TOK Math Notes

TOK Math Notes Axioms Axioms are the basic assumptions that any mathematical system relies on; they are often thought of to be self-evident truths. However, the nature of axioms prevents one from trying to prove an axiom; one would be stuck in an “infinite regress” if this was attempted. Hence “obvious” knowledge is assumed to be true. The four traditional requirements for an axiom are: 1. Consistent (The same set of axioms should always produce the same answer) 2. Independent (You should not deduce an axiom from another axiom; they are se lf-evident truths) 3. Simple (Should be as clear as possible for easy comprehension) 4. Fruitful (Have a diverse set of applications in terms of theorems produced) The fundamental problem with axioms lies in the assumption that axioms are actually true since there is not a method of verification besides “It’s obvious”. The knowledge issue that arises is, “To what extent are the assumptions behind axioms in a mathematical system invulnerable to error?” Since there are limits to the way of knowing, such as perception, reasoning and language, it is possible that the assumptions in axioms can be misrepresented and thus misused, making it vulnerable to e rror. Deductive Reasoning This is essentially the application of deductive reasoning on axioms. Theorems Theorems are concepts concepts and “rules” that arise when one applies the axioms of a formal system onto an example. Once established, theorems can be used to create even more theorems.

Euclid’s axioms were: 1.It shall be possible to draw a straight line joining any two points. 2.A finite straight line may be extended ex tended without limit in either direction. 3.It shall be possible to draw a circle with a given centre and through a given point. 4.All right angles are equal to one another. 5.There is just one straight line through a given point which is parallel to a given line. (van de Lagemaat 190).

This critical knowledge issue was brought to the mind of mathematicians when some started to question the validity of Euclid’s axioms. What if they were not true, or were true in some case s? What if there were other forms of mathematical “truth”? Prior to the 19th century, Euclidean geometry was assumed to be correct because the axioms were true since they passed both truth tests: they were logically consistent and true in the world (handout). After all, how could one doubt “self -evident truths”?

Riemann Geometry: His system of geometry challenged those who believed that non-Euclidean axioms would “lead to a contradiction and so collapse”. However, no contradictions appeared, even though he had replaced Euclid’s first, second and fifth axioms. Furthermore, Riemann was able to modify these axioms by assuming that space was like the surface of a sphere, not an infinite plane that Euclid assumed. Essentially, Riemann proved that there can be multiple approaches to describing physical reality, and that axioms were not invulnerable to criticism and interpretation. Riemannian geometry also introduced the idea that formal systems have limits. For example, Euclidean geometry would appear to work quite well for regular life, but it is extremely poor for understanding flight plans and space around stellar phenomena, such as black holes.

Godel’s Incompleteness Theorem: stated that it is impossible to prove that a formal mathematical proof is free from contradiction. In other words, we cannot be certain whether or not mathematics is free from contradictions; it cannot give us certainty.

In a proof , a theorem is shown to follow logically from the relevant axioms. Conjecture: a hypothesis that seems to work, but has not been shown to be true.

The Nature of Propositions: Analytic: a proposition that is true by definition Synthetic: any proposition that is not analytic.

How we know a proposition is true: A Priori: The proposition is true without experience. (analytic) A Posteriori: The proposition needs experience to be known. (synthetic)

Formalism: 

Mathematical truths are true by definition

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Math is a game we make up; we can change the rules at any time and everything will still make sense.

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Math is invented

Platonism: 

Mathematical truths give us a priori into the structure of re ality.

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Math controls the universe; we can’t change the rules because that would ruin the order of the world.

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Math is discovered