Geometry is racist.

I have a confession I need to get off my chest. I think Geometry is racist. Well, not geometry (the discipline of mathematics), but specifically Geometry (the course commonly taught in American High Schools). I realize that in presenting this argument I’m going to have to make some rather sweeping generalizations about human history, but I that it’s important for me to put this hypothesis into words because I don’t think there’s enough evidence to disprove it.

Fair warning: this may be a lengthy read.

I believe that the reason the curriculum of Geometry specifically references Pythagoras, Euclid, and Descartes is because this emphasis provided Whites another means by which to rationalize slavery. Greek society was a prototype for White society because it used mathematics and philosophy to moralize slavery, and Descartes provided Whites a method of reconciling this position with Christianity.

To understand this, we first need to acknowledge that the entire human history of mathematics is intertwined with human slavery. Both mathematics and slavery emerged in early civilizations after the transition from hunter/gatherer to agricultural society lead to social stratification and the development of written language. In fact, I think mathematics is a necessary precondition for slavery. The very concept of slavery depends on a life being assigned a quantifiable value. Slave labor provided the ruling class with the time and resources to work on furthering mathematical knowledge, and then advances in mathematics were used to increase the efficiency of the slave trade. This vicious cycle quickly elevated the slavers to a god-king status which they then exploited to moralize their slavery.

Our oldest mathematical texts are from Egypt and Babylon, approximately 4000 years ago. Surviving records such as Plimpton 322, the Moscow Mathematical Papyrus, and the Rhind Mathematical Papyrus show that teachers have been giving math worksheets of roughly high school level difficulty since at least 1900 BC. What’s interesting about these three texts is that all three of them show evidence of the Pythagorean Theorem — over a thousand years before Pythagoras was even born! There’s also a strange lack of evidence attributing the theorem to Pythagoras, along with indications that proofs of it may have been circulating in Mesopotamia and India several hundred years prior. The very fact that this theorem bears the name of Pythagoras despite this questionable lack of evidence should alone be sufficient evidence of Geometry’s racism, but Pythagoras is only the first of three dominoes.

When Pythagoras supposedly lived around 500 BC, the Greeks (and Romans) had lively slave trades. However, unlike the god-king theocracies of Babylons and Egypt, the Greeks fashioned themselves as a “Democracy”. The Greeks needed an alternative means of moralizing slavery and whether he intended it or not, Pythagoras’ merger of mathematics and philosophy provided them with all the justification they needed. The Greeks used philosophy and mathematics to separate themselves from the people they enslaved, which served as a prototype for White society. Those that didn’t measure up to Greece’s intellectual standards were labeled as “barbarians” and treated as less than human. To paraphrase Plato, these barbarians were in the dark and needed to be shown the light. Incidentally, the “barbarian tribes” of Northern Europe enslaved by Greeks and Romans were likely a source of genetic markers for light skin, light hair and light eyes that would later be associated with “Whiteness”.

Meanwhile, mathematics probably continued to develop in the Middle East with influence from early civilizations in India and China. However, we don’t know by exactly how much. There’s another strange lack of mathematical records from Babylonian society under the Persian Empire. What we do know is that when Alexander the Great conquered the region in 331 BC, the Greeks assimilated a wealth of astronomical data. It’s quite likely that the Greeks stole whatever mathematical data they could through military conquests and these materials ended up in the Library of Alexandria where Euclid would have had access to them. Euclid’s work is based on a set of stolen mathematical literature that Euclid was privileged to have access to. While it’s impossible to discount the probability that Euclid might have added some original ideas, I do think this fact merits taking a closer look at his work and evaluating whether or not it belongs in our school’s curriculum.

My problem with the presentation of Euclid in Geometry (the course) is the parallel postulate. To be fair, presenting Euclid’s axioms as fundamental truths about the universe made sense in mathematics education up until as recently as 1905. We now know that space is not “flat” as Euclid assumed, but that gravitational forces can bend space around a mass. My understanding is that we teach Euclidean geometry because (a) it’s a useful approximation of reality and (b) it provides an introduction to axiomatic systems. The main issue with Euclid as presented in Geometry is that its axioms are presented without proof and we have solid scientific evidence that one is not always true. How can we ever expect students to think critically about information when we don’t question the assumptions made in mathematics?

Euclid’s mathematics is not irreplaceable within the mathematics curriculum. We could just as easily teach students about axiomatic systems using first-order logic. Instead of teaching compass and straightedge constructions, we could teach origami. In fact, teaching these alternatives could not only increase the diversity of the mathematics but simultaneously increase the rigor. There is some really great math showing that we can solve problems with paper folding that are impossible to do with a compass and straightedge only. Learning origami would help prepare students for high level physics and computer science. We owe it to our students to do better than Euclid.

Approximately 300 years after Euclid, the Roman Empire was nearing its peak and Christ was born. The death of Christ coincides with the start of two centuries of peace. It’s no wonder that early people considered this a miracle. Early Christians were originally persecuted in the Roman Empire, but by 380 AD it would become the state religion. If you think about it, Christianity offered the Romans a pretty sweet deal considering their involvement in the religion’s origin. The Romans killed Jesus, and through Jesus’ death were forgiven all their sins. And that meant all their sins: slavery included. To suggest that otherwise would be heresy. Before long, Christianity had accumulated so much Power that they rewrote the entire calendar around their own beliefs.

