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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?

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    @IMSoP It's been asked and answered multiple times at mathematics.SE. Jan 29, 2023 at 11:17
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    @DavidHammen can you give an example at mathematics.SE? I searched questions there and only found math.stackexchange.com/questions/348889/is-11-2-a-theorem, which is not exactly what I asked here.
    – Qiulang
    Jan 29, 2023 at 12:32
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    @Qiulang Here's another: 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. Jan 29, 2023 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?
    – CGCampbell
    Jan 30, 2023 at 13:02
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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 Mathematics 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, 2023 at 10:34

3 Answers 3

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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.]

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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. Jan 29, 2023 at 8:16
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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, 2023 at 8:39
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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, 2023 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. Jan 29, 2023 at 18:52
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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, 2023 at 10:14
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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.

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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, 2023 at 12:27
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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, 2023 at 16:58
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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. Jan 29, 2023 at 18:55
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    When 1+1=2 is actually stated as a proposition, on page 86 of part III, 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, 2023 at 20:21
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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! Jan 31, 2023 at 4:32
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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

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    But of course, Gödel would never have got there without Russell leading the way. Jan 31, 2023 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, 2023 at 18:22

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