mean that you enjoy it for all of its traits  someone may enjoy my long winded arguments for their content despite that they are long winded. I tend to believe that if her audience enjoys her writing, then it is because of what she has to say, probably not because it rambles.
3.) Most people defend poorly written things by saying "That's what my audience wants, though". I know, I've seen this done way too many times; I've done this way too many times.
4.) She was asking for opinions on what she wrote, she was not asking for opinions on what her audience would like. It is not well written, if someone likes it for not being so, that still doesn't make it well written; a negative opinion isn't negated because someone is okay with it.
chridd wrote:Because if they have talent and creativity, then they may be able to use that to improve the quality of their daytoday life to something better than a normal person's life. Also for some people, acting normal doesn't work (e.g., due to lack of abilities, or due to it tiring them out or not getting them what they care about).
What is your point? I never said that people can't, or shouldn't, be different or themselves. My point was that "being different" does not entail greatness, nor is it, by itself, a thing to aspire to.
I'm having trouble following what being normal has to do with quality of life  I know plenty of perfectly normal people with a perfectly fine quality of life. Why would "being different" raise your quality of life? Do you mean "embracing your talent"? But that doesn't seem like "acting abnormal" to me. What is abnormal about being skilled at something or being exceptionally creative?
Being an exceptionally talented artist doesn't entail that you will act any different than any other random person, it certainly does not entail that you will be deviant. I have yet to see any evidence that this is the case, it seems to be more the case that the weird talented people stick out more in memory. For every weird genius I know, I know about ten normal one's. And for every weird person I know, one out of twenty of them have above normal talents.
Finally, I don't accept that "different" is meant in the pure statistical sense, mainly because of this
While normalcy may be the easy way out, deviancy is what creates the greatsthat is why they are great. As Syndrome in the Incredibles movie puts it, if everyone is super, no one will be. Dare to dream. Dare to be deviant. And you will stand out, and you will go far.
The statistical notion of "deviant" doesn't really have much to do with "Dare to dream. Dare to be deviant", the quote sounds a lot more like "Don't fit in, stand out, act different than everyone else. Don't conform!", and that's a bunch of bullshit, not conforming to social trends and the like is not what makes "the greats", talent is what made them and talent does not seem to be what that quote is encouraging (and how does one encourage others to be talented?).
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Re: The Ironic Paradox of Normalcy [Is my argument sound?]
...but it seems to me like people often attack works that other people like by claiming that it's somehow objectively bad.Forest Goose wrote:3.) Most people defend poorly written things by saying "That's what my audience wants, though". I know, I've seen this done way too many times; I've done this way too many times.
Again, you're assuming that there's some meaningful definition of "well written" besides what someone likes. Whether someone likes a thing is what matters (along with whether it misinforms them and probably some other things). If a work is enjoyed and doesn't misinform (and accomplishes the writer's and audience's other goals and doesn't introduce negative externalities), but is not well written, then the idea of wellwrittenness isn't very useful.4.) She was asking for opinions on what she wrote, she was not asking for opinions on what her audience would like. It is not well written, if someone likes it for not being so, that still doesn't make it well written; a negative opinion isn't negated because someone is okay with it.
If one is normal, I'd assume their quality of life is also normal (because it's based on other aspects of their life). If their quality of life is normal, then their quality of life is not better than normal. (Also, to clarify, I'm using "quality of life" to mean how good their life is in general, not specifically economicsrelated or other easilymeasurable aspects.)I'm having trouble following what being normal has to do with quality of life  I know plenty of perfectly normal people with a perfectly fine quality of life.
To me, the word talent implies being good at a task that's already been defined (e.g., playing an existing piece of music well, playing an existing sport well), perhaps making incremental improvements, rather than coming up with completely new ideas or ways of thinking and doing things (e.g., coming up with new genres, discovering major scientific theories). Or, at least, it doesn't apply to the comingupwithidea part but only the execution. Does that accurately describe the distinction you're making?The statistical notion of "deviant" doesn't really have much to do with "Dare to dream. Dare to be deviant", the quote sounds a lot more like "Don't fit in, stand out, act different than everyone else. Don't conform!", and that's a bunch of bullshit, not conforming to social trends and the like is not what makes "the greats", talent is what made them and talent does not seem to be what that quote is encouraging (and how does one encourage others to be talented?).
