“Is Soares perhaps suffering from the common misconception that words can only have a single definition?”
Personally I like words with single definitions. For example, I’d rather the mathematical objects Set, Group, Category, ECT had their own made up names than borrow words used for other things.
I haaate mathematical terms that are used isolatedly about one thing. It leads to expanding your concept-namespace by 300 for reading a single page of mathematics.
If you’re having to learn 300 new mathematical concepts, it seems to me that the cost of learning 300 new words is lost in the noise. (And even if it’s a single page, figuring out what that single page says is going to take you days, weeks, months, or years.)
Further, you’re having to learn 300 new word meanings in any case. Are you saying this is much easier when the words already have existing but completely different meanings?
Sometimes the mathematical meaning is closely related to the existing meaning. (“Set” and “category” are arguably of this kind.) In that case, using the existing words may well be a good decision. But if you tell me that somehow the concepts “ring”, “field”, “module” are easier to learn because those are existing English words, I’m skeptical.
I don’t buy that it’s much harder for most learners either, though. It’s not as if, when someone writes “let F be a field”, you’re left wondering whether perhaps they mean that it’s a patch of land suitable for agriculture.
(The ambiguities that bother me are the “internal” ones. If someone says “let R be a ring”, you may not know whether or not you’re supposed to be assuming that it has a multiplicative identity element.)
I am a bit hazy what I am hesitant about. My threat model generates that things could be worse in the form that ring multiplication might have a different word from field multiplication which could have a different word than archimedian field multiplication. And there is a way trying to understand “group has two operations that distribute over each other” which can try to get rid of having multiplication and addition as separate entities.
If one does this too much the result it so abstract it is hard to get a handle on. But on the other direction naming every single quirk forms a zoo where it is hard to see patterns and systems.
It seem a lot of times if I bother too look up what is the “definition” of a thing it ends up being a list of 5 or so axiom like things. And there is about 10-15 of axiom types or axioms that appear them. BUt the trouble is that the different combinations that appear make a combinatorial explosion. And then some people think you should be able to connect names to those definitions as a “shorthand”.
(Pedantic notes: A group doesn’t have two operations, you’re thinking of rings or fields or modules or algebras or other such things; one of them distributes over the other, but not the other way around.)
I agree that it would be annoying if we had different names for all the different things we currently call multiplication.
It sounds as if you might prefer a system where instead of saying “R is a ring” we have some concise way of saying what operations R has and what axioms it satisfies. Something like “R is an (01+-*,ACD) algebra”, meaning it has addition & subtraction (= addition and additive inverses), multiplication but not necessarily division (so not necessarily multiplicative inverses), multplication is associative and commutative, and there’s a distributive law. And you’d say that instead of “commutative ring with unity” or “commutative ring” or “ring”, so there’d be a bit more verbosity and a bit more scope for not reading carefully exactly what’s being assumed, but less ambiguity, and the terminology wouldn’t so strongly favour a smallish set of particular types of structure that have their own names. Instead of “monoid” we’d say “0+ algebra”, instead of “semigroup” we’d say “+ algebra”, instead of “abelian group” we’d say “0+- algebra”, instead of “group” we’d say “1*/ algebra”, etc.
(My notation there assumes that it’s understood that something called addition is always commutative and associative.)
There’s definitely something to be said for that. On the other hand, those structures with special names have those names because those types of structures keep coming up. “Group” is only five letters and one syllable, and a lot of things are groups.
I was thinking of fields. I am doing a pretty bad math sin (atleast for the context) on not being precise and referring to the right target with “distributes over the other”. It is too foggy in my mind, I faintly recall something about being able to project + structures into * via something, but is wasn’t as straighforward and trivial than I thought.
While there migth have been at various point paid various amounts of attention on which things should be named and which should not be I doubt that it is completely reconstructible from anonymising everything and building it back again. A thing being named is somewhat sticky and will probably push back to outstay its welcome partly because of terminological inertia. A significant reason why we call things we call them is that other know what we are referring to and that it is customary.
