In the beginning it was thought that the statement 1+1=0 is false, and necessarily so. However, with the birth of modular arithmetic, it was found that indeed, 1+1 does indeed equal to 0 (in the mod 2 setting).

Now in the sciences for example physics, it's understood that Newtonian mechanics has in a sense been falsified, in the sense that, while it remains correct in its regime of validity, there are other regimes in which it is not correct.

However, while most people would say that in the second case, we have Newtonian mechanics being falsified, we do not consider 1+1=2 to have been falsified (in its own regime of validity). Why is that?

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    1+1=2 is still true in the mod 2 setting. It just so happens that 2 and 0 are names for the same object in this setting. – Dan Staley Apr 9 at 20:14
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    Newtonian mechanics is not considered to be falsified by any working physicist that I know. Where did you get that? Or do you think QM or GR falsified it? Neither is true. It is indeed possible that GR is falsified in the future, like if there is a dilaton after all. And in a sense the original version has already been falsified with the addition of a cosmological term of the opposite sign to one that Einstein considered and rejected. QM definitely does not falsify Newtonian mechanics. QM is built on the same foundation, and for sure F=ma is still true in QM. – Kostas Apr 10 at 9:03
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    @Kostas Relativity isn't accounted for in Newtonian physics, so at high velocities strictly Newtonian physics does not properly apply. Quantum mechanics also gives rise to phenomenon that are impossible in purely Newtonian mechanics. Luboš Motl has a great explanation on physics SE physics.stackexchange.com/a/97749/127931 – JMac Apr 11 at 12:48
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    @Kostas "Newtonian mechanics is not considered to be falsified by any working physicist that I know" you don't know many - it's been falsified in quantum mechanics and by relativity; and probably others. However for most "everyday" physics that people want to deal with, it remains a sufficient approximation. – UKMonkey Apr 11 at 15:43
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    And once again, an argument arises due to vague use of terminology. Unless you agree on what the verb "to falsify" means, you cannot agree on whether or not it applies. – barbecue Apr 11 at 16:55

In math, we define stuff like numbers and operators, then we go on to prove other stuff from those premises.

When you ask: "Is 1 + 1 = 0?", a mathematician will just ask back: "With what definition of +?"

  • If you assume natural numbers and the common definition of +, then this statement is false.

  • If you assume numbers modulo 2 and + meaning XOR, then this statement is perfectly true.

You cannot say that we falsified the claim that 1 + 1 = 2; we just came up with new definitions for what + could mean.

For physics, the situation is a little different: Here we measure stuff we want to explain, then we whip up some fancy theory to explain the measurements, and finally, we test the theory by measuring more stuff, trying to falsify our theory.

In the example you gave, Newton had some measurements (like falling apples) that he wanted to explain, so he came up with his theory of force, acceleration and movement, and he could explain his measurements. We continued testing his theory until finally the Michelson-Morley Experiment produced numbers that could not be explained with Newtonian Physics anymore. So, Einstein came up with a new theory that could explain all that the old theory could explain, and which also explained the result of the Michelson-Morley Experiment.

Note that I said "could explain all that the old theory could explain". Newton's theory worked fine for small numbers, and Einstein's theory had to make the same predictions for these small numbers. More precisely, Newton's theory is nothing more or less than a convenient approximation of Einstein's theory for small numbers.

We do this a lot in physics: We know that some stuff obeys some complex rules, but we don't want to bother with deriving mathematically correct results, so we just use an approximation (and hopefully check that this approximation is indeed not too far off). The point is, in the end physicists can only falsify stuff by measurement which includes measurement errors, and it does not help our cause to calculate stuff we can't measure. But the approximations allow us to make conclusions we cannot derive with mathematical rigor.

