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Sunday, June 28, 2015

The DOLEZAL POINT: Rachel Dolezal, Political Correctness, and the 2nd Derivative

Mark Charalambous

Since everyone is riffing on Rachel Dolezal, I figured I’d get my licks in, too. After all, how could the Third Rail shy away from a story about a woman from the “Privilege” class who chooses to disguise herself as a member of the ethnic class that occupies the bottom rung of virtually every measure of societal success?

Keywords: Political Correctness, Identity Politics, Transracialism, Transethno-something?...

You get the idea. Yes, this is ripe for a ride on the Third Rail.

To the question of “Why everyone everywhere is so captivated by this story?”  well, after some false flags, the truth has actually emerged into the general consensus. It’s this:  This case of identity politics is so absurd that not even the leftist loons can avoid acknowledging it. Bill Maher gets it.   Even Jon Stewart gets it.

The Left has been indoctrinating us for decades that all identities we used to consider immutable, those that constitute conditions of birth, basically your sex and your race, are in fact. . . not so!  We’ve been wrong all along!  They are now recognized as “social constructs.”
Rachel Dolezal, now-and-then
Yep, “Society’s to blame,” in the words of Monty Python (comedic references are unavoidable and de rigueur when discussing these matters).  It is the consensus of society and your own volition that determines your sex, race, or ethnicity.

Well, the upshot of all this from the Third Rail is this: 

REJOICE!   I bring good tidings!

Rachel Dolezal is The Tipping Point.  We are finally there!  The absurdities and inherent paradoxes of political correctness have reached their zenith.  Political correctness is officially beyond parody. You cannot make up stories that are both funnier and more preposterous than this one.

What’s The Onion going to do—write a fake news story about a dad who “self-identifies” as a mother?  That’s not as funny as Rachel Dolezal pretending to be black.

Now that you’re laughing, I’m going to teach you some math. Seriously. I’m going to explain to you, mathematically, what a tipping point is. Its formal name is an inflection point.

Before we do that, just a little background and more editorializing on political correctness is in order.

Most if not all politically correct notions are built on a foundation of quicksand. Gay marriage, for example, is predicated on the ridiculous notion that procreation is not implicit and irrevocable in the meaning and definition of marriage. Transethnicity, this latest farce, is predicated on the notion that race is a social construct, not biological.  Transgenderism is woven from the same identity-politics cloth: the inherent biology of one’s sex is meaningless; what matters is what sex you feel like, your “gender.”

Political correctness had a beginning in time. I might put it at around 1970. A good case could be made that it falls within a few years in either direction of that year. This is clearly arbitrary, but bear with me. It’s as good a year as any to pick. The first Earth Day was held in 1970, the same year the EPA was created. There’s nothing intrinsically wrong with these events, but they were harbingers of a “new consciousness.”  It was the consciousness of the sixties generation beginning to be implemented, as we precious, sanctimonious folks embarked upon adulthood and our mission to remake the world in our image.

As time went on, political correctness (henceforth abbreviated to “PC-ness”) increased. There was more and more of it.  There was more in say, 1984 than there was in 1983. And the amount of that increase was greater than the increase from 1982 to 1983.  What we are talking about here is a rate; a rate of increase: how much PC-ness is increasing over a given period of time.

Time for the math.

A function is a math formula (or read: equation) for describing how something changes due to some other quantity.

For example, weekly pay could be a function of a fixed rate of hourly pay and the number of hours worked in the week. If the rate of pay was $10 per hour, and we used the variable h to count the number of hours worked, the function would look like this: P = 10•h  In math, we typically don’t write multiplication crosses or dots, and so this equation would actually be written like this:   P = 10h.

To find out your pay (P) for a given week, you multiply the number of hours worked, h, by 10. Bingo.

Figure 1.  Function over time
We now consider a function that represents a quantity changing over time. If we want to graph how a function is changing over time we first establish what we call the time axis. We usually draw a horizontal line at the bottom of the graph that represents the time axis.  It will have some scale drawn on it so specific times and time intervals can be noted or measured.

This establishes that the “picture” of the function’s behavior, typically a line or curve, is “read” from left to right indicating how the function changes going forward in time. The value of the function at the left end of the line occurs before its value at the right end, or for that matter any point on the line to its right. Refer to Figure 1.

Figure 2.  Constant function

If the line is flat (horizontal), the quantity that the line is describing (the function) is not changing over time. As you move right or left on the line (forwards or backwards in time), the line does not go up or down. The quantity is neither increasing or decreasing in time. In math we refer to this as a “constant” function. This is shown in Figure 2.

