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I read from several places that Bertrand Russell spent many pages in Principia Mathematica to prove 1 + 1 = 2, e.g. here said "it takes over 360 pages to prove definitively that 1 + 1 = 2", while here said 162 pages.

I do not believe that is the case, however, as I don't see why you'd need to prove 1+1=2 in the first place.

But Wikipedia's article for Principia Mathematica mentions:

"From this proposition it will follow, when arithmetical addition has been defined, that 1 + 1 = 2." – Volume I, 1st edition, p. 379

So did Bertrand Russell actually spend 360 pages proving that 1 + 1 = 2? What did Bertrand Russell want to accomplish by doing that?

Qiulang
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  • I found a similar question asked here https://www.quora.com/Do-you-need-300-pages-to-prove-that-1+1-2 – Qiulang Jan 29 '23 at 07:52
  • Why do I have such a strong sense of deja vu with this question? Has it been asked somewhere else on the network perhaps? – IMSoP Jan 29 '23 at 10:42
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    @IMSoP It's been asked and answered multiple times at [mathematics.SE](https://math.stackexchange.com). – David Hammen Jan 29 '23 at 11:17
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    @DavidHammen can you give an example at mathematics.SE? I searched questions there and only found https://math.stackexchange.com/questions/348889/is-11-2-a-theorem, which is not exactly what I asked here. – Qiulang Jan 29 '23 at 12:32
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    @Qiulang Here's another: https://math.stackexchange.com/questions/278974/prove-that-11-2, which is marked as a duplicate of the question you found. A Google search for "1+1=2 principia site:math.stackexchange.com" will give you even more hits. The SE search tools aren't as good as Google's. – David Hammen Jan 29 '23 at 14:22
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    Wait for it. Why is this question not closed? Where is the claim? The claim comes from the author of the question. The only outside source of the supposed "issue" is a reddit post, and I would posit it not widely known or accepted (2.2k reads is not even a drop in a bucket). Where is the notability of the claim being untrue? I thought the _rules_ of this site were a claim had to be notable and _could not be just made up by the OP_ on the spot? It's a neat fact. It's possibly even true, but why is this being allowed on [skeptics.se]? – CGCampbell Jan 30 '23 at 13:02
  • @CGCampbell "The claim comes from the author of the question", no that is not the case. I first learned it from [Logicomix](https://en.wikipedia.org/wiki/Logicomix) – Qiulang Jan 30 '23 at 14:56
  • @RobbieGoodwin I must be out of my mind if I decide to read "Principia Mathematica". BTW, I read logicmix a long time ago and then I came cross [the link](https://www.storyofmathematics.com/20th_russell.html/) I posted in my question so I asked the question. – Qiulang Jan 31 '23 at 02:19
  • @CGCampbell: that paraphrased claim has been floating around somewhat widely for decades, my high-school maths teacher (who I believe was a maths PhD) made it years ago. – smci Jan 31 '23 at 03:20
  • @CGCampbell: I agree. Skeptics.SE is for verification of alleged facts. If the question is "did R&W prove that 1+1=2 on page 360", that question is on topic, but also trivial to answer: just open the book. If the question is "*why* did R&W feel the need to prove it", then I think that becomes a [motivations question](https://skeptics.meta.stackexchange.com/questions/621/politics-beliefs-and-motivations-questions-should-not-be-allowed-here). – Nate Eldredge Jan 31 '23 at 04:37
  • @NateEldredge But I did not ask "why did R&W feel the need to prove it", I asked did they spend 360 pages in order to prove it, knowing too well that I will not read Principia Mathematica! So I believe my question is valid here. – Qiulang Jan 31 '23 at 05:55
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    @Qiulang But you already show that it took more than 360 pages in the Wiki quote you show. Why are you skeptical and about what? Your question really seems to be "why did he" and that is not on topic here. It is, in [math.se] but you'd be told asked and answered there. You can look in the PDF, to the quoted pages, and verify (as long as you are looking at the right edition, etc) that the proof is concluded where it was said to have. So, answering the title question is trivial and no skepticism is warranted. That leave only an off-topic question of, as Nate Eldredge says, motivation. – CGCampbell Jan 31 '23 at 10:34
  • @Qiulang I'm not saying it is a bad question, in the right forum or on the right exchange. But it simply, in my opinion, does not fit _here_. I don't have enough rep to VtC, but I would if I could. (I did not either up- or down-vote it) I am very much cognizant of the issues of precedent here in Stack Exchange. Topicality is something that must be guarded, with all one's might. Otherwise any given Exchange simply becomes a catch all for any question on any topic. If that's what you want, start your own forum, because that is not SE's purpose. – CGCampbell Jan 31 '23 at 10:37
  • See also the post [Is there a summary of Russell’s Principia Mathematica?](https://philosophy.stackexchange.com/questions/56969/is-there-a-summary-of-russell-s-principia-mathematica) – Mauro ALLEGRANZA Feb 01 '23 at 10:22
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    @MauroALLEGRANZA thanks for the comments and the link, especially after seeing someone said he wanted to vote to close my question. – Qiulang Feb 01 '23 at 11:40
  • This "claim" is at best a mildly amusing quip and not meant to be taken seriously. It's similar to the claim that *scientists spent centuries to prove that water is essential to life*. While the claim may be literally true according to certain narrow interpretations of what it means to "prove" something in a narrow field of mathematics or science, it's not true by most interpretations (which is also exactly why the claim might have been considered striking in the first place and so here garnered so many upvotes). –  Feb 14 '23 at 05:42
  • I'm not sure if claims on StoryOfMathematics.com (which seems to just have cobbled together a few passages from Wikipedia and elsewhere), Reddit, and Quora are "notable". –  Feb 14 '23 at 05:43

