So I have been pondering about language. By language L I just mean a series of symbols. The upper limit of this series of symbols is Aleph-zero. Yet somehow using these symbols the human is able to understand the meaning of phenomena of cardinality Aleph-one. Let us denote the meaning of phenomena as M. How is this happening (what is the possible mappings)? The cardinality of L is Aleph-zero, that of the phenomena Aleph-one and M(Aleph-one) must be Aleph-zero?
So this seems to have interesting possibilities:
- M is not a well defined operation and cannot be expressed in mathematics (rather should be taken for granted)
- There is no one to one mapping between L and M(Aleph-one). We humans do not understand meaning (solely) through language
- There is no one to one mapping between M(Aleph-one) and Aleph-one. Certain meanings are redundant.
- There can be truths about the phenomena of Aleph-one which cannot be captured in L.
Here's an example:
I have a friend Bob who speaks only in morse code. After a long time I explain him to him what rational numbers are. He understands he concept. A natural question comes does my friend understand this concept in it's entirety. Perhaps there are things he doesn't get. But I can talk more morse and he (perhaps) will get it. How do a countable set of symbols make Bob understand phenomena which is uncountable.
- Mathematics cannot ever formally describe meaning*
- Well maybe Bob used innate knowledge and was inspired by speech to bring it forth.
- While the phenomena of rational numbers can be of cardinality aleph-one the meaning behind the rational numbers reduces to phenomena of cardinality to aleph-zero.
- There are truths about the rational numbers that no amount of symbols will ever make sense on why it is so.
Are there any possibilities I have missed? What answer(s) are correct?
*Please do not ask me to formally describe meaning since I kind of subscribe to this line of attack.