A mathematical theory is a mathematical model of a branch of mathematics that is based on a set of axioms. It can also simultaneously be a body of knowledge (e.g., based on known axioms and definitions), and so in this sense can refer to an area of mathematical research within the established framework.[1][2]

Explanatory depth is one of the most significant theoretical virtues in mathematics. For example, set theory has the ability to systematize and explain number theory and geometry/analysis. Despite the widely logical necessity (and self-evidence) of arithmetic truths such as 1<3, 2+2=4, 6-1=5, and so on, a theory that just postulates an infinite blizzard of such truths would be inadequate. Rather an adequate theory is one in which such truths are derived from explanatorily prior axioms, such as the Peano Axioms or set theoretic axioms, which lie at the foundation of ZFC axiomatic set theory.

The singular accomplishment of axiomatic set theory is its ability to give a foundation for the derivation of the entirety of classical mathematics from a handful of axioms. The reason set theory is so prized is because of its explanatory depth. So a mathematical theory which just postulates an infinity of arithmetic truths without explanatory depth would not be a serious competitor to Peano arithmetic or Zermelo-Fraenkel set theory.[3][4]

Jean Dieudonne, one of the most prominent mathematicians of the 20th century, writing about Zermelo-Fraenkel set theory in his book Dieudonne, Jean (1982). A panorama of pure mathematics as seen by Bourbaki. Academic Press. p. 215. stated: "The theory of sets, so conceived, embraces all mathematical theories, each of which is defined by the assignment of a certain number of letters (the “constants” of the theory) and relations that involve these letters (the “axioms” of the theory): for example, the theory of groups contains two constants, G and m (representing respectively the set on which the group is defined, and the law of composition), and the relations express first that m is a mapping from G x G to G, and second the classical properties of the law of composition."

In other words, any mathematical theory is an extension of the set theory( some relatively novel mathematical theories such as category theory may need to use additional axioms such as Tarski's axiom - see Tarski–Grothendieck set theory).

If a mathematical theory T is obtained from ZF by adding n new constants and some axioms then a model of T is a n-tuple satisfying the axioms. For example, a model of the theory of groups is a pair(A, f), where A is a nonempty set, f is a binary operation on A, satisfying the axioms of the theory.

This definition of model differs from the definition of model in model theory but is consistent with using the term "model" in mathematics, for example, the Beltrami-Klein model of the hyperbolic geometry.

See also

References

  1. ↑ Nelson, Sam. "Theorems and Theories". www.esotericka.org. Archived from the original on 2014-08-19.
  2. ↑ Chu-Carroll, Mark C. (13 March 2007). "Theorems, lemmas, and corollaries". Good math / bad math (blog).
  3. ↑ Maddy, Penelope (2011). Defending the Axioms: On the Philosophical Foundations of Set Theory. Oxford University Press. p. 82.
  4. ↑ Maddy, Penelope (1988). "The Journal of Symbolic Logic". Believing the Axioms II. 53 (3): 762. doi:10.2307/2274569. Retrieved 2 March 2020.


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