Lambda the Ultimate

Yes, "pronounce".

If so I suggest watching online lectures for a bit, and see how professors talk while writing on the board. There are great courses available as video from schools like MIT.

I agree. I don't pronounce |- "turnstile". I don't pronounce it at all. When I see it, I think "OK, the thing on the left can prove the thing on the right".

Still, it is useful to know that it's called turnstile, for when other people call it that. So the original question still stands!

In that case, I just copy latex's math mode terminology :)

So, yeah, you can probably guess how I feel about UML.

I'm somewhat interested in the notation you settled on, could you give the details?

It's nothing terribly amazing. I "borrowed" it mostly from programming languages I had seen at the time, so it looks a lot like Lisp. I use infix notation for binary operations only, parenthesize aggressively assuming almost nothing about operator precedence, and switch to prefix notation inside parens for any operation that isn't binary (well, the parens may get omitted for unary operations on single terms).

As to the symbols themselves ... the usual thing with set theory, where capital A means a set and lowercase a means a member of a set and script A means a subset of the set, gets transformed into symbols like 'aset', 'amem' and 'asubs' respectively. I use square brackets [] to refer to parallel elements when dealing with parallel collections. For example aset[x] and bset[x] would refer to associated members of parallel sets aset and bset, usually for all x meeting some criteria.

I know the greek alphabet, but if there's any latin letter that looks even a little bit like something, I usually don't recognize it unless concentrating hard and being very careful, so Kappas, Chi's, etc, get converted to the written-out letter names. I recognize lower-case lambda for example, as text, but uppercase Lambda always gets read as "L", so I have to write it out.

When letters have grown to have a particular meaning in my mind, the way lower-case lambda means function definition for example, I absolutely will not use those letters for an unrelated operation. If a letter like lower-case lambda is being "misused" in some context, I have to rewrite it as something else, otherwise the "text" will mean the wrong thing to me.

Script versions of greek or other non-latin characters are the worst of the worst. They tend to be unrecognizable, unordered, unmemorable squiggles to me, and I can't reliably associate anything with them because they all run together in my mind. I draw them very carefully in a notebook at about triple size, write some arbitrarily-chosen label like "glork" or "zomble" under them, and then painstakingly use that as my standard translation of that particular squiggle in that particular domain. I have to refer back to the "squiggle notebook" a lot. I use different labels (different notebooks) in different domains, because sometimes the same squiggles get used differently in different domains (and of course no instructor would ever deign to *mention* explicitly that they are being used differently) and in that case using the same name would mislead me into a blind assumption that something is the same thing or at least has the same properties.

If script versions of greek letters are being used under the assumption that they will be associated with other presentations of the same greek letters, then I have to write that down in the "squiggle notebook" and use tags for them that extend the names of the greek letter. But this is hard for me to do, harder than any other aspect of the notation conversion. It's the experience most people would have when trying to recognize particular rocks presented in unfamiliar contexts, and distinguish them from other members of a large set of nondescript rocks, and remember which other members of that set of nondescript rocks they are associated with.

The math symbols that I actually use are those I whose meanings are strongly associated with a single operation in my mind. Usually that means either that I learned them in grade school and have practiced enough with them that they are "text" to me, or that I have seen them consistently and frequently used for exactly one operation and rarely for anything else.

They are + and - for addition and subtraction, ¬ for logical negation, ± for approximation bounds, ∈ for member-of, ∋ for such-that, = for equality, ∀ meaning for-all, ∃ for there-exists, × for multiplication and ÷ or / for division, ∩ for intersection, ∪ for union, for comparisons, | for choice-of, ⋁ for logical-or, ⋀ for logical-and, and maybe a few others I haven't thought of right now.

But I generally extend them with disambiguators if they refer to operations on any domain other than that which they are most closely identified with, and never use them at all for any operations that don't have exactly the same logical properties as the operations they're most closely identified with.

+ for example, is strictly addition. If we're talking about some non-numeric addition, it gets extended as, for example, vector+ or tensor+ or something. That's fine as long as + is referring to something that has the properties (associative, commutative, identity element, inverse operation in subtraction, etc) of addition on numbers.

If + meaning addition-on-numbers isn't even a relevant idea in some domain and there is exactly one operation in that domain that behaves like it, then I can stand to use the bare + sign for the other operation that behaves like it. + as a symbol for something that doesn't behave like addition, however, is _Just_Plain_Wrong_ on a level I can't even communicate, and the + sign isn't even in the symbols I use for operations like that.

× for multiplication and ÷ or / for division are, again, strictly for these operations on numbers, or for operations that behave exactly like those operations when multiplication and division on numbers aren't relevant to the domain. But these signs are among the most most frequently elided and heavily overloaded in all of math, so I'm "weird" and have to actually write the symbol between two things that are getting multiplied, or, with parens, in front of more-than-two things getting multiplied. I write extended symbols for so-called multiplications on different types or that mix different types (say, of a number by a vector or matrix) and write unrelated symbols for so-called multiplications that don't have exactly the same properties (associative, commutative, distributive, identity element, reversible by a division operation, etc) as multiplication on numbers.

I use "crossprod" never × for cartesian products; cross-products need a sub-operation for joining the elements, and nothing I would ever write with the × sign does. "(aset (crossprod ×) bset)" refers to a cross-product whose members are made by multiplying the members of the original aset by the members of the original bset, while "(aset (crossprod conc) bset)" refers to a cross-product whose members are made by concatenating the members of aset with the members of bset.

etc. Aside from diagnosing and analyzing my particular set of cognitive quirks, is this helpful?

Ray

It is very helpful because a hobby of mine is designing better systems.
For me I have no trouble recognizing arbitrary pictures, but I HATE it when they make the name of something non-arbitrary, so that alpha-renaming fails. My other big frustration is when they they make it unclear which function they are applying to which arguments. But your convention for uninary, binary and higher arity functions is very sensible.

I was reading the first half of the fist book of Knuth's "The Art of computer programming, and when he was talking about simplifying expressions containing operations on arrays of elements, I had to draw out the array and color in the sections according to how many times that area was represented in the results, before the simplifications made sense to me.

A while ago on the rgrd newsgroup you said you had some text files full of ideas for creatures and items, if it isn't too bothersome could you send a file or two to andershorn@mailinator.com ? I really enjoyed the few ideas you shared.