Flat Earth Misconceptions

I'm so tired of repeating myself to Flat-Earthers. It's like they revel in doing their math wrong, but there's not enough characters in twitter to send them the proper equations. So I'll address a few classic pics here to explain how it's done.

The distant mountain picture

This is a favorite among flat-earthers, and it's pretty easy to show they get the math wrong.  Someone told this guy that he can use a simple linear fit for earth's CURVED surface (8 inches per mile or some such tripe). I'm not sure where this approximation came from, but it obviously doesn't fit a curve very well. Anyhow, here's an example:

Here, we have the classic flat-earther example of a "mountain that's too far away to see." First, let's check the facts. After a bit of digging, I figured out that they're claiming this is Mt. Denali.  That's 140 miles away, with an elevation of roughly 20,000 ft.

It's actually 700 ft at the summit of the Hilltop Ski Resort, but it turns out that won't matter in the end.  

Anyhow, let's assume they're right and it's Denali.  First, calculate the distance and angle to the horizon from an elevation of 700 ft.  I'll let WikiHow explain how to do it. I'm using the arccos formula: 

$ d = r  \arccos(\frac{r}{r+h}) $

The horizon from Hilltop is calculated here at 32 miles.

The horizon from Denali is calculated here at 174 miles.

Suppose the picture were taken from the parking lot of Hilltop instead of the top.  The horizon is still 27.5 miles away, well within the range of Denali's summit. 

32 + 174 means you can see the tip of Denali from as far as 206 mi at an altitude of 700 ft (assuming there's no mountains in between and neglecting atmospheric lensing, of course). This is obviously not a problem for observation from 140 miles away.

Just for convenience, we could figure out how tall an object needs to be to see it from a distance, x. We simply add the horizon distance to the solution to our first equation for h:

$ \cos( \frac{d}{r} ) = \frac{r}{r+h} $

$ h = \frac{r}{cos(\frac{d}{r})} - r $

d = 140-32 = 108

h = 7700 ft tall. 

This means we should be able to see the top 12,300 of Denali. 

Time Zone Solar Models

This is another flat earther hand-waving explanation for time-zones.  

Even the most casual thought regarding this picture and you'll realize that since the sun is above a flat earth, you'll be able to see it at all times from any location on earth. Turn this model on its side and draw a vector from any point on earth to the sun and you'll see that it never sets.  The figure below shows this and explains the problems.

If there's some other way to interpret the animation above, I sure can't think of it.  Send me a comment or tweet to help me understand.
I'm left wondering if flat earthers lack all spatial reasoning. It sure seems that way.

Pilots Would End up in Space!

This one is so bizarre that it takes a little work to wrap your head around.  Essentially, the argument is that airline pilots would need to adjust their elevation by 1 degree every 6 or 7 minutes to keep from climbing higher and higher and crash into the firmament or flying into space.  There's some specious assumptions here:

• That airplanes fly along laser-like straight lines
• Airplanes will continue climbing at the same rate for a given angle of attack even as the air thins.
• That the pilot has the sensitive equipment to recognize one-degree drift over 6 minutes

In the same way as you might make minor course corrections along a straight road to keep your car centered in your lane, the autopilot (or the real pilot) is constantly adjusting to maintain a reasonably constant altitude and heading.  These corrections are happening at a rate of tens to hundreds per minute.  To the pilot or autopilot, this feels like maintaining altitude.  In reality, it's conforming the flight to the curvature of earth.  This publication described the typical roughness of a commercial flight.  I've pulled out the "rough cruise" section because the constant readjustment of altitude is more clearly apparent.  Same thing happens in "smooth cruise", but more gradually.

Where's the curve?

Simple computer models can help us understand when and where we'll be able to observe curvature of a sphere.  The key factors which affect the appearance of curvature are:

• Altitude above the surface
• Camera Focal Length (or field of view)
• Radius of the sphere

I'm sure someone could write an expression  for the apparent curvature versus these parameters, but it's easier to just show you. I'm using an open-source tool called Celestia which accurately presents the position, sizes, and velocities of celestial objects.  I highly recommend it. It's fun to play with.

Changing the Field of View

Here's the  same view as the Field of View is altered.  This is like zooming out on your camera.  The FOV is reported in the bottom-right while the location (constant) is presented in the upper-left.

Changing the Altitude

This one is a little more obvious.  As you move to higher and higher altitudes, the curvature becomes more apparent.

Note that the Distance is 30 km. That's a decimal place, not a comma.

Radius Matters Too

This one is just for fun.  Here's some spheres of different radii from the same distance.

Mythbusters are shills?

Well, as we've shown, the curvature isn't expected to be visible at low altitudes. Here's Adam Savage  at a 12 mile altitude witnessing the curvature for himself.  So I suppose he's got to be a liar now, eh?

The Moon / Spinning Earth can't be felt!