3 Answers3

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If you have only studied mathematics at school, the way it works at university/academic level can be quite alien.

By looking at the original Principia Mathematica, by Alfred Whitehead AND Bertrand Russell (e.g. this large PDF), we can confirm the claim.

It isn't until page 359 that the concept of "2" is introduced (as a "cardinal couple" - it isn't until later that they show that this is equivalent to the cardinal number, 2, that we are familiar with.)

On page 362 there is the quoted claim that Proposition 54.43 provides the basis for 1 + 1 = 2

It is worth noting: Whitehead & Russell don't spend 360-odd pages just adding two numbers together, like you were taught in school. They spend the treatise defining what was hoped to be a complete and consistent basis for all of mathematics. That means they weren't just proving that 1+1=2 (under their system of mathematics) but also defined (amongst a lot of other propositions) what "1", "2", "+" and "=" meant. They based this on a minimum set of "axioms" or assumptions. They tried to avoid allowing paradoxes and contradictions [before Kurt Gödel came along and proved that to be impossible.]

Oddthinking
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    I may need to ask another question for "It isn't until page 359 that the concept of "2" is introduced". Can it be that hard to introduce the the concept of 2 ? – Qiulang Jan 29 '23 at 07:54
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    @Qiulang Yes, it is, particularly from a set theoretic perspective, which is what *Principia Mathematica* was trying to accomplish. In the end, it's a bit futile as completeness and self-consistency are impossible in a mathematical system with sufficient complexity. All that is needed is multiplication and recursion. We still use multiplication and recursion because they're so useful. – David Hammen Jan 29 '23 at 08:16
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    @Quilang: Maths isn't structured in the way you learned it: first counting to ten, then a hundred, then addition and subtraction, then the times tables, and long multiplication, later negative numbers, etc. The book starts by defining propositions, equivalence, variables, classes, relations. These don't need any actual numbers (except maybe equivalents to 0 and 1). The idea of cardinal numbers starts in Part 2. – Oddthinking Jan 29 '23 at 08:39
  • Comments about "weed out" courses have been [moved to chat](https://chat.stackexchange.com/rooms/142427/discussion-on-answer-by-oddthinking-did-bertrand-russell-spend-360-pages-in-prin). – Oddthinking Jan 29 '23 at 09:20
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    @Qiulang One could quite early define "2" as the set $\{\emptyset,\{\emptyset\}}$ (though, from first principles, it also takes a while to write this). But no number is an island - it is quite useful to define and investigate a few more general concepts, of which "2" is just a special case, first – Hagen von Eitzen Jan 29 '23 at 18:16
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    You "confirm the claim" by saying "On page 362, it is proven that 1+1=2", but I don't see how that implies "Russell and Whitehead spent 362 pages to prove that 1+1=2". – Stef Jan 29 '23 at 18:33
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    Actually, I don't quite understand the quote to be saying that 1+1=2 is proven. The quote seems to say that on page 362 all the necessary foundations have been laid that you *could now go on to prove* 1+1=2. In fact, not even that: the quote says that on page 362 all the necessary foundations have been laid that you *could now go on to define what "addition" even means, **and then** go on to prove* 1+1=2. So, we're still not quite there yet. – Jörg W Mittag Jan 29 '23 at 18:52
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    @Stef: Yes, you are right, which is why I give context in the last paragraph explaining it isn't that simple. – Oddthinking Jan 30 '23 at 00:19
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    @JörgWMittag: Yes, you are right. I mention that the cardinal number 2 doesn't get introduced until later. I say the proposition "provides the basis", rather than proves 1 + 1 = 2, but the first link cites that proposition so I think it is what they meant by the claim. – Oddthinking Jan 30 '23 at 00:23
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    @Qiulang Math is a very well defined system, you can definitively *prove* things are true. This lets math build upon pre-existing work with total confidence; a new proof relies upon an earlier proof which relies upon an earlier proof. Math also has fundamental assumptions, called axioms, like 1 + 1 = 2. This meant that all proofs were ultimately made based on a bunch of assumptions which meant it could all be wrong, and something was wrong, they were running into contradictions and paradoxes. Russell worked to reduce those assumptions and provide a whole new way to think about math. – Schwern Jan 30 '23 at 06:31