Assuming this is the case, I agree that talent is important, but sometimes there are great things that could be done that aren't incremental improvements to what already exists, and perhaps even the opposite of current trends. Someone(s) needs to come up with the idea, and someone(s) needs to execute it correctly (which, yes, will probably take talent), and both those types of people will be going against the general trends. Trends need to start somewhere, and surely at least some of the people involved in getting a great trend to become popular (that is, they're involved when it's not popular) are doing a great thing (and thus are great). (Not necessary or sufficient for greatness, but still can be related... necessary for certain types of greatness.)
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mittfh wrote:I wish this post was very quotable...
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Re: The Ironic Paradox of Normalcy [Is my argument sound?]
chridd wrote:...but it seems to me like people often attack works that other people like by claiming that it's somehow objectively bad.Forest Goose wrote:3.) Most people defend poorly written things by saying "That's what my audience wants, though". I know, I've seen this done way too many times; I've done this way too many times.Again, you're assuming that there's some meaningful definition of "well written" besides what someone likes. Whether someone likes a thing is what matters (along with whether it misinforms them and probably some other things). If a work is enjoyed and doesn't misinform (and accomplishes the writer's and audience's other goals and doesn't introduce negative externalities), but is not well written, then the idea of wellwrittenness isn't very useful.4.) She was asking for opinions on what she wrote, she was not asking for opinions on what her audience would like. It is not well written, if someone likes it for not being so, that still doesn't make it well written; a negative opinion isn't negated because someone is okay with it.
You seem to be assuming "well written" means something very contrived. There is a general notion of what is good and what is bad, you can get a feel for that by looking at what people tend to like and what people tend to reject. If you ask me to evaluate a piece of music, I can freely say, in a general sense, that it is good or that it is bad and be meaning something without qualifying it to myself or an audience.
If we can't say a piece of writing is bad, then why do we even bother to try and become better writers? More over, why do people try to become better at anything? Because there are standards that exist and are, generally, agreed to  I'm not asserting that there is some absolute metric.
I'm not "attacking" her work, I'm pointing out that, according to most standards that most people agree with, it isn't good. Why is that an attack? It's an attack if I say she is pathetic and will never be able to do better...but I'm not saying that, I'm saying "this isn't good, this is why, you should take that into consideration".
If one is normal, I'd assume their quality of life is also normal (because it's based on other aspects of their life). If their quality of life is normal, then their quality of life is not better than normal. (Also, to clarify, I'm using "quality of life" to mean how good their life is in general, not specifically economicsrelated or other easilymeasurable aspects.)
...that's even more confusing (and I find it odd that you object to rambling writing being called bad when someone solicited opinions; but you are fine with nebulous notions of how good someone's life is sans context or measurement, but whatever).
So, are you asserting that people who aren't normal have a better quality of life or that they can attain one? And why? Not being normal, in the sense you seem to be using, should entail being deviant in some sense, not every  so why would the notnormals be deviant in that way? Do most notnormals have a substantially better quality of life? I have yet to see any evidence of that anywhere, rigorous or personally.
.To me, the word talent implies being good at a task that's already been defined (e.g., playing an existing piece of music well, playing an existing sport well), perhaps making incremental improvements, rather than coming up with completely new ideas or ways of thinking and doing things (e.g., coming up with new genres, discovering major scientific theories). Or, at least, it doesn't apply to the comingupwithidea part but only the execution. Does that accurately describe the distinction you're making?
Assuming this is the case, I agree that talent is important, but sometimes there are great things that could be done that aren't incremental improvements to what already exists, and perhaps even the opposite of current trends. Someone(s) needs to come up with the idea, and someone(s) needs to execute it correctly (which, yes, will probably take talent), and both those types of people will be going against the general trends. Trends need to start somewhere, and surely at least some of the people involved in getting a great trend to become popular (that is, they're involved when it's not popular) are doing a great thing (and thus are great). (Not necessary or sufficient for greatness, but still can be related... necessary for certain types of greatness.)
You're talking about what, the one in a million person who creates a lasting new discovery that is not incremental? Is that who the article is addressed at, those potential people? That type of person doesn't need to be told to "Dare to Dream"; nor do they need to be weird and eccentric anyway. But, even assuming they do, then what, there is an exception to my point that applies to the rarest of the rare, I fail to see how that really factors in to anything. The point still stands, in terms of what it means to "Dare to be deviant" to 99.99% of the population, it is just acting different for its own sake and won't bring about any greatness (nor will it, by itself, bring about greatness in those rare people, it will, at best, be concomitant the traits that do make them great) .