If the quantity increases as we go forward in time, the line goes up.  A line describing PC-ness would necessarily go up (to the right) because it is always increasing (over time).

If a quantity is decreasing over time, such as water evaporating in a pot, the line will go down (to the right).

From this point forward, I’ll use PC-ness as my example function. Any point on a line or curve that represents PC-ness shows how much PC-ness there is at that point in time.

Figure 3. Linear function
PC-ness has been increasing over time. If its picture were in fact a (straight) line and not a curve, it would imply that for all durations of time of the same length (for instance the time frame could be yearly, or every 5 years, or every day), the amount of increase over that time frame would be the same. In other words, it would increase by the exact same amount from 1977 to 1978 as from 1992 to 1993; or, the increase from June 1982 to July 1982 would be exactly the same as the increase from September to October 2007. We say the rate of increase is constant. We would then call the graph of PC-ness the graph of a “linear” function; because its graph is a straight line. An increasing linear function is shown in Figure 3.

If the rate of increase was not always the same, but was getting bigger all the time, the graph would not be of a line going up—it would be a curve going up, as in Figure 4. This would correspond to the situation that the increase in PC-ness from 1984 to 1985 was more than the increase in PC-ness from 1983 to 1984. And the increase from 1985 to 1986 was greater than from 1984 to 1985. The rate of increase is increasing.

Figure 4. Exponential function
There are many functions that correspond to something increasing in this manner, at an increasing rate. They are all curves that go up. One of the most common is called the exponential function. When we say that something is “growing exponentially” we are describing behavior like this, something that is getting bigger and getting bigger by bigger amounts all the time. Figure 4 shows an exponential function.

If there is nothing to arrest this increasing rate, we call this “exponential or uninhibited growth.”  Generally, population growth is exponential until there are mitigating environmental (or human) factors that throw a wrench into the works. One such example is the growth of mold on bread. There is only so much bread. The mold grows very quickly at first, but eventually it slows down as there is less bread to grow on. Eventually the mold reaches a maximum value. No more can grow. And that is what leads us to our final growth picture, shown in Figure 5.
Figure 5.  Logistic function

What if there was exponential-like behavior in some human endeavor, but something happened to derail it? A good example of this is the introduction of new technology.

When CDs were first produced the technology was new. The cost to switch over to the new music medium was expensive.  Both the CDs and the machines you needed to play them on cost more than the current music media, records and cassettes. As more and more people gradually discarded their old relics of the previous technology, dubbing-cassette boomboxes and record players and the like, the sales of CD players and CDs went up. As sales went up, the costs went down. As the cost went down, more people switched to the new media. Rinse and repeat.  What you have here my friend is an exponential-like growth pattern.

But would it go on forever?  Of course not!  The technologists never rest. When the second wave of the digital revolution arrived, the need for a physical medium for the music was replaced with software, a file of 1s and 0s that could be downloaded from the internet and played on a new device, the iPod.

What happened to the sales of CDs and CD players then?  They started to decrease. The growth in the number of CDs and CD players purchased would slow down. The rate of sales would slow down. As shown in Figure 5 on the top half of the “S-shaped” curve, eventually the sales will come to a crawl and the total amount of CDs and CD players sold will approach a maximum, fixed value. The picture of this growth pattern shown in Figure 5 describes what is called a “logistic” function.

We’re now ready to understand the tipping, or inflection point. The inflection point is the point where the picture changes from going up rapidly to a slower and slower rate of increase. Note, it is still going up, but at the far right the curve looks like it actually wants to become flat; meaning:  no change.  This is important; until a maximum value is eventually reached, no matter where we are on the curve the quantity being measured is always going up. But after a certain amount of time has passed a tipping point is reached where the curve bends toward the other direction. At this inflection point, the rate of increase slows down, until it crawls almost to an eventual stop at what is called the “carrying capacity” of the function, the point of maximum saturation, when no new CDs and CD players are purchased.

If the inflection point for PC-ness occurred in say, 1991, the increase from 1992 to 1993 would have been less than the increase from 1991 to 1992, and so on.

Granted, we sensible Americans don’t just want the rate of PC-ness to slow down and the total amount of it to eventually level off at some “acceptable” amount—we want PC-ness gone.  Period. As in zero. Ideally we’d like all the PC mischief that has plagued the body-politic over the past 45 years or so to not just stop, but reverse and eventually evaporate.

