This post is homage to the amazing power of approximation of asymptotics. A finite number of terms of a divergent series can rapidly approach the value of a function while a convergent series needs an extremely large number of terms to get to the same accuracy and precision.
is the illustrative function (chosen as mentioned in Professor Bender lecture on Mathematical Physics). The plot shows the single term Stirling approximation/function (obviously begs the question what algorithm does Mathematica used) v partial sums of power series/function.
Hoffman et al provided an interesting article on the underlying mechanism of aging. This was a clearly written and strong argument that aging is a matter of physics not biology.
This post explores the age distribution of a population (in equilibrium) with a constant hazard (age independent of mortality).
In the following, a population of 1000 “individuals” all 1 years old. The time evolution individuals are removed from the population and replaced by 1 year olds who then age. 100 cycles are shown. The red line is equilibrium distribution.
This is not the human situation as the author remarks. We are fortunate to have repair mechanisms that protect us for as long as they do.
I have looked at Mathematica Meta site and dared (somewhat apprehensively)to answer and comment.
The questions (from my viewpoint) were more philosophical in nature.
This post is motivated by a comment that prompted introspection. The comment expressed concern that users were quick to code at the expense of quality and rigor, and that this was self-reinforcing (a mutual admiration society) resulting in an overall degradation in quality. Users were offering answers beyond their depth of understanding.
I think this is a legitimate and that I certainly may be one of the users contributing to this trend. I would apologize unreservedly for that. I think, therefore, encouraging users (“pure of heart”) to sermonize (to borrow the metaphor in question title and not pejoratively) is valuable. Such users could post answers where they thought deeper explanations were valuable or that an important misconception was being promulgated. The deep reasoning (I think) helps everyone.
I am not sure if it is a trend. I do think (and have directly experienced through my own mistakes and over-reach) that expert users are very quick to identify problems and edit, comment or answer to deal with conceptual issues.
I also believe that MSE is a diverse community. The diversity includes:
- special areas of interest
- and many more
I think that is a strength. The site is primarily aimed at using Mathematica/Wolfram Language and so it is natural that “code” has primacy. However, the intersection with Mathematics, Statistics, Physics, Engineering etc. is large. Therefore, insights, constructive criticism and correction would only enhance the value of the site. I think this already occurs but perhaps not enough. I think it is a matter of balance.
I believe, with absolutely no disrespect, the site is a playground. There are novice users who want guidance. There are expert or experienced users who enjoy challenges (be it efficiency, terseness, a completely different approach). There is the dense middle that I occupy that likes to play the game. Sometimes spectacularly fail but almost all the time learning new approaches.
MSE may have a lot of academic and commercial users. It is a site where people with a shared interest can get help with specific problems and contribute all sorts of ideas. I have enjoyed the creativity. The site is not producing answers for peer-review manuscripts or commercial intent. Of course, some of the ideas and approaches may developed for such purposes. However, a clarion call to keep well above a minimum standard is important. I think that finding the balance between academic purity/integrity and just playing with the black boxes without understanding the assumptions or limitatons (with the dangerous potential of misuse) is difficult. I thought the balance was ok. However, this may be because I am blinded by the shiny black boxes and the colorful outputs. I know that I have worried about this for a long time.
I have spent 20 years teaching people “a particular set of skills”. At the start they have a lot of enthusiasm, a lot of (frequently wrong pre-conceptions) and the teaching involves long periods of “play” before they enter the “real world”.
I am not an expert and I may have contributed to some quality degradation. If so, I am sorry. I like the community. I hope I can be a better citizen.
No man is an island,
Entire of itself,
Every man is a piece of the continent,
A part of the main.
If a clod be washed away by the sea,
Europe is the less.
As well as if a promontory were.
As well as if a manor of thy friend’s
Or of thine own were:
Any man’s death diminishes me,
Because I am involved in mankind,
And therefore never send to know for whom the bell tolls;
It tolls for thee.
The post title relates to a phrase (from Andrew Gelman) quoted in “The Seven Pillars of Statistical Wisdom” by Stephen Stigler. It refers to the frequently hidden, frequently unrecognized (or recognized) decisions in the path from hypothesis to conclusion.
I enjoyed and learned a lot from this book. It provides a historical framework under-pinning the authors construction of a framework to understand statistical science. This naturally includes misconceptions, personalities and unintended sources of progress or innovation.
The seven pillars are:
Professor Stigler also hints at an emerging eighth pillar.
I particularly enjoyed this excerpt (quote from Alfred Marshall 1885):
the most reckless and treacherous of all theorists is he who professes to let facts and figures speak for themselves, who keeps in the background the part he has played, perhaps unconsciously, in selecting and grouping them and in suggesting the argument post hoc ergo propter hoc.
I was kindly made aware of Professor Stigler’s invited address which covers the seven pillars.