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    @Qiulang Turns out he failed, we now know mathematical logic has limits (Godel's incompleteness theorems are the math equivalent of "this statement is false"), but did a lot of great work in the process. Here's some pop math explanations. [Why it took 379 pages to prove 1+1=2](https://www.youtube.com/watch?v=AwbZaTjXo-s) and [The paradox at the heart of mathematics: Gödel's Incompleteness Theorem](https://www.youtube.com/watch?v=I4pQbo5MQOs), – Schwern Jan 30 '23 at 06:38
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    @Schwern "this statement is false", I will argue that it is probably more like "this statement is **unprovable**" – Qiulang Jan 30 '23 at 08:54
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    Several commenters suggest that Russell’s foundationalist programme “failed” or was shown “futile” by Gödel. This is a major misconception: Gödel’s theorems showed that *some* early hoped-for goals were unnattainable — a self-consistency proof, and a complete decision procedure — but the core of the foundationalist programme was a complete success, showing all mathematics can be based in a unified logical system, and is essential to mathematics today. Our formal systems and views of them have evolved since Russell, but the programme was unquestionably a success not a failure. – PLL Jan 30 '23 at 10:14
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    https://en.wikipedia.org/wiki/Logicism has some pretty nice words to say about all of this. – AnoE Jan 30 '23 at 11:17
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    Kurt Gödel didn’t prove avoiding contradictions to be impossible. He only proved it to be impossible to prove either the truth or falsity of every possible statement. – user3840170 Jan 30 '23 at 22:07
  • @user3840170: My understanding is he showed you can't have both completeness and consistency, via a technique that showed any such system would have... contradictions or paradoxes. – Oddthinking Jan 30 '23 at 22:15
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    How many pages were **actually* spent on the proof? The page number at which the proof occurred, could have been a 1,000,000 and the proof could have been 1-line long, right? – user1271772 Jan 30 '23 at 22:17
  • @Oddthinking, Gödel proved that any formal system sufficiently complex to support Peano arithmetic will contain statements whose truth value cannot be decided. – Mark Jan 31 '23 at 01:13
  • @Mark: Yes, that was the result. But what the technique used? Coming up with a system to generate a proposition that, if its truth value *were* decided, there would be a contradiction/paradox. (That is my understanding, anyway.) – Oddthinking Jan 31 '23 at 01:18
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    @user1271772: Yes. The Sagan quote applies here: “If you wish to make an apple pie from scratch, you must first invent the universe". Before you can make that 1-line proof, you need the infrastructure, You could argue that those 360 pages were insufficient. You need English dictionaries and English primers as well, even to start. It is clear the page number metric is vague and ill-defined. I showed the interpretation given by the claimant has an evidentiary basis, and also argued why it was misleading to someone not understanding the context. – Oddthinking Jan 31 '23 at 01:21
  • How many of those pages are needed for a "reasonable" proof (not one that begins by defining every word of the English language that's used)? – user1271772 Jan 31 '23 at 02:31
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    @user1271772: That isn't what they set out to do. They set out to provide a new basis for much of mathematics. I don't think "reasonable" has a meaning here. It depends on what parts of maths you already have defined, and can rely on. How long does it reasonably take to bake an apple pie in an electric oven? Depends if you already have all the infrastructure of apples, running water, electricity, an oven... – Oddthinking Jan 31 '23 at 03:25
  • But the question asked how many pages it took to prove 1+1=2 and counting the number of pages in the book until the proof shows up is not the correct answer. Do you know the answer? – user1271772 Jan 31 '23 at 03:28
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    @user1271772: The question didn't ask that. I showed how the conclusion of 362 was reached, and then showed why this was misleading. The real answer to "how many pages does it take?" is "Well, it depends on what part of maths you already have to rely on." I once generated a "proof" that 2 (by which I mean the successor to the successor to 0) is an even number that took a full A4 page in 6pt, but I could not rely on a modulo operator, because that hadn't been defined in the mathematical model I was using. – Oddthinking Jan 31 '23 at 03:55
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Did Bertrand Russell spend 360 pages in Principia Mathematica to prove 1 + 1 = 2?

