Does Mathematica use first-order or second-order unification?

Why did Russell and Whitehead write the incomprehnsible Principia Mathematica, & why use Isaac Newton's title?

• Isaac Newton's great masterpiece's title was Principia Mathematica. Why did Russell and Whitehead write their incomprehensible Principia Mathematica, & why use Isaac Newton's title - Just to pretend they were continuing his work? I'm sure they were not. According to the Modern Library, it was the 23rd best nonfiction book of the twentieth century, but Will Durant says it has been useless. I dislike their pretentiousness in misleadingly stealing Newton's title

• Answer:

  1) Newton's work's full title is Philosophiae Naturalis Principia Mathematica, which means literally "the mathematical principles of natural philosophy. Russell and Whitehead's work is just Principia Mathematica. The titles are both apt. Newton's work tries to describe natural phenomena (especially physical "laws") using the language of mathematics. Russell and Whitehead's work is an attempt to discuss the foundations of mathematics starting from type theory. Base on this, I think it is more proper to call Russell and Whitehead's work as Principia Mathematica (compared to the colloquial shortening of Newton's title), as the work actually discusses nothing else but the principles of mathematics. 2) Will Durant is entitled to his own opinion, but I think you are being unfair saying that Russell and Whitehead is "misleadingly stealing" Newton's title. Pretentious they maybe, but certainly not misleading. As I outlined above, the title certainly reflects accurately the scope of the work. Furthermore, paying homage to classics in the title of a book is a commonly accepted practice both in fiction and in nonfiction. Would you say Faulkner's "The Sound and the Fury" is a misleading theft of Shakespeare? 3) Just because you cannot understand the book(s) doesn't make it incomprehensible to all. But I concede that it is a difficult book to read. So why did they write the book? Remember that the book pre-dates Godel's incompleteness theorems by almost two decades. A large open question at the time is whether propositional logic is internally consistent and whether the whole of mathematics can be formulated in a completely consistent frame work. And the generic expectation is "yes". Russell had formerly discovered his name-sake paradox in naive set theory (Let A be the set of sets that do not contain themselves as an element, is A an element of A?), which signals that not every simple minded way of formalizing mathematics into propositional logic can succeed. Principia Mathematica is an attempt to show that with a properly defined axiomatic system, the corpus of mathematics can be described all as proper logical consequences of the starting axioms. Note that even in view of Godel's theorem, the Principia has not lost its usefulness. Godel's incompleteness theorem basically says that sufficiently powerful logic systems cannot be both consistent and complete. But to the best of my knowledge, no one (especially not Russell and Whitehead) claimed that the corpus of mathematics is complete! It can be that all of mathematics (at least at the beginning of the 20th century; so we can exclude competing logic systems like Zermelo-Frenkel etc.) is consistent, but not complete, in that there will be unanswerable questions that can be posed within the logical framework. 4) At the very least, one must admire this three-volume treatise as a tour de force. To just attempt to formalize all of mathematics is a job best compared against Euclids Elements.