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Although everyone is surely wondering about this question, I am asking it here though:

Why is 1+1=2?

Why is our universe build up like in that way, that such a formula is always working? (I know the Newtons basic laws: e.g., Where an "atomical" mass is can not be another at the same time, therefore it has to be: mass + mass = 2 x mass)

Are there other places in our "extistence" (universe etc.), where it can be in another way - even hypothetical? (The - possible - anti matter "world" seems to be working in that way too).

And additionally questioned: Why there are mathematical functions, which have at some point no result (better spoken: ininity?). Are they comparable to, e.g., black holes - or the event horizon?

Could it be that our universe is like mathematical/bounding "box", in which - if we could - reach the e.g. the top end, we get immediately to the bottom, e.g. like the program

move_right: x=x+1; if x> (a_value) then x= 0;

move_left: x=x-1; if x< 0 then x= a_value;

(same for y and z direction)

Before anyone here is "voting" to close this question as off-topic: I've asked it already in Astronomy, where the people told me to drop it in another forum!

Addendum 20170821: I meant it in special in materialistic way. Why couldn't materia not behave like intense way, although there are enough place betweenin (sub-)atomaric level between the nucleus and the surronding electrosn (otherwise the light quants/waves woulnd't pass eg. glass / H2O molecules etc.)

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    Welcome to Philosophy.SE. About 1+1=2: does philosophy.stackexchange.com/q/8738/2953, esp. "What [Peano] was attempting to do was find a set of axioms that accurately captures our intuition about how the integers act," answer your question? It's not clear to me what you intend to ask with your second question - why would these functions be "comparable" and in what sense? I also don't see how this question or the third one is related to philosophy as defined in the help center. As for the third question, are you looking for anything more than "Yes, it could, but why would it?"? – user2953 Jun 29 '17 at 21:23
  • Thanks for the answer. I followed the physical way about the question 1+1=2, not the [peano]. The second question I was thinking about that whether there are possible places which don't base on the phsyical formula. The third question (why there are functions which have an infinite result at some point) I was thinking about posibility that a black hole is like such a function with an inifite result inside it. Of course (but I do not ) could ask physics scientists Stephen Hawking et al. :) about that, but I would like to know whether there are other explanations and ideas about that – aprogrammer Jun 29 '17 at 21:41
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    As Einstein said "As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality". This formula is "always working" because it expresses our convention for using symbols 1, 2, + and =, not anything about the universe. For physical things it will not always be working: when we put two droplets of water next to each other they merge, so 1+1=1 in that case. The question of physics is when our made up rules for symbols approximate something real, and the reason they often do is that those that do are just more useful to us. – Conifold Jun 29 '17 at 22:22
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    First answer: we define 2 as 1+1; more precisely, we define 2 as the successor of 1 and we prove that, in general, n+1 is equal to the successor of n. – Mauro ALLEGRANZA Jun 30 '17 at 6:02
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    There is plenty of "other stuff" that does not, many quantities are intensive, like temperature, adding objects with the same temperatures does not add their temperatures, etc. Some stuff obeys it, and it happens to be useful to us, which is why we invented symbols that mimic that. The order of explanation is in reverse. – Conifold Jun 30 '17 at 12:49
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What do you mean by 1, what do you mean by 2, and what do you mean by +?

This is really the key question here -- if you're just taking a naive view of numbers as 'objects that count how many things I have' (these are really cardinal numbers), then of course 1+1=2. If I have one object and I then gain another object I have two objects total, and we have picked the symbols 1 and 2 to represent these notions with the symbol + to represent the aquisition of more objects.

But what if we are keeping track of the number of particles in a quantum system? Say we shoot 1 proton at 1 other proton at about 99% of the speed of light -- do we now have 2 protons? Not so! In particle physics we have found that if you produce enough energy to replicate a given particles rest mass and you confine the energy to small enough of an area, the particle can come into existence in exchange for the energy (this is a very rough outline). The particle collider over at CERN takes advantage of this fact to study some of the most fundamental physics in our universe by slamming two protons into each other* with a bunch of extra kinetic energy (hence 99% LS), at which point literally hundreds of new particles are created and tracked using magnetic fields and photomultipliers and many other complicated devices.

