I remember hearing of a mathematician who a rather long volume proving that 1+1=2. Apparently, he had a very strict definition of “rigor”. Did this really happen? And if it did, did anyone actually read it?
You’re thinking of a large section of Principia Mathematica by Bertrand Russell and Alfred North Whitehead, written 1910-1913 (not the book of the same title by Sir Isaac Newton). They attempted to put a rigorously strict underpinning under all of mathematics, including a famous attempt to prove for all time that one plus one equals two.
Hundreds of pages later, they still couldn’t do it and gave up. Apparently Godel’s Incompleteness Theorem got in the way.
An actual mathematician could provide a better explanation.
Ah yes, those names sound familiar.
So… did anyone actually read those hundreds of pages?
I guess if they were unsuccesful(which is actually a pretty amazing thing), then there wouldn’t be much point.
If only I knew about Godel’s Incompleteness Theorem when I was in elementary school. Maybe I could have gotten better grades in math. “I got the question wrong? Prove it!”
AFAIK, it is absolutely impossible to prove that 1 + 1 = 2 since this is an axiom: The correctness of this equation is generally assumed, but never proven. Yet nobody doubts it, so one can renounce waterproof evidence and take this axiom as an assumption for real, mathematically correct, evidence.
I’m not a mathematician either, but, it would seem to me that whether (and how easily) the proposition can be proved depends on your definition of “1” and “2” (and addition, for that matter).
For example, addition could be defined through the use of a “lookup table” combined with the familiar rules for “places” and “carrying” that are taught to first graders.
This seems a reasonable definition, and using such a definition, it’s easy to prove that 1 + 1 = 2.
I suppose that Russel and Whitehead used definitions that were more fundamental than what I just used.
The OP was indeed presumably thinking of Russell and Whitehead’s Principia Mathematica, but ElvisL1ves is incorrect in going on to suggest that they failed to prove that 1+1=2 and so gave up. It certainly took them hundreds of pages to get remotely close: Cambridge University Press still keep the start of the project in print as *Principia Mathematica to 56, the most famous passage coming on p360, after they prove theorem *54.43:
“From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2.”
So even then, they still have a bit to go. But they get there eventually. On the other hand, the whole three volumes weren’t just about doing this; I think most of the third volume (written by Whitehead alone) is deriving the basics of geometry.
Why so much effort and why were they bothering? Very roughly, the idea was that, rather than just saying that there are numbers 1,2 etc which obey certain rules, the number 1 is the set of all sets that have one thing in them. And two is the set of all sets with two members etc. Regarding sets as more “fundamental” than numbers, they wanted to be able to derive all of arithmetic (and then the rest of maths) from these sets. Talking about things like “the set of all sets that…” can be difficult and the whole apparatus of the book is to make such stuff rigorous. Now you might think that defining numbers in terms of sets is screwy, but Russell and Whitehead weren’t stupid. They had philosophical reasons for wanting to reduce maths to an applied version of logic and the project was also a natural extension of much of what was going on in maths at the time.
Nor does Godel invalidate their proof that 1+1=2, as such. His paper was called On Formally Undecidable Propositions of Principia Mathematica* and Ralated Systems* and this isn’t one of the propositions he’s refering to. What he showed is that there are other theorems in arithmetic that are true, but which Russell and Whitehead could never have proved with their methods, no matter how many hundreds of pages they used. (Assuming the system is consistent.) Since they wanted to derive all of arithmetic, at least in principle, this undercuts the whole project. Nobody now believes that this is the proof (whatever that means, post-Godel) that 1+1=2, but it’s still a valid proof that 1+1=2 in a limited (and arguably uninteresting) sense. Not so much wrong, as obsolete.
As to whether anybody’s ever read all 3 volumes, it seems possible that perhaps a dozen people did. Norbert Wiener certainly read much of it, to judge from remarks in Ex-Prodigy, but I don’t know if this was all the way through. The introduction by Russell lays out the basic ideas in about 50 pages and is a brilliant piece of technical writing, recommendable to anybody who has some knowledge of formal logic.
Thanks for the clarifications, Bonzer.
Sounds like Russell and Whitehead were reduced to saying “It’s elephants all the way down”, though.
You can do this at home.
Take one penny. Put another one next to it. The number of pennies in front of you is designated by the English language as “Two.” Thus, by adding one to one, you get two.
I don’t care what your theorems say. 1 + 1 = 2. A theorem that says that you don’t get 2 pennies in the aformentioned process is flawed.
Yeah, but this only holds for pennies. Maybe if I try this with some weird object, I’ll get a different answer. And suppose I redefine “one” and “two”? Then you’re out of luck.
lucwarm hit the nail right on the head. The definitions of “1”, “2”, and “+” (not to mention “=”) are what matter here. I’ll give a brief outline of the problem as I’ve seen it posed. I’m not sure this is how Russell and Whitehead posed it, but this is how some other famous mathematicians have. If you don’t like math, you might want to stop reading now.
