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That's not the point. The point was it was true for R[1/g] but not for D(g), so you cannot substitute, so you cannot claim that f and g are equal. As someone t

That's not the point. The point was it was true for R[1/g] but not for D(g), so you cannot substitute, so you cannot claim that f and g are equal. As someone totally outside of the maths field, it seems to me that the definition of equality needs a context. You can't say that two things are equal without specifying the larger context in which you are claiming the equality.

dkarl 4y ago
> it seems to me that the definition of equality needs a context That's precisely it. Mathematicians are often casual with terminology even when being completely precise inside their context. I did not get beyond introductory graduate-level mathematics, but I distinctly remember a professor announcing this in the third or fourth lecture of an algebra class: "From today until the end of the semester, when I say 'ring' I will mean a commutative ring, except on occasion when it will be obvious that the ring is not commutative." Some professors/authors were more fastidious about using unambiguous terminology, but many preferred to slap the easiest word possible on whatever relationship felt the most important in the context. So one professor would say "what we're really interested in is the quotients modulo 10," and another would say "12 is equal to 2" and expect you to understand the same thing from it.
rurban 4y ago
You can handwave equality, but you need to specify which equality. same with "isomorphism". same form or equal form? how is the equality specified there? canonical has a precise definition, but the paper doesn't come close. eg array-equal as in lisp, vs eqc vs equal vs eq. list of atoms with the same order and same values. or list of equal atoms. (ie same pointers). or if the values are same, or all the other atom properties need to be the same. without a precise definition of equality everything is just handwaving. in perl "" equals 0 equals !!0. so are they equal, and if so which equality? same with JavaScript, python and PHP with completely handwave equality definitions. in the end it all comes to Greenspun, all the lisp equalities. Grothendiek is lucky to ignore pointers. for him all natural numbers can be equal. 1 === 1. for us eg with python higher numbers with the same value are sometimes not eq. This would have driven Grothendiek even more nuts.

dkarl 4y ago

> it seems to me that the definition of equality needs a context That's precisely it. Mathematicians are often casual with terminology even when being completely precise inside their context. I did not get beyond introductory graduate-level mathematics, but I distinctly remember a professor announcing this in the third or fourth lecture of an algebra class: "From today until the end of the semester, when I say 'ring' I will mean a commutative ring, except on occasion when it will be obvious that the ring is not commutative." Some professors/authors were more fastidious about using unambiguous terminology, but many preferred to slap the easiest word possible on whatever relationship felt the most important in the context. So one professor would say "what we're really interested in is the quotients modulo 10," and another would say "12 is equal to 2" and expect you to understand the same thing from it.

rurban 4y ago

You can handwave equality, but you need to specify which equality. same with "isomorphism". same form or equal form? how is the equality specified there? canonical has a precise definition, but the paper doesn't come close. eg array-equal as in lisp, vs eqc vs equal vs eq. list of atoms with the same order and same values. or list of equal atoms. (ie same pointers). or if the values are same, or all the other atom properties need to be the same. without a precise definition of equality everything is just handwaving. in perl "" equals 0 equals !!0. so are they equal, and if so which equality? same with JavaScript, python and PHP with completely handwave equality definitions. in the end it all comes to Greenspun, all the lisp equalities. Grothendiek is lucky to ignore pointers. for him all natural numbers can be equal. 1 === 1. for us eg with python higher numbers with the same value are sometimes not eq. This would have driven Grothendiek even more nuts.