Subsection of "On Language"
Mathematics are the languages we use to describe nature.
They are either inconsistent or incomplete, just as other (natural) languages.
Zero and infinity are abstractions, cheats, placeholders until we find better, more truthful math.
Theory that builds heavily on something being zero (ignored) or being infinite (always there), are likely imprecise, wrong and in need for further work.
Kurt Gödel
Proving our best tools (mathematics, language) are inherently broken/limited:
- Incompleteness theorem: math is either inconsistent or incomplete
- Gödelisierung: mapping between a math "dialect" (natural number algebra) and natural languages --> effects/limits of math applies to other languages as well
Implications:
- We can't have a "world formula" describing all of the universe, inside the universe. Likely, the best description of the universe, is the universe itself
- All languages are valuable. Esp. for the parts where they don't overlap, their niches.
Examples: (certain UK local lang with various terms/grades for wetness/moisture? TODO check!), (South-)East-Asian languages having plenty of words for group association, in/out group separation etc., Latin and English having plenty of words for "killing"/"war" --> imperialistic use, German allowing for very detailed descriptions, lengthy compound nouns (--> Germany as a early philosophy, science, arts, medicine, engineering hub, over-engineering tendency as a downside), Japanese allowing for refinement, redefinition, reinterpretations of symbols, strive for perfection
OZWIP Of Zero and Infinity
We never observed "infinite something", nor "zero something".
For infinity, this is intuitive and trivial.
For zero, this may be a bit surprising just by how commonly used the number is.
OZWIP
--> Example: common argument for why 0 shouldn't be a natural number, we can enumerate the world around us in natural numbers
Examples:
- Gravity is assumed to have infinite reach for now
- Probability theory, statistics, law of large numbers: probability is assumed to converge if a random event is repeat a very large (infinite) number of times
- Any formula ignoring things (taking them as 0) is an abstraction ignoring minor, most likely irrelevant detail for easier application
Abstractions
Abstractions are formed from examples.
They leave out detail, thus becoming applicable not just to a singular occasion, but more examples.
Recurring pattern, recurring number problem
What further to take from here
- It's weird to tell kids "you are either good at math or you are good at languages", "it's fine/normal to be bad at math, at least you can speak a foreign language"