The natural numbers are defined starting with zero, which is a name given to the empty set. The successor of a number n (denoted n’) is that number unioned with the set containing that number (recall that the union of two sets is the set containing all objects contained in either set). One is defined to be 0’ and two is defined to be 1’ (or 0’’). So zero is {}, one is {{}}, two is {{}, {{}}}, and so on and so forth. For the curious, equality between numbers is set equality, and two sets are said to be equal iff they have the same members.
Addition is defined as follows:
n + 0 = n
n + m’ = (n + m)’
There are technical concerns with this definition (cf. Edmund Landau’s Foundations of Analysis). So the task at hand is to prove that 0’ + 0’ = 0’’. That’s not easy, especially when you want to give rigorous definitions of “true” and “proof” and things like that.
I hope there are no errors in what I’ve written, but it’s late and I’m tired. For an excellent introduction to the topic of set theory (which is where this arises from), I would recommend Paul Halmos’s Naive Set Theory.
You got a cite for that? Or do you expect us to just take your word for it? 
Of course you should just take my word for it. It really bothers me that Landau defines the natural numbers to start with one. Who without the qualifications to discuss these things would have feelings on that issue?
Either of the books I mentioned (Landau’s Foundations of Analysis or Halmos’s Naive Set Theory) are good sources. For the more adventurous reader, Elliott Mendelson’s Introduction To Mathematical Logic, 4th Ed. is a really good introduction to the more technical issues.
Of course, you could skip all that and just read the Principia. It’s your life, man.
Can anyone prove that 1+1 + anything BUT 2?
I think we could all spend our time a little more wisely.
Yes. Given proper definitions of 1, +, and =, I could prove that 1 + 1 = 3, 1 + 1 = 1, 1 + 1 = 0, 1 + 1 = any number you choose, or that 1 + 1 is undefined.
**
Believe it or not, there is a use for this. Math is a tool that a lot of people use. It’s rather sophisticated, and when it breaks it can be rather difficult to fix. A large part of Russell’s work was formalizing math to avoid a difficulty cleverly named “Russell’s paradox”, detailed below. Part of his work involved proving that what we wanted to be true inside of his formal system was in fact true. One of the things we wanted to be true was that 1 + 1 = 2; hence, the proof.
Russell’s paradox: If we assume that a set is any old collection of objects, we can consider sets that contain themselves as elements (this is purely abstract; you’re better off not trying to match this to physical reality). We can then consider the set of all sets which do not contain themselves as elements. A rather natural question to ask is whether this set contains itself. If it does, then by definition it shouldn’t; if it doesn’t, then by definition it should. So the assumption that sets exist leads us into trouble. It is for this reason that we must prove everything we can.
I wish I could contribute something mathematical here, but after bonzer and ultrafilter, all I can come up with is this -
I once gave a biograpical term paper on Russell for a math class. I recall reading that he said that he had a dream about Principia Mathematica. He dreamed that in the distant future, some librarian was going through a pile of books with the task of deciding what to keep, and what to throw out. The librarian picked up the last remaining copy of Principia Mathematica on earth, and paged through it for a while. After becoming bored, confused, or disinterested, the librarian closed the volume - and hefted the book in his hand, deciding it’s worth on the weight of the volume alone.
Russell says he woke up at this point, never knowing whether the volume was preserved or not.
Anyone interested in how logic and set theory are used to define arithmetic should take a look at the following:
It’s not quite what Russell and Whitehead were trying to do…they don’t have a rigourous proof that 1+1=2. They do have a rigorous proof that 2+2=4, however, with detailed proofs of all of the sub-theorems all the way down to the axioms. Each theorem is on its own page with links to the axioms and subtheorems used to prove it, and to other theorems which use it in their proofs…it’s just enormous.
Hey, that looks pretty cool. I’ll have to spend some time checking it out.
Given that:
“1” indicates a value of the set from .5 to 1.49
“2” indicates a value of the set from 1.5 to 2.49
“3” indicates a value of the set from 2.5 to 3.49
then “1”+“1” does not necessarily result in a value that falls within the set of “2”. For example, 1.4 (definable as “1”) + 1.3 (definable as “1”) would equal 2.7 (definable as “3”).
Or, in other words, 2+2=5 for sufficiently large quantities of 2.
Doesn’t 1 + 1 = 10 ?
I don’t know, Rhthm, I was always under the impression that nothing was more than the sum of it’s parts, and if you only have one of something, and you combine it with one of another, you only remain with the single combination, or one.
–Tim
Sorry Homer, I really didn’t have anything too constructive to add, so I shouldn’t have taken up the bandwidth with semantic hoohas . Oystaman asked if 1 + 1 could be anything other than 2. If I am not mistaken (a very, very likely possibility) in binary, 1 + 1 = 10 - same quantity/ value of 2, but not the digit 2. Silly Rhythm.