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Would the universe be thrown into chaos when the most fundamental equation is proved to be wrong?

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    1+1=2 is not "the most fundamental equation": it is a theorem of arithmetic, a simple consequence of arithmetical axioms and definitions. – Mauro ALLEGRANZA May 28 '18 at 6:16
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    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 ? – Mauro ALLEGRANZA May 28 '18 at 8:09
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    How do you define the symbols 1, 2, +, and = in that statement? – jpmc26 May 29 '18 at 3:29
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    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. – JakeSteam May 29 '18 at 8:04
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    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. – corsiKa May 29 '18 at 17:24

14 Answers 14

109

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.)

  • 7
    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). – H Walters May 28 '18 at 19:13
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    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. – nneonneo May 28 '18 at 20:49
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    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. – Mayou36 May 29 '18 at 8:56
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    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. – Carl Witthoft May 29 '18 at 17:16
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    @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. – corsiKa May 29 '18 at 17:22
102

As any mathematician will tell you, 1 + 1 = 2 follows trivially from definitions, and is not a theorem. Your question makes no sense.

It is as though you declared:

I define 1 fluid zounce to be exactly 30 millilitres.

But what if it turns out I'm wrong?

It is your definition. It cannot be wrong because fluid zounces, prior to your definition, simply did not exist.

  • 3
    Could one read their question, more charitably, as "what if we discover that 1+1 = 2 does not follow from Peano's postulates?", so that it retains whatever philosophical edge it has? – Schiphol May 28 '18 at 11:34
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    I would dispute that every mathematician will say that 1+1=2 is a definition. I see your point obviously but in general 2 will be S(S(0)) rather than 1+1. So there's an argument to be made that S(S(0))=S(0)+S(0) it's a trivial argument straight from the definition of +, but one that eventually ends up being sort of tricky due to the whole infinite induction you need when you want this to work in general. – DRF May 28 '18 at 11:53
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    @DRF I take the position that OP is perhaps unfamiliar with Peano arithmetic, hence the oversimplification. But I understand that one needs to define + after defining 0 and S(.) - however, as you say, it is then a trivial step to 1:= S(0) and 2:= S(1). Though I stand by the overall idea that these are all axiomatic or definitional claims that can only be refuted if you choose a different definition of +, which would not be a refutation at all. It would just be a different definition. – Sputnik May 28 '18 at 12:22
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    @Schiphol I don't mean to be overly dismissive of the question, but I do not see that it has any philosophical edge, or even necessarily that Peano needs to be brought into it. The question just seems to be based on a misunderstanding, as though a disproof of 1 + 1 = 2 could have any discernible shape, or that we would all collapse into a black hole should one come to pass. It would be another thing entirely if it were phrased as the more consequential but equivalent 'why can we safely assume 0 ≠ 1 and what are the strongest arguments to the contrary?' – Sputnik May 28 '18 at 12:38
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    @EricDuminil, Merriam-Webster literally defines "two" to be "being one more than one in number", which is exactly S(S(0)). So in this case we certainly have a definition. – AnoE May 28 '18 at 22:42
19

most fundamental equation

Your assumption is flawed. 1 + 1 = 2 is not an axiom of mathematics, but (as Sputnik points out) a consequence of the Peano axioms applied to base 10 representations of numbers.

One can easily change from decimal (base 10) to unary (base 1) and say:

1 + 1 = 11.

Or, change to binary (base 2, what your computer actually uses), and say:

1 + 1 = 10.

And for the sake of it, I can go into roman numerals:

I + I = II.

So, there are representations in which 1 + 1 is not 2 (and even systems where you don't have the glyph 1), but the universe hasn't imploded yet because of that.


Now, what if your question was more like...

What if the Peano axioms contradicts observations from the natural world?

In that case, my answer would be two-fold:

  • Mathematics based off on the Peano axioms would still be useful
  • Mathematicians would come up with another set of axioms that would fit the natural world, along with mathematics based off on those new axioms

To understand this, take for example Newtonian physics: they are a big ruleset of mathematics built on top of some axioms that nicely fit the observations from the natural world.

