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Thanks to the popular television show House MD our culture has in large part adopted the “Houseism” Everybody Lies. In retrospect it seems sort of axiomatic. Everybody has an agenda, ergo, everybody lies. There is also a great joke in one of the episodes of The West Wing that roughly goes: A politician is giving a speech and his opponent yells out, “You’re lying!” and the politician responds, “Well yes I am, but hear me out.” People like those sorts of jokes. They like to believe that all politicians lie, and that they themselves are somehow morally superior and would not lie, which is ironic since they are at that moment lying to themselves.

Which brings us to why this is mathematically important: polling data isn’t wrong, it’s just largely false. Admittedly polling respondents on the extreme ends of the political spectrum will answer honestly who they support. However, those that lie between one and two standard deviations from the mean (the centrists lie close to the mean) are going to probably answer falsely in the hopes of skewing the data set. Which explains why the data is false but not wrong. The number of false responses on either side of the mean are very probably about the same number of false responses, so they balance each other out.

How do I know this to be true without a mathematical proof? Because the poll numbers that get broadcast don’t add up to 1, or in other words, 100%. For example, polling numbers released on the news last night indicated 47% favored the President, and 47% favored his opponent, and . . . 10% were unsure. 47 + 47 +10 = 104%. This tells us several things. First, the poll questions were worded either poorly or slanted to elicit a particular response. Second, some percentage of respondents don’t understand what they are being asked. And Third, respondents lie.

Admittedly there will be more than two candidates on the ballot in November, and some of the standard error can be attributed to that. There will also be some small percentage of “write in votes.” Yet every rational human being knows the winner of the election will be one of the two major party candidates. It is a Bernoulli trial, with only two possible outcomes. And that outcome will be made entirely by approximately 10% of the voters, based solely on how they feel on exactly the day of the election. No amount of advertising, money or suddenly uncovered new information is going to change that.

Now to make it more interesting, that precious 10% are probably not going to vote for either candidate, so much as they will vote against the other candidate. The solution seems rather simple: pull all remaining ads for the remainder of the year, use the money for some warm-fuzzy public works project (a children’s cancer hospital would be a great one) and let the news media do the rest of the advertising for you for free. Sure it’s a ploy, but the 10% don’t really care. They’re just going to hate that candidate less than the other candidate, and at least something good and tangible comes out it.

 

One of the truly lovely things about not being an “expert” in a given field is having the freedom to be too stupid to know why something can’t be done, and therefore going ahead with it anyway. So here’s your Gedankenexperiment for today: tape a piece of black construction paper to the wall, walk across the room, sit down, and look at the construction paper. Is it there?

How do you know it’s there? Technically, no band of visible light is being reflected from it, so you aren’t seeing any light returning from it to your eye. What you are seeing is the absence of reflected light that you do see from the surrounding area. Yet you know the piece of construction paper is there, and it is because of this difference between the light from the surrounding area and the lack of light from the construction paper. But why don’t you see any light from the black construction paper?

There is an elaborate and highly cool looking mathematical answer to that, but the bottom line is because the black construction paper absorbs the light, the photons. Period. On some level those photons “convert” to heat energy and are radiated out at a different frequency that simply isn’t visible to your eye. However, does that have anything to do with the mass of the piece of construction paper? Not much.

Assuming you used a standard sized, letter size, A4 or the like, piece of black construction paper, and we’ll assume you used some fairly heavy stock on the order of 70# paper, well, 70# paper has a mass of about 140 grams per square meter, and an A4 or roughly letter size piece of paper is one-eighth of a square meter, then your piece of paper has a mass of about 8.75 grams or a little less than a third of an ounce! I’m going to go out on a limb here and say that 9 grams does not have enough mass to generate a gravitational field so strong that light can’t escape from it.

Therefore, it is not totally unreasonable to say that black holes are not black because their mass is so monumental, and it is, that light can’t escape from them, but that they may be so dense that light can’t pass through them and otherwise just non-reflective. This way of considering black holes, as single atoms of unimaginable size and mass, that are simply non-reflective gives rise to a way of considering the other great, and much more abundant, material in the universe that we likewise cannot see: Dark Matter. In much the same way as a black hole, dark matter is evidenced by its influences on surrounding objects that we can see.

