What Math Really Is

There is an almost ubiquitous misconception about what mathematics really is, and it’s a misconception that genuinely beckons a correction.  I would take a guess that if the average person was asked “What is mathematics?”, they would respond with something along the lines of “well, it’s a bunch of rules that help you find certain numbers”.  While this may have been a correct answer long ago, it is far from correct today.

Fortunately, what it actually is can be summarized very succinctly.  Mathematics is simply the process of making assumptions and proving what follows.  Hence, we all do math on a daily basis—either when talking to one another, or when thinking to oneself:  “given what I know, I think that…”.  This is math.

This is also why math courses through calculus are terrible—as they are absurdly misleading.  Current curriculum is libelous to the discipline of mathematics and its participants, and action needs to be taken to address this.

At this point, high school language arts and composition courses may teach more math than actual math courses.  Fundamental to mathematics is logic and its application in the context of sets.  Logic was essentially nonexistent when I was in high school.  Yet math and language arts classes implicitly assume that students have a solid understanding of it when they are asked to make arguments.  Granted, as we are logical entities cognitively, we are trivially masters of logic.  But in terms of conveying it communicably, improved training is necessary.  I feel the overlooking of this necessity is a grave miscalculation that has hindered scientific thinking (an ability from which every citizen of the world can drastically benefit) for far longer than it should have.  This needs to change.

Proposed Changes to Secondary Math and Language Arts Curriculum

It’s no controversy that education appears to be under par in comparison to what we might hope.  I claim this has not only to do with home/environmental conditions for individual students and the ever omnipresent media (and lately, social networking media as well), but also with the curriculum in the classes they take–particularly math and language arts courses.  What seems to be the case is that the math courses focus on topics that, while enhancing the mental discipline of students, fail to teach concepts that are readily applicable to everyday situations.  Language arts courses emphasize composition (and in particular, stress the ability to make effective arguments) under the premise that students are already well-versed in the grammatical aspects of language (an assumption so far from the truth that the very idea of expecting them to write cogent argumentative essays is ludicrous).

Math:

I’d recommend keeping the middle school curriculum essentially the same (introducing key concepts of elementary algebra and geometry), but I’d vehemently oppose the idea of continuing this sort of curriculum for another four years.  In 9th grade, students should be introduced to key concepts from set theory (set notations and operations, functions, equivalence relations, bijections and counting principles, and concepts from elementary number theory, including divisibility, prime numbers, and the fundamental theorem of arithmetic).  With this collection of concepts in their inventory, they can spend 10th grade covering concepts from logic and proof writing–and applying them to concepts they learned the previous year (as well as everyday phenomena).

11th grade could then be devoted to a more serious algebra course–covering concepts of operations more thoroughly than the 9th grade course, identities, concepts like associativity and commutativity, matrices, inverses, basic definitions and examples of groups, rings, and fields, polynomials, and comparisons between integers, rationals, reals, and complex numbers, and the fundamental theorem of algebra.  If students only took these courses and earned a C in them, I’d still argue they were better prepared for life than earning an A in the traditional courses.

An additional 12th grade class covering geometry more seriously could entail concepts of lengths, triangles, trigonometry, polygons, area of polygons, polytopes and hypervolume, axiomatic Euclidean, spherical, and hyperbolic geometry and basic properties of spherical and hyperbolic triangles, approximation of areas/volumes and method of exhaustion (leading up to a freshman course in real analysis (to replace calculus)).

Language Arts:

I’d recommend middle school curriculum to focus a little more on grammar and sentence structure.  A 9th grade course could solidify this with a rigorous treatment of the core concepts of language: parts of speech, morphology, syntax, and grammar.  Concepts like word families, clauses (independent, dependent, subject, and predicate), transitive and intransitive verbs, and sentence diagramming should be emphasized.

A 10th grade course emphasizing argumentative writing would appropriately accompany the 10th grade math curriculum in logic and proof writing.  I’d recommend abstaining from requiring students to read nontechnical works until the third year of language arts.  This way, rather than getting distracted by a fiction or nonfiction story as a platform for an argument, emphasis is instead placed on the concept of argumentative writing itself.  Smaller readings should be used in this course.

An 11th grade course could then more effectively do in one year what four years of language arts typically try to do.  This course could focus on work-based and research-based extended essay writing.

Curriculum comparable to the AP English Literature course could make an appropriate optional fourth year class.

