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R. Barrett and P. P. Delsanto
in principle impossible. As physics relies on mathematics for its formulation, one may well ask what implications such an observation might have for
physics. In his moving tribute to his friend, Hermann Weyl, John Wheeler
puts the question: Do we not have to say that the notion of a physical world with
a continuous infinity of degrees of freedom is an equal idealisation, an equal folly,
an equal trespass beyond strict logic? [20] In other words, shouldn’t time and
space also be quantised? The response to this question, like many others at
the advancing edge of physics, depends on who is providing the answer [29].
In earlier Chapters, we have discussed several times the disagreement
between the deterministic nature of Classical Physics and Relativity on the
one hand, and the probabilistic nature of Quantum Mechanics on the other.
If ever we are to develop a quantum theory of gravity, this conflict must be
tackled. In a recent paper [30], Flavio Del Santo and Nicolas Gisin have
cast doubt on the determinism of classical physics, claiming that its alleged
deterministic character is based on the metaphysical, unwarranted assumption of
“infinite precision”. By “infinite precision” they mean the specification of the
value of a quantity as a real number (see Chap. 4) with an infinite number of
decimal places.
As long ago as 1955, Max Born had raised similar doubts: “Statements like
‘a quantity X has a completely definite value’ (expressed by a real number and
represented by a point in the mathematical continuum) seem to me to have no
physical meaning [31].” In other words, a proposition, such as “X = 1”, is
neither true nor false, since X may equally well be equal to 0.999999999…,
with any arbitrarily large number of digits “9”.
Let us return to our question: are time and space also discrete? Apparently so, because for intervals of time below a certain threshold, it is difficult
to conceive how any physical interaction can take place. The same considerations apply to space, since time and space must be treated on an equal
footing, being part of the same four-dimensional space-time, as we saw in
Chaps. 6 and 7.
As Del Santo and Gisin point out, whatever the size of these extremely
small cells of space-time, they must be capable of carrying all the information
about local interactions. In other words, information, and all the numerical
quantities involved, must be embodied into a physical system (encoding),
allowing it to be manipulated (computation) and transmitted (communication). But since most naturally occurring numbers have an infinite number
of digits (even rational numbers such as 1/3 must be stored with an infinite string of digits), they cannot be stored in a finite (and extremely small)
cell. They must somehow and somewhere be truncated. This implies that the
mathematics becomes “quantized”, since truncated numbers are multiples of
R. Barrett and P. P. Delsanto
in principle impossible. As physics relies on mathematics for its formulation, one may well ask what implications such an observation might have for
physics. In his moving tribute to his friend, Hermann Weyl, John Wheeler
puts the question: Do we not have to say that the notion of a physical world with
a continuous infinity of degrees of freedom is an equal idealisation, an equal folly,
an equal trespass beyond strict logic? [20] In other words, shouldn’t time and
space also be quantised? The response to this question, like many others at
the advancing edge of physics, depends on who is providing the answer [29].
In earlier Chapters, we have discussed several times the disagreement
between the deterministic nature of Classical Physics and Relativity on the
one hand, and the probabilistic nature of Quantum Mechanics on the other.
If ever we are to develop a quantum theory of gravity, this conflict must be
tackled. In a recent paper [30], Flavio Del Santo and Nicolas Gisin have
cast doubt on the determinism of classical physics, claiming that its alleged
deterministic character is based on the metaphysical, unwarranted assumption of
“infinite precision”. By “infinite precision” they mean the specification of the
value of a quantity as a real number (see Chap. 4) with an infinite number of
decimal places.
As long ago as 1955, Max Born had raised similar doubts: “Statements like
‘a quantity X has a completely definite value’ (expressed by a real number and
represented by a point in the mathematical continuum) seem to me to have no
physical meaning [31].” In other words, a proposition, such as “X = 1”, is
neither true nor false, since X may equally well be equal to 0.999999999…,
with any arbitrarily large number of digits “9”.
Let us return to our question: are time and space also discrete? Apparently so, because for intervals of time below a certain threshold, it is difficult
to conceive how any physical interaction can take place. The same considerations apply to space, since time and space must be treated on an equal
footing, being part of the same four-dimensional space-time, as we saw in
Chaps. 6 and 7.
As Del Santo and Gisin point out, whatever the size of these extremely
small cells of space-time, they must be capable of carrying all the information
about local interactions. In other words, information, and all the numerical
quantities involved, must be embodied into a physical system (encoding),
allowing it to be manipulated (computation) and transmitted (communication). But since most naturally occurring numbers have an infinite number
of digits (even rational numbers such as 1/3 must be stored with an infinite string of digits), they cannot be stored in a finite (and extremely small)
cell. They must somehow and somewhere be truncated. This implies that the
mathematics becomes “quantized”, since truncated numbers are multiples of