But then Einstein noticed that some of the axioms didn't really fit (in particular when things go the speed of light), and came up with relativistic physics, which pretty much invalidate all of newtonian physics.

Even we know newtonian physics are wrong (because they are based on a model too simple), they are a tool valid for a lot of problems.

Same with Peano-based arithmetics: even if they don't fit some observation in the natural world, they would still be good tools. And as a consequence of the unfitness, another set of mathematics could be derived from that.

  • 4
    The symbol "1" would normally be defined as the multiplicative identity, and "2" would normally be defined as the sum of the multiplicative identity with itself. That 1+1=2 wouldn't be an "axiom", but would rather be implied rather directly by those definitions. If one were to define symbols differently, the equation using those symbols might not hold, but adding the multiplicative identity to itself would still yield the sum of the multiplicative identity and itself, regardless of what symbols were needed to write that fact. – supercat May 29 '18 at 14:49
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    Thank you for bringing up Newtonian physics vs Relativistic physics, because finding out 1c + 1c != 2c is exactly what happened. The math was right, but our model for adding velocities was wrong at high velocities, so we fixed the model to match observations. It must take into account the Lorentz factor at high velocities. Similar issues with classical vs quantum mechanics. – Schwern May 29 '18 at 19:45
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    You also don't see many Arabic mathematicians claiming that because they use different numerals they have therefore disproved 1 + 1 = 2. So it's a shame that the first part of this answer is wrong, because the second part is very good. – Steve Jessop May 30 '18 at 9:12
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    @SteveJessop At least partly because 1, 2, etc. are Arabic numerals. But your overall point is valid. (i.e. It's a shame that the first part of your comment is wrong, because the second part is very good.) – Ray May 30 '18 at 16:54
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    One quibble. Newtonian physics is not "wrong." It works perfectly well in the context in which it was discovered. I have never needed to use general relativity in any of my 30 years of physics related work. Newtonian mechanics have severed me well and correctly in my context. What Relativity does, is extend Newtonian physics to properly explain phenomena that occur at near the speed of light, and expand the range of contexts in which we can properly reason about gravity and light. – John May 31 '18 at 22:00
13

If 1 + 1 != 2, then 1 - 1 != 0, which means that the charge on the protons in a nucleus no longer cancel the charge on the electrons. Thus all atoms acquire a net electric charge and all macroscopic bodies are attracted (or repelled) to (from) each other with an incredible force - 36 orders of magnitude stronger than gravity. This would mash the whole universe into a sub-atomic pulp in rather short order...

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    Sure, but then it would also not do that. – Sneftel May 28 '18 at 15:34
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    Total protonic reversal? Crossing the streams is bad, Ray. – Mazura May 28 '18 at 17:35
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    This is actually the only answer I've read here that presents a theory on the "what would happen" part of the question. Bravo, Oscar. – CPHPython May 28 '18 at 17:35
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    "If 1 + 1 != 2, then 1 - 1 != 0" I don't get it. How is that conclusion made? – rus9384 May 28 '18 at 21:07
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    @CPHPython That could happen if 1+1=2 is false (and if electric charge obeys the rules of +). But if it's disproved, that just means the way we make disproofs is broken. – immibis May 28 '18 at 23:17
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What would happen is conceptually very simple. The paper proving "¬1+1=2" will be retitled "Zermelo–Fraenkel Set Theory is inconsistent" and published.

From there, it gets harder. Depending on how the proof works, we should end up with a new, weaker set theorem resulting in consistency being restored. Or something worse; the Peano Axioms could be invalid with the consequence of, well, I frankly don't know. Some operation we're used to having goes away, but it won't be addition. Integer addition can't be disproved in the finite realm (thanks science!) so something else on the path to the counterproof is thrown out. Maybe the handling of infinity is wrong in all math. Maybe something else. I'm sorry if this sounds like speculation. The speculation is in fact in the question. It kinda depends on how big of a hole you want to punch.

On the practical side, we already know what happens. 1+1=2 will still be true for any reasonable domain and use case so we will continue to use it. After awhile, the failure mode will be understood and carefully (or not so carefully) excluded like we do in Computer Science for overflow now.