Update of Language Definition

Note that I have removed the factorization requirement from the definition of a language in Fundamental Knowledge Part 1;  so we will just have $\mathcal{L}_{F,T,W}=F[T[W]].$  This will remove some triviality in examples of fuzzy logic systems in the upcoming post.  The original motivation behind the factorization was that traditionally compound terms are considered formulas, but terms themselves are not considered formulas.  I don’t really see why we can’t let terms be formulas;  let us assume “substitutions” have already been made.

I have also removed the requirement that $\varphi(\phi)=\varphi(\psi)$ for all $\phi,\psi$ in a theory $X$ where $\varphi$ is a logic system.  Instead I have defined a logic system that satisfies this condition as a normal logic system.

Lemmas, Theorems, Corollaries, and Propositions

I thought I’d attempt to come up with a consistent convention of classifying mathematical statements.  To do so we will simply think of the collection of statements as a partially ordered set where $\phi\leq\psi$ is interpreted as $\phi\Rightarrow\psi$.  And for each statement/node/point there will be two $\mathbb{N}$-valued functions that represent the number of incoming and outgoing edges (i.e. immediate antecedents and consequences).  A point will be called a corollary if it has one incoming edge.  A point will be called a theorem if its set of successors is a “large” subset of the poset (even if it is a corollary).  A point will be called a lemma if at least one of its immediate successors is a theorem.  A point will be called an axiom if it has zero incoming edges.  And a point will be called a proposition if it is none of the above.

Fundamental Knowledge-Part 2: Models

The next task is to absorb the traditional area of mathematical logic.  One key missing ingredient is a model.  Let us recall the traditional setup (taken from [1]).

Definition 2.1.  Let $S$ be a set (of symbols).  An $S$-structure is a pair $\mathfrak{A}=(A,\mathfrak{a})$ where $A$ is a nonempty set, called a universe, and $\mathfrak{a}$ is a map sending symbols to elements, functions, and relations of $A$.  An assignment of an $S$-structure $(A,\mathfrak{a})$ is a map $\beta:S\to A$.  An $S$-interpretation is a pair $\mathfrak{I}=(\mathfrak{A},\beta)$ where $\mathfrak{A}$ is an $S$-structure and $\beta$ is an assignment in $\mathfrak{A}$.

For shorthand notation, the convention (with some of my modifications) is to write:  $c^\mathfrak{A}=\beta(c)$, $(f(t_1,...,t_n))^\mathfrak{A}=\mathfrak{a}(f)(\beta(t_1),...,\beta(t_n))$, and $(xRy)^\mathfrak{A}=\beta(x)\mathfrak{a}(R)\beta(y)$.  These are the terms.  Formulas are then built from the terms using traditional (although this can be generalized) logical connectives.

The notion of a model is then defined via induction on formulas.

Definition 2.2.  Let $\mathfrak{I}=(\mathfrak{A},\beta)$ be an $S$-interpretation.  We say that $\mathfrak{I}$ satisfies a formula $\phi$ (or is a model of $\phi$), denoted $\mathfrak{I}\vDash\phi$, if $\phi^\mathfrak{A}$ holds, where $\phi^\mathfrak{A}$ is defined via its components and $\beta$ and $\mathfrak{a}$ where necessary.

Formal languages in convention are built up from the formulas mentioned above, which are nothing more than special cases of Alt Definition 1.3.  A model for a language is hence nothing more than an $A$-interpretation into a structure, where $A$ is an alphabet (provided it is equipped with a logic system).  This is precisely what I have constructed in Part 1;  the symbols of $W\subset A^*$ are mapped to the universe $\mathcal{L}_{F,T,W}$.  The next thing to establish is that every model is a language model.  This is trivial since a model by definition satisfies a set of formulas as well as compounds of them (i.e. it must satisfy a language).  Hence we have no need to trouble ourselves with interpretations and may simply stick to the algebra of Part 1.

While we have absorbed model theory, there are a few more critical topics to absorb from mathematical logic. We return to the language of Part 1 (no pun).  Let $X$ be a theory of $\mathcal{L}_{F,T,W}$ and $\varphi:F[X]\to V$ be a binary logic system.  A formula $\phi\in\mathcal{L}_{F,T,W}$ is derivable in $X$ if it is a proposition (i.e. is in $F[X]$).  We may write $X\vdash\phi$.  This definition is in complete agreement with the traditional definition (namely, there being a derivation, or finite number of steps, that begin with axioms and use inference rules);  it is nothing more than saying it is in $F[X]$.  Similarly $\phi\in\mathcal{L}_{F,T,W}$ is valid if $\varnothing\vdash\phi$, or equivalently, it is derivable in any theory.  In our setup this would imply $\phi\in F[\varnothing]=\varnothing$.  Hence no formula is valid.