The spread of Christianity throughout Europe is relevant to our discussion because the next major event in mathematical history happens to coincide closely with the birth of Islam. Around 800 AD, Muhammad ibn Musa al-Khwarizmi developed the branch of mathematics we now call “Algebra” and started a new mathematical revolution in the Middle East. This presented Christianity with a threat because geometry was always considered sacred and the followers of Islam were creating new mathematics for another god. The bible says “Thou shalt have no other gods before me“. By 1100 AD, Christianity had come to the conclusion that since they couldn’t disprove the Muslims mathematically, the only other option was through force. The Christians took to war against the Muslims and with each mosque they burned they destroyed part of that mathematical progress. Remnants of this “Holy War” persist through the present day.

It’s because of this religious tension between Muslims and Christians that our story leads to Rene Descartes. Another essential component of the Geometry curriculum is the idea of Cartesian geometry which is an merger of algebraic principles with Euclidean geometry. The very name would have you believe that Descartes invented it, but that honor should probably be attributed to Fermat. No, I think there’s another reason we attribute it to Descartes, and that is because of his work in ontology. Most people are familiar with “I think, therefore I am“, but that statement is really the first axiom in a system that Descartes constructed in order to develop a mathematical argument proving the existence of God. Descartes is one of a few mathematicians that attracts cross-curricular study because of this. We don’t even provide his argument a fair critique. Like his Greek counterparts, we treat Descartes’ work as gospel when in reality he was just repackaging the ideas of others. In this case, Descartes had taken the Muslim-sounding “algebra” and re-packaged it in form that would be more acceptable for Christians.

The philosophical assertion “I think, therefore I am” takes on another meaning when we put it in context of mid-1600s White society. By associating “thinking” with “existing”, Whites were able deny Blacks basic human rights by labeling them as “unthinking savages”. It’s the very same strategy employed by the Greeks to justify slavery, only the vocabulary had shifted. The same idea would later manifest itself through literacy tests, and persists today through standardized testing practices. Whites retain Power by defining a culturally biased set of standards.

A few weeks ago I listened to a podcast from Freakonomics that described America’s Math Curriculum as a “Geometry Sandwich”. Ever since I taught it I’ve been asking myself the same kinds of questions:

Why “sandwich” Geometry in the middle of two years of Algebra?
Why are Euclid, Pythagoras and Descartes the only three names that students explicitly learn in Geometry?
Why is the name al-Khwarizmi not given the same level of treatment in Algebra?

The most probable explanation I can come up with to answer all three of these questions is that the American Math Curriculum is racist. It is probably not the intent of the Curriculum developers to be racist, but racism is not about the intent — it’s about the consequences of the behavior. There is a non-zero probability that cultural biases exist within our Math Curriculum and they produce racist consequences. Mathematics is a just tool, but people decide how that tool is used. We can either continue to use that tool to enable racism, or we can use that tool identify and dismantle it.

For me, I’ve decided that the probability of the Math Curriculum being racist is too high for me to ignore. I’ve decided on the following course of action:

I plan to start substituting “Right Triangle Theorem” in place of “Pythagorean Theorem”.
I plan to start substituting “Rectangular Coordinate System” in place of “Cartesian Coordinate System”.
I plan to pay closer attention to situations where Euclidean geometry is implicitly assumed and question whether or not this assumption is warranted.
I plan to make a consistent effort to present a diverse range of mathematicians in my classrooms.
I plan to actively look for other artifacts of racism in the curriculum and address them accordingly.

I realize that my argument here needs work and that I’m personally having difficulty disentangling race and religion. Even if I haven’t yet convinced you that “Geometry is racist”, I think there’s sufficient evidence here to conclude that it’s definitely not anti-racist. That fact alone is reason enough for me to act. If you’re also a teacher of mathematics, I hope I’ve convinced you enough to join me in these actions.

I used to identify my race as “Decline to state”

Recently I’ve read a couple of books (namely, White Fragility and How to be an Antiracist) that have made me reexamine certain aspects of myself through a lens of racial privilege.

For a significant period of my life, I refused to identify my race as “White” on any survey I completed. I’ve since realized that my doing so was an act of racism and I apologize. While I can’t change the past, I hope that sharing my story will serve as a token of my promise to make amends in the future.

The new information which led me to this conclusion was this idea that the white race begins with slavery. I had previously defined “White” as a ~2000 year old construct when in reality it began ~400 years ago. This redefinition caused all sorts of cognitive dissonance until I learned about a defensive mechanism that white people often have called channel-switching, where we redirect discussions about racism to other factors. I discovered I had been subconsciously conflating the “White Race” with “Christianity” because I identified them both with the same structure of “White Power”. I had falsely assumed that since “White Power” was inherently “Christian” that the “White Race” through which “White Power” manifested was also “Christian”.

