# Is the mathematical truth 1+1=2 analogous to the conservation of energy? [closed]

They seem to express the same concept in different fields.

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## closed as off topic by Manishearth♦, dmckee♦Dec 4 '12 at 19:13

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1+1=2 is a statement about natural numbers. Depending on how deep you like your axioms it can actually be taken as a reasonable definition of the number 2. What exactly is the "concept" similar to energy that you feel this expresses? – Robert Mastragostino Dec 4 '12 at 16:11

1+1=2 follows from the axioms of mathematics. There are different sets of axioms, one example is Peano axioms: http://en.wikipedia.org/wiki/Peano_axioms .

Mathematically, the conservation of energy can be seen as a consequence of the symmetry of physical laws under shift in time. That said, a theory without time shift symmetry doesn't conserve energy. The conservation of energy can be derived from a more general concept, the principle of least action which is usually taken as a postulate.

So no, at least I don't see any connection between the two statements.

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No.

1+1=2 is basically a trivial truth that is by definition true (In the correct context of course. You know what I/O.P. means. Please don't be picky here). An important point is that both quantities on the two sides of the equations are completely static and cannot be changed.

However, the conservation of energy is far from trivial and expresses a conservation of an important physical quantity, relating between initial and final potential and kinetic energies, and can be utilized to conduct very non-trivial calculations.

And conservation of energy is not even true in certain fields or theories of physics(say, GR), or has unclear meanings.

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1+1=2 is not trivial at all from abstract point view (but it is from intuitive point of view), it needs hundreds of pages to be proven from ZFC axioms! – TMS Dec 4 '12 at 16:23
@TMS Hundreds of pages? I think that's an exaggeration. Just notice that applying the successor operation on 1 (1 + 1) is equal to the second integer (2) by identifying elements. Hundreds of pages!? – namehere Dec 4 '12 at 16:27
@TMS: I don't quite see how that link supports you argument. How long you take to "prove it" depends on the axioms, and (I'm pretty sure) in ZFC you get it fairly straight away. – NikolajK Dec 4 '12 at 16:44
@TMS No. Energy isn't conserved in GR. Basically in general manifolds there is no appropriate notion of energy as there are no timelike killing vectors. The local energy-momentum tensor conservation includes additional terms from the connection so I don't accept that either. – namehere Dec 4 '12 at 16:52
@MoziburUllah: Let's raise this question at MathSE to be sure. – TMS Dec 4 '12 at 17:21

In the first place, 1+1=2 is not a "truth", but a statement valid only under certain algebras. For binary numbers 1+1=10 whereas 1+1=11 for algebras satisfying the axiom A+B=AB.

The statement of conservation of energy is a law of nature and does not depend on the axioms/algebra chosen. Moreover, the statement 1+1=2 is also valid for non-conserved quantities such as entropy. For instance "1" could be (adding the correct units) the production of entropy in a given system and the other "1" (adding the correct units) the flow of entropy received from surrounds. Then the total variation of entropy would be "2" (adding the correct units), but entropy has not been conserved.

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anyone care to put forward an experiment DISproving 1+1=2 as a physical constant? :P – danny Dec 5 '12 at 1:33

I would agree. The key word is 'analogous'. He isn't positing an identity.

1+1=2: Both sides are/look different, yet they are expressions of the same thing. A 'change' has been accomplished that moves 1+1 to 2, (or from 2 to 1+1), yet they remain the same.

Conservation of mass, is built on the principle that 1+1=2. If mass is made of indivisible atoms, this simply expresses it.

After Einstein, we now know that this should be mass-energy conservation. The counting has become more complex.

Its really more of a philosophical/historical/intellectual psychology kind of question.

One of the reasons as to why we view the principle of conservation of mass-energy, as of fundamental importance (in addition to its use as an explanatory/calculational device) is because as some of the other posters have pointed out, in the right axiomatic framework, 1+1=2 is tautology. It helps underline the fundamental position that conservation principles take.

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