2. Why do 2 + 2 = 4?

This question refers to the more general problem of necessity. Why are there necessary truths such as those of mathematics? From where do the constraints of logical necessity come from and where do they reside even when there is no physical reality to instantiate them?

• In your view, is there an answer to this question?
• Is it important and why?
• Any suggestions for an answer, however outlandish?

The Parmenideum's Reply

The human concept of number and the necessary nature of a priori mathematical truths are a manifestation of a law of constancy. Specfically in this case, of the law of conservation of energy.

We suggest that the necessary truths do not in fact exist independent of material reality. Though it is at first sight absurd, abstract logical truths could be the way they are as a result of physical reality being the way it is (which of course raises a further question). This dependency is not on any one particular instantiation of reality – say one with repulsive gravity, non Euclidean space-time and so on. The proposition considers only the fact that there is a physical reality without considering any of its properties.

Indeed, the proposition argues that all abstract and physical properties of reality are contingent on only one necessary truth: the conservation of energy. Put another way, it argues that it is only a result of the fact that energy must necessarily be conserved that there are what appear to be necessary abstract truths. As such, the laws of abstract logical necessity both exist and are the way they are because there is a material reality that obeys the conservation of energy. In a sense it could be claimed that necessary truths such as the sum of 2 + 2 equalling 4 are only necessary because they rest on an even more fundamental necessary requirement: that energy be conserved. In a kind of absurd logic, necessary abstract truths are "contingent" on the fundamental law of constancy, and turn out as being necessary only because constancy is necessary.

This raises an interesting notion. Without constancy their could be no reality (see below). If there is no reality then there would be no necessary laws of mathematics or logic. In which case we would not need to agonise over where exactly the "eternal" laws of abstract necessary truth reside if all of what we recognise as physical reality were to disappear. They would not have to reside anywhere if they themselves resulted from the existence of physical reality in accordance with the law of constancy. We immediately dispense with Platonism, which has caused everyone not just one or two problems!

The justification for positing the conservation of energy as the basis from which all necessary truths arise comes from the fact that quantum physics allows the creation of energy ex nihilo, provided that the total sum is zero. Zero total energy is a necessary stipulation, otherwise there would be something to explain. It may be asked why something should even start to come from nothing, and that is of course the problem addressed in the question here. Nevertheless, given that energy must be conserved, and given that all of what exists is ultimately constituted of energy, what ever exists will necessarily be compelled to conform to this constancy. It is suggested that number as well as all necessary a priori truths are a manifestation of this respect for constancy.

One difficulty (among many): it may be possible to show how the laws of arithmetic may be explained in this way, but what about Pi, the properties of triangles and so on?

A tentative attempt to explain this concept more fully will be posted here one day, but just to wet the appetite and emphases that the discussion must necessarily sail close to the absurd to get anywhere at all in these waters, the article will be entitled "Waah Okasterar" (and come accompanied with instructions of how to perform hand stands and such).

The Parmenideum.