Wikipedia:Reference desk/Archives/Mathematics/2023 May 17

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May 17

How do we know 1 + 1 = 2?

Who told us this was the case and why did we so readily believe them? ___ 216.168.139.240 (talk) 14:36, 17 May 2023 (UTC)

I can demonstrate that to you right now.

Here is one dot: ⋅

Here are two dots: ⋅⋅

If I take one dot: ⋅ and i put another dot: ⋅ next to it, I get two dots: ⋅⋅

Therefore, one and one is the same thing as two. Q.E.D.. Thus endeth the lesson. --Jayron32 14:39, 17 May 2023 (UTC)

The proof in Whitehead and Russell's Principia Mathematica (second figure in the article) is only slightly more complicated... --Wrongfilter (talk) 14:44, 17 May 2023 (UTC)

Explain how those two dots are in any way related to each other. I see two seperate instances of dots connected by nothing, explain how those two dots which still maintain their individual forms end up creating 2? Mathematics seems like subjective bologne is what I'm trying to say 216.168.139.240 (talk) 14:50, 17 May 2023 (UTC)

This isn't the place to play silly games. Please stop if you aren't interested in being useful or asking serious questions. --Jayron32 14:53, 17 May 2023 (UTC)

Oh I'm sorry, "Jayron32" I seem to have missed the section of the Reference Desk Guide that says that questions have to benefit anyone other than the asker. 216.168.139.240 (talk) 14:56, 17 May 2023 (UTC)

I've yet to be convinced that numbers aren't just social constructs... AndyTheGrump (talk) 14:51, 17 May 2023 (UTC)

I'm pretty sure that in the Peano axioms, this is because 2 is shorthand for the successor function of 1 (i.e. 2 = S(1)), and 1 + 1 = 1 + S(0) = S(1 + 0) = S(1) = 2. In other words, 1 + 1 = 2 because 2 is precisely what succeeds 1. This is just one way of axiomatizing the natural numbers though, and you may find different axiomatizations that have different ways of attaining 1 + 1 = 2. Generally speaking, however, I feel that in most systems, 1 + 1 is 2 precisely because whatever 1 + 1 is, needs to have a shorthand, for which "2" is most logical. GalacticShoe (talk) 17:07, 17 May 2023 (UTC)

Writing this with Chinese characters kind of drives the point home: how do Chinese schoolchildren know that adding  一  to  一  gives  二 ?  --Lambiam 18:27, 17 May 2023 (UTC)

I'm not so sure that's such a convincing argument after looking at the jellybeans in The Secret Mumber. ;-) NadVolum (talk) 08:22, 18 May 2023 (UTC)

My contribution was not an argument. Given that 一 + 一 needs a shorthand, the point I was making is that 二 is a convenient symbol to serve as that shorthand, as it more or less announces itself as being just that. The real argument is that the 2 from the original question is a symbol for representing 1 + 1, whatever that is.  --Lambiam 10:58, 18 May 2023 (UTC)

Way back in the 1960s, when I went on a Maths Camp at the age of 16 (yes, I was a nerd), I learnt that 1 + 1 = 10, in binary. That knowledge served me well in a long IT career. HiLo48 (talk) 08:42, 18 May 2023 (UTC)

You could add base -2 where 1+1 = 110 and balanced ternary to that where 1+1 = 1T. Both of those were used in early computers! NadVolum (talk) 11:40, 18 May 2023 (UTC)

I learned that on dad's 29th birthday. Mom put five candles on his cake and lit four. This was in 1967 so the gag had not yet been done to death, and it took him a moment to get it. —Tamfang (talk) 16:50, 18 May 2023 (UTC)

