The Third Realm

 

 

Plato on Math

Although Plato stresses the importance of mathematical training throughout Republic, in Book VI, he gave us a metaphor for understanding how mathematics is a gateway for a deeper understanding of reality: the divided line. Today's reading reminds us about just how important mathematics was to Plato. In fact, mathematician and historian of mathematics Morris Kline sees that mathematics is central to Plato’s entire conception of knowledge and points out that Plato may have even been part of the quasi-religious sect of mathematicians headed by Pythagoras (see Kline 1967: 62-63). Notice, for example, the cryptic language that Plato uses when discussing the subject matter of mathematicians:

Recall that, as a metaphor for how he conceived of the world and the objects within, Plato gave us The Divided Line. What Plato has laid out for us is his hiearchy for the reality of objects. In other words, Plato is organizing the types of objects that exist from less fundamental to more fundamental. At the bottom are mere copies of things: reflections in the mirror, paintings, and the like. These are only copies of the real thing; for example, your reflection is a mere copy of the real you. The next level up is the realm of physical objects. This is where you and I live, along with all the physical things that we interact with on a day-to-day basis. We might think that this is the ultimate level of reality, but Plato disagrees. Upon reflection, we might come to think that Plato might have a point. After all, the reality that we see with our senses can't be the ultimate reality. We know that physicists tell us about a world of atoms and smaller subatomic particles that we can't see with the naked eye.

As you can see in the diagram, the next level up consists of mathematical objects. This means that all mathematical objects (like numbers), as well as mathematical relations (like equality) and functions (like addition, subtraction, and all the more complicated functions) exist in this realm, independent of all human thought. In other words, numbers are real and they are more fundamental than the reality we inhabit. This shouldn't sound too strange if you believe that mathematics has the power to help us understand our world: mathematics has this power because it is upon mathematics that our world is ordered. This is why mathematics is always involved in the sciences and in any other enterprise that involves knowing our world in a deeper way.

Above mathematical objects are The Forms on which our reality is based. These are the actual properties of the universe upon which everything we see is based. To see this realm is to have a "god's-eye-view". In other words, it is to understand reality as it really is—something that might come in handy if you are a Guardian. Try to imagine combining all the knowledge of all the disciplines and then extending that knowledge until it is complete, where everything that can be known is known. That is what it is to know The Forms. And of course, The Forms are ordered too, just like everything below it. At the top of the hierarchy is The Good.

And so Plato believed that this realm of The Forms could be understood via the realm of mathematical objects. One way we could interpret this is that if we want to understand reality as it really is, we have to go through mathematics. All of our knowledge, then, will ultimately be based on mathematics; mathematics is the medium—or the gateway—through which we can understand the basis of all reality (The Forms). As such, understanding some basics about math, including some of the jargon, is extremely useful for understanding the world we live in.

Argument Extraction

 

 

 

Mathematical Thinking

 

Stats Basics

Understanding the basics of statistics is key for a critical thinker. Many arguments are sometimes couched in statistical language, and if you are not fully cognizant of what is being stated, then you might call an invalid argument valid. Some of the most basic jargon has to do with the distinction between associated values (also referred to as dependent values) and independent values. Associated values have some kind of connection. For example, perhaps they are both caused by some third factor, or perhaps they are just regularly correlated. Take a look at the scatter plot pictured. Clearly there is a relationship here. It appears that cities with a high percentage of homeownership are cities with a low percentage of multi-unit buildings. This makes intuitive sense. If a majority of people in a city own a home, then there won't be much demand for apartment buildings.

Whereas we've previously discussed so-called natural experiments (or observational studies), statisticians can also experiment. Researchers use these experiments to check if there is a causal connection between two variables, namely to see if an increase (or decrease) in one variable (the explanatory variable) causes an increase (or decrease) in the other variable (the response variable). Experimentation includes various forms of sampling. As long as the experiment utilizes random assignment, we are allowed to make causal conclusions, i.e., random assignment allows us discover relationships between explanatory variables and response variables. For example, if a new treatment is being used to treat disease X at some hospital and the staff chooses which patients with disease X get the new treatment via a coin flip, then this would be random assignment (Diez et al. 2019: 32). However, if one would want to generalize some medication for the general public, then random assignment is not enough. This is because for generalizability we need random sampling, i.e., we need it to be the case that everyone in the population had an equal chance of being included in the study. This is not the case in this example, since the coin toss was only performed for those with disease X—clearly not random sampling.

Things to Watch Out For

Misusing Regression Analysis

If you're familiar with statistics or have taken a stats course, then you are probably familiar with regression analysis. Regression analysis is a set of statistical processes for estimating the relationships between a dependent variable and one or more independent variables (see Diez et al. 2019, chapter 8; or watch this helpful video. There is a potential problem with regression analysis: you can essentially have a computer run this analysis on data but draw the wrong inferences.

