Additions:
Even today, strictly speaking, mathematics does not model a physical continuum. The most it can do is talk about an approximation. We cannot write down a typical [[http://en.wikipedia.org/wiki/Real_number real number]], like ""π"" or ""e"", but only the symbols for the number, or we can write down as many decimal places as time or space permits. [[http://en.wikipedia.org/wiki/Empiricism Empiricism]] in science does not strictly allow us to obtain the limit for such quantities. There is a mathematical idea of a limit, through successive approximation, and in many cases we can calculate what the result of successive approximation would be, but we cannot actually carry out an infinite sequence of successive approximations, because it would require infinite time and resource to do so. We know that there is no such thing as a perfect mathematical line in Nature; a mathematical line has no width, but that is not true of any line we can draw. Likewise we know that more precise measurement of the position of a particle does not enable us to identify it's position, but merely leads to the uncertainties of quantum theory. In order to understand those uncertainties, rather than base science mathematical assumptions which cannot be empirically justified, we should re-examine the foundations of mathematics and of physics to see what can be justified.
<<""Definition: A measurement is a count of the units of a measured quantity, where the definition of a unit of the quantity invokes comparison between the matter under study and some specially, but arbitrarily, chosen reference matter""
Deletions:
Even today, strictly speaking, mathematics does not model a physical continuum. The most it can do is talk about an approximation. We cannot write down a typical [[http://en.wikipedia.org/wiki/Real_number real number]], like ""π"" or ""e"", but only the symbols for the number, or we can write down as many decimal places as time or space permits. [[http://en.wikipedia.org/wiki/Empiricism Empiricism]] in science does not strictly allow us to obtain the limit for such quantities. There is a mathematical idea of a limit, through successive approximation, and in many cases we can calculate what the result of successive approximation would be, but we cannot actually carry out an infinite sequence of successive approximations, because it would require infinite time and resource to do so. We know that there is no such thing as a perfect mathematical line in Nature; a mathematical line has no width, but that is not true of any line we can draw. Likewise we know that more precise measurement of the position of a particle does not enable us to identify it's position, but merely leads to the uncertainties of quantum theory. In order to understand those uncertainties, rather than base science upon mathematical assumptions which cannot be empirically justified, we should re-examine the foundations of mathematics and of physics to see what can be justified.
Additions:
Even today, strictly speaking, mathematics does not model a physical continuum. The most it can do is talk about an approximation. We cannot write down a typical [[http://en.wikipedia.org/wiki/Real_number real number]], like ""π"" or ""e"", but only the symbols for the number, or we can write down as many decimal places as time or space permits. [[http://en.wikipedia.org/wiki/Empiricism Empiricism]] in science does not strictly allow us to obtain the limit for such quantities. There is a mathematical idea of a limit, through successive approximation, and in many cases we can calculate what the result of successive approximation would be, but we cannot actually carry out an infinite sequence of successive approximations, because it would require infinite time and resource to do so. We know that there is no such thing as a perfect mathematical line in Nature; a mathematical line has no width, but that is not true of any line we can draw. Likewise we know that more precise measurement of the position of a particle does not enable us to identify it's position, but merely leads to the uncertainties of quantum theory. In order to understand those uncertainties, rather than base science upon mathematical assumptions which cannot be empirically justified, we should re-examine the foundations of mathematics and of physics to see what can be justified.
Deletions:
Even today, strictly speaking, mathematics does not model a physical continuum. The most it can do is talk about an approximation. We cannot write down a typical [[http://en.wikipedia.org/wiki/Real_number real number]], like ""π"" or ""e"", but only the symbols for the number, or we can write down as many decimal places as time or space permits. [[http://en.wikipedia.org/wiki/Empiricism Empiricism]] in science does not strictly allow us to obtain the limit for such quantities. There is a mathematical idea of a limit, through successive approximation, and in many cases we can calculate what the result of successive approximation would be, but we cannot actually carry out an infinite sequence of successive approximations, because it would require infinite time and resource to do so. We know that there is no such thing as a perfect mathematical line in Nature; a mathematical line has no width, but that is not true of any line we can draw. Likewise we know that more precise measurement of the position of a particle does not enable us to identify it's position, but merely leads to the uncertainties of quantum theory. In order to understand those uncertainties, rather than base science mathematical assumptions which cannot be empirically justified, we should re-examine the foundations of mathematics and of physics to see what can be justified.
Additions:
Additions:
Leucippus placed his atoms in //the void//. Exactly what was meant by the void is subject to some debate, but if the theory was proposed as an answer to Zeno then the void was not a space continuum, as it was later understood by the Epicureans. It may have been the void in the sense of ""Parminedes"", meaning a complete absence of properties. Parminedes had criticised this notion of the void, saying that something with no properties cannot be said to have existence. If this is what Leucippus intended, then Democritus corrupted the notion by saying that “size” is a property of an atom. This invokes Parminedes’ paradox - if the void has no properties then atoms would all be jammed together so that motion would be impossible.
"" In relational quantum gravity, Parminedes’ paradox is avoided because matter is described in terms of elementary point-like, or sizeless, particles, in the sense that a point is that which has no size, and size cannot have meaning in the absence of the properties of space. The notion of the void is useful, not as a physical entity, but for visualisation. Relational quantum gravity describes a structure consisting of elementary particles, their interactions, and nothing else. To understand the properties of that structure we can draw diagrams on paper, but it must be understood that the geometrical properties of the paper have no bearing on the properties of the structure drawn. Mathematically these diagrams are graphs, in which only the nodes and lines have meaning. |
""
Deletions:
Leucippus placed his atoms in //the void//. Exactly what was meant by the void is subject to some debate, but if the theory was proposed as an answer to Zeno then the void was not a space continuum, as it was later understood by the Epicureans. It may have been the void in the sense of ""Parminedes"", meaning a complete absence of properties. Parminedes had criticised this notion of the void, saying that something with no properties cannot be said to have existence. If this is what Leucippus intende, then Democritus corrupted the notion by saying that “size” is a property of an atom. This invokes Parminedes paradox - if the void has no properties then atoms would all be jammed together so that motion would be impossible.
""| In relational quantum gravity, Parminedes paradox is avoided because matter is described in terms of elementary point-like, or sizeless, particles, in the sense that a point is that which has no size, and size cannot have meaning in the absence of the properties of space. The notion of the void is useful, not as a physical entity, but for visualisation. Relational quantum gravity describes a structure consisting of elementary particles, their interactions, and nothing else. To understand the properties of that structure we can draw diagrams on paper, but it must be understood that the geometrical properties of the paper have no bearing on the properties of the structure drawn. Mathematically these diagrams are graphs, in which only the nodes and lines have meaning. |
""