There is nothing especially mystical or difficult to understand about matter that we cannot see. From early childhood we all experience wind. We can’t see it, but we can feel it and see its effects. It is, barring any other substances in the air, invisible, transparent, but we know it is there. Yet while most people have no problem grasping the concept of transparency, there seems to be something more difficult about opaque and non-reflective.

This is not the answer to what a black hole is, or what dark matter is. It is another way of looking at the problem. Nobody, and I mean nobody, not even the guy you’re thinking of, has the math skills to definitively explain what a black hole is, but, it is not a geometrical single point of zero length, width or depth with infinite mass. It just doesn’t reflect or give off any light.

I will concede that a black hole may be such a massive single atom that it is capable of the reverse process of how a photon is made (degraded along with a neutrino from the conversion of a proton into a neutron) and that the mass may be capable of crushing a photon and a stray neutrino back into a neutron to form a proton, because if mass can be converted to energy, then the reverse of the process is algebraically true.

Somehow I suddenly have the overpowering feeling that two groups of grad students, one at Cambridge and one at Stanford have just been ordered to produce a Voodoo doll of me. (If you get that joke, consider yourself officially a nerd.)

I read not too long ago that philosophy is dead, that science has completely usurped philosophy and therefore philosophy is no longer viable in science. If that is true, then I am a crackpot, and what I’m about to write is just because I don’t understand the complexities of the advanced mathematics. Let me take a shot at it anyway, even at the risk of being just another crackpot: 0=0, 1=1, 2=2 and ∞=∞.

Now I realize that the idea of infinity is a bit difficult to conceptualize, and Georg Cantor eventually drove himself nuts trying to prove a series of infinite sets, but whatever infinity is, it must by definition equal itself! This presents a significant philosophical logic problem for the definition of cosmic black holes for a variety of reasons. So let’s start with a common definition of what a black hole is and what’s inside. A black hole is are “area” in space where the gravitational strength is so strong that nothing can escape its grasp, not even light, whence the name black hole. Current theory holds that a black hole is caused (in adverse to created, which we won’t bother with) by a “singularity” at the center, which is a collection of matter and energy that has been crushed down to a single point of infinite density and infinite gravitational attraction, implying that they would have infinite mass.

There are at least two significant problems with using infinity in any definition of a black hole. The first, and perhaps most notable, is that if a black hole had infinite anything, especially gravity, it would instantaneously swallow the entire universe, and very probably every universe to infinity. Again, that may seem conceptually difficult to imagine, but gravity has no spatial limitations, and an infinite gravity source would pull in everything instantly. That is the problem with understanding what “infinite” or “infinitely” means.

If the entirety of existence in all universes being sucked out of existence in a time equal to zero is too much for you, let’s scale it down to something more reasonable. If the singularity at the center of a black hole is infinitely dense and has an infinite gravitational field, why are they different sizes? It is not because of observational distance. Black holes come in all sorts of different sizes, including what are called “super massive black holes” found at the center of many galaxies. If all black holes are powered by a singularity with an infinitely dense and infinite gravitational field, this would suggest from the definition that there are different values for infinity! While there is no way to calculate what infinity is, if it is part of mathematics then it must be equal to itself.

Therefore, the gravitational field of a singularity is not infinite, but a real number. Admittedly that value is unimaginably big, and will undoubtedly take Graham Notation to eventually calculate, but it is not infinite. Which leads us to a second problem, if the differences in sizes of black holes means they have different gravitational strengths, then the singularity at the center also are neither infinitely dense nor infinitely massive. Again, that does not mean they are not unimaginably dense and massive, just not infinitely.

