Mdthbs

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?

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.

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.

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 :-)