Relative Consistency, or What Gödel Might Have to Say About ‘Big Data’

Posted in Audience Series, Blog

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Susan Zhang
By Susan Zhang

In the early 1900s, David Hilbert set out to prove the consistency of mathematics by reducing all mathematical statements into a formal language, from which we could deduce all mathematical statements.

Hilbert believe that derived statements would be consistent with one another. There would be no method of derivation in which we can obtain, from the same set of axioms, “1 + 1 = 2” in one case and “1 + 1 ≠ 2” in another.

One hundred years later, we sit on more data points about human behavior than ever. “Data-driven” is the go-to phrase for making decisions using statistical inference and complex computations. In digital marketing, utilizing these data points can help drive consumer outreach, illustrate trends in consumer behavior, and shed light on patterns that would have otherwise gone unnoticed.

The ways in which choose to “utilize” this data can vary tremendously. How then can we choose the “best” model?

In order to determine which method yields better results, some metric of measurement is needed from which error can be minimized. Unfortunately, these “true sets” or “true values” are not necessarily present or obvious.

Take, for example, the task of describing everyday human behaviors.

Do people who shop at one grocery store also frequent the nearby fast food chain? Do people with a higher income behave differently than the unemployed? In each case, the point of the investigation is to determine the “truth set” – what people are actually doing, how they should be classified, and what this classification implies about the state of the world. Sure, we can create our own target sets with pre-defined socio-economic biases, but then our algorithms would merely strive to confirm such biases within the entire population, not develop them independently from the raw data itself.

In 1931, Kurt Gödel published two incompleteness theorems establishing the impossibility of Hilbert’s claims. His second incompleteness theorem can be paraphrased:

Given a set of axioms and all statements derived from these axioms, there cannot exist a statement within this set that proves the consistency of this system. If such statement exists, then this system is inconsistent.

You can almost think of this like defining a word in the dictionary using the word itself: the self-referential nature negates the explanation.

The idea behind Gödel’s second incompleteness theorem closely mimics the limitations seen in the task of defining human behavior. We need some “truth set” to base an algorithm upon, but at the same time, any method used to obtain an audience’s true behavior, which simultaneously proves its own consistency, would be in violation of Gödel’s theorem.

While there may not be a method of deriving the absolute state of the world and knowing its degree of consistency, there is a way we can build ourselves up, layer by layer, using relative consistency.

Let’s start with describing people who own cars. Suppose we have a data set where 10% of the population consists of 18 to 23-year- olds. Our car ownership algorithm determines that 2% of all car owners are 18 to 23 years old.

This makes sense since young adults may be less capable of buying a car than older adults. The 2% number, when compared to the 10% number, appears to be accurate. But if the algorithm determined that 80% of all car owners are 18 to 23 years old, we would have a problem. The 80% number, when compared to the 10% number, does not appear to be anywhere near accurate.

In this case, the inconsistency in the results points to a potentially flawed algorithm or a corrupted input data set that is not representative of the true population. A check for the relative consistency of the results would tell us where a problem might exist, and prevent us from further iterations on a flawed algorithm and data set.

Like the processes of quality assurance in a manufacturing plant and ongoing maintenance for the structural base of a skyscraper, these consistency checks are fundamental to the iterative process of extracting meaning from big data. While we rely on complex algorithms to augment human intelligence and intuition, we must also question the integrity of the algorithms themselves to ensure that inconsistencies are rooted out as early as possible.

Gödel’s theorems may only be applicable in a particularly esoteric branch of mathematics, but they still illustrate a lesson that we can all benefit from: it is better to iterate with relative consistency than to settle for inconsistent systems.