Introduction to Abstract
Mathematics
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Understanding Mathematical Truths and Formal Logic Systems, Study notes of Mathematics
An introduction to the organization of mathematics classes, the nature of mathematical truth, and the concept of formal logic systems. It explains the importance of participating in mathematics and reading carefully, and discusses the subjective and formal aspects of mathematical truth. The document also introduces the MIU System as an example of a formal logic system and explains how theorems are derived from axioms using the rules of production.
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2021/2022
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Introduction to Abstract
Mathematics
I. General Class Organization
● explain homework assignments ● exams and quizzes ● how to study mathematics reading carefully is the key to success you must participate to learn ... mathematics is NOT a spectator sport
Mathematical truth is a formalism.
● .... true statements are those which follow from the axioms by means of logic. ● This is what provides mathematics with its greatest strength - truth can be demonstrated, there is no ambiguity about a mathematical truth; and its greatest weakness - mathematics can not prove anything that is not mathematical.
Logic vs. Meta-logic
Meta-logic is the language and thinking process we use to talk about logic. It is not itself logic, rather it is a stepping outside of the logical system in order to examine that system. We do this in other areas as well, for instance, we learn English grammar in foreign language classes.
A formal logic system
For instance, MUIU is a theorem. MI (2)→ MII (2)→ MIIII (3)→ MUI (1)→ MUIU. ● MUIIU is also a theorem. MI (above)→ MUIU (2)→ MUIUUIU (4)→ MUIIU. 1. xI ⇒ xIU 2. Mx ⇒ Mxx 3. xIIIy ⇒ xUy 4. xUUy ⇒ xy A theorem is a proposition which can be obtained from the axioms by applying the rules of the grammar.
A formal logic system
But clearly U is not a theorem.
Every theorem must be derived from the axiom MI, but there is no rule which removes an M, so all theorems must start with an M. (This is meta-logical arguing, we are talking about the rules of production, not using them!!)