Now if you’re a chemist and thought you dodged the bullet because I chose to pick on the astronomers, physicist and mathematicians, think again. The super-heavy elements, such as the Lanthanides and Actinides can only be formed (naturally) in the supernova of a giant star because of the immense pressures required to form these elements. However, even if a black hole is not infinitely dense, it is immeasurably dense, so as lighter elements pass over the event horizon and toward the center of the “singularity” they will ultimately be crushed into heavier and heavier elements. Again, this does not imply that the periodic table is therefore infinitely long, it does imply that the number of periods is an enormously larger real number than 8 or 9! Furthermore, due to the completely different gravitational environment they exist in, their half-life is likewise much longer. So inside of the event horizon and approaching the “singularity”, Flerovium’s half-life may go from 2.8 seconds to 2.8 million years. Admittedly that last point may be viewpoint dependent, since in theory time may cease to exist inside of the event horizon.

There will come a point when someone, like Steven Hawking, or Andrew Wiles or Ronald Graham will calculate the actual density, mass and gravity of a black hole of a given size, and from those calculations someone will calculate the maximum “heavy element” possible for that given size. Indeed, the center of a black hole may not be an infinitely dense singularity at all, but a single atom of a massively super heavy element. Perhaps it’s element number 3↑↑3 on the extended periodic table, which massive as it may be, is still a real number and not infinitely dense.

 

While I largely consider myself a mathematician, people are far more passionate about their politics than they are about their math skills, and I need readers, so I’m going to push your political hot-button ever so slightly and see what falls out. But yes, I’m going to use mathematics for the stick I poke you with. I’ve probably lost half my readers right there.

For the sake of simplicity I’m going to assume that if you are reading this you are a conscientious individual that meticulously votes at every opportunity, federal, state and local. Put another way, I’m going to presume you care and want your vote to count for something. Well, you’d be correct: your vote does count. Not very much, but it does count. I’m not trying to disparage your vote, it counted as much as anyone else’s vote. There were 131,257,328 votes cast in the 2008 federal election[i], meaning your vote accounted for 1/131,257,328, or about 7.619 x 10-7% of the total popular vote. (If you’re rusty on scientific notation, move the decimal point seven places to the left to get the numerical percent, or nine places to the left to get the actual decimal fraction.)

Okay, now that you’re feeling insignificant and wounded, let me finish you off: The number of representatives in Congress, the Executive, or Judicial branches has not changed since . . . 1960. And that was only in the Senate with the addition Hawaii. The number of representatives in The House of Representatives hasn’t changed since 1912! So, in one-hundred years, exactly, the population has increased while your number of representatives has not.

Welcome to the Rules of Exponentiation. Let’s say that population growth is 1% on average, which  is relatively close to the actual number in the preceding decade. I am going to spare you the logarithmic explanation of why this works, but it means that if you divide the growth rate by roughly 70 you end up with the amount of time it takes something to double. Assuming our 1% population growth number, that means the population doubles every seventy years. (Historically over a longer period of time the population growth rate has actually been much higher than 1%, but using the last decade seems reasonable here.)

So, for those of you celebrating your one-hundredth birthday, when you were born your Representative was responsible for just over 212,000 of you, and today, you are one of about 722,000 represented by that same one Representative. Which means that, roughly, you have only 30% the representation in government that you had a hundred years ago. For those of us a little younger the math is relatively simple, but looks pretty ugly to the uninitiated, because it is not simply a matter that your representation gets diluted 4% over four years (assuming a 1% population growth), but actually slightly more, roughly 5.7% due to exponential population growth in one presidential election cycle, and 11.4% in just two cycles.

So with each passing year, or with each passing election, your vote counts for less, and your representation becomes also less. What’s the solution? (I’m expecting some comments in the comments area below.) Given the voluminous amount of derogatory rhetoric thrown around about our congress, I am guessing that there won’t be a large faction pressing for tripling the number of representatives in the House of Representatives, but that’s just a guess. There exists, conceivably, some legitimate mathematical arguments for doing exactly that. I suspect there may even be some reasonable legal arguments for doing exactly that. But still, I will be extremely surprised if anyone presents them in anything approaching a lucid form. Now there is a correct mathematical solution, but again, I doubt very much it will appear in the comments.

Go.


