Changing How the World Thinks

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A universe of nothingness

The mathematical fabric of reality

Peter Atkins mathematical reality metaphysics

There is no charge in the universe, and no energy. Mathematics is the key to understanding how something spring forth from this nothing, and is the thread from which the fabric of reality is woven.

What is the ultimate fabric of reality? My own position is quite clear: I am a materialist while being a scientist but a mathematical realist when being a speculator. Materialism, the view that the fundamental particles, including their agglomerations into atoms and molecules, together with the forces that bind them into interesting entities, are all that is necessary for all the properties of the world. There is nothing beyond these entities.

Scientists are hewers of simplicity from complexity: they chip away at the world around them, seeking to reveal the ultimate entities and at the same time stand in delight at the complexity that their interactions can yield. The reductionist program is the easy part. Much more difficult is the reverse journey from the identified simplicities up to the emergent properties that give us the natural world, up to and including consciousness. 

So much for being a scientist. As a speculator, imagination can spring out from the bonds of evidence and wonder whether there is an ultimate reality beyond the material.  Speculation (some might call it philosophy, but ‘speculation’ is the more honest term) is merely that, but might turn into science one day if evidence is accrued that supports it.

I think it possible to argue that there is no energy in the universe. Yes, there are positive and negative contributions to the total energy, but I suspect that the total is zero, just like the total charge.

Speculation relies upon hints, hints that might one day be regarded as evidence. What are the hints that litter the universe? One hint suggests that there is nothing here at all. Take electric charge. There are obviously positive and negative electric charge (for instance, of protons and electrons, respectively), but the total charge of the universe is zero. How do we know that? Because the strength of the electromagnetic interaction is so much greater than the strength of gravity, that any imbalance of charge would result in the universe being blasted apart as soon as it was formed, with no opportunity for weak gravity to bind entities into galaxies, stars, and planets. More contentiously, I think it possible to argue that there is no energy in the universe (with mass being another manifestation of energy, through  E = mc2). Yes, there are positive and negative contributions to the total energy, but I suspect that the total is zero, just like the total charge.

So, my first point is that there is nothing here at all, and the incipience of the universe was an event in which absolutely nothing (not even spacetime) turned into a more interesting form of nothing, where opposites became distinguished. Such a view greatly simplifies, without solving, the problem of what happened at the Creation. Absolutely nothing turned into more interesting net nothing. How that revision of nothing occurred is an entirely different question, which for the purpose of this article I shall set aside.

The other hint perhaps worth noting is the extraordinary aptness of mathematics for describing the physical world (the only world, I might add). This aptness was first noticed publicly by E.P. Wigner, and has given rise to some speculation. Science can proceed without mathematics (the theory of natural selection was formulated without mathematics, but has been greatly enriched by the incorporation of mathematics to unfold its consequences). But fundamental physical theories, such as quantum mechanics and general relativity would not exist without mathematics.

It might be that mathematics ‘works’ merely because it is an emotion-free, logical mode for the unfolding of the consequences of a hypothesis or theory, and has no deeper significance. But could it be that its aptness springs from a structural resonance? Could it be that mathematics is the universe, and that our feeble scratchings at the surface of phenomena are exposing this deep structure of reality? I once called this resonance ‘deep structuralism’, and see no reason why to draw back from the term.

Could it be that mathematics is the universe, and that our feeble scratchings at the surface of phenomena are exposing this deep structure of reality?

There are hints that deep structuralism might be a profitable speculation. For instance, it is possible to create the integers from the empty set, and once you have the integers then, as Kronecker once indicated, by making them do things for which they were not originally intended, you have mathematics. Thus the rich fabric of mathematics springs form the empty set, just as the present universe sprang from absolutely nothing.

A question that immediately arises is how mathematics becomes tangible. We inhabit, we think, a world of tangible objects. How can that be, if the world is some kind of manifestation of numbers? I have no idea; but I can speculate.

All perception, I think, amounts to touch. Touch is obviously touch, and relies on sensors in the skin that respond to the presence of incident matter. Hearing is touch. In its case, the touch if the impact of the molecules of air on the diaphragm of the ear and its transmission into sensors within the ear. Taste and smell are touch, in so far as they arise from molecules fitting into receptors. Vision is touch, in the sense that light causes a change in shape of the photoreceptive retinal molecule so that it can no longer sit in the pocket of a protein, is expelled, and the resulting change in shape of that protein initiates a signal to the brain. Within the dark caverns of the brain, these different touches are interpreted as different varieties of perception: touch itself, hearing, taste, smell, and sight.

Where does touch stem from? It stems from the inability of one piece of matter to occupy the same space as another piece of matter. And from where does that inability spring? It springs from the deep quantum mechanical principle known as the Pauli principle.

Where does touch stem from? It stems from the inability of one piece of matter to occupy the same space as another piece of matter. And from where does that inability spring? It springs from the deep quantum mechanical principle known as the Pauli principle, a statement about the mathematical symmetry of wavefunctions. So touch, and therefore all perception, springs from mathematics. I realise that that account of perception and sensation might raise more questions than it answers, and will leave no one with a visceral sense of comprehension; but it might be a hint about our ability to perceive mathematics as matter.

Science is wonderful, exposing as it does the material infrastructure of the world. It elucidates wonders. Speculation is fun. It wonders about elucidation. Maybe, perhaps, who knows, speculation might on occasion by chance light on truth.

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