While it downloads, speaking about Lambek, in Lambek-Scott they prove the equivalence of CCC's and simply typed lambda calculus. I suffered some functional programming exposure via Lisp, and came later to see that in the core of lisp, the lambdas, were formalized by (you guess it), lambda calculus. This idea migrates also to Haskell, so if one is pressed to give a formal Haskell semantics, that is a good bootstrap. But that is equivalent to CCCs (cartesian closed categories) as per Lambek-Scott, where the essential feature of CCCs is having functions of functions, higher order functions. And that is not just self-pleasure, look at Conal Elliot's "compiling to categories". Applications I think I've heard of: probabilistic programming and automatic differentiation.
Lambek-Scott: Introduction to Higher-Order Categorical Logic
> even though French is only a semigroup and not a ring ... and addition has not been defined, it is possible to multiply two matrices if ... each row of the first meets each column of the second in at most one non-empty place
What's interesting with French verb conjugation is that speaking them is far easier to learn than writing them. Many French conjugations sound essentially identical to each other when spoken. If all you focus on is speaking, you don't need to worry about the spelling aspect, which is where a lot of the confusion arises from. When speaking, the meaning is clear from the use of the verb rather than finding the word on a list of conjugations. Once you understand how the words are used, then you can focus on the spelling (and always think of the conjugation with the pronoun as a joined unit).
Unfortunately, the same feature makes listening to French difficult. The phrases "il parle" and "ils parlaient" (he speaks/they speak) sound almost identical. The latter ends in five silent letters.
It's usually clear from context, and easily understood by a native speaker. But a non-fluent speaker may be having trouble following the context, and lacking the additional clue of conjugation isn't helping.
I do a decent job at following written French, but spoken French remains a challenge (unless specifically over-emphasized for small children and stupid foreigners). I watch French television with subtitles on.
« parle » definitely does not sound like « parlaient », only the 3 last letters are silent. The former sounds sorta like an English speaker would pronounce « parhl », the latter like « pahrleh ». It also doesn’t translate to « they speak » but « they were speaking ».
You might have gotten confused with « parle » and « parlent » which do sound the same (3 silent letters again) and translate to what you said!
Probably generalizable to the Portuguese conjugation system which is the most complicated of big Latin derived languages in the number for forms still in use. (Except for maybe Romanian)