[i] Results according to the Federal Elections Commission, http://www.fec.gov/pubrec/fe2008/2008presgeresults.pdf

Nerd Politics

An interesting thing happened this morning: A friend of mine, who has historically been less than enthusiastic about math, posted a chart of Spending Growth by Presidency. It was interesting because it at least implies that, like it or not, some voters this fall will have to use math to some degree to make their decisions. Put another way, Nerds will decide the election no matter how you feel about the candidates. So here’s your chance to understand what’s really going on. The graph you’re looking at is called a Normal Standard Distribution Density Curve. We’re going make some assumptions here, because mathematicians love to make assumptions . . . well, statisticians do. We’re going to assume that the electorate (voters) are relatively evenly distributed between the parties.

So what you’re looking at, without a lengthy explanation that the lower case sigma means standard deviation, or what a standard deviation means, is a graph of the electorate . . . that’s you. To stick with conventional practice, Republicans are on the right in red, and Democrats are on the left in blue. (If you’re wondering why the whole thing adds up to only 99.8% of the electorate it’s because of a thing called significant digits, and in this case we are only using one place past the decimal.) Everything in brown are the people that will actually determine the election: The Nerds. This means, that 15.8% of people will vote Republican and 15.8% of people will vote Democrat even if the parties put up Satan versus Lucifer. In other words, 31.6% if voters will tow the proverbial party line because they are culturally programmed to support that agenda. The remaining 68.2% of voters will vote based on what they understand to be true at the time of the election.

This is not to say that the Nerds are not an emotionally charged bunch, because they are, and they don’t all come to the same conclusions based on the available data. If that were true then previous elections would have not had such close decision results. Which brings us to the fact that, the closer you are to the center, the more likely it is that you will decide the election. When a couple of percent of the centrist Nerds swing one way or the other the decision goes with it. And centrists like math. Centrists are a lot more concerned with numbers than they are with political hot-button issues. Sure centrists have opinions and feelings about political hot-button issues, but they tend to be more ambivalent about those issues than they are about whether they will have a job next week or next year.

Prior to the age of the internet, and instant access to numbers and data, it was easier for political candidates and parties to obfuscate what was really happening in the world. Voters had to rely on whatever information was published in the mainstream media, which arguably might be slanted to the agenda of the publishing company. Now however, virtually every department in the government, both federal and state, has the most recent data on their area publically available, and historical data as well. The voters looking directly at that data just happen to be the Nerds who are only a fraction of one standard deviation either side of the center, and they will be the ones who consider the numbers, mathematically analyze them, and decide the election.

If you’re looking for the point to all of this, here it is: Be a part of the actual decision. Tune out all of the rhetoric, ignore the moral hot-button issues, brush up on what is roughly eight or ninth grade mathematics, and start looking at the data from all the various agencies on the internet. Whichever side you ultimately come down on, your decision will have the virtue of being significantly more rational, and statistically much more likely to make an actual difference.

Number 10: “How many square feet are there in an acre?” Now this in and of itself is not actually a stupid question. Indeed it is a perfectly legitimate question. The reason it makes my Top Ten list is because the person asking the question was apoplectic when I actually gave him the number! He literally went into a sort of physical palsy. When I, foolishly, inquired what the problem was, he responded that he didn’t think I knew the answer off the top of my head, but I might know where to look it up. In a kneejerk reaction I said, “You didn’t ask me if I knew where to look it up, you asked how many square feet there are in an acre. If you thought I didn’t know the answer, why the hell did you ask me?”

In retrospect I wish I had answered, “Well, you know there are 640 acres in a square mile right? Then it’s easy, 5280 feet in a mile, 5280 squared divided by 640 is 43560. Come on, you’re just screwing with me right?”

Number 9: “How do you know that?” This question makes my list out of sheer numbers, as it is the most commonly asked stupid question. Often the circumstances are like those in number 10, where I’ve been asked a question and when I answer the question “How do you know that?” is the very next thing out of the person’s mouth. However, there is one incident that sticks out in my memory above all the rest.

A friend of mine was doing a crossword puzzle and came to me quite agitated, swearing up and down that the creator of the puzzle had “screwed up this time.” Like an idiot I unthinkingly opened my mouth and asked what the point of contention was. “The clue is “Box Elder” but the line starts with an N.” This was long before the days of the internet so he couldn’t do a search, but had to rely on his own personal pool of knowledge.

Without thinking I said, “Negundo.” It’s just one of those odd bits of information you pick up over a lifetime that sticks with you for no particular reason.