What bothers me most was that I knew that race and religion were not the same, and was careful not to invoke my atheism as a minority defense. I’d even openly identify as “Caucasian”, but the term “White” triggered in me a storm of rage and “Decline to state” was the calm. I thought that distancing myself from that label made me “not racist”, but I was wrong. The only way to engage in antiracism is through accurate statistical measurements of racial disparities. I wasn’t thinking about the potential harm that mislabeling myself could do, placing my own individuality over the welfare of others, and I’m sorry.

I’m going to try to continue down this path of antiracism, but I need help. I’m trying to look at this as a data analysis problem and realized that I don’t have enough information to properly disaggregate race from religion. I don’t know how to authentically engage in antiracism without also being antireligious. This presents a problem because I’m “in the closet” at work. I carefully avoid revealing my thoughts on religion because I fear that doing so could get me fired. In order to learn how to navigate this space, I need more information about this intersection of race-religion.

The first step I can take on my own. Quite frankly, I need to learn about how and why so many blacks adopted the religion of their oppressor. I believe that the best way for me to acquire this knowledge is through the narratives of black thinkers who are critical of the role religion plays in the white power structure. I plan to seek out black atheists, listen to their stories, and lend whatever weight I can to their voices. I’ve known for a long time that black atheists were underrepresented in the atheist community and it’s time I did something about it.

The second step is going to require feedback. I accept the premise that there is a non-zero probability with which I will commit acts of racism in the future. I need friends of color to call me out when this happens and engage me in an honest discussion about why. If you do this publicly, I will make my best effort to model antiracist behavior in response, but do so with the understanding that my response will likely be influenced by my antireligious views.

If you are reading this and are someone I work with, I hope you can understand the thin line I’m attempting to walk. I propose we establish a hidden signal: invoke each others’ first names when calling out acts of privilege (in whatever form that may take). This will serve as a reminder that what follows is being said as a friend and that we’re both together in this fight for equality as human beings.

Measuring Rational Behavior

Is “rationality” a measurable quantity?

In a previous blog post, I discussed some common logical errors that often arise in political discourse. This led to a rather interesting discussion on Twitter about political behaviors and how to model them mathematically (special thanks to @mathguide and @nesa_k!). One of the questions that came up this this discussion was how to define “rational behavior” and whether or not this is a measurable quantity. What follows is my hypothesis on “rational behavior”: what it is and how to measure it.

Please keep in mind that this is just a hypothesis and I don’t quite have the resources to verify these claims experimentally. If anyone has evidence to support or dispute these claims, I would certainly be interested in hearing it!

Defining “rational behavior”

Before we can begin to measure “rationality”, we must first define what it means to be “rational”. Merriam-Webster defines “rational” as “relating to, based on, or agreeable to reason”. The Online Etymology Dictionary describes the roots of the word in the Latin rationalis, meaning “of or belonging to reason, reasonable”, and ratio, meaning “reckoning, calculation, reason”. It’s also worthwhile to mention that ratio and rational have a distinct mathematical definition referring to the quotient of two quantities. Wikipedia suggests that this usage was based on Latin translations of λόγος (logos) in Euclid’s Elements. This same Greek word lies at the root of “logic” in English.

Based on these definitions and etymology, I think its fair to define rational behavior as “behavior based on a process of logical reasoning rather than instinct or emotion”.

Even this definition is far from perfect. In the context of game theory, “rational behavior” often defined as the process of maximizing benefits while minimizing costs. Note that by this definition, even single celled organisms like amoeba would be considered to exhibit “rational behavior”. In my opinion, this minimax-ing is a by-product of evolution by natural selection rather than evidence of “reason” as implied by the typical usage of the word “rational”.

I should also clarify what I mean by “logical reasoning” in this definition. In trying to quantitatively measure rational behavior, I propose that it makes sense to use a system of fuzzy logic rather than Boolean logic. By using the Zadeh operators of “NOT”, “AND”, and “OR”, we can develop an quantitative measure of rationality on a scale of 0 to 1. In logic, we say that an arguement is considered sound if it’s valid and its premises are true. Since we’re using the fuzzy “AND” in this model, the rationality measure is the minimum truth value of the logical validity and base assumptions.

Using this definition, we can also define irrational behavior as “behavior based on an invalid logical argument or false premises”. I’d like to draw a distinction here by defining arational behavior as “instinctive behaviors without rational justification”, to cover the amoeba case described above. An amoeba doesn’t use logic to justify its actions, it just instinctively responds to the stimuli around it.

Rationalism and Language

There’s an implicit assumption in the definition of “rational behavior” that I’ve used here, and that is that this requires some capacity for language. First-order predicate logic is a language, so the idea that “rational behavior” is language dependent should come as no surprise. In fact, the same Greek word “logos” from which “rational” is derived was also used as a synonym for “word” or “speech”. The components of language are necessary for constructing a formal system, by providing a set of symbols and rules of grammar for constructing statements. Add a set of axioms (assumptions) and some rules for inference, and you’ll have all the components necessary to construct a logical system.

A Dynamic Axiomatic System Model of Rational Behavior

A this point we can start to develop an axiomatic system to describe rational behavior. Using the operators of fuzzy logic and the normal rules of first-order logic we can create an axiomatic system that loosely has the properties we would expect of “rational behavior”. It’s very unlikely that the human mind uses the exact rules of fuzzy logic, but it should be “close enough”. We also have to consider that the basic beliefs or assumptions of a typical person vary over time. Thus, it’s not enough to model rational behavior as an axiomatic system alone, we must consider how that system changes over time. In other words, this is a dynamic system.

