Consistency, Completeness, and Geometry
The preceding Dialogue is explicated to the extent it is possible at this stage. This leads back to the question of how and when symbols in a formal system acquire meaning. The history of Euclidean and non-Euclidean geometry is given, as an illustration of the elusive notion of "undefined terms". This leads to ideas about the consistency of different and possibly "rival" geometries. Through this discussion the notion of undefined terms is clarified, and the relation of undefined terms to perception and thought processes is considered.
While you read
- Why does DRH keep apologizing about his use of the word "isomorphism"?
- Give an example of when a word can have multiple meanings. When you say something, how do you communicate the intended meaning?
- Draw pictures of Euclid's five postulates. Do they seem like safe truths to assume?
- Do you consider mathematics to be the same in all possible worlds? Consider carefully what you consider to be mathematics.
- Do you consider logic to be the same in all possible worlds? What, then, constitutes a possible world?
- Answer the question on p. 102 about the record and the record player. How can they both pass their respective tests?
Euclidean geometry is consistent and complete
Something that you'd think would be mentioned based on the title: Euclidean geometry is an interesting example of a formal system that we know is both consistent and complete. It's not powerful enough for Gödel's Incompleteness Theorem to apply.
To really be able to say that, you have to formalize things a bit more than Euclid with his undefined terms, which modern mathematicians have done. See, for example, ➟ Tarski's axioms (which are slightly simplified), or ➟ Hilbert's axioms.
Further reading on Wikipedia: ➟ Euclidean geometry
Examples of non-Euclidean geometry
Spherical geometry is useful for modeling the surface of a planet such as the Earth (which is what "geometry" was originally supposed to mean anyway). In spherical geometry, Euclid's undefined term "line" refers to a great circle -- any circle dividing the sphere exactly in half, such as the equator of an idealized spherical Earth.
Every pair of lines meet in a "point", as defined by Euclid -- so this means the undefined term "point" actually refers to two locations across the globe from each other.
Consider the great circle A formed by the prime meridian and the 180° meridian, and the great circle B formed by the 90°E meridian and the 90°W meridian. These two "lines" meet in a "point" known as the North and South Pole. Meanwhile, A, B, and the equator form an equilateral triangle where every angle is a right angle.
Meanwhile, in ➟ hyperbolic geometry, there are many parallel lines through a point not on the given line. Lines curve away from each other.
Hyperbolic geometry can be modeled as a disk where things become smaller and smaller as they approach the edges -- which happens to be illustrated in Escher's Circle Limit III.
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- How to represent meaning
- This quotation from p. 88 has perhaps affected me more than most readers:
- Every word which we use has a meaning to us, which guides us in our use of it. The more common the word, the more associations we have with it, and the more deeply rooted is its meaning. Therefore, when someone gives a definition for a common word in the hopes that we will abide by that definition, it is a foregone conclusion that we will not do so but will instead be guided, largely unconsciously, by what our minds find in their associative stores.
- On top of ideas in Marvin Minsky's Society of Mind, this paragraph has affected the representation of the knowledge representation project that I maintain, called ConceptNet, a semantic network where meaning is represented as an aggregate of connections to other meanings, instead of by computationally precise definitions.
- The semantic network of GEB itself on p. 370 is a good visualization of what this representation looks like. If someone blanked out one of the bubbles, you may very well be able to infer what it was from its connections, once you've finished the book. --rspeer