So, a physicist with a quartz-clock, a ruler, and a scale will simply use Newtonian Mechanics to predict their measurements. A physicist listening in to gravitational waves does not have this luxury, they must use General Relativity to derive their predictions. The first physicist simply uses an approximation (Newtonian Mechanics) to an approximation (Special Relativity) to a theory (General Relativity) which we do not know what theory it approximates yet (String Theory? Loop Quantum Gravity? Something else?)

In this sense, no physical theory is fully correct or incorrect. For some theories we know where they start producing numbers that actually disagree with our experiments, and for others we may not have discovered those places yet. But that does not reduce their usefulness when we apply them to problems where we know that they yield sufficiently precise results. In the end, any physical theory is just an approximation of reality.

  • Comments are not for extended discussion; this conversation has been moved to chat. – Geoffrey Thomas Apr 12 at 7:15
  • Very nicely put. I've bookmarked this so I can refer others to it in future. – Neil_UK Apr 12 at 10:49
  • For example we use Pi. We rarely use every digit of Pi. – candied_orange Apr 12 at 12:12
  • Awesome! You should title your post something like "induction vs deduction". – Joshua Ronis Apr 12 at 14:24
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    @candied_orange we never use every digit of Pi, and never will. :D – Cássio Renan Apr 12 at 20:54

The hypothesis 1+1=0 is false in the domain of natural numbers. If the domain is the finite field of the integers mod 2, then one is no longer in the domain of the natural numbers and the statement 1+1=0 would be true in that domain.

The question is why do we not consider these to be falsifications of each other?

These are not contradictions or falsifications if we view these statements in their separate domains. The domain of the natural numbers is not the domain of the integers mod 2. Although the statements may look the same, they are statements derived or not from different domains and so are different.

Falsification would be involved in mathematics by assuming something with the intent to arrive at a contradiction. If one can arrive at the contradiction, then one can conclude that what one assumed true was false. One name for this inference rule would be proof by contradiction.

Wikipedia describes it as follows:

In logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by first assuming that the opposite proposition is true, and then shows that such an assumption leads to a contradiction. Proof by contradiction is also known as indirect proof, proof by assuming the opposite, and reductio ad impossibile.

Wikipedia contributors. (2019, March 26). Proof by contradiction. In Wikipedia, The Free Encyclopedia. Retrieved 16:55, April 9, 2019, from https://en.wikipedia.org/w/index.php?title=Proof_by_contradiction&oldid=889548940


1 + 1 = 0 is false.

Meanwhile, (1_2) +_2 (1_2) = 0_2 is true. Here +_2 is a different operation than +, and 1_2 and 0_2 are different things than 1 and 0. So it's not surprising that one equation is true while the other is false.

The problem is that we do not like to write "_2" everywhere, so we often write 1 + 1 = 0 when we mean 1_2 +_2 1_2 = 0_2. This can possibly lead to confusion, though hopefully the author (or context) will make it clear what is meant by the equation 1 + 1 = 0, whenever it is written, so that ambiguity is avoided.

I would not say that Newtonian physics is "false", but I would say that it does not accurately predict certain observations we make about our universe, while Einstein's Relativity does seem to predict these observations quite well. So Newtonian physics is apparently not the best theory for our physical universe.

However, since there are no other universes(?), physicists frequently omit the phrase "for our physical universe" for convenience.