Moon: The gravitational acceleration of the moon on the surface of earth is given by $ a = G m_{moon}/r^2 $ or roughly $ 3.6 \times 10^{-5}  m/s^2  $ compared to the $ 9.8  m/s^2 $ I measured in high school for earth's gravity.  That's less than one part in a million.  A hard thing to measure.

Rotation: The acceleration of an object on the equator due to circular motion from the rotation of the earth is given by: $ a = \omega^2 / r $.  This also tiny at $ 0.034 m/s^2 $. This effect (0.34%) MIGHT be measurable by exceptionally sensitive equipment and a skilled scientist, but these are the sorts of people the Flat Earth crowd seems to consider untrustworthy.

Edit: I fixed my math above. Rotation is actually much more important than I originally calculated. Thanks to @TheOlifant for catching my error:

@haplesspete @AtheistEngineer Thanks for the link! That's a great read. 1 q.: Where is my mistake in this? pic.twitter.com/3M6QtVGG37

— The Olifant (@TheOlifant) May 2, 2016

Mentality

The mentality of flat-earthers seems to be very similar to that of anti-vaxers and deeply religious.  The believer thinks they've figured out that most of humanity is wrong, and that their answer is the right answer. They often tell you to "research it," and couple commands with insults "stupid" or "dummy" or "sheep." 

These believers think they've figured out what "they" don't want you to know.  The "they" varies between people, but it seems to be illuminati, the government, or the Free Masons. For devout Christians or Muslims, the "they" is Satan, heretics, or demons.  

These believers pride themselves in being different. They think they're visionaries for knowing the truth when everyone else has it wrong.  Despite having no formal training in the specific scientific claims they reject, they feel sure that all the professional scientists have been deceived by the "they."
What's particularly interesting is that these people seem to blindly follow (IMO obvious) quacks. Some guy with a YouTube channel is seen as more reliable than all the world's scientists. They wave off these brilliant scientists by presuming they've never actually TESTED any of the claims they learned in science text books without seeming to notice that:

1. The YouTube quack has never tested his flat earth claims. At best their "evidence" seems to be that they find actual physics hard to understand or inconsistent with scripture.
2. Scientists actually do verify the basics. They build more complex experiments on top of them, so if the basics weren't right, nothing would work.

Towards an Objective Assessment of Scriptural Prophecy

Introduction

I want this neon sign!

I've poked at prophecy in the past. I assert that it's better explained as post-hoc favorable interpretation than as legitimate future-telling. A Christian asked me to expand on how we can apply Bayes' Theorem to prophecy.  On the surface, this seems like a trivial task, but it turns out to become pretty murky since the likelihoods of the various events are very hard to quantify. Nevertheless, I think it's instructive to consider.

Background

Bayes' Theorem is a theorem in probability and statistics which computes the likelihood of related events given some assumptions.  In its simplest form, it states:

$ P(A|B) = \frac{P(B|A) P(A)}{ P(B) }  $

Where P(x) is the probability of x being true, and P(x|y) is the probability of x being true, assuming that y is true.  x and y being events or observations.

Method

For starters, let's consider what I think is the meat of prophecies: That they support the claim of divine and infallible nature of a holy book. Let us ask Bayes, "What is the probability that a holy book is divine ($div$) given that a prophecy it contains is true ($pro$)." That is to say:

$ P(div|pro) = \frac{P(pro|div) P(div)}{ P(pro) }  $

I'm not sure if we'll be able to get very far, but let's dive in, shall we? 

Divinity Yields Accurate Prophecy?

$P(pro|div)$

What is the chance that a prophecy will be true in a divine book?  That depends on who you ask and what you assume for the nature of the divinity which inspired the book.  That being said, be careful what you apply to this probability.  Too low and the $ P(div|pro) $ approaches zero.  Too high and a failed prophecy will prove the Bible isn't divine.

Likelihood of Divinity

$P(div)$

What is the chance that a particular holy book is divine without any other assumptions?  Again, this depends dramatically on the incoming assumptions about the holy book in question. 

• Many presuppositionalists enter the discussion with $ P(div)  = 1 $. 
• If you assume that one book is divine among all titles ever written, your value for $ P(div) $ is very close to 0.  
• If you assume that of the four holy books (Torah, Bible, Quran, and Book of Mormon), one and only one is divine, you get $ P(div) = 0.25 $ 

To be honest, I'm not satisfied with any of these answers.  All of them are pretty arbitrary given that we have no way of knowing if or how often books are divine.  

Likelihood of Prophecy being True

$ P(pro) $ 

What is the likelihood of the prophecy coming true without any assumptions about the divinity of the book in question? In evaluating this likelihood, it's important to consider:

1. If people believing the book is divine will make the prophecy more or less likely to come true
2. The likelihood that the event would happen anyways

In other words, to determine if the prophecy is true, we must consider it in the context of the events we know to be true

$P(pro|evt) = \frac{P(evt|pro) P(pro)}{ P(evt) }  $

$P(evt)$ in the denominator means that if the event would be likely to happen anyways ("There will be wars and famine and disease!!!") then the prophecy isn't likely to be true.