Eventually we’d like to see the curve that represents the amount of PC-ness become a flat line at the very bottom of the graph. But before we get there, the rate of growth of PC-ness has to start slowing down. The curve has to bend.

Question: Where is the inflection point?

I assert that Rachel Dolezal is that very inflection point. The rate of increase of political correctness that infects our world is going to begin to slow down now. Why do I make this claim?

One of the properties of the logistic function is that at the inflection point the rate of increase is at its maximum value (actually, just before the inflection point the growth rate is virtually exponential).  The tipping point for PC-ness will come at the point of its maximum madness.

I think we are at that point. The absurdities have reached hilarious proportions. Days before the Dolezal story broke the media was abuzz with accounts of Bruce Jenner’s sex change. For the past 8 years Jenner has been a reality TV whore. But ESPN, the nation’s preeminent sports network, chose to celebrate the “heroic” action of this Olympic athlete of 40 years ago by awarding “her” their Arthur Ashe Courage Award.

As this goes to press, the President has used the latest Islamist beheadings as an opportunity to once again lecture the nation on the dangers of “Islamophobia;” the White House has been bathed in rainbow colors celebrating the Supreme Court’s decision legalizing homosexual marriage; and the latest list of college “microaggressions” is making the rounds on Facebook.  I know it’s a tired cliché, but you can’t make this stuff up.

After the fallout form Dolezal from such disparate quarters—not just the expected conservative talking heads but also sources on the Left such as Maher and Stewart—it’s hard to believe that everything will soon go back to “normal.” I look for increasing ridicule as “Check out this latest case of political correctness...” becomes a frequent (and welcomed) narrative.

I hope it becomes an issue in the presidential debates on the Republican side. Some of the candidates should recognize that bashing PC-ness will be a vote-getter. To date, the candidate that has showed the most proclivity for attacking PC-ness head-on is Ben Carson.  People forget that while speaking at the National Prayer Breakfast in 2013, the event where he made his political bones by attacking Obamacare right under the nose of the President sitting a few feet away, he spent the first five minutes of that speech railing against PC-ness.

Once the Democratic race opens up after the recalcitrant Democrats realize that the “Hillary coronation” is a Jedi mind trick, watch for a candidate to test the waters by dropping a relatively innocuous but carefully contrived “spontaneous” joke or two mocking PC-ness.

If someone were serious about mathematically analyzing PC-ness to see if in fact it could be modeled by a “logistic” growth model or some other mathematical model, from which the inflection point could then be determined, two things are necessary.

First, a mathematical algorithm would have to be developed to actually quantify PC-ness. We would have to be able to measure it.  That is, we would need to produce a specific number for say, 2005, that measured the amount of PC-ness in that year, and the corresponding numbers for all the years since our starting point, 1970.

A disclaimer and cautionary note here: The notion of actually quantifying PC-ness in a mathematically rigorous way is, quite frankly, ridiculous. It depends on mutually agreed upon definitions and is inherently subjective. This wouldn’t stop academicians in the behavioral sciences from pursuing nonsense like this, however, as they’re always seeking ways to justify the “hard science” of their discipline.   Someone who believes that a differential equation from fluid dynamics can be applied to a measure of how positive, affirming statements can lead to better decision-making in the boardroom, is not going to balk at measuring PC-ness...

We, on the other hand, are just doing this mainly for fun and the opportunity to explain some cool math.

Figure 6.  Comparison of Exponential and Logistic function 
The second thing needed to demonstrate that PC-ness follows the logistic (or other) growth model is an indication that a turning point has indeed passed and the rate of PC-ness has begun to slow down.

Perhaps two years from now our hypothetical measuring system will show the increase in PC units from 2016 to 2017 is less than it was from 2015 to 2016. If no such evidence is forthcoming, then unfortunately we are still in an uninhibited growth model (Figure 4) with no end in sight—or we are still on that bottom half of the “S” curve. A side-by-side comparison of exponential and logistic growth is shown in Figure 6.
If these two conditions were met it would be possible to see if the data fits a logistic function. Once the actual function was found we can determine several things. We could find out exactly when the inflection point occurred; we could predict what the growth rate will be in a given year; and we could predict the maximum value at which PC-ness will eventually top off (mathematically speaking, the “carrying capacity”... of how much PC-ness we Americans can stand before, hopefully, starting the cleanup operations...).

I am now going to discuss in a more general way the mathematics involved in finding the rate of change and the inflection point of a function. These are standard operations of calculus and are not specific to logistic functions.