As we go through life, we “try out” different sets of beliefs and construct hypotheses about how the world works. These form the “axioms” of our “axiomatic system”. Depending on whether or not these assumptions are consistent with our experiences, we may decide to keep those axioms or reject them. When this set of assumptions contains contradictions, the result is a feeling of discomfort called cognitive dissonance. This discomfort encourages the brain to reject one of the conflicting assumptions to reach a stable equilibrium again. The dynamic system resulting from this process is what I would characterize as rational behavior.

One particularly powerful type of axiom in this system is labeling. Once a person takes a word or label and uses it to describe him or herself, the result is the attribution of large number of personal characteristics at once. The more labels a person ascribes to, the more likely it is that a contradiction will result. Labeling also has powerful social effects associated with it as well. Ingroups and outgroups can carry with them substantial rewards or risks depending on the context.

Rather than rejecting faulty axioms when confronted with cognitive dissonance, some individuals develop alternative methods of reducing the discomfort. The general term for pattern of behavior is called cognitive bias. This behavior can take a variety of different forms, but the one that is most relevant to this discussion is the confirmation bias. One of the ways in which the human brain can reduce the effects of cognitive dissonance is by filtering out information that would result in a contradiction with the base assumptions. Another relevant bias to consider is the belief bias, or the tendency to evaluate the logical validity of an argument based on a pre-existing belief about the conclusion.

Whatever form it may take, cognitive bias should be taken as evidence of “irrational behavior”. Not all cognitive biases are of equal magnitude, and some arguments may rely more highly on these biases than others. The goal here is not a Boolean “true” or “false” categorization of “rational” and “irrational”, but more of a scale like the one used by PolitiFact: True, Mostly True, Half-True, Mostly False, False, Pants on Fire. The method of applying truth values in fuzzy logic makes it highly appropriate for this purpose.

Examples in Politics

Consider this clip from The Daily Show. Using this clip may seem a little biased, but it’s important to remember that John Stewart is a comedian. Comedians have an uncanny knack for walking the fine line between “rational” and “irrational”, providing an interesting perspective to work with.

In the first example, we have the issue of Rick Santorum and JFK. After reading JFK’s speech on religious freedom, Santorum says that it made him want to throw up. In order to defend this statement, Santorum uses a good ole fashioned straw man argument by claiming that JFK was saying “no faith is not allowed in the public public square” when in fact JFK was saying “all faiths are allowed”. I think Santorum’s behavior here is a prime example of irrational behavior. Taking this position may very well earn him some votes with the deeply religious, but it’s clear that Santorum has some problems finding consistency between his personal beliefs and the First Amendment. His position is not based on a valid logical argument, but on a physical response to the cognitive dissonance resulting from his conflicting beliefs. This example also shows the power of deeply held self-labeling behaviors like religion.

Mitt Romney made some headlines with his “NASCAR Team Owner” blunder. It would appear that Mitt Romney had gone to Daytona to try and score some points with “average Americans”, but a slip of the tongue showed how out of touch he really is. To Romney’s credit, his behavior here is about half-rational. His assumptions are probably something like this:

I want people to vote for me.
People vote for someone they can relate to.
Most people know someone who likes NASCAR.
I know someone who likes NASCAR.

It makes sense from a logical standpoint, but it turns out that the person who Romney knows that likes NASCAR just happens to be a
“team owner” instead of a “fan”. This small detail makes it unlikely that people will relate to him, but at least the foundation of a logical argument is there.

This brings us back to Rick Santorum again. This time, Santorum calls President Obama a “snob” for “[wanting] every American to go to college”. Not only is this comment blatantly false, but he’s employing an ad hominem attack in lieu of a logical argument. This example draws a nice dichotomy between President Obama and Rick Santorum. The President is making a rational argument in favor of higher education which is well supported by evidence. By opposing this rational argument on a faulty premise, Santorum comes out of this situation looking mostly irrational. His behavior makes sense if you consider the effects of confirmation bias. Santorum believes that the President is trying to indoctrinate college students to become liberals. He believes it so thoroughly that he simply filters out any evidence that would contradict it. While most observers can hear the President say “one year of higher education or career training“, Santorum doesn’t. He hears the part confirms his beliefs and filters out the rest. I’d imagine that for Santorum, listening to President Obama speak sounds something like the teacher from the Peanuts cartoons: “one year of higher education wah wah-wah wah-wah-wah“. To Santorum’s credit, at least he had the mind to retract his “snob” statement — even if only partially. This shows that the underlying mechanisms for rational behavior are still there, despite his frequent leaps of logic.

Conclusion

I hope I’ve at least managed to present a definition of “rationality” that’s a little more precise than the everyday use of the term. I’m sure some people out there might disagree with the way I’ve rated the “rationality” of these behaviors. Different people have different experiences and consequently have different assumptions about the world. If we were to use multiple “rationality raters” and average the results, perhaps we might have a decent quantitative measure of rationality to work with.