$P(pro)$ in the numerator means you have to guess at the likelihood of the specific prophecy being true. I'm not sure how to estimate this value. Perhaps this could be done by comparing it to other prophecies in the book?

$ P(evt|pro) $ is the probability of the specific events, given that the prophecy is true. Again, this is very subjective, but it means you must clearly define what your "event" is and assess its likelihood given the prophecy being true.  If there are other ways of fulfilling the prophecy, they reduce this likelihood.

Discussion

I'm not going to drag you all the way through a specific example.  My intention in this post is to communicate the various terms that need to be considered when assessing the likelihood of a book's divinity given that a prophecy is true. Hope this helps.

Extra Credit

Suppose we believed that $P(div) = 1$ and $P(pro|div) = 1$. Show that $P(pro)$ must be equal to unity and that therefore ANY prophecy which can be shown to be false proves the assumptions are wrong -- either the Bible is not divine or the divine agent produces false prophecies.

Self-Identity, Beliefs, and Emotional Logic

Introduction

How does this guy feel when his beliefs are challenged? Do you suppose he's able to fairly consider ideas which conflict?

It's human nature at times to tie beliefs to our self-identity.  Religion is the most common example, but politics are a close second.  When we do this to ourselves, it makes us emotional about any challenge to our beliefs, as such challenges are also a threat to our self-identity.

For example, if a person says your belief system (be it atheism, Catholicism, or Islam) lead mankind to commit atrocities, the implication is that you are personally capable of and inclined to commit those atrocities. As the anger wells up in your chest, you become unable to discuss the assertion rationally. Our human minds are wired to respond to threats, and the fight-or-flight response responds to social threats as well as physical. Given the relative security and anonymity of the internet, it's a whole lot easier to choose to fight. When we chose to fight while emotional rather than to proceed with deliberate and thoughtful responses, things get ugly and we wind up looking like the "angry atheist".

Such attacks (i.e. the Hitler attack) are common on Twitter, and are blatant attempts to provoke an irrational, emotional response from the opponent.  This behavior should be called out for its immaturity or dismissed out of hand. Do not give the attacker the pleasure of seeing you mad.

Often, the provocation is much more subtle and we can miss the emotional reaction it induces in ourselves.  When a theist asserts that atheists have no morals or purpose, does that make you angry? Notice the biological reaction in yourself and use that awareness to take a deep breath and calm down before you respond.

A Solution

Sexy Lady Justice!

The best way I've found to help minimize my personal sense of threat or social hostility is to dissociate my beliefs from my sense of self. Though I identify as an atheist in this anonymous social-media persona, it's not a defining feature of my identity in real life. I've come to terms with my limited ability to know things. My senses and cognition are human, no more. I have made mistakes big and small, and I will continue to do so. It's okay to admit this fact and continue to do my best moving forward.

This approach to truth and honesty allows me to evaluate any claim that's made fairly and honestly. I'm not in any way set on maintaining my atheist belief. But I am keenly aware of the human cognitive biases which can lead to false beliefs. After all, I've fallen for many of those cognitive bias errors myself.

These cognitive biases are often mental shortcuts and assumptions we all make to simplify the decision-making process. The apply to all of us, and not just in our evaluation of religious beliefs.  I've made that mistake in many areas. Some good examples are irrational fear of flying and our tendency to make snap judgements of other people's motivations. It takes meticulous deliberation to think through beliefs and assumptions carefully, and each of us is liable to make that mistake when we're quick to reach a conclusion.

I often tell theists that I "Will Convert for Evidence", and I mean that with all sincerity. I believe I have drawn the most reasonable conclusion possible given the evidence available to me. If I discover new evidence that leads me to conclude a god actually exists, I will change my belief. Thus far, all the evidence I've seen is better explained by failures of human cognition such as group-think, wishful thinking, and emotional decision making.

The best I've seen from theists seems to be "promising" me that they "know" it's true. While I don't doubt the sincerity of their convictions, I understand the ways that people reach the wrong conclusions, then double-down on those conclusions rather than re-evaluating them.

I generally ask with sincerity how they know their particular god is real and how they selected it from among all the other religious beliefs. Most haven't considered any other religious beliefs. Those who have tend to draw comparisons like, "So which is most plausible? A mad prophet on a flying horse or humble Jesus on a donkey?" [link]

Such statements make it clear just how fair the evaluations of other beliefs were. The fear of damage to self-identity leads people to tip the scales in their identity's favor. The end result is a less impartial assessment of alternative explanations and a greater chance of missing the correct interpretation.