The "Buenos Aires constant" is a humorous misinterpretation of mathematical notation. It stems from a misunderstanding of how definite integrals are represented. Someone saw the integral of a function with respect to x, evaluated from a to b, written as ∫ₐᵇ f(x) dx and mistakenly believed the b in the upper limit of integration was a constant multiplied by the entire integral, similar to how a coefficient might multiply a variable. They specifically misinterpreted ∫₀¹ x² dx as b times some constant and, upon calculating the integral's value of 1/3, assumed b = 1 and therefore the "Buenos Aires constant" was 3. This anecdotal observation highlights how notational conventions can be confusing if not properly understood.
The blog post "Kelly Can't Fail" argues against the common misconception that the Kelly criterion is dangerous due to its potential for large drawdowns. It demonstrates that, under specific idealized conditions (including continuous trading and accurate knowledge of the true probability distribution), the Kelly strategy cannot go bankrupt, even when facing adverse short-term outcomes. This "can't fail" property stems from Kelly's logarithmic growth nature, which ensures eventual recovery from any finite loss. While acknowledging that real-world scenarios deviate from these ideal conditions, the post emphasizes the theoretical robustness of Kelly betting as a foundation for understanding and applying leveraged betting strategies. It concludes that the perceived risk of Kelly is often due to misapplication or misunderstanding, rather than an inherent flaw in the criterion itself.
The Hacker News comments discuss the limitations and practical challenges of applying the Kelly criterion. Several commenters point out that the Kelly criterion assumes perfect knowledge of the probability distribution of outcomes, which is rarely the case in real-world scenarios. Others emphasize the difficulty of estimating the "edge" accurately, and how even small errors can lead to substantial drawdowns. The emotional toll of large swings, even if theoretically optimal, is also discussed, with some suggesting fractional Kelly strategies as a more palatable approach. Finally, the computational complexity of Kelly for portfolios of correlated assets is brought up, making its implementation challenging beyond simple examples. A few commenters defend Kelly, arguing that its supposed failures often stem from misapplication or overlooking its long-term nature.
Summary of Comments ( 1 )
https://news.ycombinator.com/item?id=43095943
Hacker News commenters discuss the arbitrary nature of the "Buenos Aires constant," pointing out that fitting any small dataset to a specific function will inevitably yield some "interesting" constant. Several users highlight that this is a classic example of overfitting and that similar "constants" can be contrived with other mathematical functions and small datasets. One commenter provides Python code demonstrating how easily such relationships can be manufactured. Another emphasizes the importance of considering the degrees of freedom when fitting a model, echoing the sentiment that finding a "constant" like this is statistically meaningless. The general consensus is that while amusing, the Buenos Aires constant holds no mathematical significance.
The Hacker News post titled "The Buenos Aires Constant" has generated several comments discussing the mathematical coincidence explored in John D. Cook's blog post. Many commenters engage with the idea of "coincidences" and the human tendency to find patterns.
One compelling thread discusses the nature of such mathematical curiosities. A commenter points out that if you allow enough flexibility in defining your constants and operations, you can essentially create any relationship you want. They suggest the "Buenos Aires constant" is less a fundamental mathematical truth and more a playful exploration of numerical relationships. This idea is echoed by other commenters who mention the infinite monkey theorem – the idea that given infinite time, a monkey randomly hitting keys on a typewriter will eventually produce any given text. Similarly, with enough manipulation of numbers and operations, seemingly significant relationships can be fabricated.
Another commenter explores the concept of "almost integers," numbers that are remarkably close to integers but not quite. They point to examples like the Ramanujan constant (exp(pi*sqrt(163))) and discuss the underlying mathematical reasons for their existence. This ties into the broader discussion of the surprising appearances of certain constants like pi and e in seemingly unrelated areas of mathematics.
Further discussion delates into the base-dependent nature of the observation. The "Buenos Aires constant" relies on the decimal representation of pi. The apparent "coincidence" would disappear if a different base were used. This reinforces the idea that the observation is more of a numerical curiosity tied to our specific representation of numbers, rather than a fundamental mathematical property.
Some commenters appreciate the lighthearted nature of the blog post and the subsequent discussion, acknowledging the fun in finding such numerical oddities. Others express skepticism, labeling it a mathematical triviality. One commenter humorously suggests that Buenos Aires should adopt this constant as its official motto, highlighting the playful tone of the conversation.
Overall, the comments on Hacker News reflect a mix of perspectives, from those fascinated by the numerical coincidence to those who view it as a relatively insignificant mathematical observation. The discussion delves into broader mathematical concepts like almost integers, the base-dependence of numerical representations, and the inherent human tendency to find patterns, even where they may not truly exist.