So if 1 and 2 and 3 and so on are counting the number of particles we have and + represents putting them together with great force, then 1+1=500 or something like that. We could instead choose to use numbers for keeping track of the scalar quantity we call energy throughout the above process and argue that this number does indeed follow regular addition laws, but then you have specified entirely what you mean by 1 and 2 (and +)! This is really the heart of the answer -- 1+1=2 whenever the quantities you have defined to be 1 and 2 together with the relation + satisfy this equality.

*the protons do not actually collide -- they are forced to within incredibly small distances of eachother, at which point vector bosons can mediate short range interactions between the momentum states of the protons to effectively produce a 'collision'.

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1 + 1 = 2 is a mathematical statement. To say it is "always working" in the universe is a bit tricky. There's a selection bias hidden in that statement. A more reasonable phrase would be "1+1=2 always works in our universe for topic which we have found it to always work for." It turns out that we attribute great importance to the facets of the universe where the laws of addition apply, but there's plenty of cases where it doesn't work out that well.

Consider random walks as an example. In random walks, 1 + 1 does not equal 2. In fact, if you're doing your random walk in one dimension, there' a 50% chance that the two random walks will be in opposite directions, suggesting 1 + 1 = 0!. Random walks happen all the time in nature. That's why air molecules can collide with each other billions of times every second and generally stay in the same place.

Also, there are gestalt properties. Gestalt is actually a concept built around 1 + 1 not equaling 2. The official phrasing for a gestalt is "a system where the sum of the parts is not the same as the whole." I'm quite certain that you would not be comfortable with me cutting you in half, and arguing that two halves are just as good as the whole was. It's not a place where you would like to have to argue 1/2 + 1/2 = 1.

In religions, we very often find marriage phrased as "two flesh become one." That's certainly not 1 + 1 = 2. That's 1 + 1 = 1!

So really, the universe is full of all sorts of different properties. Some of them can be modeled using the laws of arithmetic, some cannot. If you find that the universe "always works" using the laws of addition, then that suggests that your preferred ways of viewing the world select for those laws.

As for your torroidal universe (which is the technical name for your universe in a box), the counter question would be "why couldn't the universe be torroidal?" What fundamental laws does the universe uphold that would prevent it? The joy of hypotheticals like that is you never have to prove that the universe is torroidal, just that there's nothing preventing it. As for what it would mean? It turns out that the laws of physics you use work just fine on a torroid. They don't require a flat space (which is good, because relativity suggests space is not locally flat!)

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Whole volumes could be written about the various attempts, starting with Leibniz and not yet ending with Russell, to prove from the axioms that 1 + 1 = 2. Of course the answer why 1 + 1 = 2 will be answered in different ways by the experts of different scientific branches, depending on their points of view. But there is a very simple and, as far as myself and many friends and students are concerned, also very satisfying answer.

Mathematics, first in its basic structures including counting, and the four basic operations, and later in many productively evolving branches, has been shaped by the physical environment. Therefore it is in no way surprising that it describes this environment. Whether you take two apples or two sheep or two equal masses: The result is always twice of the single property. Even in modern mathematics numbers have been identified with cardinalities of sets. Three is the cardinal number of all sets that contain three elements. (It is not allowed to talk about the set of all such sets, but we say class, and everything is fine.) Three is the number of trees in my garden. And 7*7 = 49 according to the tables which everybody has learnt in school and which they can check with a handful of coins.

Therefore there is absolutely no reason to brood about the unreasonable effectiveness of mathematics in the natural sciences. Mathematics simply has been constructed that way. Therefore it is not at all necessary to "prove" that 1 + 1 = 2. And if mathematicians would choose a different axiom system where 1 + 1 = 3 could be proved, then this kind of mathematics would simply become irrelevant for any practical purpose as we can observe it with other theories which are completely decoupled from reality.