  • "Zermelo–Fraenkel Set Theory is inconsistent" -- or an even better title, if the proof didn't require all the ZF axioms. – Steve Jessop May 30 '18 at 9:08
  • Pudlak theorizes that if a contradiction was found in the Peano Axioms, we would start restricting the induction axiom to "small" formulas, for some definition of small. This would likely restore consistency. – PyRulez May 31 '18 at 13:35
  • And this sort of already happened once with Russel's Paradox. (Except I don't know that Cantor's set theory was commonly considered a good foundation for all of mathematics at the time like ZF[C] is now.) – aschepler Jun 3 '18 at 0:13
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1+1=2 is a necessary truth---roughly, a statement that is true in every possible world. Your question, thus, is asking for true counterfactual conditionals with impossible antecedents. These are sometimes called counterpossibles (e.g., section 5.1 here).

The traditional view used to be that all these counterpossibles are trivially true. According to this view, "if one plus one were not two, then q" would be true for arbitrary q. More recently, several philosophers have argued that making sense of science and everyday reasoning requires a semantics for counterpossibles that does not trivially entail their truth. See references to this debate in the last SEP entry linked to above.

In any event, rest assured, one plus one necessarily equals two.

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    "in every possible world". This is debatable. There can be a world we can't understand and even imagine since it's logical laws (and arithmetical if they even exist there) are completely different. – rus9384 May 28 '18 at 11:11
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    @rus9384 the consensus among theorists working on this topic is that logical truths are necessary. Assuming here that the OP is not interested in contesting the truth of the Peano axioms, then 1+1 = 2, which follows from these axioms, is necessary. In the possible-world construal of necessity, being necessary just means being true in every possible world. Because, as you say, we sometimes need to reason about impossible states of affairs, some theories work with a notion of impossible world for exactly this purpose. – Schiphol May 28 '18 at 11:17
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    So, that world is impossible, because we can't think of it? Blind people can't see, but that's not the issue. There are colors other animal perceive which we won't perceive (unless technology will advance much enough). It is just so that our sense of logic does not allow perception of other logical systems. And we can't be sure Peano axioms really work in our world. Even 1+1=2 may be contested on quantum level. – rus9384 May 28 '18 at 11:41
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    Well, let's say this: possibility is a useful notion, in that not every well-formed sentence in the indicative represents a possible state of affairs. Take a sentence that expresses one of those not-possible thingies. How should we reason about them? Some say: by postulating extra worlds in which per impossibile such thingies are true. – Schiphol May 28 '18 at 11:44
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    @rus9384 I don't think 1+1=2 can be contested on any level. What you might contest is that the Peano axioms model the world well at the quantum level. That doesn't make 1+1=2 not true given the Peano axioms though. – DRF May 28 '18 at 11:56
9

The proof must have been carried out in some kind of formal system, otherwise it's not so much a proof as a persuasive argument. So, we have a proof in some system of the statement 1 + 1 != 2.

Philosophers in the subject of logic, and mathematicians, would look closely at the details of this proof. Since all formal systems that anyone is interested in prove the opposite of this statement, also proving this statement demonstrates that whatever system was used, is inconsistent. So that system could no longer be used for serious work. Therefore, logicians would have learned something extremely important about that particular logical system, and they would want to know what other systems the same technique will prove inconsistent.

The universe could not be "thrown into chaos" unless one believes in some kind of (dare I say it: magical?) effect by which the motion of stars in the Andromeda galaxy is significantly affected by what markings you make on a piece of paper on Earth. A solipsist might, I suppose, believe that the universe is sustained solely by their personal belief in logical consistency, and hence that the universe would be fundamentally altered by their reading this proof. Most people have enough faith in the existence of an external reality, not to believe that the universe has any interest in what proofs humans do or do not produce.

I expect that philosophers not interested in logic and formal proof systems would mostly ignore the result, at least until the logicians explained to them exactly under what conditions they (the non-logicians) are actually using the same flawed system that proves 1 + 1 != 2, and therefore what reasoning it is they need to stop using.