The algebraic form of a logistic function is fairly complicated, and the typical reader would not have the math background to understand it. It has a time variable in it along with several special numbers, along with exponents and cool things like the number e

So rather than tackle an actual logistic function to demonstrate the actual math to do these things, I am going to use as my function something understandable to anyone familiar with basic algebra.

Here is the algebraic expression that I am going to use as my function for PC-ness:
5x3 – 3x2 + 7x + 1

A logistic function does not look like this (though there are similarities), but we are going to use this simple algebraic expression to illustrate the mathematics of finding the inflection point. The process is the same.

Let’s pretend that this expression is the function that represents the amount of PC-ness in years after 1970. If you wanted to find out how much political correctness there was in 1970 you would replace the variable x with 0 (1970 is 0 years after 1970), and then do the math. We “plug in” 0 to the expression.
5(0)3 – 3(0)2 + 7(0) + 1 = 1

The answer would be 1. That would be the amount of political correctness in 1970. We might use that as a benchmark and name it 1 PC unit, or PC1.

If we wanted to find out how much PC-ness there was in 1971 we would replace the variable x with 1 (1971 is 1 year after 1970). Then we’d “plug it in” and eventually arrive at the value of 10.  We might say the amount of political correctness in 1971 was 10 PC units or 10PC1, that is, 10 times the amount of political correctness in 1970.

Now, here’s where it gets interesting. What follows is one of the things you learn how to do in calculus. Yes, calculus is needed to find inflection points. It is higher math.

What if you wanted to find out the rate of change at any of the years?  Or, for that matter, at any point in time. (Note, with calculus you can find the rate of increase at any day, hour, minute—any instant—of your choice.)

What you have to do is what we call “take (or find) the 1st derivative” of the function. When a derivative is taken, what you get is another function which describes the rate of change of the original function. This is something you learn how to do in calculus, and obviously I’m not going to show the various rules for doing this here, but I will show you what it is in our example so that you see it just produces another (occasionally simpler) mathematical expression.

The first derivative of 5x3 – 3x2 + 7x + 1  is 

15x2 – 6x + 7

By the way, if you’ve never seen calculus before but you’re good with numbers, you might be able to discover the pattern that produced the 1st derivative here. Go ahead, try.  I won’t tell. You’ll have another opportunity in a minute.

Now that you have the 1st derivative, if you wanted to find how much PC-ness was increasing in 1971, you would replace x with 1 (again: because 1971 is 1 year after 1970, and we established 1970 as being Year 0). Then, plug it in to the 1st derivative,  15x2 – 6x + 7 . 

The rate of increase in PC-ness in this model would give 16. You would interpret that as “PC-ness was increasing at a rate 16 PC units per year in 1971.”

We’re almost there. Remember, what we actually care about is how fast the rate of increase is changing. We are looking for something that will indicate that it has begun to slow down; that is, that the rate of the rate of increase in PC-ness is slowing down. As stated above, a derivative of a function gives the rate of change of the function. When you take the derivative of the 1st derivative you are finding a function that describes the rate of change of the 1st derivative—yes, a rate of change of a rate of change! 

Guess what we call the derivative of the 1st derivative? You guessed it:  the 2nd derivative, of course! The first derivative we found above was the rate of change of the function, specifically: how much the amount of PC-ness was changing over time. To find out how much that quantity changes, we are taking its derivative. We take the derivative of the 1st derivative, or in other words, the 2nd derivative of the (original) function.

Here it is, the 2nd derivative. The derivative of the 1st derivative 15x2 – 6x + 7  is:  ­­­­­­­­­­­­­
30x – 6

(If you think you figured out the pattern for taking derivatives before, this may confirm it for you; or, give you another clue with which to work.)

Driving: Distance, Speed and Acceleration

Another example of functions and derivatives
An everyday experience that is often used to explain the concept of the derivative, a fundamental concept in calculus, is driving.

You go for a drive. You drive a certain distance. The (main) function corresponds to your position.  How many miles had you driven at some particular point in time on the trip? It will be a line/curve going up. As the line moves to the right we are going forward in time, and we are moving, so our overall distance traveled is always increasing.

The 1st derivative of this position function gives the rate of change of the distance traveled over time, the rate of our motion: our speed. How many miles per hour were you driving at that particular time?

The 2nd derivative of the position function is the derivative of the 1st derivative; the rate of change of the 1st derivative. The rate of change of your speed. That’s acceleration.  Acceleration is the 2nd derivative of the position function. When you find the 2nd derivative you can plug in a number for the time variable and it will tell you what your acceleration was at that instant in time, how many miles per hour per hour, or miles per hour2. This will be a positive number if you are speeding up or a negative number if you are braking or slowing down.

