patternModerate
What precisely differentiates Computer Science from Mathematics in theoretical context?
Viewed 0 times
sciencewhattheoreticalcomputercontextpreciselymathematicsfromdifferentiates
Problem
I am a university level student of Computer Science having a great passion to study Mathematics. I have a firm belief that Computer Science or Theoretical Computer Science is a direct branch of Mathematics and Logic and also of the opinion that a Computer Science degree has always to be Math oriented as a matter of fact. Please correct me if I am wrong.
I frankly feel that there isn't a lot of difference in the 2 subjects to be frank as every "computation" involves "calculation", although every "calculation" might not be a "computation". Again please provide substantial information and evidence and do update me if I am mistaken here. Thank you
I frankly feel that there isn't a lot of difference in the 2 subjects to be frank as every "computation" involves "calculation", although every "calculation" might not be a "computation". Again please provide substantial information and evidence and do update me if I am mistaken here. Thank you
Solution
Theoretical computer science is what theoretical computer scientists do; and mathematics is what mathematicians do. Other than that, there is no accepted definition of either. One might argue that theoretical computer science is a particular branch (or branches) of mathematics, influenced (at least originally) by the problem of efficient computation.
Many areas of mathematics are clearly not theoretical computer science, say functional analysis, category theory, algebraic geometry, algebraic number theory, and many others. However, sometimes these areas are applied to theoretical computer science. Parts of them might then form part of theoretical computer science, if there is a community of theoretical computer scientists who decided to work on them.
On the other hand, computability theory is not necessarily part of theoretical computer science, depending on which community of theoretical computer science one takes as reference. Theoretical computer science (or at least its part known as "Theory A") is traditionally about what can be efficiently computed rather than about what can be computed at all.
Many mathematical theorems have no computational content, and in some cases this can be made precise. One example is Kawamura's result that ODEs are hard to solve in general. It is not true that every mathematical proof has constructive content – indeed there are mathematical techniques which are inherently non-constructive, for example the use of compactness or the axiom of choice in any of its other guises.
If you are interested in the computational content of mathematical statements and proofs, you might want to look into reverse mathematics and bounded arithmetic (sometimes known as feasible mathematics or feasible arithmetic).
Many areas of mathematics are clearly not theoretical computer science, say functional analysis, category theory, algebraic geometry, algebraic number theory, and many others. However, sometimes these areas are applied to theoretical computer science. Parts of them might then form part of theoretical computer science, if there is a community of theoretical computer scientists who decided to work on them.
On the other hand, computability theory is not necessarily part of theoretical computer science, depending on which community of theoretical computer science one takes as reference. Theoretical computer science (or at least its part known as "Theory A") is traditionally about what can be efficiently computed rather than about what can be computed at all.
Many mathematical theorems have no computational content, and in some cases this can be made precise. One example is Kawamura's result that ODEs are hard to solve in general. It is not true that every mathematical proof has constructive content – indeed there are mathematical techniques which are inherently non-constructive, for example the use of compactness or the axiom of choice in any of its other guises.
If you are interested in the computational content of mathematical statements and proofs, you might want to look into reverse mathematics and bounded arithmetic (sometimes known as feasible mathematics or feasible arithmetic).
Context
StackExchange Computer Science Q#64371, answer score: 18
Revisions (0)
No revisions yet.