Sort of. But the phrasing of the claim (either as you stated it, or in the version "it takes over 360 pages to prove definitively that 1 + 1 = 2" in the web page you linked to) is misleading. The truth is more nuanced.

I'll try to present arguments going in both directions to convey what I think is the most accurate point of view, which is that the claim is both somewhat true and somewhat false.

The main argument supporting "yes":

  • It is true that Russell and Whitehead prove a claim on page 362 of the Principia Mathematica (using the page numbers of the edition linked to in @Oddthinking's answer) about which they state "From this proposition it will follow, when arithmetical addition has been defined, that 1 + 1 = 2." This implies that by that point in the book, they consider the claim that 1+1=2 to still not be proved. And they consider the claim proved on that page to be a main step forward toward proving that 1+1=2. (As Wikipedia states, the proof is only completed in Volume 2 after arithmetic addition is defined.)

Arguments supporting "no":

  • Just the fact that an author proves a claim on page 362 of their book does not imply that they "spent 362 pages to prove" that claim. It is quite possible that much of the preceding 361 pages were "spent" doing things that are tangential or even completely unrelated to the claim proved on page 362.

    Indeed, this appears to be true in the current example of the Principia Mathematica. To take a random example, Chapter III, spanning pages 66 to 84, concerns the topic of "incomplete symbols". I've never studied the Principia in detail so cannot authoritatively claim that there isn't anything in this chapter that's relevant to the proof of the claim on page 362, but it does seem unrelated to me, and at least the first couple of pages of chapter III have rambling philosophical-sounding discussions about the meaning of statements such as "Socrates is mortal", "Scott is Scott", "Scott is the author of Waverley", etc, which clearly have nothing to do with the claim that 1+1=2.

  • The claim that Russell and Whitehead "spent 362 pages to prove 1+1=2" is misleading in another way, since it suggests not only that the proof on page 362 relies in a logical sense on everything that precedes it (which as I said appears to be false), but also that proving this claim is the goal of all the preceding developments. In other words, it is a claim about the motivation that Russell and Whitehead had when writing the work leading up to the infamous 1+1=2 claim. It makes it sound like they spent a stupid amount of effort with their only (or main) goal being to prove something completely obvious that every child knows is true. But that's false. Their actual goal (discussed in the Wikipedia article and many other places) was much more ambitious, although, to their misfortune, we now know that that goal was unattainable thanks to the work of Gödel.