“Huh? What the %^$&# is a negundo?”

“It’s the specie name of the genus Acer which is commonly called the Box Elder.”

“How do you know that?”

“Well, it’s not a secret.”

Number 8: “How big is a fifty-five gallon drum?” This one was especially interesting because my supervisor at an engineering firm came to me nearly in a panic with this question. My obvious answer was something along the lines of, “Oh, I’m guessing 55 gallons.” After some flailing of arms and stammering incoherently it turned out he actually wanted to know the dimensions of a 55 gallon drum. This turned out to be the antithesis of Number 9, since I didn’t happen to know the exact dimensions of said drum off the top of my head, which left my supervisor more than a little crest-fallen. Literally. In an endeavor to cheer him up I suggested that he could easily calculate the dimensions by using the formula for the volume of a cylinder, convert the 55 gallons into cubic units, then use a calculus derivative to determine the minimum values for diameter and height, because logically a drum manufacture would want to use the smallest amount of material for the given volume. I was young and naïve, and did not realize that using the word “calculus” in a sentence is akin to making a pejorative remark about one’s mother, apparent. He didn’t speak to me again for two weeks.

Number 7: “What is your greatest strength, and what is your greatest weakness?” More than a few times I have been asked this set of questions in an interview. And more than a few times it has been the reason I didn’t get the position.

“What is your greatest strength?” Not being so narcissistic that I sit around thinking about what my greatest strength is.

“What is your greatest weakness?” Not knowing what my greatest weakness is. If I knew what my greatest weakness was, don’t you think I would have done something to correct it?

Number 6: “Boating? We’re not going to do any boating, are we?” Allow me to explain, this was not because the person misheard me, it happened because the person I was talking to is a moron, and I don’t really care if that term is politically incorrect. There was an event about to happen that involved everybody from the department getting together for an outing. One of my colleagues ask me if I intended to go, and I said no. When he asked me why, I said, “I’m not sure. I just have this terrible feeling of foreboding.” No, he did not mishear me, because after repeating it as clearly as possible, I was then compelled to explain what foreboding meant.

Number 5: “What do you mean? Was he cool or was he a dick?” Now I will admit that I do commonly use a vocabulary in normal conversation that makes me sound like the King of the Nerds, but I don’t go out of my way to do it. It’s just my manner of speech. However, there are some people you will encounter in your life where it is just impossible to dumb-it-down enough. There was a friend of mine that was having trouble dealing with another colleague, and asked me to take some materials to this colleague so that he wouldn’t have to deal with him. Upon my return, by which time I was long since thinking about something else, my friend as me, “So how was he?” After a moment trying to understand the vagueness of my friend’s question I realized what he was asking and said, “Oh, he was perfectly congenial.” My friend contorted his face into a most painful looking expression and asked me, “What do you mean? Was he cool or was he a dick?” It took me a moment to recover from shock, but I manage to get out, “He was cool.”

Number 4: “Viet Nam, where was that fought?” At the onset of Desert Storm there was as you might expect quite a lot of talk at work the following day about it, and various references to Viet Nam which was still fresh in our minds at that time. One colleague, a high school graduate, and I swear to God a genuine Eagle Scout, asked, “Viet Nam, where was that fought?” When the response came, “Viet Nam.” he apparently thought it was inflected as a question asking for clarification. To which he responded, “Yeah, Viet Nam, where was that fought.” By inserting a vulgar explicative between Viet and Nam, and ending with, “you idiot!” it started to coalesce. Again, I swear this is true, his response was, “Oh, that’s the name of a place? I thought it was just the name of the war.”

Number 3: “How many square inches are there in a cubic foot?” This may seem like an innocent slip of the tongue, where it not for the fact that the person asking me this question had a freshly minted Masters Degree in Electrical Engineering! I literally had to explain to him why one could not answer the question because of the difference between two dimensional units and three dimensional units. At which point he was now quite indignant, but willing to play along. “Okay, so how many cubic inches are there in a cubic foot?” Being equally indignant by this time I was unwilling to just tell him the number, so I rather flippantly said, “Oh, maybe twelve cubed?” Apparently solid geometry is not a requirement in electrical engineering, even at the Masters level, because he responded, genuinely surprised, “Oh, is that how that works?”

