# How “Principia Mathematica” builds foundations

I understand that "Principia Mathematica" tries to build foundations of mathematics. For comparison ZFC achieves same task. From what I understand ZFC are axioms formalized in First Order Logic. Question is what is Principia based on and why it took so many pages to prove 2=1+1. I suspect that this is less tedious task in ZFC.

• I also have to say that I'm not sure what your question is. Yes, proving $1+1=2$ in $\sf ZFC$ is quite easy if you agree on how to interpret integers as sets, and how to define their addition; and if you already have all the rules of logic behind you. If you don't then you will find this to be equally tedious. – Asaf Karagila Jun 25 '13 at 15:40
• It might interest you to know that Russell once said roughly "there are only 6 people in the world who've read through all the volumes of Principia Mathematica; 3 Poles and 3 Texans." I can guess who the Poles were, but who were the Texans???? – Doug Spoonwood Jun 25 '13 at 19:21

If you're really interested in this, it would be worth your while to take a look at the actual Principia and see what it does and how it does it. There are a few reasons why it takes so long to prove $1+1=2$.

One reason is that the Principia starts with considerably less than the ZF axioms; it starts with only five logical axioms along the lines of $p\lor p \implies p$. It does not start with any axioms about sets or relations or numbers.

Another reason is that the foundational machinery was not fully developed in 1911. In 2013, we would define an ordered pair as a certain sort of set (typically, $\langle x, y\rangle {\buildrel \text{def}\over \equiv } \{\{x\}, \{x,y\}\}$) and then define a relation as a set of ordered pairs. This approach hadn't been invented yet at the time Whitehead and Russell were writing. So there are several chapters of Principia Mathematica that develop properties of sets, and then several more chapters that develop properties of relations which, from a 21st-century point of view, are completely redundant. Similarly there is a chapter on propositional functions of one variable, and then a very similar chapter about functions of two variables.

But the biggest reason for the length is that the Principia proves everything in extremely small steps. For example, an important precursor to the $1+1=2$ theorem is $$∗54\cdot 43.⊢((α,β∈1)⊃((α∩β=Λ)≡(α∪β∈2)))$$ which says that if $\alpha$ and $\beta$ are sets that each have one element, then they are disjoint if and only if their union has exactly two elements. The steps in the proof are tiny. They start with the previously-proved $\ast 54\cdot 26$, which states that $\{x\}\cup\{y\}$ has two elements if and only if $x\ne y$. To get from there to $\ast 54\cdot 43$ takes about twelve steps.

The hypotheses at one point of the proof include that $\alpha$ and $\beta$ are sets with one element each. They invoke a theorem to conclude that $\alpha = \{x\}$ and $\beta=\{y\}$ for some $x$ and $y$. Most treatments would gloss over this point. Then they prove, via a previous theorem $\ast51\cdot231$, that $\{x\}\cap\{y\}$ is empty. Most treatments at this point would take as proved that $\alpha\cap \beta$ is empty, but in Principia Mathematica this is a separate, explicit deduction, justified by theorem $\ast13\cdot12$, which states that if $x=y$, then a predicate $\psi$ is true of $x$ if and only if it is true of $y$.

All these tiny steps quickly add up to a lot of paper.

Finally, Principia Mathematica does not take 300 pages to only prove $1+1=2$. There is a great deal of explanation and discussion, and a great deal more is proved besides.

• So if I understand correctly main difference is that PM starts with fewer axioms than ZFC? – Trismegistos Jun 26 '13 at 7:01
• Yes, but that is not relevant. What is relevant is that they much weaker axioms. – MJD Jun 26 '13 at 13:44
• So why this approach is discontinued? Did it proved to be to too tedious? Another question is what the power of such system comparing to ZFC is. – Trismegistos Jun 26 '13 at 14:12

Perhaps the first point to grasp is that Principia is a type theory rather than a set theory -- indeed the authors officially do not believe in sets, thinking that talk purportedly of such things needs to translated away.

Now, the obvious places one might first look at for an account of Russell and Whitehead's type theory and the overall project of Principia are unfortunately both rather unhelpful (I mean Wikipedia and the Stanford Encyclopedia of Philosophy, whose respective entries -- here and here -- are not up to the best standards). But they may help a bit, though here are two more serious suggestions:

For a philosophical discussion of the project, see Ch.5 of Michael Potter's fine book Reason's Nearest Kin (OUP, 2000)

For an excellent paper on the early history of type theories by Fairouz Kamareddine, Twan Laan, and Rob Nederpelt which puts things in perspective, see here.

But neither of those is exactly easy. I too would be interested to know if, somewhere out there on the web, someone has posted a decently reliable ten-page exposition of what's going on in Principia that you can point students to.

• about PM, I think that two recent books are useful : Gregory Landini, Russell's Hidden Substitutional Theory (1998) and Bernard Linsky, The Evolution of Principia Mathematica: Bertrand Russell's Manuscripts and Notes for the Second Edition (2011). The first one contains a lot of material and reflections about the philosophical background of logicism, but is hard to follow (for me). The second one is more historical, and more readable, but gives a lot of informations about the process of transformation of math log from PM to the "modern" one. 1/2 – Mauro ALLEGRANZA Jan 24 '14 at 15:07
• Regarding shorter papers, I may suggest some of the papers collected into Peter Hylton, Propositions Functions and Analysis Selected Essays on Russell’s Philosophy (2005), mainly chapter 3 : Logic in Russell’s Logicism, and some of the chapters of Nicholas Griffin (editor), The Cambridge Companion to Bertrand Russell (2003), mainly : Russell and Frege (Ch.4, by M.Beaney), Mathematics in and behind Russell’s Logicism” (Ch.1, by I.Grattan-Guinness), Bertrand Russell’s Logicism (Ch.5, by M.Godwyn & A.D.Irvine) and The Theory of Types (Ch.8, by A.Urquhart). 2/2 – Mauro ALLEGRANZA Jan 24 '14 at 15:14

You are referring to the Russell--Whitehead "Principia" (rather than Newton's); see here. Russell's was a logicist program aimed at placing symbolic logic at the foundation of mathematics. ZFC is a competing program that placed set theory at the foundation, instead. The logicist approach was initiated by Frege and was much more popular at the beginning of the 20th century than it is today. Perhaps one of the reasons is the number of pages it took to prove 2=1+1 :-)

• So this distinction between logic approach and set theoretic approach is exactly what I wanted to ask about. I do not get what is the difference because ZFC is formulated in FOL, which is logic after all. Is the difference in lesser number of axioms or is it something very different? – Trismegistos Jun 26 '13 at 7:00
• First order logic serves as a tool to formulate ZFC, but the fundamental object is the concept of a set in this approach, as well as the fundamental relation $\in$. Meanwhile, in the logicist approach, the fundamental entities are in the realm of symbolic logic. – Mikhail Katz Jun 26 '13 at 7:04
• I would venture the following comment which may or may not reflect accurately your views but here it goes. Due to the success of ZFC, many mathematicians view "sets" as being built into the structure of reality itself. They are conditioned by their professional training to such an extent that it is very hard to imagine the world before Cantor & Co. This is what makes it difficult even to understand what it could possibly mean that the logicist program was not based on sets. – Mikhail Katz Jun 26 '13 at 7:46