The difference between a function and a lambda

I think the discussion regarding objects vs. functions is a red herring. If the question is, "Is a lambda a function or an object?" the answer should be yes.

That's the point of first-class functions: they aren't treated differently than any other type. Java already manages to mostly ignore the differences between Object and primitive types (and Scala does even better), so whether a lambda is a subclass of Object or is a new primitive type or something else isn't really important for the language. What is important is that you can put your lambdas in collections, pass them around to be called, call them, and do anything else you might want to do with either an object or a method.

Scala accomplishes this by using a wrapper object that has a method called apply which can be invoked with just (), making it look just like a method call. This works splendidly; much of the time you don't need to even care whether you have a method or are calling the apply of a function object.

Yes, it's simply a recurring phrase in the title of several papers, starting from a couple in the 70s, in which Sussman and Steele demonstrate the use of lambda calculus for programming, by means of a minimalist Lisp dialect named "Scheme" they devised for the purpose. You can find the papers themselves here; they're interesting and surprisingly relevant.

I'm not sure if this is ever explicitly stated, but it's clear (from context, having read the papers, and knowing the general background and research interests of the authors) that the phrase is simply a catchy slogan for their contention that lambda abstractions, as a computational primitive, are not only universal in the formal sense (of being able to encode any program in some fashion, however awkward), but universal in a practical sense that any and every construct present in other languages, even those that are baked-in from the ground up, can be reimplemented in a lambda-based language in a way that is both effective and natural to use.

The repeated phrase leads to the obvious generalized form "for all X, lambda is the ultimate X", which is the sense I've generally taken "Lambda the Ultimate" to mean as the blog name, noting that LtU is concerned with programming language design and theory. Ironically, LtU would probably also be one of the best places to find someone who could tell you about something for which lambda is not the ultimate implementation. :]

Note also that Sussman is one of the authors of SICP, a very influential textbook that also uses the Scheme language and spends a fair amount of time introducing lambda abstractions as a concept.