God created the real numbers

> Given their non-constructive nature "real" numbers are unsurprisingly totally incompatible with computation.

It is funny you say that when Turing defined Turing machines to compute real numbers (like π for example). In its original definition, a number was computable if its Turing machine did not stop. Which makes sense since π does not have a finite decimal expansion.

Today, we usually define Turing machines to decide problems and a problem is decidable if for every input its Turing machine stops with a ``yes'' or ``no'' answer. I guess this is what makes people think what you said in the quote above. Maybe this definition is more intuitive but this conclusion from it could not be more wrong.

Think about it for a second, if the computable numbers were countable there would be no uncomputable problem (Turing actually used the classic cantor diagonal argument to prove that there were uncomputable numbers)

Energy is continuous, not discrete.

Because energy is action per time, it inherits the continuity of time. Action is also continuous, though its nature is much less well understood. (Many people make confusions between action and angular momentum, speaking about a "quantum of action". There is no such thing as a quantum of action, because action is a quantity that increases monotonically in time for any physical system, so it cannot have constant values, much less quantized values. Angular momentum, which is the ratio of action per phase in a rotation motion, is frequently a constant quantity and a quantized quantity. In more than 99% of the cases when people write Planck's constant, they mean an angular momentum, but there are also a few cases when people write Planck's constant meaning an action, typically in relation with some magnetic fluxes, e.g. in the formula of the magnetic flux quantum.)

Perhaps when you said that energy is discrete you thought about light being discrete, but light is not energy. Energy is a property of light, like also momentum, frequency, wavenumber and others.

Moreover, the nature of the photon is still debated. Some people are not convinced yet that light travels in discrete packets, instead of the alternative where only the exchange of energy and momentum between light and electrons or other leptons and quarks is quantized.

There are certain stationary systems, like isolated atoms or molecules, which may have a discrete set of states, where each state has a certain energy.

Unlike for a discrete quantity like the electric charge, such sets of energy values can contain arbitrary values of energy and between the sets of different systems there are no rational relationships between the energy values. Moreover, all such systems have not only discrete energy values but also continuous intervals of possible energies, usually towards higher energies, e.g. corresponding to high temperatures or to the ionization of atoms or molecules.

That presentation is like all the research that has been published in this domain, i.e. it presents some ideas that might be used to build an alternative theory of space-time, but no such actual theories.

There are already several decades of such discussions, but no usable results.

Time and space are primitive quantities in any current theory of physics, i.e. quantities that are assumed to exist and have certain properties, and which are used to define derived quantities.

Any alternative theory must start by enumerating exactly which are its primitive quantities and which are their properties. Anything else is just gibberish, not better than Star Trek talk.

However, the units of measurement for time and length are not fundamental units a.k.a. base units, because it is impossible to make any physical system characterized by values of time or length that are stable enough and reproducible enough.

Because of that, the units of time and length are derived from fundamental units that are units of some derived quantities, currently from the units of work and velocity (i.e. the unit of work is the work required to transition a certain atom, currently cesium 133, from a certain state to a certain other state, i.e. which is equal to the difference between the energies of the 2 states, while the unit of velocity is the velocity of light in vacuum).

The set of computable numbers is actually countable (see ref. linked above). It has to be by definition because the set of finite computer-programs is itself countable.

This is the whole point of the un-reality of "real" numbers: "all" of it (= measure 1) is uncomputable except a "tiny" measure-0 set.

> The universe [...] doesn't require infinite precision,

Doesn't it though?

What happens when three bodies in a gravitationally bound system orbit each other? Our computers can't precisely compute their interaction because our computers have limited precision and discrete timesteps. Even when we discard such complicated things as relativity, what with its Lorentz factors and whatnot.

Nature can perfectly compute their interactions because it has smooth time and infinite precision.