Jean-François COLONNA

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(CMAP28 WWW site: this page was created on 12/15/1997 and last updated on 08/05/2024 19:02:41 -CEST-)

- The radius of the visible Universe is approximately 4.6e+10 light-years (not to be confused with its age [1.37e+10 years] and not forgetting its expansion),
- One light-year is approximately 9.5e+15 meters,
- The Planck length (the limit beyond which gravitational effects become as important as quantum effects) is 1.6e-35 meter. It is considered by some to be a limit similar to the speed of light.
- What is the distance between two points, and how is it measured?
- Given three points {A,B,C} aligned in this order (for the sake of simplicity) where the distances d(A,B) and d(B,C) are known, what is d(A,C)?
- How do you calculate the product of two distances (representing, for example, an area)?
- Do the infinitely large (singularities in space-time, or the energy density at the "moment" of the
*Big Bang*) and the infinitely small (in the mathematical sense of these two terms) exist "in nature"? - etc...

So, by taking the Planck length as the unit of length, the radius of the visible universe is approximately 2.7e+61. Thus, when limiting ourselves to measuring distances, fewer than seventy decimal digits are needed. What, then, is the purpose of Real Numbers and the infinite precision they allow, especially considering that all measurements are tainted with errors (are they the "numbers of nature")? The first answer that seems to come to mind is the irrationality of certain constructions -potentially elementary ones-: for example, the length of the diagonal of a unit square. But does this measurement have a physical meaning (this is not about doing "pure mathematics")? It would seem that the answer lies in the necessity of transitioning from discrete to continuous representations in order to perform calculations (for example, necessary for predicting new phenomena) that would otherwise be impossible.

Along the way, other questions, sometimes seemingly trivial, may be posed:

(The first question currently arises in the context of research conducted on

T(x,t+Dt) - T(x,t) 2 grad(T(x+DX,t)) - grad(T(x,t)) -------------------- = k .-------------------------------- Dt DxThis formula has no practical use. To derive anything from it, it is necessary to use differential calculus, which requires a "

dT(x,t) 2 --------- = k .div(grad(T(x,t))) dtSome remarks and questions arise:

- Temperature is a macroscopic notion, so what physical sense does it make to consider the function 'T(x,t)' for very small "volumes" 'Dx'?
- While this is a completely classical physics problem, as 'Dx' and 'Dt' tend towards 0,
they will inevitably lead us into the quantum universe and its fluctuations (particularly regarding positions and thus lengths).
When reaching the Planck length and time (1.6 10
^{-35}meters and 0.5e 10^{-43}seconds respectively), gravity can no longer be ignored. What happens physically when crossing these "barriers"? What physical sense does this passage to the limit have? How is it possible that a process which steps out of its domain of validity gives rise to an equation that provides complete "satisfaction"? Note that (even though the previous example does not fall under Quantum Mechanics) Heisenberg's uncertainty relations practically forbid infinitely precise values, for instance, for the speed and position of a particle. Despite this, infinitesimal quantities related to these still appear in the equations of physics (differential and partial differential equations). - What sense does the relation R have when 'Dx' and 'Dt' cross the threshold of the physically forbidden values mentioned above?

Despite these remarks and questions left (temporarily...) unanswered, the heat equation thus obtained seems to be a good classical model of this phenomenon. There is thus an

It should be noted in passing that certain constants of nature are known with very high precision, currently close to that of double-precision floating-point numbers (64 bits) in our computers. Therefore, their least significant digits risk being ignored by our machines... This is, for example, the case with the Rydberg constant, which appears in the spectroscopy of an atom with a nucleus of infinite mass: its value given by the NIST - National Institute of Standards and Technology - is 10973731.568525(73) m

If infinity (the infinities...) did not exist in the Universe, could we then dispense with Real Numbers in mathematical physics (note that, obviously, this question is posed without forgetting, for example, the irrationality of the square root of 2, but this is not about doing mathematics, but physics and in this discipline, what is the square root of 2...)? Shouldn't a new arithmetic (with "new" numbers and new elementary operations -to evaluate, for example, the 'd(A,C)=d(A,B)+d(B,C)' mentioned in the introduction-) adapted to physics be imagined?

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