One plus one never equals two

David Chik (Plymouth University)

Published in physic.philica.com

Abstract
No physical entities in the universe fulfil the statement “1+1=2”.

"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.

Reference

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.

Peer-review ratings as of 07:12:30 on 16th Dec 2017 (from 1 review, where a score of 100 is average):
Originality = 54.26, importance = 6.25, overall quality = 26.11

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.

In physics, for example, we have plausible experimental evidence dating back to the 19th century that electrical charges can cancel each other out: thus 1 + (-1) = 0, which is much the same proposition. Again, Dalton’s breathrough in chemistry was essentially based on the finding that chemicals combined in ratios of (low) integers, often at 1:1, and the empirical discovery of atomism has been an approach to this asymptote of precision.

But perhaps more importantly, the question of whether or not abstract mathematics is a valid description of the real world has indeed been ‘examined carefully’ since long before Plato, and produced a very rich literature. Epicurean atomism describes a number of crude but empirical experiments focused on resolving whether matter is infinitely divisible, or made up identical particles. And I do not need to remind the author of the discussions in Newton’s Principles and Einstein’s Physical meaningof geometrical propositions, to name a few sources.

In short, this ground has been covered very thoroughly both by physicists and abstract philosophers. 1+1=2 is, at least, an extraordinarily robust generalization for dealing with physical phenomena. Yes, it is probably a “mere illusion produced by neural activity,” but so is this article, and so is this review.

Author comment 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 GĂ¶del’s incompleteness theorem may be irrelevant in physics ?

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