As mentioned above, a proof is given in Principia Mathematica. But a proof does not exist in isolation since it depends on commonly used axioms, definitions, and rules of inference. The goal of Principia Mathematica was to be a kind of reboot mathematics from as fundamental level as possible, starting with the rules of logic. A statement like 1+1=2 is actually meaningless until you define the concepts involved" What is 1? What is 2, What is equality? What is addition? After actually answering all these questions, Whitehead and Russell went ahead and proved the statement given their interpretation of the concepts involved. Peano's program was a bit different in that he created a system of arithmetic based on axioms similar to the way Euclid created a system of geometry based on a different set of axioms. For example one of Peano's axioms might be stated "1 is a number," where the concepts of 1 and "number" are left undefined, just as Euclid's axioms leave the concepts of "point" and "line". (At least in modern interpretation of Euclid; Euclid actually did include "definitions" but they could more reasonably be described as aids to intuition. Definitions define concepts in terms of other concepts, so a certain set of concepts must be left undefined in order to avoid circularity. Peano, as was the custom at the time, started his number system at 1, but the more natural starting point is 0 and, after modifying the definitions accordingly, his axioms work just as well.) I'd recommend Edmund Landau's Foundations of Analysis for a readable exposition of numbers and arithmetic starting from Peano's axioms and continuing to complex numbers. It will hopefully provide an clearer understanding of where arithmetic actually comes from starting from a solid foundation. (My favorite quote from the book, and actually one of my favorite quotes ever: "Please forget what you have learned in school; you haven't actually learned it." It's a reminder that you should maintain a healthy skepticism even when attending something as uncontroversial as a mathematics class.) Whitehead and Russell were more ambitions in that they wanted to put all of mathematics on a more solid foundation. In their approach though, they just kept digging until they struck not bedrock, but lava, which is probably a bit deeper than you'll want to go. --RDBury (talk) 14:36, 18 May 2023 (UTC)

Whitehead and Russell also never got there, Principia Mathematica is famously unfinished, and likely unfinishable, though the task and its unfinishableness is likely what led to some of the more modern metamathematical advances like Gödel's incompleteness theorems and Zermelo–Fraenkel set theory, and the like. Which is to say that PM is foundational to modern mathematics, but not in any way in the way that Russell and Whitehead intended. --Jayron32 15:23, 18 May 2023 (UTC)

At least in modern interpretation of Euclid; Euclid actually did include "definitions" but they could more reasonably be described as aids to intuition. – we might also add that whether the "definitions" are original to Euclid is a subject of controversy, with many scholars judging them to be a later interpolation. –jacobolus (t) 15:58, 18 May 2023 (UTC)

In the context, we may read "Euclid" as short for the collection of authors of the Elements as the work has been transmitted to us, just like "Homer" is "the" author of the Ilias and Odyssey, and Bourbaki of the Éléments de mathématique.  --Lambiam 19:46, 18 May 2023 (UTC)

At a fundamental level, the answer does not only depend on the meanings of 1 + 1, = and 2, but also on what it means when we say that we "know" something. One may even ask how we know that we know something, or for that matter anything. I think that most people spontaneously develop an abstract notion of the number of objects in a collection of sufficiently equal objects and can verify mentally, rather than experimentally, that joining a pair of singleton collections results in a collection containing one pair. This intuitive method can be formalized, but it takes a mathematical mindset to verify that this formalization faithfully reflects the intuition. This verification can not itself be formalized; it can only be done by intuition and be supported by experimental observation.  --Lambiam 20:15, 18 May 2023 (UTC)

I asked Bard this question and got a hilarious proof in response:

Here is a simple proof:

• Assume that 1 + 1 is not equal to 2.
• Then, 1 + 1 must be equal to something else.
• Let x be the number that 1 + 1 is equal to.
• Then, 1 + 1 = x.
• But we also know that 1 + 1 = 2.
• Therefore, x = 2.

This proof shows that 1 + 1 must be equal to 2, no matter what x is.

--Lambiam 20:21, 18 May 2023 (UTC)

Because of ring theory and abelian groups. EvergreenFir (talk) 20:38, 18 May 2023 (UTC)

My favorite is the old one:

Teacher: Suppose x is the number of sheep ... ...

Pupil: Yes I can see that follows okay. But what happens when the number of sheep is not x?

NadVolum (talk) 22:03, 18 May 2023 (UTC)