What would happen if suddenly, 1+1=2 is disproved?
Would the universe be thrown into chaos when the most fundamental equation is proved to be wrong?
1+1=2 is not "the most fundamental equation": it is a theorem of arithmetic, a simple consequence of arithmetical axioms and definitions.
**IF** 1+1=2 is a sort of "law of the universe", and thus universally TRUE, you cannot disprove it. **IF** we can imagine to disprove it, it is because it is in some sense a human convention or mental construction: if so, why do think that the universe will care about us ?
If you're interested on the effects on an individual, try Division By Zero (by the same author as Arrival). It covers this exact scenario.
If you want some historical grounding in this topic, look to what happened as we discovered relativity and how it "invalidated" Newtonian mechanics. We still use Newtonian mechanics to build bridges and buildings, but it's not the only tool in the toolbox for calculating orbital paths. Newtonian mechanics were disproved (or rather, shown to be good approximations but incomplete) and yet they are still incredibly useful tools.
If you "prove" that 1+1=2, then it would mean that you're using a different axiomatic frame... so... you must have created a new arithmetic! Good for you! Now... what applications would it have? Maths are not fundamental to the workings of the universe, but to our understanding of those workings... you'll need to work on how this new arithmetic is useful... but that's all. Good luck changing everybody's mindset
@Barranka It is also possible that the proofs that 1+1=2 are all wrong, and everybody is under some illusion.
@Ovi LOL... it could be... as a friend of mine said once: "there must be a space where zero millions is... a lot of money!" It would be nice to prove that! It would be interesting if a consistent system allows to disprove both "1+1=2" and "0 is nothing"
1 + 1 = 2 cannot be disproved because it is true by definition. The set of integers is defined by the quantities 0,1 and the rule that for any value n in the set of integers, the next value will be n + 1 (or conversely, the previous value will be n - 1). The idea of 1 + 1 will be 10 and not 2, if you are counting in base 2 is specious in the context of this question; numeric base is merely a system of representation - a choice. There is also the idea that 2 != 2, but rather 1.9999...; this is also specious because it is about approximation and the relationship between real and integer numbers.
Is the world in chaos now? Because one plus one is not equal to two, at least not all the time.
Take one liter of water and one liter of sand. Add them together. What do you get? Wet sand, but certainly not two liters of it.
Take one rabbit and add one rabbit. Add them together. You have a reasonable chance of ending up with quite a bit more than two rabbits, if you wait a sufficient amount of time.
Even in the realm of pure mathematics one plus one is not necessarily equal to two. If you're working with modulo two arithmetic, 1 + 1 = 0. If you're dealing with modulo two arithmetic and 1 + 1 = 2, you've done something very wrong. -- Also, it's not like modulo two arithmetic is an obscure side-note - your computer is using it right now in the form of "bitwise xor", and modern computers could not function without it. (Though admittedly, modulo two arithmetic is rather simple in its properties, so there's not a lot of mathematicians that bother to study it.)
Mathematics is based on axioms - assumptions about the properties of a system - and the implications that follow logically from those systems. If one of those implications is found to be "counter-factual", then either the logic was invalid, or one of the axioms was incorrect for that system. - For that system is an important bit. Just because something is counter-factual for one set of axioms doesn't mean that it's counter-factual for a different set of axioms.
Take Euclid's parallel axiom. Include those with the rest of Euclid's axioms, and you get Euclidian geometry. This is the "standard" geometry which you and I are familiar with, and with which a substantial fraction of mathematicians operate. However, you can set up different geometries where this doesn't hold. In fact, modern physics tells us that we're actually living in a non-Euclidean geometry - advanced physics would not function in a true Euclidean geometry where the parallel axiom holds.
Now does that mean that Euclidean geometries and the parallel axiom are wrong? No. It's a perfectly valid mathematical construct which hundreds of thousands of mathematicians and engineers - and physicists - use daily. The fact that Euclidean geometry has axioms which produce results inconsistent with the observed world doesn't mean Euclidean geometry is invalid, it just means that those axioms don't apply to the system you're observing. It doesn't mean that they won't apply - or even that they aren't the best ones to use - in some other situation.
So 1+1=2 is a very convenient observation, and holds in many cases. But not all. Sometimes 1+1=0, or some other number. Just because the axioms of standard, natural number arithmetic don't hold for a particular system doesn't mean they're invalid, it just means they're not applicable to that system, and you have to come up with another set and another arithmetic system.