How can this happen? One scenario where this might occur is if you are using regression analysis on data sets that do not have a linear relationship. For example, you might decide to analyze the relationship between K-12 funding and gross domestic product (GDP). It might be the case that both (1) having a high GDP opens up funds for more education funding and (2) having a well-funded k-12 system tends to promote GDP growth. However, this relationship might be non-linear. In other words, the system of which both k-12 funding and GDP are a part of might be a system in which the change of the output is not proportional to the change of the input of a given variable. Put bluntly, the system is too complex and regression analysis is not the right tool.¹

 

Breaking the cardinal rule: correlation does not equal causation.

What would happen if you tried to find a correlation coefficient for the rise in autism in the US and the rise of the GDP in China? You’ll find one if you look for one, but you know this isn’t a theoretically sound relationship. Don't go thinking the rise in autism caused the rise in China's GDP!

 

The Biased Sample Fallacy

The biased sample fallacy occurs when an arguer draws a conclusion from an inadequate sample pool; i.e. it is a fallacy that occurs when there is an insufficient amount of evidence to satisfactorily draw a conclusion.

One of my favorite examples of this is a conversation between two survivors of a shipwreck. One says to the other, "It looks like everyone that survived prayed to be rescued. So, that must mean that prayer saved them." The other person responds, "But what about all the people who prayed and died anyway?" This second individual is, of course, aware that they are looking at a biased sample. Anyone who prayed and still died isn't there to report that prayer didn't work!

Some common sampling biases in statistics:

1. 1. Convenience sampling: This occurs when researchers only gather data that is easy to gather but that isn't representative of the general population.
2. 2. Non-response: This occurs when researchers employ some survey where a large proportion of those surveyed did not or could not respond. This is what happened in the shipwreck example above.
3. 3. Voluntary response: This occurs when the method of survey allows for too much self-selection. For example, in a voluntary survey on capital punishment, perhaps only those with strong feelings (either for or against) will take the time to actually respond, thereby skewing the data.

Another interesting example of biased sampling is that of the poll done by The Literary Digest of its own readership (plus registered car owners) for the 1936 presidential election between FDR and Alf Landon. The poll suggested that Landon would win; however FDR won by a landslide (60% to Landon’s 36%). Of course, the sample for the poll was biased. You must recall that it was the Great Depression and so those who could afford subscriptions to literary digests and cars were of higher wealth. As such, they were less likely to feel the economic hardship of the times (and hence more likely to vote Republican).

 

Omitted Variable Bias

This happens when you don't note the presence of a very important variable in some statistical relationship. For example, consider a study that links playing golf to increased risk of heart attack. If the study didn’t carefully control for age (since older people tend to play more golf), then the study is methodologically shoddy!

 

Too many variables!

Your analysis should only include the variables that are necessary. For example, if you're trying to explain some rot in the plants in your garden, you really shouldn't include details about how much you spend on Starbucks every month. It's just not relevant. That might be a silly example, but the point stands: keep it simple. “Any regression analysis needs a theoretical underpinning. Why are the explanatory variables in the equation? What phenomena from other disciplines can explain the observed results?” (Wheelan 2013: 147).

 

The Recall Bias

The recall bias occurs when participants do not remember previous events or experiences accurately or omit details. In other words, the accuracy and volume of memories may be influenced by subsequent events and experiences. Put bluntly, what you're experiencing in the moment affects how you remember the past, sometimes distorting memory completely. For example, Wheelan (2013, chapter 6) reminds us that getting diagnosed with breast cancer changes one’s retrospective assessment of their eating habits. In particular, it makes them remember eating more high-fat foods when compared to those who were not diagnosed with breast cancer. In other words, getting diagnosed with a disease makes you more likely to remember your unhealthy habits!

 

 

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FYI

Suggested Reading: David Diez, Mine Cetinkaya-Rundel, and Christopher Barr, OpenIntro Statistics, Chapter 1

TL;DR: TED-ED, Why do people fear the wrong things? - Gerd Gigerenzer

Supplemental Material—

• Video: Crash Course, What is Statistics?

• Video: Crash Course, Mathematical Thinking

Related Material—

• Video: Santa Fe Institute, What is Complexity Science?

• Link: Santa Fe Institute Home Page

• Video: Complexity Explorer, Agent-Based Modeling: What is Agent-Based Modeling?

Advanced Material—

• Chapter: David Diez, Mine Cetinkaya-Rundel, and Christopher Barr, OpenIntro Statistics

 

Footnotes

1. These kinds of systems sometimes referred to as complex adaptive systems, and there's a whole field dedicated to their study.