Another argument supporting "yes":

  • I think ultimately the claim does contain a kernel of truth, in the sense that this bit of history of mathematics lore is often cited to highlight the absurdity of Russell and Whitehead's efforts. They did in fact go to absurd lengths to formally prove things everyone considers obvious. And the 1+1=2 claim is probably the most extreme, easy-to-digest illustration of this aspect of the Principia, and one that unfortunately hurts the public image of mathematics and mathematicians to some extent, by giving the incorrect impression that we mathematicians (I am myself a mathematician, by the way ;-)) are obsessed with trivialities and with pointless formalism. This impression is common enough that even @Oddthinking, in his otherwise excellent answer, says "If you have only studied mathematics at school, the way it works at university/academic level can be quite alien". No! Even to most professional mathematicians working at universities the Principia seems "quite alien".

    The point is that if the saying that Russell and Whitehead "spend 360 pages to prove 1+1=2" is misleading and portrays these great thinkers in a worse light than they deserve, well, they did kind of do something to invite a bit of criticism and ridicule. They had noble aims of course, and a proper understanding of the context within which they were doing this work (as discussed, for example, here) makes what they were doing seem quite a bit more reasonable than the criticism makes it out to be. But ultimately, from a modern perspective I have to admit it seems pretty ridiculous.

Dan Romik
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    Interestingly, Newton's *Principia* also seems "quite alien" to professional physicists, and also to professional engineers, who rely upon Newtonian mechanics to a much greater extent than do professional physicists. Newton didn't use calculus in his *Principia*. (Actually, he did use it; he just went to great lengths to hide the use of it.) Getting back to mathematics, the Axiom of Choice also is quite alien. – David Hammen Jan 29 '23 at 10:26
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    What I think is alien to high school mathematicians but natural to academic mathematicians is thinking about "addition" as "any operator defined by having these properties..." rather than "putting the numbers in a column with the ones, tens and hundreds columns lined up, and using a specific algorithm to calculate it - don't forget to include the carry!". Similarly, 1 is defined by Peano's Axioms or Church Numerals or (in this case) Sets/Relations with a single element. I think that is natural to anyone who has passed Maths 101, but not the general public. – Oddthinking Jan 29 '23 at 12:27
  • I think I first got this idea from reading [Logicomix](https://www.amazon.com/Logicomix-search-truth-Apostolos-Doxiadis-ebook/dp/B0117S8JSS/ref=sr_1_1?keywords=Logicomix&qid=1674997233&sr=8-1), a graphic novel about Russell's life. So your words "the 1+1=2 claim is probably the most extreme, easy-to-digest illustration of this aspect of the Principia" just gets to the point. lol – Qiulang Jan 29 '23 at 13:06
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    @DavidHammen Even if Newton had been more explicit in his use of his calculus it is likely that it still would have appeared "alien" to modern physicists as modern calculus is actually based much more on Leibniz's formulation, symbology and notation than Newton's. (National rivalries being what they were, English-speaking calculus students generally are not made aware of this). – RBarryYoung Jan 29 '23 at 15:47
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    @DavidHammen the writings of Euler and Gauss also seem alien in many ways to someone reading them from a modern perspective. That’s not quite what I meant. I think Russell’s Principia is _more_ alien, because he works so hard to do something that we now understand to be basically pointless. That is not the case with Newton, Euler, Gauss etc. – Dan Romik Jan 29 '23 at 16:58
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    @DanRomik Newton used synthetic geometry rather than algebra and calculus in his *Principia*. While maybe not quite alien, that synthetic geometry is obscure (at least nowadays). If you've read his *Principia*, you would know what I mean. I'd call it "alien". What Newton used is no longer used for teaching or using Newtonian mechanics. – David Hammen Jan 29 '23 at 17:30
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    @Qiulang: I think what all answers so far have failed to point out is that the footnote quite clearly is tongue-in-cheek and proves that the authors are well aware of what their efforts must look like to the casual observer. In other words: the footnote on page 362 is meant as a joke by the authors. – Jörg W Mittag Jan 29 '23 at 18:55
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    "at least the first couple of pages of chapter III have rambling philosophical-sounding discussions about the meaning of statements such as "Socrates is mortal", "Scott is Scott", "Scott is the author of Waverley", etc, which clearly have nothing to do with the claim that 1+1=2." While not strictly necessary for proof, "Scott is Scott" is a very relevant example of equivalence, which absolutely must be established in order to prove 1+1=2. – 16807 Jan 29 '23 at 19:08
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    @JörgWMittag good point. My own sense is that the authors did know the humorous effect of the remark, but that it was actually meant in earnest. So it’s maybe half-joke and half a serious comment. – Dan Romik Jan 29 '23 at 19:23
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    When 1+1=2 is actually stated as a proposition, [on page 86 of part III](https://quod.lib.umich.edu/cgi/t/text/pageviewer-idx?c=umhistmath&cc=umhistmath&idno=aat3201.0002.001&frm=frameset&view=image&seq=126), it's accompanied by the comment "The above proposition is occasionally useful", which seems to me to clearly be tongue-in-cheek. – Micah Jan 29 '23 at 20:21
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    @Micah it’s tongue in cheek to say it’s occasionally useful, but that doesn’t mean that including 1+1=2 as a theorem was a joke. My impression is they were trying to make a serious point, but the tongue in cheek commentary was just an added humorous flourish typical of the writings of English intellectuals at the time (and also today). – Dan Romik Jan 30 '23 at 01:39
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    I think the source of confusion is that when we say "they proved that 1+1=2", it sounds to a layperson that there was reason to *doubt* that one plus one actually equals two, in some real-world sense. No mathematician doubts that. But if PM is to be an appropriate foundation for all of mathematics, one had better be able to derive from it any standard mathematical fact, such as 1+1=2, and so R&W verified that they could. So to me, it's not really that they proved 1+1=2, but rather that 1+1=2 can be derived from their axioms. The latter statement is far from obvious! – Nate Eldredge Jan 31 '23 at 04:32
  • The contents of the Principia are not vastly different in style from what you might find in a modern late undergrad/postgrad set theory course (other than the hideous dot-based precedence notation, which makes all the mathematical expressions basically unreadable). Of course it's going to look alien to mathematicians not familiar with foundations, but that entails neither that the book was fundamentally flawed and had no value, nor that the authors' effort is deserving of ridicule. – mudri Jan 31 '23 at 15:54
  • @mudri maybe a good way to look at it is that the Principia is basically a scientific dead end. Scientific research operates at the boundary of human knowledge, and sometimes our explorations take us in a direction that ends up being unproductive. There is no way to predict when that will happen, and no shame in it when it does. Nonetheless, I think it’s not unfair to say the Principia _is_ fundamentally flawed. It may still have some value, but considerably less than what its authors hoped it would have. – Dan Romik Jan 31 '23 at 18:57
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Why do we need to prove 1+1=2 in the first place?