Of course it also depends to an extent on what you mean by disproving that 1 + 1 = 2. One might imagine a "physical proof" rather than a formal logical one. If you mean that someone has proven that they can place one orange in an empty bowl, and then place another orange in the same bowl, and no other oranges have been added or removed, and that the bowl now contains some number of oranges other than 2, you might say they've proved 1 + 1 != 2. But everyone's expectation is that actually, some kind of previously-unknown physical process involving oranges is involved. So while you've discovered something that really changes our notions of the nature of reality, that's not because of the "most fundamental equation" being logically wrong, it's because oranges (or physical objects in general) apparently don't obey arithmetic any more, and therefore the equation is no longer applicable to them. Naturally, this would be extremely troubling, because humans rely all the time on being able to count things, and so human society might well be throw into chaos.

8

Maybe relevant to the discussion is Inconsistent Mathematics :

it is the study of commonplace mathematical objects, like sets, numbers, and functions, where some [emphasis added] contradictions are allowed.

And see the discussion about Arithmetic :

An inconsistent arithmetic may be considered an alternative or variant on the standard theory, like a non-euclidean geometry.

The standard axioms of arithmetic are Peano's, and their consequences—the standard theory of arithmetic—is called PA. The standard model of arithmetic is N = {0, 1, 2, ...}, zero and its successors.

The consistent non-standard models are all extensions of the standard model, models containing extra objects. Inconsistent models of arithmetic are the natural dual, where the standard model is itself an extension of a more basic structure, which also makes all the right sentences true.

Inconsistent arithmetic was first investigated by Robert Meyer in the 1970's. There he took the paraconsistent logic R and added to it axioms governing successor, addition, multiplication, and induction, giving the system R#.

In 1975 Meyer proved that his arithemtic is non-trivial, because R# has models. Most notably, R# has finite models with a two element domain {0, 1}, with the successor function moving in a very tight circle over the elements.

Such models make all the theorems of R# true, but keep equations like 0 = 1 just false.

So what ? Maybe we can survive to a (limited ?) amount of inconsistency.


But consider this thougth-experiment, based on an intuitive example derived from Graham Priest analysis of the general structure of models of inconsistent arithmetic :

imagine the standard model of arithmetic, up to an inconsistent element

n = n + 1.

This n is suspected to be a very, very large number [emphasis added], "without physical reality or psychological meaning." Depending on your tastes, it is the greatest finite number or the least inconsistent number. We further imagine that for j, k > n, we have j=k.

If in the classical model j≠ k, then this is true too; hence we have an inconsistency, j=k and j≠ k. Any fact true of numbers greater than n are true of n, too, because after n, all numbers are identical to n.

No facts from the consistent model are lost.

But now consider the case that n is very very big but not "without psychological meaning" and imagine your bank account adding to an amount of n USD (or GBP or whatever).

From that moment on the bank account will not grow any more, whitout any "disruption" in the usual laws of arithmetic.

Are we allowed to consider it as a case of "the universe be thrown into chaos" ?

4

Gödel's Theorem says roughly that any sufficiently useful mathematical system is either incomplete or contradictory, that is either there are statements that cannot be proved or disproved, or there are statements that can be proved both true and false.

There are many statements that we haven't been able to prove true or false (but that might be because we were not clever enough), and no contradiction has been proven (but that might also be because we were not clever enough), so it is not unconceivable that "1 + 1 ≠ 2" could be proven. 1 + 1 = 2 would then be simultaneous true and false.

What would happen? A lot of swearing among mathematicians would happen. A lot of discussions would be going on how we can ignore this fact and be left with useful mathematics. The universe wouldn't change.

Considering the question: "1 + 1 = 2" cannot and will not ever be disproved (meaning the proof, which is not much more than simple application of axioms, is proven to be incorrect). What is remotely possible is that on top of the proof that it is true, there might be also a proof that it is false.

1

Mathematics and/or science would improve.

Mathematicians seek and use patterns to formulate new conjectures; they resolve the truth or falsity of conjectures by mathematical proof (from wikipedia). We might argue that 1+1=2 steems from definition not from proof making the question moot or ill formed. But your question is still valid in a wider sense. A mathematical proof can be wrong. It has already happened. This mathoverflow question is full of historic proofs and conjetures which are not correct. When such an error is discovered nothing universe-shattering happens. We just stop being wrong and become right, we have improved our knowledge of mathematics.

