This article explores a particular kind of abstractness in mathematics, especially the construction of numbers and the cardinalities of infinite sets. It is all very interesting indeed.
However, the kind of abstractness I most enjoy in mathematics is found in algebraic structures such as groups and rings, or even simpler structures like magmas and monoids. These structures avoid relying on specific types of numbers or elements, and instead focus on the relationships and operations themselves. For me, this reveals an even deeper beauty, i.e., different domains of mathematics, or even problems in computer science, can be unified under the same algebraic framework.
Consider, for example, the fact that the set of real numbers forms a vector space over the set of rationals. Can it get more abstract than that? We know such a vector space must have a basis, but what would that basis even look like? The existence of such a basis (Hamel basis) is guaranteed by the axioms and proofs, yet it defies explicit description. That, to me, is the most intriguing kind of abstractness!
Despite being so abstract, the same algebraic structures find concrete applications in computing, for example, in the form of coding theory. Concepts such as polynomial rings and cosets of subspaces over finite fields play an important role in error-correcting codes, without which modern data transmission and storage would not exist in their current form.
When I was learning me a Haskell I had a great time when I realised that as long as my type was a monoid I could freely chain the operations together purely because of associativity
However, the kind of abstractness I most enjoy in mathematics is found in algebraic structures such as groups and rings, or even simpler structures like magmas and monoids. These structures avoid relying on specific types of numbers or elements, and instead focus on the relationships and operations themselves. For me, this reveals an even deeper beauty, i.e., different domains of mathematics, or even problems in computer science, can be unified under the same algebraic framework.
Consider, for example, the fact that the set of real numbers forms a vector space over the set of rationals. Can it get more abstract than that? We know such a vector space must have a basis, but what would that basis even look like? The existence of such a basis (Hamel basis) is guaranteed by the axioms and proofs, yet it defies explicit description. That, to me, is the most intriguing kind of abstractness!
Despite being so abstract, the same algebraic structures find concrete applications in computing, for example, in the form of coding theory. Concepts such as polynomial rings and cosets of subspaces over finite fields play an important role in error-correcting codes, without which modern data transmission and storage would not exist in their current form.