Part of the problem with measuring rationality is the speculative nature of trying to determine someone else’s assumptions. We can generally use what a person says as an indication of what they believe — at least for the most part. It’s also important to consider not only the statement, but the context in which the statement is made. In political discourse, we implicitly assume that politicians are being honest with us. They might be wrong about the facts, but this idea that they are honestly representing their own views is something that voters tend to select for. Perhaps this is why Romney is still struggling against Santorum in the primary. Santorum may have problems getting his facts straight and presenting a logical argument, but he has a habit of saying what he believes regardless of the consequences. Romney, on the other hand, says what he thinks will win him the most votes. Many voters do not vote “rationally”, they vote according to how they “feel” about the candidates. Romney may be more “rational” than Santorum, but his calculated responses cause him to lose that “feeling of honesty” that Santorum elicits from voters.

In the next article, I’ll attempt to explain the origins of rational and irrational behavior. I think the key to understanding these behaviors lies in evolution by natural selection. I would argue that both rational and irrational behaviors contributed to the survival of our species, and this is why irrationality persists into the present. Stay tuned!

Wake up Virginia District 4! Stop the hate!

/rant on

Earlier this week, the Virginian Pilot published an article entitled Forbes versus LeGrow: In God, only one trusts. Rather than focusing on the candidates’ stances on political issues, the article focuses solely on the candidates’ differing religious beliefs. Not only is this coverage thoroughly distasteful, but some of the comments added by readers demonstrate a sickening level of ignorance and intolerance. Voters in Virginia’s 4th Congressional District need to look past religion this November. To do otherwise is to reinforce a culture of bigotry and hate that has plagued this great nation for far too long.

Allow me to start by correcting Mr. Forbes on the language used in the Declaration of Independence. The “Creator” referred to in the Declaration of Independence is not “God” as used in the Christian sense of the word. Rather, the word “Creator” is used here as a metaphor for “Nature”. The Treaty of Tripoli clarifies this, explicitly stating that the US “is not, in any sense, founded on the Christian religion”.

Secondly, Mr. Forbes swore an oath to uphold the Constitution of the United States. Included in the 1st Amendment of the Constitution is the following:

Congress shall make no law respecting an establishment of religion

Mr. Forbes has sponsored two bills which, if passed, would violate this Amendment:

H.Con.Res.274 attempts to reaffirm “In God We Trust” as a national motto
H.Res.397 falsely characterizes the founding of this nation as being religious in nature

By proposing this legislation, Mr. Forbes has made it clear that he has no intention to adhere to his oath to uphold the Constitution and is therefore unfit to hold office. Mr. Forbes also started a “Congressional Prayer Caucus”, further blurring the line between church and state.

That’s not to mention the fact that Mr. Forbes also participated in Glenn Beck’s rally on 8-28-10, an event which was coincidentally held on the anniversary of Martin Luther King Jr.’s famous “I have a dream” speech and at the same location. Mr. Forbes attendance at this event is an implicit endorsement of Beck’s platform. The issues with such an endorsement are too numerous to list here, so instead I’ll point to this clip from The Colbert Report and leave it at that.

The real issue that I want to address here, are the reactions from the VA Democrats in the Pilot article. State Delegate Lionell Spruill says “I can’t take him to churches as an atheist… That would hurt me.” Really? Contrary to popular belief, atheists do not spontaneously combust upon entering churches. Spruill is not in danger of being physically hurt by bringing Dr. LeGrow into a church. Instead the issue seems to be that Spruill is afraid that supporting an atheist for office may harm his chances for re-election. This behavior is inconsistent with the Democratic party’s platform, which says that the party is committed to “[e]nding racial, ethnic, and religious profiling” (emphasis mine). Heck, even the Republican platform condones this type of behavior. The US Constitution explicitly states that “no religious Test shall ever be required as a Qualification to any Office or public Trust under the United States”. It doesn’t get much clearer than that.

Part of the problem is with constituents like Rev. Jake Manley Sr., who says in the Pilot article: “I could not vote for a man who doesn’t believe in some power higher than his.” Really, this is just a euphemism for “I could not vote for an atheist”. This is religious profiling, and not even very subtle at that! If he had instead said “I could not vote for a Black/White/Mexican/Asian/Christian/Jew/Muslim” there would be public outrage! But for some reason, people think it’s okay to engage in blatant discrimination against atheists. It’s not.

My message to my fellow voters in VA-4 is to not let religion cloud your vision this November. Here we have an opportunity to replace an incumbent who has ignored his Congressional oath with a doctor who cares about providing people with medical care, better education, and a clean energy future. Vote with reason, your nation needs it right now.

/rant off

An Open Letter To Barack Obama On National Day of Prayer

Dear President Barack Obama,

I’m writing today to urge you to reconsider your position on the National Day of Prayer. I was most displeased to hear that you will continue to acknowledge the National Day of Prayer, despite the recent Supreme Court Ruling of its unconstitutionality. I feel that Judge Crabb’s ruling in this case was correct. While the White House argues that this ruling does not prevent you from issuing a Presidential Proclamation recognizing this day, doing so ostracizes a significant body of your constituents and contradicts the spirit of the Constitution.

