$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 1.2: Naïvely

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

Let us begin by talking informally about mathematical structures and mathematical languages.

There is no doubt that you have worked with mathematical models in several previous mathematics courses, although in all likelihood it was not pointed out to you at the time. For example, if you have taken a course in linear algebra, you have some experience with $$\mathbb{R}^2$$, $$\mathbb{R}^3$$, and $$\mathbb{R}^n$$ as examples of vector spaces. In high school geometry you learned that the plane is a "model" of Euclid's axioms of geometry. Perhaps you have taken a class in abstract algebra, where you saw several examples of groups: The integers under addition, permutation groups, and the group of invertible $$n \times n$$ matrices with the operation of matrix multiplication are all examples of groups - they are "models" of the group axioms. All of these are mathematical models, or structures. Different structures are used for different purposes.

Suppose we think about a particular mathematical structure, for example $$\mathbb{R}^3$$, the collection of ordered triples of real numbers. If we try to do plane Euclidean geometry in $$\mathbb{R}^3$$, we fail miserably, as (for example) the parallel postulate is false in this structure. On the other hand, if we want to do linear algebra in $$\mathbb{R}^3$$, all is well and good, as we can think of the points of $$\mathbb{R}^3$$ as vectors and let the scalars be real numbers. Then the axioms for a real vector space are all true when interpreted in $$\mathbb{R}^3$$. We will say that $$\mathbb{R}^3$$ is a model of the axioms for a vector space, whereas it is not a model for Euclid's axioms for geometry.

As you have no doubt noticed, our discussion has introduced two separate types of things to worry about. First, there are the mathematical models, which you can think of as the mathematical worlds, or constructs. Examples of these include $$\mathbb{R}^3$$, the collection of polynomials of degree 17, the set of 3x2 matrices, and the real line. We have also been talking about the axioms of geometry and vector spaces, and these are something different. Let us discuss those axioms for a moment.

Just for the purposes of illustration, let us look at some of the axioms which state that $$V$$ is a real vector space. They are listed here both informally and in a more formal language:

Vector addition is commutative: $$\left( \forall u \in V \right) \left( \forall v \in V \right) u + v = v + u$$.

There is a zero vector: $$\left( \exists 0 \in V \right) \left( \forall v \in V \right) v + 0 = v$$.

One times anything is itself: $$\left( \forall v \in V \right) 1v = v$$.

Don't worry if the formal language is not familiar to you at this point; it suffices to notice that there is a formal language. But do let us point out a few things that you probably accepted without question. The addition sign that is in the first two axioms is not the same plus sign that you were using when you learned to add in first grade. Or rather, it is the same sign, but you interpret that sign differently. If the vector space under consideration is $$\mathbb{R}^3$$, you know that as far as the first two axioms up there are concerned, addition is vector addition. Similarly, the 0 in the second axiom is not the real number 0; rather, it is the zero vector. Also, the multiplication in the third axiom that is indicated by the juxtaposition of the 1 and the $$v$$ is the scalar multiplication of the vector space, not the multiplication of third grade.

So it seems that we have to be able to look at some symbols in a particular formal language and then take those symbols and relate them in some way to a mathematical structure. Different interpretations of the symbols will lead to different conclusions as regards the truth of the formal statement. For example, if we take the commutivity axiom above and work with the space $$V$$ being $$\mathbb{R}^3$$ but interpret the sign $$+$$ as standing for cross product instead of vector addition, we see that the axiom is no longer true, as cross product is not commutative.

These, then, are our next objectives: to introduce formal languages, to give an official definition of a mathematical structure, and to discuss truth in those structures. Beauty will come later.