Encouraging people to be different because some amazing people were different, to me, seems to have the same problem as encouraging selfesteem because competent happy people seem to have it. Perhaps the amazing people are different because of how amazing they are, perhaps the competent happy people have high self esteem because they are competent and happy. Encouraging selfesteem, for its own sake sans basis, seems to lead to egotistical people  encouraging people to be different for its own sake doesn't seem to lead anywhere any better.
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Re: The Ironic Paradox of Normalcy [Is my argument sound?]
People can try to become better writers because they want to become better at achieving their goals in writing (which might not be the same as other people's goals, including goals of the people who agree on the standards). Saying that "the audience wants it" is not an excuse for "bad" writing sound like you're invalidating the goal of appealing to that particular audience, rather than giving advice of how to reach the goal that she does have. (Whether it's likely the case that the audience likes or dislikes rambling, I can't tell without knowing the audience better.)Forest Goose wrote:If we can't say a piece of writing is bad, then why do we even bother to try and become better writers? More over, why do people try to become better at anything? Because there are standards that exist and are, generally, agreed to  I'm not asserting that there is some absolute metric.
That comment wasn't specifically about you, more about the general vibe I get from people who treat badness as objective.I'm not "attacking" her work, I'm pointing out that, according to most standards that most people agree with, it isn't good. Why is that an attack? It's an attack if I say she is pathetic and will never be able to do better...but I'm not saying that, I'm saying "this isn't good, this is why, you should take that into consideration".
Not only the people who create the new discoveries, but also early adopters of the new ideas before they become popular, before it becomes obvious to most people that they're great. There are probably also smallerscale nonincremental changes, and if so, more people are likely to come up with them. Also, it's not always possible to know whether an idea is great when one first comes up with it.You're talking about what, the one in a million person who creates a lasting new discovery that is not incremental? Is that who the article is addressed at, those potential people? [...]
Also, I believe being different from the norm (at least in the relevant area) is important for a person to be great to me—that is, for them to make a positive difference in my life. Most of the problems in my life have been caused by people not solving the right problem or trying to solve the problem in the wrong way (but are doing what society says to do), rather than by people doing the right thing poorly or doing what others around them say not to do. Merely talented people probably aren't going to make that much of a positive difference in my life, and they might make a negative difference if they're talented at doing something that hurts me or is opposite to my goals. Something similar is true for creative works: the works I've come across that I dislike are more often in the wrong genre or about the wrong topic, rather than poorly made (in fact, I've noticed times when someone says a work is particularly good or particularly bad and I don't see how it's any better or worse than other works in a similar genre). A great person to me then would likely be someone who does things that help me or that I like, despite what general rules in society say. I doubt I'm the only person like this.
There are most likely people for whom the opposite is true—where their needs match up well with what lots of people are trying to provide, and what some people can provide, and have come across lots of people trying the same thing with varying levels of talent in ways that actually matter—and for them, talent would be important for greatness. ...which is perfectly fine, but not everyone's like that.
Could this be the reason for our disagreement?
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mittfh wrote:I wish this post was very quotable...
Re: The Ironic Paradox of Normalcy [Is my argument sound?]
chridd wrote:Most of the problems in my life have been caused by people not solving the right problem or trying to solve the problem in the wrong way (but are doing what society says to do), rather than by people doing the right thing poorly or doing what others around them say not to do. Merely talented people probably aren't going to make that much of a positive difference in my life, and they might make a negative difference if they're talented at doing something that hurts me or is opposite to my goals.
Do you work for a criminal organization?
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.
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Re: The Ironic Paradox of Normalcy [Is my argument sound?]
No. Why the heck would you think that?z4lis wrote:chridd wrote:Most of the problems in my life have been caused by people not solving the right problem or trying to solve the problem in the wrong way (but are doing what society says to do), rather than by people doing the right thing poorly or doing what others around them say not to do. Merely talented people probably aren't going to make that much of a positive difference in my life, and they might make a negative difference if they're talented at doing something that hurts me or is opposite to my goals.
Do you work for a criminal organization?
(Edit to add: I should probably clarify that I'm interpreting social conventions to be things that are generally less serious to violate than laws. I would not consider "murdering people" or "stealing things", for instance, to be a typical example of a social convention, and I would not consider things like that to be a good way to violate norms. Also, when I said "something that hurts me" above, I wasn't referring to physical pain.)
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mittfh wrote:I wish this post was very quotable...
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