I would like to say that I supported you in the 2008 election. During your campaign, you presented yourself as a man of reason and principle. Having moved from California to Virginia earlier that year, I felt like my vote made a difference for the first time in my life. This feeling was reaffirmed during your acceptance speech when you specifically thanked “non-believers” among other groups. As a atheist, this was the first time I had heard any President speak of “non-believers” in a positive light. I felt a glimmer of hope that CHANGE was possible.

Since then, that glimmer of hope has been gradually dying out. You promised to end the war in the Middle East, were awarded the Nobel Peace Prize, and yet the military occupation continues. You promised to end the arrests of medical marijuana users acting in accordance with state laws, and yet the DEA raids have continued. Dreams of single-payer health care were reduced to hopes for a public-option, and eventually turned into “be happy you got any health care reform at all”. You promised an environmentally conscious energy policy, but shifted your stance to support offshore drilling about a month before the BP oil spill. You continue to proclaim support for ending “Don’t Ask, Don’t Tell”, but I’m beginning to doubt that this will go through either.

In the times of old, people prayed for things that were beyond their control. People prayed for rain in periods of drought. When that didn’t work, they offered virgin sacrifices. Nowadays these practices are largely obsolete. Instead of praying for rain, we build aqueducts and irrigation systems. Instead of praying for the sick to improve in health, we intervene with medical treatment. While some people continue to pray in times of desperation, I am not one of them. For myself and others like me, the act of prayer is considered an ineffective method for bringing about change. Actions consistently provide better outcomes than prayers. This is my request to you: instead of a Day of Prayer, proclaim May 6th as a Day of Action. The American people didn’t elect you to office to “pray for change”, they elected you to “act for change”.

Make no mistake, such a declaration would undoubtedly draw heat from the religious community. Bear in mind that we atheists suffer through this discrimination everyday of our lives. Hate mail and death threats are no strangers to atheists who speak their minds. The separation of church and state is one of the founding principles of this nation, set forth in the Constitution that you have sworn to uphold, and I hope that you can set aside your personal views to uphold the rights of the “non-believers” who helped elect you to your present position. To pursue an appeal of Judge Crabb’s decision is a waste of government resources. There are more pressing matters that need your attention.

Please Mr. President, use May 6th to bring us a real moment of “peace and goodwill” by withdrawing our nations troops from their posts overseas. Prove your commitment to treat everyone with “dignity and respect” by ending “Don’t Ask, Don’t Tell”. Show that the right “to love one another” extends to everyone, including those in the LGBT community, by making a motion to repeal the “Defense of Marriage Act”. Set an example for what it means “to understand one another” by not alienating non-believers with a “Day of Prayer”. Do these and show that CHANGE is not beyond our control.

Denialist Misrepresentations of Math and Evolution

This is so me right now.

I generally try to avoid flamebait, but I saw this article linked off of Twitter. I should have stopped reading after the first section where it’s clear that the author is a troll. Evolution and science denialism aside, the misrepresentation of mathematics in the article is inexcusable.

After attacking Darwin and scientific thought in general, an appeal to emotion, he proceeds into a second hand quote from a philosopher on the subject of “fallacies”. It’s kind of ironic that the inclusion of this quote would serve as an appeal to authority.

Next, he goes into intelligent design saying:

we could find incontrovertible evidence that reality, matter, life, has been designed, but that interpretation of the evidence would be discarded because naturalism dictates the exclusion of anything which might lead outside of a naturalistic explanation.

This is absolutely false. Scientific theories are necessarily falsifiable. If the evidence implied a “design”, that’s what the scientific theory would be. The fact is that the evidence points to the contrary. Biology shows a picture of “unintelligent design”, consistent with a process of genetic mutations occurring over time. The naturalistic explanation is the one that the evidence supports.

Then he claims that Gödel’s Incompleteness Thereom proves this.

He managed to get Gödel’s basic background information right, but incorrectly describes the Incompleteness Theorem.

From the article:

Any system that is consistent is incomplete.

The consistency of axioms (axioms=assumptions that cannot be proven) cannot be proved from within the system.

The real Incompleteness Theorems:

Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true,^[1] but not provable in the theory (Kleene 1967, p. 250).

For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent.

Notice how the part about “basic arithmetic” is conveniently left out of the definition? That’s because the author doesn’t want you to know that there can exist axiomatic systems which are both complete and consistent. First-order predicate logic was proven to be both complete and consistent by none other than Gödel himself. Furthermore, saying that the Incompleteness Theorem “utterly [destroyed] atheist Bertrand Russell’s logical system for arithmetic” doesn’t give Russell the credit he deserves. Gödel’s technique was based on the same idea as Russell’s Paradox to begin with. Despite its incompleteness, the development of Russell’s work into Zermelo-Fraenkel set theory was an important building block in the foundation of later mathematics. By referring to him as “atheist Bertrand Russell”, it’s clear that the author is more concerned about religion than the actual mathematics.

Next we have a very weak analogy. He describes three items on a table and says:

Now draw a circle around those items. We will call everything in that circle a system. Gödel’s first theorem tells us that nothing in the circle can explain itself without referring to something outside the circle.