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If you want to find out how much the rate of increase in political correctness was itself increasing, in say, 1984, you would replace x with 14 (1984 -1970 =14), and get 414 PC units. (The actual dimensions of the 2nd derivative of a function measuring total PC-ness over time would be PC units per year per year, or PC units per year2.) 

And now, finally, the magic is revealed. How do we find out where the inflection, or tipping point is, i.e., when it occurs?  How do we find the date when the rate of increase begins to decrease?

What is actually happening at the inflection point?  Look again at Figure 5.  Look at the rate of increase to the left of the inflection point. If we cut the graph at the inflection point and throw away the top of the “S” part, we would have a graph similar to the one of exponential, uninhibited growth (Figure 4). Getting bigger and bigger all the time.

Just before the inflection point the rate of change of PC-ness has its maximum value!  Just beyond the inflection point the change in PC-ness has a slightly lower value. What has happened in between is that the rate of change of the increase in PC-ness has gone over from a tiny positive number to a tiny negative number. It has crossed the threshold of zero change in the rate of increase in PC-ness

This is analogous to stepping on the gas pedal to accelerate when passing a car on the highway. After you’ve passed the car you let up on the pedal. You are then decelerating. The moment where you let up on the pedal was the inflection point. In that instant you were neither increasing your speed nor slowing down. Your acceleration was zero. (Refer to the sidebar, Driving:  Distance, Speed, and Acceleration, for further insight into the driving analogy.)

The point of inflection comes at the point of maximum increase of the function.  That is at the most extreme point of increase in PC-ness. Which I claim is right now!

To find out exactly when it is, you have to find the point where the second derivative is exactly zero. You solve an equation. In our little example, we would have to solve the equation
30x – 6 = 0

The answer to this, the solution to this equation, is the fraction 1/5, or 0.2 as a decimal number.

Please remember that the algebraic expression we have been using is not a logistic model that describes the behavior we are talking about here, either sales of CDs or PC-ness. It was provided as a simpler example to show you how derivatives work. Clearly, sometime in March 1970 (the date corresponding to x = 0.2 when 1970 = 0) could not be the tipping point of PC-ness!  
(However what we did is sound, and there is an inflection point in the graph of  5x3 – 3x2 + 7x + 1 at  x = 0.2 .  If you’re curious, you can enter the function into a graphing calculator and look.)

But if PC-ness could be quantified, then a function for it could be formulated. And if that function is logistic, if it fits the “S” pattern, then an inflection point could be found (See NOTE 1). And it might very well have been in June 2015.

Henceforth: the Dolezal Point.

There is one irrefutable argument to be made in support of the contention that PC-ness must turn around. Has there ever been a time when one generation so enthusiastically pursued and promoted the tastes and value system of its parents'?  Whatever happened to “Question authority,” my generation’s watchword?  The blowback is coming. 

# #

In reality PC-ness would not follow the strictly logistic model. The inflection point would be higher up on the “S”.  In a logistic function the inflection point lies exactly midway between the minimum and maximum values of the function. 

If Dolezal were the tipping point in a logistic model where we established 1970 as the starting point for PC-ness, we would have another 45 years of gradually diminishing new PC-ness to endure before the maximum value would be reached!  We would only be halfway to our collective “carrying capacity” for PC-ness. Now that’s a scary thought!

A better fit can be found from a function that coincidently is of the same form as our simpler example:  a cubic polynomial function.  See Figure 7.  Its leading coefficient would be negative, and we would restrict its domain to begin at its first local minimum, which should also be a zero of the function, to a point on the graph that lies somewhere beyond the local maximum and before the next (and last) zero.
Figure 7. Cubic Polynomial function

I have tried to fit some speculative data values to such a model using cubic regression, using 0 as the beginning value of PC-ness in 1970 (x = 0), an inflection point value of PC-ness of 400 occurring at 2015 (x = 45), and setting a maximum value of PC-ness of 550 occurring in 2020 (x = 50); yes, the eternal optimist...

Here is the model it produced:
–0.0176x3 + 1.44x2 – 19.17x + 37.34

(after some rounding which had negligible effect on the graph). The model fails before about 1985, and from that point does only a fair job of matching the speculative points that I created for 5- or 10-year intervals from 1970 to 2030.

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Copyright 2015  Mark Charalambous / The Third Rail

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