| One way to approach your wishes to learn "mathematics from first principles" is to see modern mathematics as the study and application of formal systems: http://en.wikipedia.org/wiki/Formal_system A formal system has several components: An alphabet of symbols from which sequences or strings of symbols are constructed. Some of these strings of symbols can be well-formed according to some formal grammar, which is the next component. Next we have a collection of basic assumptions, called axioms, which are supposed to reflect the obvious truths about whatever we want to formalize in the formal system. And then we have some rules of inference. They allow us to derive conclusions from premises. An example would be the rule of modus ponens: If we have "If A then B" and "A", we can conclude "B". An example of a formal system is ZFC set theory which can be regarded as a formalization of one concept, the concept of a set: We take "classical predicate logic" as a background formal system, it already has logical symbols, like symbols for AND, OR and "IF ... THEN ..." and quantifiers "FOR ALL ..." and "THERE EXISTS ...". We enhance this logic with one non-logical symbol, the binary element-of-symbol ∈. With it we want to express the idea that something is an element of something, for example x ∈ y is supposed to mean that x is an element of y. Of course this is a bit simplified, but now we can build expressions (with symbols from the alphabet, according to the grammar for logical formulas plus the element symbol) which talk about the element-of-relationship between individuals. Next, we sit together at a round table and discuss which properties about sets and element-of or membership of a set we see as self-evident - there is room for discussion and there can be many different intuitions. For example, as in ZFC set theory, we may want to have some existence axioms. They guarantee us that in this formal system certain objects do exist. An example is the axiom of the empty set: There exists a set which has no elements. This statement can be written in our formal language. Other axioms may have a more constructive meaning. Instead of telling us that something exists, they say that given the existence of some objects we know the existence of further objects. An example would be the axiom of set unions: Given some arbitrary sets A and B, there exists a set C, which contains all the members of A and all the members of B as its elements. Another axiom asserts the existence of an unordered pair of any two given sets, from this we can define the concept of an ordered pair, which is very important. ZFC is one example of a set theory, there are many different set theories. You could exchange classical logic with intuitionistic logic and arrive at some formal system for intuitionistic or constructive set theory. You can drop certain axioms, because maybe they do not appear as self-evident to you (for example the axiom of choice, which contributes the "C" in ZFC, is not accepted by some people). You may add further axioms to arrive at a possibly stronger theory. One interesting aspect about set theory is that the concept of set is very powerful and expressive, because many concepts from modern mathematics can be build up from sets: natural numbers 0,1,2,3,... can be constructed from the empty set, functions can be represented through ordered pairs of sets. Sometimes set theory is regarded as "the foundation of all mathematics", but feel free to disagree! Just because natural numbers can be modelled as sets it is not certain that natural numbers are indeed sets. The basic pattern above is the formalization of an intuitive or natural concept, something from everyday life. We try to capture the essentials of this concept within a formal system. And then we can use the deductive power of the formal system to arrive at new and hopefully interesting conclusions about whatever we wanted to formalize. These conclusions are theorems. Not all theorems are interesting, some are even confusing, paradox and disppointing. Formalization is used to arrive at new insights about the original concept. Interesting in this context is Carnap and his idea of explication of inexact prescientific concepts: http://en.wikipedia.org/wiki/Explication What I want to express with this is that it is really possible to start your journey into mathematics at a beginning. |