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5 days ago
2
In my post about the myth that Logicians are crazy I mentioned in passing that Whitehead and Russell spend 300 pages proving 1+1=2 (but were both sane). Two people privately emailed me:
Are you sure Russell and Whitehead weren't a few axioms short of a complete set? How could they take 300 pages to prove 1+1=2. Isn't it... to obvious to be worth proving?I responded by saying that they had to define 1, +, =, and 2 rigorously. One of them responded Are you a few limit points short of Banach space? That aside, there are some questions the 1+1=2 proof brings up:
- How did they spend 300 pages proving 1+1=2?
- Is it easier in ZFC?
- How important is or was Principia Mathematica? Wikipedia says PM is widely considered by specialists in the subject to be one of the most important and seminal works in mathematical logic and philosophy since Aristotle's Organon. The Modern Library places it 23rd in a list of the top 100 English-Language nonfiction books of the twentieth century. Here is the list they are referring to. The other books look... readable.
- I had thought that nobody reads PM anymore; however, its entry on amazon says it has a rank of roughly 294,000. This is far better than a book that truly nobody reads. For example this book has an Amazon rank roughly 5,300,000.
- While more people are buying it than I thought, are people actually reading it? Did they ever? My guess is no and no, but I really don't know.
- Can a book be influential if few people read it? Yes if they are the right people. Godel read it and I think it inspired him. (Its mentioned in the title of his Incompleteness paper.)
- PM was an early attempt to formalize all of math from the ground up. This may be one of those tasks that you are almost destined to do in a clunky way before doing it smoothly.
- I am talking in a vacuum here, having never read it. If any of my readers have actually read it and want to comment on what it was really like, you are more than invited to do so.