Or, you could redefine your system such that the axioms do hold. (That's what the people frantically typing "But if you ..." comments below are doing. "If you keep them in separate containers, if they're both female, if we ignore modulo arithmetic ..." If you redefine things such that the axioms hold, the logical consequences of those axioms logically follow.)
A more compelling example would be to mix 1 liter of water with 1 liter of alcohol (neither the sand/water thing nor the rabbit thing give me a good impression of violating 1+1=2).
Nitpicks: In modulo-two arithmetic, 2 ~= 0 (they're in the same "equivalence class"), so you can validly say 1+1 = 2, or 1+1 = 42, or 1+1 = -9002. You haven't done anything wrong if you say 1+1=2 in mod 2. Second, although modulo two arithmetic is simple, the resultant mathematics can be decidedly nontrivial. Polynomials over GF(2)) underlie a significant amount of modern cryptography and error-correcting codes, even making an appearance in those ubiquitous QR codes.
"advanced physics would not function in a true Euclidean geometry where the parallel axiom holds" Out of curiosity, are you talking about relativity, quantum, field theory, or something else?
Your answer seems quite confusing to me as it contains so many small mistakes I think. 1 + 1 = 2 is either a mathematical statement, in which case your answer misses the point that this is not a fundamental truth, or about real world things. In this case, what you wanna say is: 1 + 1 is not = 2, sometimes it may be but it's by far not a fundamental truth. If you argue in the second way, please state that your answer is non-mathematical and leave the math beside.
Decibels work on a logarithmic scale so adding a 1Db sound source to another 1Db sound source then the result of 1Db + 1Db is approximately 4Db (10Log(10^(1/10)+10^(1/10)) = 4.0103Db).
A real-world example of modulo arithmetic is in anything periodic. For example, if I stand facing west, and rotate by 180°, I am facing east. Rotate by 180°, and I am facing west again. Indeed, it's rather *rare* finding examples of where `1 + 1 = 2`; and these contrivances are well described in the laws of numbers - they only apply to the "tangible world" in so far as you keep mind of how the real world *differs* from this ideal.
What utter nonsense! In the absence of explicit annotations, `1+1=2` is a pure mathematical equation. If you want to wander into chemistry, you have to say so first. Ditto for modulo arithmetic, or for numbers which turn out to be logarithms.
@CarlWitthoft But that's the point, it's not nonsense. You have some implicit assumptions. If someone found that `1+1 != 2` that would mean one of the assumptions was flawed. You can deal in those areas where those assumptions don't apply all you want, you just have to state them. In fact, exactly this happened as we transitioned from Newtonian mechanics to relativity.
@CarlWitthoft - This answer may be riddled with nonsense – but so is the question! _"Would the universe be thrown into chaos if 1+1 = 2 were disproved?"_ If you don't like this answer, I'd like to see you write one that better salvages the inquiry – especially from a philosophical perspective. As for me, I think 1 nonsensical question + 1 nonsensical answer = 1 heckuva clever retort.
Umm, perhaps before we have this discussion we need to define our terms. Mixing a unit of sand with a unit of water is not the mathematician's definition of "plus". Math students routinely get confused when they try to add 3x + 2y and say the answer is 5xy or some such, which leads math teachers to routinely say, "You can't mix apples and oranges". Likewise, unless someone says otherwise, mathematical expressions expressed in English are understood to be base 10, operating on integers, I'm sure a mathematician could list dozens of other implied assumptions. ...
... So sure, if you say, "When I write the symbol 1, I mean 'a dog', and when I write the symbol 2, I mean 'my sister's left hand', and by + I mean 'is looking at' and by = I mean 'transmogrifies into', then 1+1=2 is false", one can only say, well, sure. So what?
*Take one liter of water and one liter of sand. Add them together. What do you get?*, **Wet sand, but certainly not two liters of it.** Really? By how much? These material are quite incompressible, aren't they?
@Surb Unless it's very compact sand, the water fills in the microscopic gaps in the sand.
"You have a reasonable chance of ending up with quite a bit more than two rabbits, if you wait a sufficient amount of time." - This is questionable. If you pick two rabbits at random, you might get `male-male`, `male-female`, or `female-female`. Only one of those configuration might produce additional rabbits. So there's a good chance that no matter how long you wait, you'll only have two rabbits (until eventually you have one, and then eventually none).
@Surb - Neither is compressed. The individual grains of sand don't pack together perfectly. There are tiny pockets or air around each one. The water displaces the air, allowing the total volume of water + sand to be less than two liters. However if you track the total volume of water, sand, _and_ air, that will stay _closer_ to two liters.