“Is Soares perhaps suffering from the common misconception that words can only have a single definition?”
Personally I like words with single definitions. For example, I’d rather the mathematical objects Set, Group, Category, ECT had their own made up names than borrow words used for other things.
I haaate mathematical terms that are used isolatedly about one thing. It leads to expanding your concept-namespace by 300 for reading a single page of mathematics.
If you’re having to learn 300 new mathematical concepts, it seems to me that the cost of learning 300 new words is lost in the noise. (And even if it’s a single page, figuring out what that single page says is going to take you days, weeks, months, or years.)
Further, you’re having to learn 300 new word meanings in any case. Are you saying this is much easier when the words already have existing but completely different meanings?
Sometimes the mathematical meaning is closely related to the existing meaning. (“Set” and “category” are arguably of this kind.) In that case, using the existing words may well be a good decision. But if you tell me that somehow the concepts “ring”, “field”, “module” are easier to learn because those are existing English words, I’m skeptical.
I don’t buy that it’s much harder for most learners either, though. It’s not as if, when someone writes “let F be a field”, you’re left wondering whether perhaps they mean that it’s a patch of land suitable for agriculture.
(The ambiguities that bother me are the “internal” ones. If someone says “let R be a ring”, you may not know whether or not you’re supposed to be assuming that it has a multiplicative identity element.)
I am a bit hazy what I am hesitant about. My threat model generates that things could be worse in the form that ring multiplication might have a different word from field multiplication which could have a different word than archimedian field multiplication. And there is a way trying to understand “group has two operations that distribute over each other” which can try to get rid of having multiplication and addition as separate entities.
If one does this too much the result it so abstract it is hard to get a handle on. But on the other direction naming every single quirk forms a zoo where it is hard to see patterns and systems.
It seem a lot of times if I bother too look up what is the “definition” of a thing it ends up being a list of 5 or so axiom like things. And there is about 10-15 of axiom types or axioms that appear them. BUt the trouble is that the different combinations that appear make a combinatorial explosion. And then some people think you should be able to connect names to those definitions as a “shorthand”.
(Pedantic notes: A group doesn’t have two operations, you’re thinking of rings or fields or modules or algebras or other such things; one of them distributes over the other, but not the other way around.)
I agree that it would be annoying if we had different names for all the different things we currently call multiplication.
It sounds as if you might prefer a system where instead of saying “R is a ring” we have some concise way of saying what operations R has and what axioms it satisfies. Something like “R is an (01+-*,ACD) algebra”, meaning it has addition & subtraction (= addition and additive inverses), multiplication but not necessarily division (so not necessarily multiplicative inverses), multplication is associative and commutative, and there’s a distributive law. And you’d say that instead of “commutative ring with unity” or “commutative ring” or “ring”, so there’d be a bit more verbosity and a bit more scope for not reading carefully exactly what’s being assumed, but less ambiguity, and the terminology wouldn’t so strongly favour a smallish set of particular types of structure that have their own names. Instead of “monoid” we’d say “0+ algebra”, instead of “semigroup” we’d say “+ algebra”, instead of “abelian group” we’d say “0+- algebra”, instead of “group” we’d say “1*/ algebra”, etc.
(My notation there assumes that it’s understood that something called addition is always commutative and associative.)
There’s definitely something to be said for that. On the other hand, those structures with special names have those names because those types of structures keep coming up. “Group” is only five letters and one syllable, and a lot of things are groups.
I was thinking of fields. I am doing a pretty bad math sin (atleast for the context) on not being precise and referring to the right target with “distributes over the other”. It is too foggy in my mind, I faintly recall something about being able to project + structures into * via something, but is wasn’t as straighforward and trivial than I thought.
While there migth have been at various point paid various amounts of attention on which things should be named and which should not be I doubt that it is completely reconstructible from anonymising everything and building it back again. A thing being named is somewhat sticky and will probably push back to outstay its welcome partly because of terminological inertia. A significant reason why we call things we call them is that other know what we are referring to and that it is customary.