  • Hmmm. I appreciate the effort @Eric Towers. Yet, when I wondered how other universes could matter to me, I wasn't referring to "mechanisms of action," but rather about how another universe could have any impact on my experiences, my purposes, my plans or my actions. I deal with whatever gravity affects me in any case, and when I do need to figure it into my calculations (very rarely!) its origin isn't relevant. I hope that's clear enough... – Rortian Apr 11 at 1:10
  • @Graipher, I understand your point, and I appreciate your participation. "Maybe how you commute to work in the year 3000 depends on FTL travel exploiting that. " Who? Me???? I don't think so...doesn't that seem rather unlikely to you? I won't be looking forward to that! I'm concerned about more immediate and pressing matters than considering theoretical aspects of abstract descriptions of events. "fundamental research...might lead to something new." I get that. I'm simply more concerned with people than I am with physics. – Rortian Apr 11 at 11:19
  • @Rortian Even as such, you ignore the possibility of the butterfly effect. It may be the case that due to some essentially untraceable chain of events linked to a slight perturbation in an alternate universe, your interactions with your friends is vastly different. I don't see your point. This sounds like the "I don't get it, so it doesn't matter" argument. – Don Thousand Apr 12 at 4:38
  • @DonThousand I appreciate your thoughts on this. I’m not ignoring chaos and complexity, I’m dealing with it. I’m simply focussing on what seems to be much more important to me and the people around me, rather than spending my (precious) time (I’m much older than you!) considering the theoretical effects of other possible worlds on our current activities. I’m not a physicist and I don’t aspire to be one. … “It may be the case”…Do you believe that anything is possible? If so, then the notion of possibility would lose its meaning… – Rortian Apr 12 at 9:52
  • @DonThousand I think I “get” what you’ve said; can you tell me why I ought to believe that it’s more important to me than it is right now? I’m working to promote educational philosophy and educational psychology, critical thinking and ethics. Would you care discuss any topic within those categories? I'd be happy to! – Rortian Apr 12 at 9:53

Your example from mathematics shows: To assess a mathematical statement one should first fix the context, the domain of validity of the symbols. Because in the context of natural numbers the statement 1+1=0 is false. While in the context of Z/2Z the statement is correct. Except the rare case of undecidable questions, in mathematics one can prove or disprove the correctness of a statement.

In natural science, e.g. in physics, one can never prove a general statement. All „laws of nature“ have the status of hypotheses. One can confirm a hypothesis or one can disprove the hypothesis by a counter example. But one cannot prove it.

IMO Newton’s laws are a good approximation in a broad domain, i.e. for small energy and medium dimension. One can deduce Newton’s laws as first order approximation from the field equations of General Relativity. But anyhow, Newton’s basic concepts like space, time and gravitation have quite a different meaning in Einstein’s theory of curved spacetime.

To say that a statement turns out as a first approximation is a refined and better version than saying that the statement is false. I prefer the refined version :-)


Hmmm. What about 1 + 1 = 10 ?

Is that equation, expressed in binary arithmetic, "false in the domain of natural numbers"?

My grounding in math and logic isn't very strong, but I understand the Wikipedia entry...I just don't think that the notions of truth and falsity can coherently apply to inductive inferences (abstract descriptions of unobservable things).

I've also heard people (philosophy professors!) say that Einstein disproved Newton's stuff, but that seems incorrect to me in the postmodern age of philosophy. Newton wasn't mistaken, his principles describe his observations very well. His theoretical model wasn't poor or wrong, and scientific proof seems to me to be an oxymoron. People who have faith in that idea reject the notion of fallibilism, which is commensurate with the postmodern approach:

Fallibilism is the epistemological thesis that no belief (theory, view, thesis, and so on) can ever be rationally supported or justified in a conclusive way. Always, there remains a possible doubt as to the truth of the belief. Fallibilism applies that assessment even to science’s best-entrenched claims and to people’s best-loved commonsense views.

Stephen Hetherington, Internet Encyclopedia of Philosophy

I've learned (is this correct?) that true/false distinctions are properties of formal languages ( arithmetic and logic), where axioms and operations are strictly defined and the symbols are unrelated to observable phenomena (only to each other).

As for discourses in natural languages, the idea of epistemic truth (especially the theory of correspondence to reality) has been quite thoroughly discredited...or so I've been told...

I'm new here; I wouldn't be happy to hear that I (and my wisest philosophy instructors) have misinterpreted Kant, Kuhn and Popper, but if a wise expert disagrees I'll listen. I believe that it's theoretically impossible to denote the absolute truth about unobservable phenomena or complex abstractions, but I'm still learning...

