The chasm between reality and its representation

Daniel Alroy

1.

Maps and territories. It is not mathematically possible to project 3D curvature onto a 2D surface without distorting areas or angles. This fact poses a challenge to map makers. The common Mercator cylindrical projection map distorts areas. As a result, such maps starting from either pole to 70 degrees latitude are virtually unusable. Thus, maps of the earth are necessarily incorrect; they are also inherently incomplete when contrasted with the represented territory. In conclusion, maps of the earth are necessarily incorrect and incomplete. By contrast, it is meaningless to ask whether a given territory is complete or correct. These attributes are, in principle, inapplicable to territories. Thus, there exists an unbridgeable chasm between maps and the territories they represent.

2.

Mathematics and science. Mathematics rich enough to contain arithmetic is subject to limitations in regards to completeness and consistency. In physics, representations are made by use of mathematics. As a consequence, the limitations of mathematics become an ineliminable part of physics. The same is true for any other scientific discipline that is mathematically formulated.

3.

Completeness, consistency, and reality. The attributes of completeness and consistency are not applicable to reality. Hence, there exists an unbridgeable chasm between mathematically formulated theories and the reality they represent. The dictum ‘the map is not the territory’ applies to physical theories as well as to cartography. Statements of the form ‘the map is identical with the territory it represents’ are tautologically false.

4.

Universal Turing Machines. Alan Turing has formulated a procedure that makes explicit the notion of computation. That abstract formulation is now called the Universal Turing Machine (UTM). John von Neumann modeled the general-purpose digital computer after the UTM. Both the UTM and the general-purpose computer are means for mathematical representation. Hence, it is tautologically false to state the UTM or the computer is identical with any aspect of reality they may represent.

5.

Can the brain be a Universal Turing Machine? Current intellectual fashion suggests that the biological cell, the brain, and nature itself, are UTMs. Such suggestions are based on the tautologically false premise that a representation is, or can be, identical with the reality it represents. The cell, the brain, and the universe are not universal Turing machines.