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Perhaps better, is there an accessible version of the Principia? I am looking for a summary that would summarize and clarify Russell’s reasoning behind his famous conclusion that 1 + 1 = 2.

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    Have you tried the wikipedia article? Or the SEP article? And by "reasoning behind", do you mean you want to know why Russell thinks that 1+1=2 or do you want an explanation of what axioms/definitions/rules of deductions are used in PM? Because the conclusion that 1+1=2 might be famous, but it isn't due to Russell (or Whitehead), or any particular person of history for that matter. – Not_Here Nov 10 '18 at 2:22
  • Here's something that might help but mainly to understand *54.43. I have only skimmed it: blog.plover.com/math/PM.html The three volumes of the Principia Mathematica are on the Internet archive: archive.org – Frank Hubeny Nov 10 '18 at 4:06
  • @Not_Here Well, frankly and probably unrealistically, I was hoping for something that would spoon feed the proof to me. Thanks to you and Frank Hubeny for the citations. – Mark Andrews Nov 10 '18 at 5:31
  • Since you seem to be asking for a logic/philosophy reason, are you sure you want Russell's reasoning in particular? As Mauro said, other proofs are much simpler, and the system used in PM is not really used by others. – Mark S. Nov 10 '18 at 18:07
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For an introductory exposition, you can see : Richard Zach, Principia Mathematica and the Development of Logic (2010).

A more detailed exposition is into : Ivor Grattan-Guinness, The Search for Mathematical Roots, 1870-1940, Princeton UP (2000), Ch.7.

The issue is that the "standrad" proof of 1+1=2 from Peano's axioms is quite simple: it needs very few lines starting with the definition of 1 and 2 and the axiom for +.

In PM, instead, the Peano axioms are themselves derived from more basic principles and definitions, and this is much more longer.

As you can see from the reprint of the first chapters : Alfred North Whitehead & Bertrand Russell, Principia Mathematica to #56, Cambridge UP (2nd ed 1927), the definition of 1 is at page 345 : Def 52.01.

The definition per se is quite simple :

1 = the class of all classes α such that α = { x } for some x (Def).

{ x } is a singleton, i.e. a class with the only element x. Thus 1 is defined as the class of all classes with exactly one member.

The first result proved from it is :

52.1 a class α ∈ 1 iff α = { x } for some x.

As you can see, the level of details is very high.

Then we have (page 358) the definitions of 0 and 2; from them several further results are proved :

54.101 a class α ∈ 2 iff there are x,y such that x≠y and α = { x } ∪ { y }.

54.102 a class α ∈ 0 iff α is the empty class.

Finally (page 360) we arrive at :

54.43 if classes α, β ∈ 1, then α ∩ β is empty iff α ∪ β ∈ 2.

From this proposition it will follow, when arithmetical addition has been defined [emphasis added], that 1 + 1 = 2.

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