Faith And Infinity

Infinity is an elusive concept for many of us. Is it a number? Can you do things like divide it by 2? Is there anything bigger than infinity? These questions provoke some very interesting proofs, for instance, the infinity that represents the size of the set of real numbers (the largest set that includes numbers like 0, 1, 80.2, e, π , 300243434.344325245425) is “bigger” than the infinity that represents the size of the familiar counting numbers (i.e. {1, 2, 3, …}). There are a number of popular books on the topic that you may want to read if you find that sort of thing interesting, like this one.

Infinity is an idea that pulls people in, often at the elementary school level. In my experience, few teachers are equipped to provide students with good answers (what did your teacher tell you when you asked “what happens when you divide a number by 0?”) and students are left to wander in a haze of mathematical mysticism. If the idea of infinity makes you feel uncomfortable, you are in good company. E.T. Jaynes laid it out nicely in “Probability Theory: The Logic Of Science”:

…we have to recognize that there are no really trustworthy standards of rigor in a mathematics that has embraced the theory of infinite sets. Morris Kline (1980, p. 351) came close to the Jeffreys simile: “Should one design a bridge using theory involving infinite sets or the axiom
of choice? Might not the bridge collapse?” The only real rigor we have today is in the operations of elementary arithmetic on finite sets of finite integers, and our own bridge will be safest from collapse if we keep this in mind.

(emphasis mine)

There is a small sect of mathematicians who in fact take this to a somewhat extreme view, the ultrafinitists who deny the existence of very large numbers. The comic is really just a depiction of an account given in “Mathematics Under The Microscope”, of an Alexander Yessenin-Volpin lecture. It is a fun idea to deny the existence of those very big numbers. I doubt most people think big number denial is very mathematically nutritious, but I find it quite sweet and tasty.

Surprised that we don’t have an exact number for the number of legal arrangements of chess pieces on a board? The enumeration is a challenging problem (the number presented is far far bigger than the number of stars in the observable universe) and you can find some insightful discussion here.

An important point to consider is that a chess diagram doesn’t encode all the information of the game state (the chess position), since you can’t see things like whose turn it is or if one can castle.

It turns out that the most recent upper bound is obtained by using a (almost frighteningly) succinct Haskell program, seen here.

Oh and those chess diagrams were picked up from here.

The Universal Property Of Yoga

Go ahead, think of a word, any word, then add “yoga” in front of it. There is a good chance that the compound word you just thought of is already generating incoming cashflows for someone out there.

This comic was inspired by a link from a friend of mine, who, incidentally is an amazing artist. See the copyheart at the bottom of the page? I could explain it to you, but she does it much better.

Ever heard the story about cakemix makers, and how their product sales boomed after they took out the powdered egg, so that homemakers could feel good about themselves when baking? Turns out that there isn’t much evidence behind the idea.

I wanted to reimburse the authors’ who made the nice samarkan font in the last panel, but it turns out the emails in the readme don’t exist anymore. I would love to send them their modest requested fee for use of their awesome work, if anyone can find them, let me know.

A Fallacy Of Taste

How many of us have had this ‘tongue map’ presented to us in grade school?

Allow to me elaborate, in case you are unfamiliar. As a child, it was not uncommon to observe demonstrations of the taste senses mapped to specific areas of the tongue. The same experiments seen on “kid science” shows would often be replicated in the classroom setting.

Children would gingerly follow instructions, dabbing Q-tips soaked in aqueous solutions of sugar, salt, lemon juice and unsweetened cocoa on their tongues. Stickers would be placed over a tongue-diagram, which indicated how strongly they sensed the corresponding taste (sweet, salty, sour and bitter) in different areas.

Full marks would be awarded to the students who could replicate the canonical result shown in the comic. Some children would not be able to observe any patterned spatio-lingual mapping of taste, a teacher would likely affirm that their failure to reproduce the canonical map lies in them. The ‘success’ of the former set of children was not due to their refined sense of taste, but rather, some earlier exposure to the experiment and the ability to quietly ignore their own perceptions when it is expected by an authority figure.

The truth is that this map is a complete fallacy. The original paper by Edwin G. Boring in 1901, reported subtle differences in taste sensitivities (not the discrete sensation, as it is often implied in current demonstrations). Virginia Collings took a critical look at this and reported that taste sensitivity is pretty uniform across the tongue…in 1974. A testament to the canard’s tenacity can be seen by looking at a relatively recent Nature paper, where modern biology is used to explicitly map taste receptors and the authors present a diagram whose caption reads: “…contrary to popular belief, there is no tongue map”.

Perhaps I am exaggerating, or even mis-remembering, how the tongue map was presented to me as a child. We’ve long since cleaned up our collective understanding of this basic concept…let’s do a little googling:

Oh, alright..just one website, not too bad.
A University of Washington “neuroscience for kids” website? <-- Their protocol involves toothpicks instead of Q-tips. Bravo. Et tu, PBS? <--- Apparently an NSF funded website 🙁 I am too frightened to look further.

This is one “experiment for kids” that needs to be repurposed as a demonstration of our embarrassingly fallible perception in the context of perceived authority, and the profound inertia of ignorance.

Kulfiophile

It is odd, but somehow unsurprising, that our visual perception is so intertwined with our perception of taste.

There are recent studies that show how the colour of a bowl can affect the taste of popcorn, how the ambient light can affect on the taste of wine, even how ‘orangeness’ beats brand, expensiveness and sugar content (!) as a factor in which orange juice consumers prefer. Beware, however, there is plenty of nonsense out there when it comes to perception. Audiophiles and vinophiles are notorious for claiming to perceive nuances that don’t exist. I am not a violinist, but if there is a difference between a Stradivarius and a comparable quality violin, no one has been able to measure it.

The effect of colour on taste probably has a lot to do with why we started eating and drinking out of transparent glass in the first place, and why food manufacturing pays so much attention to appearance and packaging.

We’ve been colouring food for at least as long as we have had saffron, I’ll guess that natural dyes were the very first specific organic molecules that humans used to indirectly alter their perception. Perhaps it is no surprise that the modern drug industry is very much a successor to the dye industry.