I don't think anyone else has fully addressed this part of the question. Before this time there had been assumptions by many that arithmetic and symbolic logic were done, complete, and unquestionable. It was assumed that any well-formed statement in these disciplines could be proved to be true or false within the language of the disciplines. Russell was trying to lay the formal foundations to demonstrate this idea, and in 1910 - 1913 published his Principia Mathematica

The Problems of Philosophy is an introduction to the discipline of philosophy, written during a Cambridge lectureship that Russell held in 1912. In it, Russell asks the fundamental question, “Is there any knowledge in the world which is so certain that no reasonable man could doubt it?” Russell sketches out the metaphysical and epistemological views he held at the time, views that would develop and change over the rest of his career. https://www.sparknotes.com/philosophy/russell/section2/

Written as a defense of logicism (the thesis that mathematics is in some significant sense reducible to logic), the book [Principia] was instrumental in developing and popularizing modern mathematical logic. It also served as a major impetus for research in the foundations of mathematics throughout the twentieth century. https://plato.stanford.edu/entries/principia-mathematica/

Unfortunately this certainty was disrupted when Kurt Gödel in 1931 published his incompleteness theorems. He showed that there were limits to provability in formal axiomatic systems such as arithmetic and logic. https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

chasly - supports Monica
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    But of course, Gödel would never have got there without Russell leading the way. – Michael Kay Jan 31 '23 at 17:36
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    Gödel proved that within any axiomatic system there were statements that were unprovable. However that doesn't negate the need for axiomatic systems. Mathematics is all based on defining a set of assumptions and then deriving a conclusion. It's only natural to ask what are the minimal assumptions you can make, i.e. axioms. Also, most relatively useful statements are provable, the unprovable ones tend to be rather obscure. – Kidburla Jan 31 '23 at 18:22