Let $F$ have a unary operation $\lnot$ and $\varphi:F[X]\to V$ be a logic system on a theory $X$.

If we assume $\lnot$ to be idempotent ($\lnot\lnot\phi=\phi$), then since $\varphi$ is a homomorphism, we have $\varphi(\phi)=\varphi(\lnot\lnot\phi)=\lnot\lnot\varphi(\phi)$.  That is, the corresponding unary operation in $V$ must also be idempotent on $ran(\varphi)$.

Definition 2.3.  A unary operation $\lnot$ (not necessarily idempotent) is consistent in $\varphi$ if for all $\phi\in F[X]$, $\varphi(\phi)\neq\varphi(\lnot\phi)$.

If we assume $\lnot$ is consistent in $\varphi$ and that $\varphi$ is a binary logic system, then the corresponding $\lnot$ in $V$ is idempotent since

$\varphi(\phi)=0\Rightarrow\varphi(\lnot\phi)=1\Rightarrow\varphi(\lnot\lnot\phi)=\lnot\lnot\varphi(\phi)=0$.

Again, proofs in a binary system are independent of the choice of valence.   If we assume consistency and idempotency, then we have a nonidentity negation which is idempotent on the range.  The case for assuming binary system and idempotency yields either a trivial mapping of propositions (all to $0$ or all to $1$), or that $\lnot$ is consistent and idempotent on $V$.  And lastly if we assume all three (idempotency and consistency of $\lnot$ together on a binary system), we obtain a surjective assignment with idempotent negation in $V$.

Let $\varphi:F[X]\to V$ be a binary logic system where $V$ is a boolean algebra.  Then the completeness and compactness theorems are trivial.  Recall these statements:

Completeness Theorem.  For all formulas $\phi$ and models $\mathfrak{I}$,

$\mathfrak{I}\vDash\phi\Rightarrow X\vdash\phi$

where $\mathfrak{I}\vDash X$.

Compactness Theorem.  For all formulas $\phi$ and models $\mathfrak{I}$,

$X\vdash\phi\Rightarrow\mathfrak{I}\vDash\phi$

where $\mathfrak{I}\vDash X$.

Traditionally these apply to, what we would call, a binary logic system $\varphi:F[X]\to V$ where $V$ is a boolean algebra (hence $F$ has a consistent, idempotent negation) under traditional operations, and in particular this fixes the operational/relational structures of $F, T$, and $W$ , but $X$ is arbitrary.  In this setup, all “formulas” (or what we would hence call propositions since they are generated by a theory) are trivially satisfiable since they have a language model.  Hence Compactness is true.  Moreover since they are propositions in a binary logic system, they are in some $F[X]$ for a theory $X$ and are hence derivable; so we have Completeness.

Lastly we wish to address Godel’s Second Incompleteness Theorem;  recall its statement:

Godel’s Second Incompleteness Theorem.  A theory contains a statement of its own consistency if and only if it is inconsistent.

We have only defined what it means for a unary operation in a logical system to be consistent.  Hence we can say that a binary logic system with a unary operation is consistent if its unary operation is consistent.  But all of these traditional theorems of mathematical logic are assuming a binary logic system where $V$ is a boolean algebra , $\lnot$ is idempotent, and the map $\varphi:F[X]\to V$ is surjective.  Hence $\lnot$ is consistent (from above discussion), and the consequence in the theorem is false.

The weakest possible violation of the antecedent of Godel’s theorem is to use a structure to create itself (i.e. that it is self-swallowing), which makes no sense, let alone using it to create a larger structure within which is a statement about the initial structure.  That a binary logic system with unary operation could contain a statement of its own consistency is itself a contradiction, since the theory itself, together with the statement $\phi$, are in a metalanguage.  It is like saying that one need only the English language to describe the algebraic structure of the English language.  As we previously said at the end of Part 1, one can get arbitrarily close to doing this–using English to construct some degenerate form of English, but you can never have multiple instances of a single language in a language loop.  Another example would be having the class of all sets, then attempting to prove, using only the sets and operations of them, that there is a class containing them.

Hence the antecedent is also false.  So both implications are true.

[1]  Ebbinghaus, H.-D., J. Flum, and W. Thomas.  Mathematical Logic.  Second Edition.  Undergraduate Texts in Mathematics.  New York: Springer-Verlag.  1994.