One plus one never equals two
Published in physic.philica.com
"1+1=2" has long been considered as an axiom in mathematics or symbolic logic. However, no one has examined it carefully from an "empirical" perspective. A central concept in science (or scientific method) is that all evidence should be reproducible by experiment or observation (rather than purely based on so-called reasoning). A recent laser experiment has demonstrated noncommutativity and bizarre arithmetic (Parigi et al., 2007). It leads me to re-think about the validity of conventional arithmetic.
If the statement "1+1=2" has a physical meaning, "1" and "2" should correspond to some real physical entities, but I am going to argue that no physical entities in the universe fulfil this statement.
We need to think about what "things" satisfy this statement. First, these things should belong to the same class. It is meaningless to ask "one apple plus one doctor equals what ?" Now suppose we have two apples. Does it fulfil the statement "1+1=2" ? The problem is, this would suggest that two big apples are identical to two small apples. OK. We are close. We need two identical entities. Can we find two things which are identical to each other ? Perhaps so, but we cannot put them together (due to Pauli Exclusion Principle). If they are not being placed together, we need to measure them separately to make sure that they are identical, but now we have the problem of Uncertainty Principle. As a last resort, you may take out a ruler and say, "OK this is the standard ruler and all length units should be based on it." We know that the length of this ruler will change according to different temperatures. The space itself may also change depending on gravitation.
As a conclusion, there is no empirical proof that "1+1=2" is a physical reality. It is very likely that this axiom, together with many other similar axioms, are mere illusions produced by neural activities. Unfortunately, most scientific theories are based on this inaccurate premise. The implication is profound.
Parigi, V., Zavatta, A., Kim, M.S. & Bellini, M. (2007) Probing Quantum Commutation Rules by Addition and Subtraction of Single Photons to/from a Light Field. Science 317:1890-1893.
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Published on Wednesday 5th March, 2008 at 12:34:24.
Peer review added 6th March, 2008 at 14:17:55
This is a very brief, clever, article. Perhaps its a flaw of such cleverness that the read is asked to take quite seriously an argument that, if seriously pursued, would be much more complex.
added 19th April, 2008 at 14:03:42
It seems to me that “robust generalisation” is different from “definition”. Natural number is “inferred” rather than “defined” from the physical world. One implication is that Gödel’s incompleteness theorem may be irrelevant in physics ?
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