It’s true that Gödel’s theorem succeeded in “stepping out of basic arithmetic”, but here’s where that omitted condition of a “formal system capable of basic arithmetic” comes into play. Are a half-full cup of coffee, a fishing pole and a jacket capable of arithmetic? If the answer is no, then Gödel’s theorem doesn’t apply. Capable of self reference? Maybe if the coffee mug says “I’m a half full cup of coffee” on it.

The analogy of a computer is a much better example. Computer programs are capable of basic arithmetic. What Gödel’s theorem implies for computers is that there exist certain programs which are computationally irreducible. The only way to determine the output of such a program is to run it. If we think of Nature like a computer program, the only way to be certain of the future “output” is to let Nature run its course. This result does not prevent science from making conjectures about the structure of Nature, but requires that science adopt a Black-box testing procedure which entails experimentation and observation. There are certainly unanswerable questions in science, such as the precise position and momentum of elementary particles, but evolution isn’t one of them. The evidence for evolution is incontrovertible.

The final second shift the analogy to the universe and the claim is that what’s outside the universe is unknowable. Just because we can’t see what’s outside the universe, which would be white-box testing, doesn’t mean we can make and test hypotheses about it as a “black-box”. The Many-worlds interpretation of quantum theory is one such example which predicts that our universe is but one of many possible universes. Similarly, M-theory predicts the existence of hidden dimensions beyond space and time. Just because some questions are unanswerable, doesn’t mean all questions are.

The article ends by claiming that evolution and naturalism are “fallaciously circular”, but here’s the real circular fallacy:

Author misinterprets Gödel’s theorem to imply that all axiomatic systems are incomplete or inconsistent.
Author mistakenly assumes that science is an axiomatic system.
Based on this misinterpretation, author concludes that science must be incomplete or inconsistent.
Since author concludes that complete scientific knowledge is incomplete or inconsistent, author ceases to look for empirical evidence of scientific claims.
Since author ceases to look for evidence, author does not find any evidence.
Since author does not find any evidence, author concludes that scientific knowledge is incomplete.
As a consequence, the author’s incomplete knowledge becomes a self-fulfilling prophesy.

This whole article is a Proof by Intimidation. The “average Joe” doesn’t know enough about contemporary math and science to go through and verify each detail. The use of mathematics vocabulary in the article is deliberately being used to distract the reader from the real issue — the overwhelming evidence for evolution. The references to Gödel’s Incompleteness Theorem are nothing more than a red herring, and the author even misstates the theorem to boot.

The Misunderstood Generation

I picked up Mark Bauerlein’s The Dumbest Generation: How the Digital Age Stupefies Young Americans and Jeopardizes Our Future * or Don’t Trust Anyone Under 30 (DG) yesterday and have been up all night reading it. Not because I enjoyed it, but because it made me angry. I should have anticipated this, considering how I’m 27 and the sub-title of the book is “Don’t Trust Anyone Under 30”. I’m a part of the generation Bauerlein is talking about, and I consider this book a biased pseudo-scientific misrepresentation of myself and my peers.

It’s important to note that I probably represent a fringe case within the generation. I read regularly and tend towards non-fiction literature. The fact that I bought this book in the first place is evidence that I’m an outlier. I play several musical instruments, saxophone and guitar being my favorites. I taught myself how to program in high school and designed websites for local businesses. I started out as a Math Major in college, but eventually double majored in Mathematics and Psychology because I was fascinated with learning how the human mind works. After graduating, I pursued another love of mine, video games, and landed a job as a programmer at a game studio. After a few years, I decided that I wanted to make video games that fostered the development of critical thinking skills. I enrolled myself in graduate school and started teaching remedial math. I’m also a complete technophile, love the latest gadgets and gizmos, and can’t stand more than a day without being connected to the Net.

Bauerlein talks quite negatively of video games, and I don’t think this criticism is well founded. There’s a substantial amount of mathematics that can be found in video games. Gamer communities like the Elitist Jerks (http://elitistjerks.com) use spread sheets and simulation programs to mathematically optimize stats and equipment in World of Warcraft. These massively multiplayer online games are complex mathematical systems, complete with virtual economies and social interaction. “Casual” players might not experience the same depth of content, but the “hardcore” players participate in a substantial amount of meta-gaming and often reflexively analyze their performance to foster continued improvement. I think its unfair to devalue competitive video gaming as simply a leisure activity; I consider such play to be equally as intellectual as playing Chess or Go. I would also note the considerable amount of mathematics, science, and art involved in making the video game itself. From my personal experience, learning to play and create video games directly contributed to my interest in math, science and engineering. There are a myriad of video games that are trivial and superficial, but there are also games I would call “higher art” that challenged my perceptions about storytelling in an interactive medium. Bauerlein doesn’t even address the topic of video games as “higher art”. He treats the entire medium as if it were completely devoid of any social value altogether.

What kinds of media does Bauerlein suggest in video games’ place? A variety of gems including Harry Potter, Dante, Milton, A Christmas Carol, Rush Limbaugh, Fox News, and the Bible. Bauerlein tries to portray the problem as a cultural war, but these repeated references to religiously themed works also reveal an ideological difference. These were probably intended as generic books and news sources, but the choices used show a pattern of right-wing religious bias. The whole argument is framed like a dichotomy between the conservative-religious-elders and the liberal-secular-youth, as personified by technology. It appears like Bauerlein is more upset about students not reading his culturally biased list of literature than he is about the real faults of our nation’s education system.