So, let's say that we are working with axioms which don't include 1+1=2. And that we get to 1+1=2 through mathematical reasoning and stablish a mathematical proof for it. And let's say, for the sake of argument, we later on discover that such proof is wrong, actually 1+1=3. No, that would not throw the universe into chaos. The universe was what it was before humans got to the concept of 1+1=2 (or so I assume, I was not really there to observe it but we have many good evidences which help us know how it was). And every time a mathematical proof has been proven incorrect the universe has not been thrown into chaos. What changed was our understanding of mathematics. It is reasonable to assume it would be the same for 1+1=3.

There is one thing which would be thrown into chaos. Mathematicians. Now that we know that 1+1=2 is false every proof which depends on it is flawed. Flawed, not exactly wrong. The statements validated by proofs that depend on 1+1=2 may still be true, but the old proofs would not serve to stablish that truth. Lots of material would need being revised and rewritten, much discussion would ensue. But we would come out wiser out of that chaos.

What about scientific theories which depend on 1+1=2?. Like what is described in another answer to this question. No, this would not mash the whole universe into a sub-atomic pulp in rather short order. The universe was what it was before we discovered 1+1=3 and would continue to be so (I assume since such has happened for other disproved proofs). Since we would have discovered that the old scientific theories do not properly explain the universe better models would be developed.

1

If such elementary things are cast into doubt, then so a fortiori are much less elementary things, such as the steps of reasoning needed to prove that one and one do not add to two. Thus it would be reasonable to doubt any such proof. In fact, I would ignore the proof—along with the dozen or so other incredible claims I encounter every day—as (I suspect) would most other people.

As a result, I would expect the proof to have as much effect on the world as a new demonstration of Euclidean angle-trisection (such as has been submitted many times before). That is, it would temporarily occupy the relatively few people who chose to look at it.

0

Short answer: Yes. If you could prove that such an elementary and seemingly obvious statement is false, then that would call into question a great deal of what we think we know about mathematics, and probably a lot of other things about the universe.

So what? Unless you have some evidence that this statement is false, it's a pointless hypothetical. Indeed, I've had lots of conversations where someone presented me with some hypothetical about a complex subject, like, "What if it was proven that this political policy that you support doesn't work?", or "What if God commanded you to do something evil?", etc. And my response is generally to say, "I don't think that hypothetical situation you describe is likely to happen. What if someone proved that 1+1=2 is false?"

In a strict mathematical sense, I don't see how you could prove 1+1=2 false because it is true by definition. The definition of "2" is "1+1". At least that's what I was taught in number theory class. Given the complexity of modern mathematics, there are probably other definitions in other branches. But you can't prove a definition false. It's true by ... definition.

-1

Nothing would happen to reality - it would stay as it is. However, we would then require a change in our theory of counting, which would reverberate through other mathematical theories that are built on counting. Since this equation of arithmetic is effectively a definition of two (see e.g., the building of arithmetic in mathematical axiom systems), a proof that this equation is wrong would mean that we cannot validly add one and one (or more precisely, any axiom system that allows us to add one and one is logically inconsistent). That would require us to formulate alternative axiom systems of mathematics that avoid the inconsistency. Reality would keep chugging along just as normal while we tried to figure that out.

-1

You can't disprove an axiom, and Peano's axioms state that 1+1=2.

Context switching, in boolean logic + means something else and 1+1=1.

  • I'm fairly sure that's circular logic. you essentially said that it's an axiom because it's in a list of axioms. – Ruadhan2300 May 31 '18 at 15:04
  • @Ruadhan2300 The Peano axioms are the usual axioms of logic. You might consider it dogmatic, but it is as trivial as "Every number has a successor." – Niklas Rosencrantz May 31 '18 at 16:31
  • Not denying that the Peano axioms are definitely a highly credible source, but "it's true because it's true" is still a weird argument to make. – Ruadhan2300 Jun 1 '18 at 8:15

protected by Philip Klöcking May 28 '18 at 16:45

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