Number 2: “Would you like another glass of Jägermeister?” I’m not sure if the question or the answer qualified as the most stupid. In fact I’m not sure of much after that. I do seem to recall that even though I don’t speak Swedish, that it was around that time that it no longer sounded foreign.

Number 1: “Is “got” a real word?” It’s a Monday morning, and I roll into work, two hours late, exhausted from a hour and a half drive, probably hung-over, and not especially thrilled to be working for people that think IQ is a place that sells ice cream, when I am accosted by my friend from Number 5. In a highly animated fit he asked me, “Got is a real word isn’t it?” Given my state of mind at that particular moment I was confused why this guy would be asking me a theological question in German, when it dawned on me he meant “Got” and not “Gott”. When I explained to him that it was in fact a real word, and attempted to explain it was a past tense of the verb to get, he became even more animated, yelling, “I knew it. I knew it.”

Now I had already come to know from previous encounters that vocabulary was not this fellow’s forté, but “got” was just too much to believe, and in a fit of curiosity-wins-over-lucidity I had to ask what the hell he was talking about.

He was happy to explain. “On Friday my girlfriend and I got in this big fight, and she says got isn’t a real word!”

Now while I am thinking, ‘God, please don’t let them breed,’ I actually asked, “Don’t you own a dictionary?”

Tentatively he said yes. “And you waited from Friday evening until Monday morning to ask ME if got was a real word instead of looking in the dictionary because of what now?”

“I just thought it would be easier.”

To paraphrase comedian Ron White, “You can fix looks, but stupid, stupid is for-ev-er.”

Now that’s harsh! Even in my family. Relax. It’s fifteen generations, not fifteen years. Still, for those of you that like to believe that your ancestors came over on the Mayflower in 1620, I’ve got some potentially disconcerting news for you. Before you come to burn me at the stake, first dust off your high school algebra and biology texts, because we’re going on a little trip.

First let’s establish some ground rules. Genealogists commonly use 30 years for the unit of a generation. Anthropologists typically use 25 years for the unit of a generation. Since anthropology in a real science, and genealogy is a social science, and the social sciences are largely hokum, we will use 25 years as the unit to designate a generation.

So, going back fifteen generations from today, 2012. would land you in 1637. Big deal, you can easily trace your heritage back further than 1637. Heritage, yes, heredity, no. This is because of two sets of numbers that operate in opposite directions . . . exponentially: your number of progenitors (ancestors) and your number of genes. For our purposes we will assume you know the difference between a chromosome and a gene. You have 46 chromosomes, but spread along those 46 chromosomes you have 23,000 genes. (Yes, there are 3 x 109 base pairs, but that’s not enough to have genetic connection, whence we use the word gene.)

You also have two progenitors, which you probably call parents, or something more pejorative depending on your relationship with them, but two progenitors none the less. Each of which contributed 23 chromosomes, i.e. 11,500 genes to what makes up you. With each successive generation before you, grandparents (4), great grandparents (8) the number of progenitors doubles. Mathematically over fifteen generations that looks like this:

 

This means that after fourteen generations, at the fifteenth generation, before you the number of progenitors you have, 32,768, exceeds the number of genes you even have, 23,000. On the other side of the equation, with each successive generation before you the number of genes contributed to you is halved:

 

In other words, by the fifteenth generation before you the number of genes contributed to your makeup by each potential progenitor is less than one! Put another way, you do not share an entire common gene with anyone more than fourteen generations back. Now wait a minute before you throw yourself off Plymouth Rock to a watery grave in a fit of depression! This does not take into account statistical probability. Any given gene has a fifty percent chance of being passed along. It’s called a Bernoulli Trial, and it means not only does any given gene have an equal chance of being passed along, but that chance remains equal with each successive passing. Although collectively over the generations the probability continues to halve, and therefore the odds (probability) become increasingly less, it can happen.

In fact it does happen. Which is why about 5%-8% of your genetic makeup is actually from ancient endogenous retrovirus DNA. There, now doesn’t that make you feel better?