These are bold claims, but there are good reasons to be skeptical of DG. The information is all second-hand, and no new research is presented. The data that is presented is not even organized into a coherent framework. It reads like series of disconnected statistics are piled on, one after the other, with no consistency in procedure. In themselves, they each sound like reasonable results. However, the data is mostly tangent to the central thesis about the role technology in producing these trends. It gradually turns into “proof by verbosity”, focusing largely on differences in cultural and ideological values which are not scientifically falsifiable hypotheses to begin with. The book repeatedly references “tradition” as an authority, as if the previous generation has some mysterious source of ancient wisdom. Science is conducted in the open. Clinging onto ideas out of tradition alone is not the way to foster progress.

There are a couple of points in particular that seem suspect. First, the inconsistency between falling rates of factual recall and increasing averages on IQ tests. Memorization skill and Intelligence are two entirely separate constructs. The obvious explanation for this phenomena is that the collection of information worth memorizing has changed but general problem solving ability hasn’t. The largest drop in the included performance statistics seemed to take place after the turn of the millennium, which is also a bit suspicious given 2001 passage of the No Child Left Behind Act of 2001. It’s difficult to compare data from before and after a major legislative change which mandates changes in how student performance is assessed and how teachers teach. There is not enough data here to rule out the interaction of other changes in the educational process as an alternative explanation. In a scientific study, the data should speak for itself. The data presented in DG shows that there is significant need for improvement in education, but it’s not enough to indict technology as the singular cause of the problem.

Another point worth making is that DG suffers from a combination of selection and actor-observer biases. In defending Generation M, I’m partially guilty of this myself. I’m an intellectual person and tend to associate with like-minded people. Thus, I have a tendency to generalize the behaviors my peer group appears to the generation as a whole. I think Brauerlein is guilty of this also. He probably tends to associate with the intellectual types and may therefore incorrectly generalizes this intellectualism to his generation as a whole. The second fallacy here, is that there is also a tendency to attribute observed behaviors to personality traits instead of the situation. As Brauerlein acknowledges, it’s not unusual for teens to go through a rebellious phase, and the technology usage might just be an expression of this. Consider another option: What if Gen M-ers are being honest when they say the information they’re being taught isn’t relevant to their lives? Certainly these questions merit additional consideration.

This is a commercial product, which is intended to sell copies, rather than a peer-reviewed study in a scientific journal. The reviews on book cover are all from popular media sources rather than the scientific community. Some of Bauerlein’s statistics are certainly interesting, but I don’t think they demonstrate anything close to a causal relationship between technology usage and intelligence. He doesn’t bother to define “intelligence” and tends to use it interchangeably with “knowledge”. I would have also liked to see an effort to normalize the data and plot it over time in comparison to technology usage rates. He cites plenty of sources showing a deficiency in these skills, but there are still too many external factors to point to technology as the source of the problem. The fact that learners process web information differently than print materials just shows that the two mediums need different approaches.

The language of the book is highly emotionally charged and features numerous stereotypical persuasive devices. It identifies a common enemy for the readers to rally against, uses cultural references to which older readers would relate closely, and tries to make the readers feel like a part of something larger than themselves. Even the choice of title and cover art seems like it was designed to trigger an emotional response rather than promote rational intelligent discourse. I found it particularly interesting how Bauerlein tries to present jazz as a higher art form in opposition to modern rap and rock. The irony is that jazz was all about “breaking the rules”, reversing the established chord progression, and eventually laid the foundation for the modern music which Bauerlein seems to despise so thoroughly.

During my undergraduate study, there were times where I found myself relearning subjects from new perspectives. Gödel’s Incompleteness Theorem completely changed how I thought about mathematics and computing. The theorem states that any fixed formal axiomatic system will have statements that are neither provable or disprovable in that system. Math ceased being about prescribed procedures and memorization and turned into an exploration of how different sets of hypothetical rules might behave. It stopped being about blindly following the rules and instead tolerated the bending or even breaking of them. Part of me wished math could had been that way from the beginning. I wanted to provide the “past me” with a variety of different sets of rules and allow me to explore how they work in a controlled environment. That’s precisely why I think Games have such potential as a educational medium. They don’t need to be Video Games. Board, Card, Dice and Pen & Paper Games have very beautiful and complex mathematical structures lurking just below the surface of the rules.

My active rejection of the traditional values is different from a passive indifference as implied by DG. I might be a statistical anomaly in this cohort, but I don’t think I’m alone. Brauerlein might reject the notion that the problem is in the situation and not the students, but my experiences showed me that many things presented as “facts” in middle/high school were quickly replaced by better models in college. Newtonian Physics became M-theory, Math became Meta-Math, and Technology Use evolved into Software Engineering. DG suggests that the curriculum is not “hard” enough, so maybe we just need to stop diluting the truth? I wish I had Logic and Set Theory as topics in grade school. I want “past me” to be allowed the opportunity to build a solid foundation for the “real” math I’ll encounter in the “real” world. I don’t want to “learn the wrong way now, learn the right way in college”. Why should I trust an authority figure that routinely hides the truth from me because “its too hard”?