Why do we Need Real Numbers?
CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, École polytechnique, Institut Polytechnique de Paris, CNRS, France
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Abstract: Measurements in physics are approximate; on microscopic scales, our universe seems quantized,
so what is the use of the infinite precision offered by Real Numbers? Moreover, and in general,
they cannot be represented in our computers. Forgetting this can lead to insurmountable problems."
Keywords: Real Numbers.
Contents of this page:
1-INTRODUCTION:
The aim of physics is to describe the mechanisms of the Universe at all scales (the "how? " question).
Let us provide some orders of magnitude:
- 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.
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:
- 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...
(The first question currently arises in the context of research conducted on quantum gravity)
where the points mentioned are points belonging to physical space and not to abstract and "mathematical" spaces.
2-TAKING THE LIMIT:
To understand this, let's take the simple example of the heat equation in a one-dimensional medium.
Let 'T(x,t)' be the temperature at the point with coordinate 'x' at time 't'.
Let 'Dx' denote a sufficiently small interval of space such that the temperature remains constant within it.
Furthermore, let 'Dt' denote a duration, also sufficiently small, so that we can write the following proportionality relation R:
T(x,t+Dt) - T(x,t) 2 grad(T(x+DX,t)) - grad(T(x,t))
-------------------- = k .--------------------------------
Dt Dx
This formula has no practical use.
To derive anything from it, it is necessary to use differential calculus, which requires
a " taking the limit" process achieved by letting the previously mentioned quantities 'Dx' and 'Dt' approach 0.
This yields the heat equation:
dT(x,t) 2
--------- = k .div(grad(T(x,t)))
dt
Some 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 a posteriori justification
for the passage to the limit, but nothing demonstrates its validity a priori! There is indeed a paradox here...
3-SOME CONSEQUENCES:
Thus, Real Numbers seem indispensable.
Unfortunately, the study of mathematical physics equations generally cannot be done without using computers.
By definition, these machines only handle quantified and finite information.
Therefore, Real Numbers cannot be stored or manipulated in general terms: Real Numbers are not computable (in the mathematical sense of the term).
Forgetting this fact can have dramatic consequences!
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-1.
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?
Copyright © Jean-François COLONNA, 1997-2024.
Copyright © France Telecom R&D and CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 1997-2024.