  • "Whether you take two apples or two sheep or two equal masses: The result is always twice of the single property." This is at best simplified and at worst fundamentally incorrect. Adding speeds doesn't behave like that at all for example. Adding masses doesn't once you consider quantum physics. Saying mathematics was purposefully constructed to describe these things far beyond even our current intuition is also historically incorrect and epistemologically absurd. – MM8 Aug 18 '17 at 15:17
  • On top of that, 1+1=2 is not axiomatic as you seem to imply, it's a definition (those are not the same thing). Writing "2" literally just means 1+1, or S(1) or whatever you prefer. It's just notation, there is no mathematical knowledge in 1+1=2 which needs to be proven. "2" is just a name for the natural number after 1. "+1" is the operation which assigns to each natural numbers its successor. The real question that should be asked here is why do we define N a certain way and not another (and this is a fascinating subject, despite you seemingly trying to play it down as inconsequential). – MM8 Aug 18 '17 at 15:22
  • o + o = oo is not an axiom but a law taken from the reality of bodies in the world. 1 + 1 = 2 is not an axiom but a definition of the meaning of 2 like all statements in the tables from 1*1 = 1 to 10*10 = 100 and beyond. Axioms are sometimes useful but not really necessary in mathematics. Why do we define |N the way we do? The answer is simple: Because everything else would lead to contradictions when being applied. (Your examples with velocities etc, do not belong to this realm - like 1 + 1 = 3 in biological reproduction.) – Heinrich Aug 18 '17 at 16:28
  • You're completely oversimplifying the issue and you're also completely wrong. Non-standard models of arithmetic have been constructed which generate number sets that you'd find very, very alien, yet still work for every single practical application like counting and the trivial addition you've used as your example. The notion that the natural numbers are the only way to get this done simply isn't true. See en.wikipedia.org/wiki/Non-standard_model_of_arithmetic . – MM8 Aug 19 '17 at 2:22
  • You are completely wrong. There is no practical application of any part of a non-standard system that differes from the natural numbers. These systems are as useless for mathematics as astrology or vodoo. I will prove that when you tell me what you think is a practical application. – Heinrich Aug 19 '17 at 12:11
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Alfred North Whitehead and Bertrand Russell wrote Principia Mathematica wherein they tried to lay out the foundations of mathematics. After years of intensive research, they eventually published it in three volumes in the years 1910-1913.

On page 362, they finally got around to proving that 1 + 1 = 2 :

enter image description here

Unfortunately there's but a handful of people in the world with both the intelligence and the patience to get to page 362 and be able to understand their conclusion. I'm not one of these people, so I guess I'll just have to take their word for it! Maybe you can figure it out now you know where to find it?

Anyway, I did read Logicomix, which is a graphic novel that describes Russell's pursuit for knowledge... including his work on the Principia Mathematica. I highly recommend it for anyone who wants to learn more about the history of the philosophy of mathematics... who wants to learn more about Bertrand Russel's contributions in particular... or who wants to get a better understanding of why it's nearly impossible to prove something as fundamental as 1 + 1 = 2.

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    This answer doesn't really server to answer the question. All you've done is state that Russell and Whitehead attempted to create a foundational system in PM and they give a proof of 1+1=2, but you don't attempt to explain their proof, explain the nature of PM or type theory, explain who they are, or explain any of the details related to what you've posted. The OP asked "why does 1+1=2?" and all you've done is say that two people answered that question but you personally find their answer confusing. – Not_Here Jul 27 '17 at 19:13
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    Not sure a graphic like that helps anything whatsoever. – jeffreysbrother Jul 27 '17 at 19:28
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    You don't have to read every single page of PM to understand it or to learn about type theory, most CS undergrads could handle everything in that book if it was written with modern symbols and they paid attention in discrete math class. The fact that your answer just claims that they had a proof but its too esoteric to understand so oh well maybe someone else can understand it is really just driving home the point that it isn't a useful answer at all. – Not_Here Jul 27 '17 at 19:47
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    I completely maintain that he is being hyperbolic. Even the wikipedia page for PM has an easy to understand explanation of the symbols and they're all symbols that anyone who has taken a class that covers propositional and predicate logic understand. The SEP even has an awesome article specifically about the notation in PM, it is not some arcane hieroglyphic ridden tome, it's a standard layout of type theory using outdated symbols. – Not_Here Jul 27 '17 at 20:05
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    @jeffreysbrother : I see. I replaced the screenshot with a better quality screenshot! Not that it helps me understand what's on that page, though ;-) – John Slegers Jul 28 '17 at 8:18
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The principle of Maths is to use them in order to solve logical problems. In another universe, if you have one apple and I give you another one apple, how many apples will you have? If you will have 3 apples, then 1+1 will be 3, because the people in that universe have been developed an arythmetic system which correspond to their realities and needs.

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