Sunday, September 30, 2018

Rambling About Sorites

[Any resemblance between this graphic & the author of this post is purely coincidental]

Suppose a man starts off with100,000 hairs on his head, but on Wednesday he loses 50 hairs. Is he now “bald.” Of course not. But say you don’t see him for a decade and when you next run into him he has only 50 hairs left on his head. Is he now effectively “bald.” Yes. But when, in time, did he “become” bald. This is just one of many ways of stating the ancient “Sorites” paradox, also known as the “heap paradox” or trying to describe how many grains of sand constitute a “heap” of sand.

This all came to my mind a bit ago when Mike Lawler tweeted out, “It is almost impossible to imagine -> 4:40 per mile pace for the entire marathon,” while referencing a newly-set record for a marathon race. Indeed, I do find it impossible to imagine… yet, it was accomplished:

I’m always amazed at how, over time, so many track-and-field records keep falling. There are of course improvements made in nutrition, training, equipment, etc. but still limits to human performance must, in the end, rule — there is never going to be a 5-second 100-yard dash, nor a 1-hour marathon (at least not as humans are currently constituted). Yet finding that ‘boundary’ which can be asymptotically-approached but not crossed seems a difficult task.
Famously, Roger Bannister barely broke the 4-minute barrier for the mile-run in 1954, following a century of efforts by others. In the 60+ years since, the record has gradually dropped to almost 3:43. How much lower can it go (can it break the 3:40 level)? Read the history/progression for this track event here:

Of course the longer the race, the more likely there is room for improvement: easier to imagine shaving a second off a marathon or 10K run (or even a mile) than the 100-yard dash. How about the pole vault, the long jump, the hammer throw, the shot put?... how easy to keep setting records there?
[One philosophical approach to Sorites is to argue that a definite boundary exists, but that it is unknowable. Perhaps a similar take exists for athletic activities: there are human (physical/physiological) limits, but it's unknowable exactly what they are...?]

...In logic, the law of the excluded middle, is both a staple, but also controversial. Claiming that a statement can’t be both true and false, the law seems simple and innocent… except that in normal discourse meaning and language are rarely so binary. Is it true that John is tall, or smart, or fast, or…. Obviously, it depends on how you define “tall…etc.”, but moreover no definition will likely neatly fit precisely all cases (especially since you also enter into issues over measurement, precision, and context). As applied in math and logic the law is somewhat more clean, but still controversial, and Sorites, with its boundary-ambiguity, gives some indication of its ongoing murkiness.
Loosely, this all also reminds me a bit of mathematical “surreal numbers” and “Dedekind cuts” where it is the ‘boundary' or middle ground that again becomes all-important. Like many ancient paradoxes, the Sorites paradox has a lot of depth.
"Fuzzy logic" and other multi-valued logics (which include 3 or more truth-values) are one alternative to the classical logic of two truth-values.
...In the meantime, there are enough variables at work in running a marathon that record-breaking can probably go on for quite awhile!

No comments: