Principia Mathematica: The Choose Your Own Adventure Story

Hi! *You* look like you want to get `lost` in an old book. [[I guess?->Right!]] [[Right!->Right!]] [[I know about *Principia* and I'm ready to get lost-take me to ∗1 already!->PM1]] [[I know about *Principia* and I'm not ready to get lost-take me to the table of contents!->Anywhere]]

Right! I **knew** you did. Well, here is an old book just for you! [[Look at the book->Book]]

See? That's an old book! <img src="images/pm-1.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[It doesn't *look* like a book to me.->Principia]] [[I guess?->Principia]]

You aren't sure? Does this picture make it look better? <img src="images/PMV11910.jpg" width="1447" height="2048" alt="Principia Volume 1" /> [[That is just a monochromatic picture...who says the book is new?->New]] [[Who wrote this book?->Who]]

This is ∗2 of *Principia*! <img src="images/pm-2.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Why is that one blue dot surrounded by so many purple ones?->PM2.03]] [[Go to *3!->PM3]] [[Go to *4!->PM4]] [[Go to *5!->PM5]] [[Go to *9!->PM9]] [[Go to *10!->PM10]] [[Go to *11!->PM11]] [[Go to *13!->PM13]] [[Go to *14!->PM14]] [[Go to *20!->PM20]] [[Go to *21!->PM21]] [[Go to *22!->PM22]] [[Go to *24!->PM24]] [[Go to *30!->PM30]] [[Go to *31!->PM31]] [[Go to *33!->PM33]] [[Go to *34!->PM34]] [[Go to *35!->PM35]] [[Go to *37!->PM37]] [[Go to *38!->PM38]] [[Go to *40!->PM40]] [[Go to *50!->PM50]] [[Go to *51!->PM51]] [[Go to *52!->PM52]] [[Go to *53!->PM53]] [[Go to *54!->PM54]] [[Go to *55!->PM55]] [[Go to *56!->PM56]]

This is ∗3 of *Principia*! <img src="images/pm-3.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Go to *4!->PM4]] [[Go to *5!->PM5]] [[Go to *10!->PM10]] [[Go to *11!->PM11]] [[Go to *13!->PM13]] [[Go to *14!->PM14]] [[Go to *20!->PM20]] [[Go to *21!->PM21]] [[Go to *22!->PM22]] [[Go to *24!->PM24]] [[Go to *30!->PM30]] [[Go to *34!->PM34]] [[Go to *35!->PM35]] [[Go to *36!->PM36]] [[Go to *37!->PM37]] [[Go to *40!->PM40]] [[Go to *51!->PM51]] [[Go to *52!->PM52]] [[Go to *54!->PM54]] [[Go to *55!->PM55]] [[Go to *56!->PM56]]

This is ∗4 of *Principia*! <img src="images/pm-4.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Why are they all so packed together around one blue one?->PM4.22]] [[Go to *10!->PM10]] [[Go to *11!->PM11]] [[Go to *13!->PM13]] [[Go to *14!->PM14]] [[Go to *20!->PM20]] [[Go to *21!->PM21]] [[Go to *22!->PM22]] [[Go to *24!->PM24]] [[Go to *30!->PM30]] [[Go to *31!->PM31]] [[Go to *32!->PM32]] [[Go to *33!->PM33]] [[Go to *34!->PM34]] [[Go to *35!->PM35]] [[Go to *36!->PM36]] [[Go to *37!->PM37]] [[Go to *40!->PM40]] [[Go to *41!->PM41]] [[Go to *42!->PM42]] [[Go to *43!->PM43]] [[Go to *50!->PM50]] [[Go to *51!->PM51]] [[Go to *52!->PM52]] [[Go to *53!->PM53]] [[Go to *54!->PM54]] [[Go to *55!->PM55]] [[Go to *56!->PM56]]

This is ∗5 of *Principia*! <img src="images/pm-5.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Go to *10!->PM10]] [[Go to *13!->PM13]] [[Go to *14!->PM14]] [[Go to *22!->PM22]] [[Go to *24!->PM24]] [[Go to *32!->PM32]] [[Go to *33!->PM33]] [[Go to *35!->PM35]] [[Go to *37!->PM37]] [[Go to *40!->PM40]] [[Go to *51!->PM51]] [[Go to *52!->PM52]] [[Go to *53!->PM53]] [[Go to *54!->PM54]] [[Go to *55!->PM55]] [[Go to *56!->PM56]]

This is ∗9 of *Principia*! <img src="images/pm-9.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Why are most of these dots blue? In other maps, most of them are purple!->PM9-10]] [[Go to *10!->PM10]] [[Go to *11!->PM11]]

This is ∗10 of *Principia*! <img src="images/pm-10.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Go to *11!->PM11]] [[Go to *13!->PM13]] [[Go to *14!->PM14]] [[Go to *20!->PM20]] [[Go to *21!->PM21]] [[Go to *22!->PM22]] [[Go to *24!->PM24]] [[Go to *30!->PM30]] [[Go to *31!->PM31]] [[Go to *32!->PM32]] [[Go to *33!->PM33]] [[Go to *34!->PM34]] [[Go to *35!->PM35]] [[Go to *36!->PM36]] [[Go to *37!->PM37]] [[Go to *38!->PM38]] [[Go to *40!->PM40]] [[Go to *41!->PM41]] [[Go to *42!->PM42]] [[Go to *50!->PM50]] [[Go to *51!->PM51]] [[Go to *52!->PM52]] [[Go to *53!->PM53]] [[Go to *54!->PM54]] [[Go to *55!->PM55]] [[Go to *56!->PM56]]

This is ∗11 of *Principia*! <img src="images/pm-11.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Go to *13!->PM13]] [[Go to *14!->PM14]] [[Go to *20!->PM20]] [[Go to *21!->PM21]] [[Go to *31!->PM31]] [[Go to *32!->PM32]] [[Go to *33!->PM33]] [[Go to *34!->PM34]] [[Go to *35!->PM35]] [[Go to *37!->PM37]] [[Go to *40!->PM40]] [[Go to *41!->PM41]] [[Go to *42!->PM42]] [[Go to *43!->PM43]] [[Go to *50!->PM50]] [[Go to *52!->PM52]] [[Go to *53!->PM53]] [[Go to *54!->PM54]] [[Go to *55!->PM55]] [[Go to *56!->PM56]]

This is ∗1 of *Principia*! <img src="images/pm-1.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[What do these colors mean again?->Color]] [[Why is *1 skipped?->Skipped]] [[Go to *2!->PM2]] [[Go to *3!->PM3]] [[Go to *4!->PM4]] [[Go to *5!->PM5]] [[Why are numbers between *5 and *9 skipped?->Skips]] [[Go to *9!->PM9]] [[Go to *10!->PM10]] [[Go to *11!->PM11]] [[Why is *12 skipped?->Reducibility]] [[Go to *13!->PM13]] [[Go to *14!->PM14]] [[Go to *20!->PM20]] [[Go to *21!->PM21]] [[Go to *22!->PM22]] [[Go to *24!->PM24]] [[Go to *30!->PM30]] [[Go to *31!->PM31]] [[Go to *32!->PM32]] [[Go to *33!->PM33]] [[Go to *34!->PM34]] [[Go to *35!->PM35]] [[Go to *37!->PM37]] [[Go to *38!->PM38]] [[Go to *40!->PM40]] [[Go to *41!->PM41]] [[Go to *43!->PM43]] [[Go to *50!->PM50]] [[Go to *51!->PM51]] [[Go to *52!->PM52]] [[Go to *53!->PM53]] [[Go to *54!->PM54]] [[Go to *55!->PM55]] [[Go to *56!->PM56]] [[Why does *1 have so many links?!->Many]]

This is ∗12 of *Principia*! <img src="images/pm-12.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Why are there so few dots here?->Few]] [[Go to *13!->PM13]] [[Go to *14!->PM14]] [[Go to *20!->PM20]] [[Go to *21!->PM21]]

This is ∗13 of *Principia*! <img src="images/pm-13.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Go to *14!->PM14]] [[Go to *20!->PM20]] [[Go to *21!->PM21]] [[Go to *24!->PM24]] [[Go to *30!->PM30]] [[Go to *33!->PM33]] [[Go to *34!->PM34]] [[Go to *37!->PM37]] [[Go to *40!->PM40]] [[Go to *41!->PM41]] [[Go to *43!->PM43]] [[Go to *50!->PM50]] [[Go to *51!->PM51]] [[Go to *52!->PM52]] [[Go to *53!->PM53]] [[Go to *54!->PM54]] [[Go to *55!->PM55]] [[Go to *56!->PM56]]

This is ∗14 of *Principia*! <img src="images/pm-14.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Go to *20!->PM20]] [[Go to *21!->PM21]] [[Go to *30!->PM30]] [[Go to *31!->PM31]] [[Go to *32!->PM32]] [[Go to *33!->PM33]] [[Go to *34!->PM34]] [[Go to *35!->PM35]] [[Go to *37!->PM37]] [[Go to *38!->PM38]] [[Go to *40!->PM40]] [[Go to *41!->PM41]] [[Go to *50!->PM50]] [[Go to *51!->PM51]] [[Go to *52!->PM52]] [[Go to *53!->PM53]] [[Go to *55!->PM55]] [[Go to *56!->PM56]]

This is ∗21 of *Principia*! <img src="images/pm-21.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Go to *30!->PM30]] [[Go to *31!->PM31]] [[Go to *32!->PM32]] [[Go to *33!->PM33]] [[Go to *34!->PM34]] [[Go to *35!->PM35]] [[Go to *37!->PM37]] [[Go to *50!->PM50]] [[Go to *55!->PM55]] [[Go to *56!->PM56]]

This is ∗20 of *Principia*! <img src="images/pm-20.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Go to *21!->PM21]] [[Go to *22!->PM22]] [[Go to *24!->PM24]] [[Go to *32!->PM32]] [[Go to *33!->PM33]] [[Go to *34!->PM34]] [[Go to *37!->PM37]] [[Go to *40!->PM40]] [[Go to *50!->PM50]] [[Go to *51!->PM51]] [[Go to *52!->PM52]] [[Go to *53!->PM53]] [[Go to *55!->PM55]] [[Go to *56!->PM56]]

This is ∗22 of *Principia*! <img src="images/pm-22.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Go to *24!->PM24]] [[Go to *32!->PM32]] [[Go to *33!->PM33]] [[Go to *34!->PM34]] [[Go to *35!->PM35]] [[Go to *36!->PM36]] [[Go to *37!->PM37]] [[Go to *50!->PM50]] [[Go to *51!->PM51]] [[Go to *52!->PM52]] [[Go to *53!->PM53]] [[Go to *54!->PM54]] [[Go to *56!->PM56]]

This is ∗23 of *Principia*! <img src="images/pm-23.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Go to *31!->PM31]] [[Go to *32!->PM32]] [[Go to *33!->PM33]] [[Go to *34!->PM34]] [[Go to *35!->PM35]] [[Go to *41!->PM41]] [[Go to *50!->PM50]] [[Go to *55!->PM55]] [[Go to *56!->PM56]]

This is *25 of *Principia*! <img src="images/pm-25.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Wait, why is this map in a hierarchical arrangement?->Hierarchy25]] [[Go to *31!->PM31]] [[Go to *33!->PM33]] [[Go to *34!->PM34]] [[Go to *35!->PM35]] [[Go to *50!->PM50]] [[Go to *55!->PM55]] [[Go to *56!->PM56]]

This is ∗24 of *Principia*! <img src="images/pm-24.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Go to *33!->PM33]] [[Go to *34!->PM34]] [[Go to *35!->PM35]] [[Go to *37!->PM37]] [[Go to *40!->PM40]] [[Go to *50!->PM50]] [[Go to *51!->PM51]] [[Go to *52!->PM52]] [[Go to *53!->PM53]] [[Go to *54!->PM54]] [[Go to *55!->PM55]] [[Go to *56!->PM56]]

This is ∗30 of *Principia*! <img src="images/pm-30.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Go to *31!->PM31]] [[Go to *32!->PM32]] [[Go to *33!->PM33]] [[Go to *34!->PM34]] [[Go to *37!->PM37]] [[Go to *38!->PM38]] [[Go to *50!->PM50]] [[Go to *51!->PM51]] [[Go to *52!->PM52]] [[Go to *53!->PM53]] [[Go to *55!->PM55]]

This is ∗31 of *Principia*! <img src="images/pm-31.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Go to *32!->PM32]] [[Go to *33!->PM33]] [[Go to *34!->PM34]] [[Go to *35!->PM35]] [[Go to *37!->PM37]] [[Go to *41!->PM41]] [[Go to *50!->PM50]] [[Go to *51!->PM51]] [[Go to *55!->PM55]]

This is ∗33 of *Principia*! <img src="images/pm-33.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Go to *34!->PM34]] [[Go to *35!->PM35]] [[Go to *36!->PM36]] [[Go to *37!->PM37]] [[Go to *40!->PM40]] [[Go to *41!->PM41]] [[Go to *42!->PM42]] [[Go to *43!->PM43]] [[Go to *50!->PM50]] [[Go to *51!->PM51]] [[Go to *52!->PM52]] [[Go to *53!->PM53]] [[Go to *55!->PM55]] [[Go to *56!->PM56]]

This is ∗32 of *Principia*! <img src="images/pm-32.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Go to *33!->PM33]] [[Go to *34!->PM34]] [[Go to *37!->PM37]] [[Go to *40!->PM40]] [[Go to *51!->PM51]] [[Go to *53!->PM53]]

This is ∗34 of *Principia*! <img src="images/pm-34.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Go to *35!->PM35]] [[Go to *36!->PM36]] [[Go to *37!->PM37]] [[Go to *41!->PM41]] [[Go to *43!->PM43]] [[Go to *50!->PM50]] [[Go to *54!->PM54]] [[Go to *55!->PM55]]

This is ∗35 of *Principia*! <img src="images/pm-35.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Go to *35!->PM35]] [[Go to *36!->PM36]] [[Go to *37!->PM37]] [[Go to *41!->PM41]] [[Go to *43!->PM43]] [[Go to *50!->PM50]] [[Go to *55!->PM55]]

This is ∗36 of *Principia*! <img src="images/pm-36.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Why is this map hierarchical?->Hierarchy36]] [[Go to *37!->PM37]] [[Go to *41!->PM41]] [[Go to *50!->PM50]]

This is ∗37 of *Principia*! <img src="images/pm-37.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Go to *38!->PM38]] [[Go to *40!->PM40]] [[Go to *41!->PM41]] [[Go to *42!->PM42]] [[Go to *43!->PM43]] [[Go to *50!->PM50]] [[Go to *52!->PM52]] [[Go to *53!->PM53]] [[Go to *54!->PM54]] [[Go to *55!->PM55]] [[Go to *56!->PM56]]

This is ∗38 of *Principia*! <img src="images/pm-38.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Go to *40!->PM40]] [[Go to *41!->PM41]] [[Go to *55!->PM55]] [[Go to *56!->PM56]]

This is ∗40 of *Principia*! <img src="images/pm-40.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Go to *41!->PM41]] [[Go to *42!->PM42]] [[Go to *43!->PM43]] [[Go to *53!->PM53]]

This is ∗41 of *Principia*! <img src="images/pm-41.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Go to *42!->PM42]] [[Go to *43!->PM43]] [[Go to *53!->PM53]] [[Go to *55!->PM55]] [[Go to *56!->PM56]]

This is ∗42 of *Principia*! <img src="images/pm-42.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Why is this one hierarchically arranged?->Adams]] [[Why can't I go anywhere from *42?->Oops!]]

This is ∗50 of *Principia*! <img src="images/pm-50.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Go to *51!->PM51]] [[Go to *55!->PM55]] [[Go to *56!->PM56]]

This is ∗43 of *Principia*! <img src="images/pm-43.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Just for fun, could I see a hierarchical version of *43?->PM43h]] [[Go to *50!->PM50]] [[Go to *55!->PM55]]

This is ∗51 of *Principia*! <img src="images/pm-51.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Go to *52!->PM52]] [[Go to *53!->PM53]] [[Go to *54!->PM54]] [[Go to *55!->PM55]] [[Go to *56!->PM56]]

This is ∗53 of *Principia*! <img src="images/pm-53.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Go to *55!->PM55]]

This is ∗52 of *Principia*! <img src="images/pm-52.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Go to *53!->PM53]] [[Go to *54!->PM54]] [[Go to *55!->PM55]] [[Go to *56!->PM56]]

This is ∗54 of *Principia*! <img src="images/pm-54.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Why is *54 hierarchical? The others are bundles of nodes.->Hierarchy54]] [[Go to *55!->PM55]] [[Go to *56!->PM56]]

This is ∗55 of *Principia*! <img src="images/pm-55.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Go to *56!->PM56]]

This is ∗56 of *Principia*! <img src="images/pm-56.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Why is *56 hierarchical? The others are in bundles.->Hierarchy56]] [[Oh! We reached *56. That's the end, right?->End?]]

This is ∗43 of *Principia*, but with a **hierarchical** arrangement! At 106 nodes-32 of which are from ∗43-it is not exactly easy to read, but it is a nice example of a medium-sized chapter. <img src="images/pm-43h.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[That *was* fun, thanks!->PM43]] [[Go to *50!->PM50]] [[Go to *55!->PM55]]

End? My goodness, *no*! Or haven't you been paying attention? [[I guess not?->Lesson]] [[I *thought* I was...->Lesson]]

How many endings does *Principia*-or any story!-have? Not one, certainly! It has a *last* page, but this is not the *end* of *Principia*'s story. Better to call it an *epilogue*. [[Why is this last page not the end?->Why]] [[Can I just have that epilogue you mentioned?->Epilogue]]

My fellow traveler, feel free to click below to get lost in a small part of *Principia*'s logical forrest by jumping to any number below. [[Go to *1!->PM1]] [[Go to *2!->PM2]] [[Go to *3!->PM3]] [[Go to *4!->PM4]] [[Go to *5!->PM5]] [[Go to *9!->PM9]] [[Go to *10!->PM10]] [[Go to *11!->PM11]] [[Go to *12!->PM12]] [[Go to *13!->PM13]] [[Go to *14!->PM14]] [[Go to *20!->PM20]] [[Go to *21!->PM21]] [[Go to *22!->PM22]] [[Go to *23!->PM23]] [[Go to *24!->PM24]] [[Go to *25!->PM25]] [[Go to *30!->PM30]] [[Go to *31!->PM31]] [[Go to *32!->PM32]] [[Go to *33!->PM33]] [[Go to *34!->PM34]] [[Go to *35!->PM35]] [[Go to *36!->PM36]] [[Go to *37!->PM37]] [[Go to *38!->PM38]] [[Go to *40!->PM40]] [[Go to *41!->PM41]] [[Go to *42!->PM42]] [[Go to *43!->PM43]] [[Go to *50!->PM50]] [[Go to *51!->PM51]] [[Go to *52!->PM52]] [[Go to *53!->PM53]] [[Go to *54!->PM54]] [[Go to *55!->PM55]] [[Go to *56!->PM56]]

It is not the last page because there are two-and-a-half more volumes to do! *Principia Mathematica* has hundreds more chapters to do. *Principia Mathematica* to *56 is just the abbreviated paperback edition-and that is all we have lost ourselves within here. [[That seems like a dumb reason.->Reason]] [[Isn't there a deeper, more philosophical reason why the last page is not the end?->Reason]]

Oh, yes, there is! When you adventured through even a small part of *Principia*'s logical forrest, you noticed a lot of gaps in numbering and skipping around. That was partly because theorems are needed in different places, but not everywhere; they can go unmentioned in many chapters at a time. [[So are Whitehead and Russell going to fill those gaps?->Dead]] [[Why didn't Whitehead and Russell fill those gaps?->Dead]]

Whitehead and Russell are definitely too dead to fill in the gaps of *Principia*! But even when they lived, they saw that their work was not the last word on Logicism or the foundations of mathematics. They knew that further deductions would need to be carried out by future generations of logicians, that mathematics would grow in new and unpredictable ways, and that there might be future improvements in the foundations of mathematics. In the words of Bernard Linsky and Andrew David Irvine: <blockquote>[["Establishing logicism would be an ongoing project, as open-ended as mathematics itself."->https://plato.stanford.edu/entries/principia-mathematica/#TherNoConcEndPM]]</blockquote> [[Wait, does that mean *I* am supposed to fill these gaps?->Epilogue]]

Yes, my dear traveler! Anyone who wants to contribute in any direction-to developing mathematics, to enriching the foundations, or to logically recapturing the deductions-can do so! The open-textured and open-ended character of Logicism, as laid out in *Principia*, aligns to mathematical practice in just that way. [[I'd like to help!->Help]] [[Can I get a list of all this book's pages?->Anywhere]]

If you would like to help, then you can get in touch with your adventure guide by clicking the [[Contact button here->https://www.principiarewrite.com/]]! [[Can you list all the book's pages?->Anywhere]] [[Can I go to *1 again?->PM1]]

It is [[*Principia Mathematica*->https://archive.org/details/principiamathema01anwh/page/n7/mode/2up]] by [[Alfred North Whitehead->https://plato.stanford.edu/entries/whitehead/]] and [[Bertrand Russell->https://plato.stanford.edu/entries/russellsep /]], published as three volumes (1910, 1912, 1913) and running 1,992 pages. [[Why is it so many pages?->Pages]]

"New"? My fellow traveler, this book is from 1910! It is [[*Principia Mathematica*->https://archive.org/details/principiamathema01anwh/page/n7/mode/2up]] by [[Alfred North Whitehead->https://plato.stanford.edu/entries/whitehead/]] and [[Bertrand Russell->https://plato.stanford.edu/entries/russellsep /]], published as three volumes (1910, 1912, 1913) and running 1,992 pages. [[Why is it so many pages?->Pages]]

Fun thing about that-*Principia* took Whitehead and Russell about ten years to write! There were numerous false starts and revisions made, especially on the foundational portions and on cardinal arithmetic in Volume II. But it is so long not just because logic is fun - but because Whitehead and Russell wanted to support Logicism. [[This is a logic book?->Logicism]] [[What is logicism?->Logicism]]

*Logicism* is the view that all mathematical truths are logical truths. Whitehead and Russell wrote this logic book, *Principia*, to show that many core mathematical truths are logical ones. The went about it by showing that there is at least one deductive system wherein many core mathematical truths can be proved using logical principles alone. [[Deductions?->Deductions]] [[What has this got to do with the weird map you showed me?->Map]]

*Deductions* are sequences of formulas, one after another, such that each step accords with some rules of inference. Deductions on a page are similar to the thinking we do in our heads when we [["use our minds, take a step at a time"->https://youtu.be/0PSym4A-km8?t=55]]. But sometimes deductions are very careful about not skipping steps or making inferences, like Whitehead and Russell attempt to do in *Principia*. Here is a sample of deductions in *Principia*-the proof of theorem ∗20.703. Notice how other starred numbers are mentioned in the proof? Those are citing earlier propositions in the book and using them in the proof. <img src="images/20.702-20.71.png" width="862" height="657" alt="20.702-20.71" /> [[What does that * refer to?->Stars]] [[What does this have to do with Logicism again?->Evidence]]

Now here is the map again! <img src="images/pm-1.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> You noticed in that proof I showed you that various numbers are cited, right? Here is a map showing all the places that any starred number-a primitive postulate, a definition, a theorem, or a rule of inference-from Chapter 1-that is, ∗1-is cited! [[Why is *1 cited almost everywhere?->Postulates]] [[What do these different colors mean?->Colors]]

The little asterisk refers to chapters of *Principia*. So "∗1" is Chapter 1, while "∗1.1" is (Theorem, Definition, or Postulate) refers to the starred number (proposition) ∗1.1, which is of course in (Chapter) ∗1 (the number before the decimal indicates the chapter a proposition occurs within). [[What does this have to do with Logicism again?->Evidence]]

Whitehead and Russell supported Logicism by showing that ordinary mathematical truths, like "1+1=2" and such, could be recovered using only logical principles-axioms, inference rules, and so on that are all part of logic. Think of it like a game-a skeptic says "that math isn't part of logic!" and then Whitehead and Russell say, "Oh, really? Here is it using logic alone!" That, they think, would provide strong evidence for Logicism. Look! There's 1+1=2 now, right in *Principia*. <img src="images/110.64-110.643.png" width="688" height="327" alt="PM110.64-110.643" /> [[Neat! But what about those maps you showed me?->Map]]

Well, ∗1 is cited so much because there are primitive postulates that are laid down at the start. Then those postulates are used to prove all sorts of other theorems that occur in the book-9,944 propositions occur in all throughout *Principia Mathematica*. Fun fact: all 16 primitive postulates of *Principia* are contained in just three chapters! These primitive postulates are some rules of inference (*modus ponens*) and five propositional logic axioms in ∗1, seven quantifier logic axioms in ∗9, and two axioms of (impredicative) comprehension in ∗12. But you'll see all those when you get lost inside the logic forest that is *Principia*! [[What about the map? Will that keep me from getting lost?->Colors]] [[Is all of *Principia* here?->to56]]

This map shows the *dependencies* between different propositions in *Principia*. The **blue** dots are all starred numbers from the current chapter-in this case, ∗1-and the purple dots are all starred numbers in any chapter of *Principia* up through ∗56 of Volume 1. This map of ∗1, and the other ones for the first chapters of *Principia*, gives you a sense of the deductive structure of *Principia* itself! When you click around, you are following links between whole chapters (to keep it intelligible!). So if some theorem from ∗2 is cited in ∗3, then you can get to ∗3's page from the ∗2 page. Otherwise, you have to go back to ∗1 and find your way to ∗3 a different way. <img src="images/pm-1.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[So this map will definitely get me lost?->Lost]] [[This map will definitely get me lost, then.->Lost]]

No, this book only covers the abbreviated paperback, *Principia Mathematica* to ∗56! We don't want you to get *too* lost on your first logic adventure! <img src="images/PMto56.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[How will I get lost when there is a map?->Colors]]

Yes! You will get lost. That is the point! You can choose different chapters to go to starting from ∗1. From there you can *only* go forwards-to get lost in a different way, you will need to go back to the beginning! Fun, huh? [[It is?->Point]] [[Is there a point to getting lost-I mean, to this "fun"?->Point]] [[Won't the sequence just take me through the chapters in numerical order? Why would this be a "choose your own adventure"? That sort of book jumps around.->Jump]]

No! Chapters of *Principia* jump around like a choose-your-own-adventure book. Depending on what theorem you want to prove or what topic in math you are dealing with, some earlier chapters will be relevant and others will be totally irrelevant. You can skip around and jump ahead, but there is no way to go from start to finish in numerical order. Since there is no way to follow all the chapters from start to finish in numerical order (∗4 skips ∗5, for starters), you will literally get lost in the deductions-just like Whitehead and Russell did! That means you will get lost in a good way. [[I think I have heard enough-how do I start?->PM1]]

The point is that the deductive structure of *Principia* jumps around like a choose-your-own-adventure book. Depending on what theorem you want to prove or what topic in math you are dealing with, some earlier chapters will be relevant and others will be totally irrelevant. Since there is no way to follow all the chapters from start to finish in numerical order (∗4 skips ∗5, for starters), you will literally get lost in the deductions-just like Whitehead and Russell did! That means you will get lost in a good way. [[I think I have heard enough-how do I start?->PM1]]

Most chapters have one or more starred numbers that refer to something in them. For instance, ∗2.06 is proven using ∗2.05, so there would be a link between ∗2 and itself in our book. The book only links between chapters, so we just omitted it from the ∗2 page since, well, you are already looking ∗2 if you are on the ∗2 page! [[That makes sense...->PM1]] [[Right, thanks!->PM1]]

This map shows the *dependencies* between different propositions in *Principia*. The **blue** dots are all starred numbers from the current chapter-in this case, ∗1-and the purple dots are all starred numbers in any chapter of *Principia* (up through ∗56). This map of ∗1, and the other ones for the first chapters of *Principia*, gives you a sense of the deductive structure of *Principia* itself! When you click around, you are following links between whole chapters (to keep it intelligible!). So if some theorem from ∗2 is cited in ∗3, then you can get to ∗3's page from the ∗2 page. Otherwise, you have to go back to ∗1 and find your way to ∗3 a different way. <img src="images/pm-1.jpg" width="1920" height="968" alt="A bunch of connected dots?" /> [[Got it, thanks!->PM1]] [[Ready to get lost again...->PM1]]

Whitehead and Russell saw that their work was not the last word on Logicism or the foundations of mathematics. They knew that further deductions would need to be carried out by future generations of logicians, that mathematics would grow in new and unpredictable ways, and that there might be future improvements in the foundations of mathematics. In the words of Bernard Linsky and Andrew David Irvine: <blockquote>[["Establishing logicism would be an ongoing project, as open-ended as mathematics itself."->https://plato.stanford.edu/entries/principia-mathematica/#TherNoConcEndPM]]</blockquote> So there are gaps because Whitehead and Russell knew that *Principia* would not be complete! [[Let me get back to the map.->PM1]] [[Didn't Kurt Gödel prove something of that sort with his Incompleteness Theorems?->Godel]]

∗12 is a chapter with just two axioms! So there are no proofs there-nothing from any other chapter is cited there. [[Take me back to *1.->PM1]] [[*12 sounds interesting, take me there.->PM12]]

Remember, ∗1 has a bunch of axioms! So these get used all over the place. Lots of the early chapters get cited in many later ones. As one goes on, many of the chapters get cited in specific parts of different volumes rather than by nearly every preceding one. [[Got it, thanks!->PM1]] [[You talk to much, let me go back to my logical adventures!->PM1]]

You sound like [[Theatetus and his reading Protagoras "again and again"->http://classics.mit.edu/Plato/theatu.html]]! You are impressive. But there is a lot of confusion about Gödel's theorems and their relationship to the Logicist project in *Principia*. Let me explain the relationship more carefully than you may have seen it discussed. Kurt Gödel's (First) Incompleteness Theorem shows that if a system is - consistent, - recursively axiomatizable, and - able to prove a sufficient amount of arithmetic, then that there are some truths of that system which will not be provable *within* that system. What does this have to do with *Logicism*? *Principia* develops one formal system. It is sufficient to prove many arithmetic truths. But it cannot prove every arithmetic truth. That does not mean *Principia* is bad evidence for Logicism. *Principia* still shows a great deal of mathematics can be done using 16 primitive postulates of logic. This may strike you as strong evidence for Logicism. More importantly, Gödel only showed that no single system with the above properties proves every formula that is true *in that system*. This leaves open that there may be other logical systems to capture specific mathematical truths that *Principia* is unable to prove (or just has not yet proved). So Gödel's Incompleteness Theorem does not prove that Logicism is false, but it does show that if the evidence for Logicism is supposed to be that *all* mathematical truths are demonstrated in a *single* deductive system, then *Principia*'s system cannot be a complete argument for Logicism-no system can do that, in fact. [[This is boring, let me go back to getting lost.->PM1]] [[Thanks for the quick lesson! I'm ready to get lost again.->PM1]]

You noticed! ∗2.03 is a proposition from ∗2 that gets cited *tons*-438 times, in fact, and all the way up to ∗374.12 in Volume III! That is why it is packed in so tightly. [[Neat, thanks!->PM2]] [[Take me back to *2.->PM2]]

You noticed! ∗4.22 is a proposition from ∗4 that gets cited *tons*-858 times, in fact, and all the way up to ∗374.12 in Volume III! That is why it is packed in so tightly. [[Neat, thanks!->PM4]] [[Take me back to *4.->PM4]]

You noticed ∗9 is a funny chapter! Whitehead and Russell have two chapters presenting quantifier theory axioms. But Whitehead and Russell supply ∗9 as being of "purely philosophical interest". The technical development of *Principia* restarts in ∗10, where they reprove many of the same numbers from ∗9. Except for ∗11, all citations of quantifier theory in later chapters rely on ∗10 instead of ∗9. So this is a rare case where there are more propositions proved than there are uses of the propositions later on. [[Cool, thanks! Take me back to *9.->PM9]] [[Take me to *10, if that's where the real action happens!->PM10]]

∗12 contains just two axioms, the Axioms of Reducibility. These starred numbers only get cited in 4 chapters. ∗12.1 is cited 10 times, and ∗12.11 is cited 7 times. Why so few occurrences? Because the theorems they are used to prove are then leveraged to do much more of the mathematical work in *Principia*. In particular, ∗12 has axioms that underwrite the "no-classes" theory of classes and "no relations" theory of relations. [[What the heck are you talking about-"no-classes" classes?->NoClasses]] [[Does "no-classes" mean classes are canceled?->NoClasses]] [[Just take me back to *12 already.->PM12]]

Whitehead and Russell do not have an ontology of classes and do not posit the existence of classes. This is very different from modern set theories where there are axioms assuring the existence of classes. Whitehead and Russell prefer to rely on an ontology of relations. The axioms of reducibility are essentially an existence posit that does much of the work classes are usually used to do, but without taking classes as entities at all. Reducibility basically assures us of the following: <blockquote>For any (well-formed) formula of *Principia*, there is a relation whose extension is all and only those things satisfying that formula.</blockquote> It turns out that, with this axiom, you can do without classes entirely. And you might think this is a plus if you think classes are suspect in light of [[Russell's Paradox->https://plato.stanford.edu/entries/russell-paradox/]]. [[Thanks! I'm ready to go back to *12.->PM12]] [[Just take me back to *12 already.->PM12]]

You are very observant! ∗25 is the first one of these chapters where the propositions are few enough and cited infrequently enough that a hierarchical arrangement looks good! Hierarchical arrangements of hundreds of nodes and links would be hard to read, but in this case there are just 95 nodes-30 of which are from ∗25. Since this arrangement makes sense for ∗25, I thought that it would be a nice treat for you! You might be weary after being lost so long, my fellow traveler! [[That was a good break, thanks!->PM25]] [[I guess? I'll just go back to being lost.->PM25]]

You are very observant! ∗36 is the second one of these chapters where the propositions are few enough and cited infrequently enough that a hierarchical arrangement looks good! Hierarchical arrangements of hundreds of nodes and links would be hard to read, but in this case there are just 64 nodes-20 of which are from ∗36. Since this arrangement makes sense for ∗36, I thought that it would be a nice treat for you! You might be weary after being lost so long, my fellow traveler! [[That was a good break, thanks!->PM36]] [[I guess? I'll just go back to being lost.->PM36]]

You are very observant! ∗42 is in a hierarchical arrangement is because 42 is the answer to "the Ultimate Question of Life, the Universe, and Everything", of course! [[No, really, why is it hierarchical?->Hierarchy42]]

Oops! You noticed that we haven't finished the book for all three volumes. I mean, YOU CANNOT HANDLE THE WHOLE FOREST YET, YOUNG WANDERER! IF YOU SURVIVE THIS PATCH OF *PRINCIPIA*'S LOGIC FOREST, THEN MAYHAP YOU WILL WANDER THE REMAINDER! [[OK...weirdo. Just say you haven't done it yet. Not a big deal.->PM42]] [[Yes, let me get back to wandering! I can prove myself worthy!->PM42]]

∗42 is the third one of these chapters where the propositions are few enough and cited infrequently enough that a hierarchical arrangement looks good! Hierarchical arrangements of hundreds of nodes and links would be hard to read, but in this case there are just 15 nodes-7 of which are from ∗42. Since this arrangement makes sense for ∗42, I thought that it would be a nice treat for you! You might be weary after being lost so long, my fellow traveler! [[That was a good break, thanks!->PM42]] [[I guess? I'll just go back to being lost.->PM42]]

You are very observant! ∗54 is fourth one of those chapters (or fifth if you count [[∗43->PM43]] where the propositions are few enough and cited infrequently enough that a hierarchical arrangement looks good! Hierarchical arrangements of hundreds of nodes and links would be hard to read, but in this case there are just 91 nodes-34 of which are from ∗54. Since this arrangement makes sense for ∗54, I thought that it would be a nice treat for you! You might be weary after being lost so long, my fellow traveler! [[That was a good break, thanks!->PM54]] [[I guess? I'll just go back to being lost.->PM54]]

You are very observant! ∗56, which is the last chapter of our book (so far!), is fifth one of those chapters (or sixth if you count [[∗43->PM43]] where the propositions are few enough and cited infrequently enough that a hierarchical arrangement looks good! Hierarchical arrangements of hundreds of nodes and links would be hard to read, but in this case there are just 72 nodes-50 of which are from ∗56. Since this arrangement makes sense for ∗56, I thought that it would be a nice treat for you! You might be weary after being lost so long, my fellow traveler! [[That was a good break, thanks!->PM56]] [[I guess? I'll just go back to being lost.->PM56]] [[Wait, the last chapter? So am I done being lost? That's the end, right?->End?]]