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This is what makes math math :p You can make up any concept you like, and if it doesn't contradict itself its perfectly valid to analyze it. Later someone has a

This is what makes math math :p You can make up any concept you like, and if it doesn't contradict itself its perfectly valid to analyze it. Later someone has a hairy problem and a mathematician points to how the made up thing he studied is isomorphic to the problem and solves it.

skissane 10y ago
> You can make up any concept you like, and if it doesn't contradict itself its perfectly valid to analyze it Why limit yourself to ideas that don't contradict themselves? http://www.iep.utm.edu/math-inc/ http://www.iep.utm.edu/math-inc/ https://plato.stanford.edu/entries/mathematics-inconsistent/ https://plato.stanford.edu/entries/mathematics-inconsistent/ https://ir.canterbury.ac.nz/bitstream/handle/10092/5626/12633603_Real%20Analysis%20in%20Paraconsistent%20Logic_a.pdf https://ir.canterbury.ac.nz/bitstream/handle/10092/5626/1263...
- dozzie 10y ago
  Because then you can prove any theorem (by applying modus ponens), which makes the theory uninteresting.
  - skissane 10y ago
    In paraconsistent logic, no you can't prove any theorem from a contradiction. Inconsistent mathematics is built upon the foundation of paraconsistent logic. https://plato.stanford.edu/entries/logic-paraconsistent/ https://plato.stanford.edu/entries/logic-paraconsistent/
  - andrewflnr 10y ago
    Or, maybe not. From the second link: Inconsistent mathematics is the study of the mathematical theories that result when classical mathematical axioms are asserted within the framework of a (non-classical) logic which can tolerate the presence of a contradiction without turning every sentence into a theorem.
FabHK 10y ago
That reminds me of the hairy ball theorem: You can't comb a hairy ball smoothly (without at least one cowlick (<- word I just learned...)). https://en.wikipedia.org/wiki/Hairy_ball_theorem https://en.wikipedia.org/wiki/Hairy_ball_theorem
lacker 10y ago
Yeah, and this particular case is even more specific than "make up any concept you like" - it's specifically inventing a new category of number that is the solution to formulas expressed in existing notation. I think I missed a fourth example of the same pattern: fractions. Originally fractions were not considered to be "numbers", they were just a way of comparing numbers with each other. https://nrich.maths.org/2515 https://nrich.maths.org/2515

skissane 10y ago

> You can make up any concept you like, and if it doesn't contradict itself its perfectly valid to analyze it Why limit yourself to ideas that don't contradict themselves? http://www.iep.utm.edu/math-inc/ http://www.iep.utm.edu/math-inc/ https://plato.stanford.edu/entries/mathematics-inconsistent/ https://plato.stanford.edu/entries/mathematics-inconsistent/ https://ir.canterbury.ac.nz/bitstream/handle/10092/5626/12633603_Real%20Analysis%20in%20Paraconsistent%20Logic_a.pdf https://ir.canterbury.ac.nz/bitstream/handle/10092/5626/1263...

dozzie 10y ago
Because then you can prove any theorem (by applying modus ponens), which makes the theory uninteresting.
- skissane 10y ago
  In paraconsistent logic, no you can't prove any theorem from a contradiction. Inconsistent mathematics is built upon the foundation of paraconsistent logic. https://plato.stanford.edu/entries/logic-paraconsistent/ https://plato.stanford.edu/entries/logic-paraconsistent/
- andrewflnr 10y ago
  Or, maybe not. From the second link: Inconsistent mathematics is the study of the mathematical theories that result when classical mathematical axioms are asserted within the framework of a (non-classical) logic which can tolerate the presence of a contradiction without turning every sentence into a theorem.

FabHK 10y ago

That reminds me of the hairy ball theorem: You can't comb a hairy ball smoothly (without at least one cowlick (<- word I just learned...)). https://en.wikipedia.org/wiki/Hairy_ball_theorem https://en.wikipedia.org/wiki/Hairy_ball_theorem

lacker 10y ago

Yeah, and this particular case is even more specific than "make up any concept you like" - it's specifically inventing a new category of number that is the solution to formulas expressed in existing notation. I think I missed a fourth example of the same pattern: fractions. Originally fractions were not considered to be "numbers", they were just a way of comparing numbers with each other. https://nrich.maths.org/2515 https://nrich.maths.org/2515