In any case, to me coherency is the gold standard of human understanding, not truth. Subjective beliefs and public discourses may be assessed according to so-called objective criteria: the reliability of relevant evidence and the justification for one's presumptions. I think that that's one thing upon which scientists and philosophers might agree.


Two understand why this is so, you have to understand the distinction between "false" and "incorrect".nYour question is caused by conflation of the two. I must admit that my usage of the two terms is not universal, but the distinction is common even when people use other terms for it (confusingly enough, textbooks on formal logic tend to use "false" for what I call "incorrect" here!).

All branches of science make models, which are representations of reality. These models contain statements. A statement is correct or incorrect with regard to a model, and true or false with regard to reality. Correctness here means a good fit to the rest of the model (usually, being derivable within the model, since scientists are usually interested in models as means of deriving predictions), and trueness means being a good fit to reality.

As a first simple example, take the model of classical conditioning, demonstrated in the famous experiment of Pavlov. The model makes the prediction that "the dog will salivate when the light goes on", and this statement is both correct within its model, and true in reality.

Note that a statement can be correct without being true, and vice versa. Imagine a situation in which a person of average build is standing upright in the middle of a room. I walk up to the person and push them forcefully on the sternum. A physics model predicts that the person will topple over. A model of interpersonal psychology predicts that the person will remain upright, and will slap me. Each of these statements is correct within its own model, but at most one is true in a given situation. This is not just about the domain of the model - a second model of interpersonal psychology can predict that I will get shouted at, but not slapped. They are simply different models, producing different predictions (statements). And note that, while in most real-life situations the psychological models are more likely to be true, one could imagine situations where the physics model is true, for example performing in a theater.

Sometimes people naively assume that science is (or should be) looking for the truest models, that is, the ones which fit reality the closest. This is not true (no pun intended). Scientists value parsimony, which means that they prefer the simplest model whose prediction is true enough for their intended use case. As a famous statistician quipped, all models are wrong, but some are more useful than others. This is why we are using widely both Newtonian physics and quantum mechanics - each of them is the preferred model for a given application.

It is also entirely possible that a model which describes one situation in reality perfectly (it produces true statements in that situation) is not at all applicable to other situations. The inverse square law is a true model when you consider the intensity of light, and not a true model when you describe the speed of a ball rolling on a horizontal plane.

Now on to your mathematics example. Mathematical models are very abstract. They can be used to represent some situation present in reality, or they can be used as mere mental constructs without any claims to be representative of a real situation. Let's first use your examples as representations of reality. In situation A, you get some sheep delivered and put them in a paddock. You want to predict the number of sheep you will have at the end. In that case, the model of addition of natural numbers will provide you with a true statement, while the second model, that of modulo arithmetic, will provide you with a false statement. Note that both statements are correct within their own models - an incorrect statement would be "1 + 1 = 1" in modulo 2 arithmetic. But one is true and one is false in your use case. If your neighbour is getting sheep delivered to two paddocks and has to ensure equal number of sheep between the two paddocks, then I also have a model which will predict how many sheep she has left over - here the modulo 2 arithmetic produces the true statement, and the addition over natural numbers produces a false one.

But if there is a pure mathematician who has no aspiration of applying your mathematics to real situations, the question of "is it true or false" doesn't even arise. Since true/false is about fitting a model to reality, and she is not fitting your model to any reality, she cannot measure the statement's trueness anymore than she can measure its physical weight or its radioactivity - it just doesn't have this quality. All she can decide about the statement is whether it is correct within its model or not.

If you think that there is just "truth", it is easy to see why you would compare the two situations you described, "1 + 1 = 2" being false* in modulo 2 arithmetic, and "each electron is in one place at a time" being false in quantum mechanics. But in fact, the two statements are not false, but incorrect in their respective models. Besides, the second one is false in the physical reality of the universe we live in. The first one is not traditionally attached to any specific use case in reality, so cannot be said to be true or false before you also provide such a use case.

* OK, not strictly false, but my point still holds with examples that are strictly false.

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