Tag Archives: mathematics

The Case for Activism

So I watched the video Innocence of Muslims that supposedly caused the outrage ultimately leading to novel US protests throughout the Middle East.

First of all, it was clearly a satire.  The issue is that of blasphemy, which is illegal in many of the countries with these protests.  Blasphemy is often recognized as unacceptable within the religion itself, versus being illegal in a country that predominantly constitutes of those in said religion.  The latter likely implies the former (i.e. being illegal likely means it is unacceptable in the dominant religion).  Although perhaps ironically, the Quran and hadith do not mention blasphemy, whereas Christianity condemns it.[1,2]  In the case in question, the issue was about the legitimacy of Muhammad’s teachings. Given the laws within the countries and the video, it’s no surprise that the protests have erupted.  What I personally find puzzling however, is why in the 21st century blasphemy is still taken seriously.

Blasphemy essentially involves questioning that which is assumed to be true.  There is an inherent anxiety in this, for if the assumptions are without merit, the whole religion falls apart.  Now of course, if the assumptions are with merit, then questioning them is meaningless, so why punish individuals in this case?  Won’t their absence of finding anything troubling with the religious assumptions and corresponding time wasted in the process be punishment enough?

Now take two other institutions that have assumptions that are questioned: law and mathematics.  In law, the assumptions are the laws themselves.  Yet for ubiquitously accepted reasons, we permit our legislative representatives to amend and refine laws so that they are compatible with the time period.  Similarly in mathematics, one may start with some axioms and prove something from them (or analogously, start with some laws and note an acceptable behavior), then later the mathematician may change the axioms to prove something more general or more specific.

Thus for the same reasons, I’d argue that the questioning of the assumptions within a religion could only be beneficial to it–except perhaps to those who benefit from the doctrine at a given time.  Religion, like law, may particularly benefit some members at a given time, and such members may be reluctant to welcome change.  This is something of which I claim all members of the institution need to be aware.  For if you fail to question the authority and dogma, you simultaneously give permission to sustain what may amount to a select few disproportionately benefiting.

[1] Saeed, Abdullah; Hassan Saeed (2004). Freedom of Religion, Apostasy and Islam. Burlington VT: Ashgate Publishing Company. pp. 38–39. ISBN 978-0-7546-3083-8. (via Wikipedia: “Blasphemy”)

[2] Mark 3:29

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.

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.

Some Mathematical Misconceptions

1.  Math isn’t useful.
Quite the contrary.  As math is fundamental to all knowledge, developments in it trickle down to economics, physics, and computer science.  From there they make their way into business and political decision making, chemistry, biology, engineering, and then onto the social sciences.  Thus it may not be immediately useful; rather, it is a long-term investment whose payoff is slowest in return, but greatest in magnitude.

2.  Math is just arithmetic:  numbers and computation.
It’s no surprise that this conception exists given high school and lower division curriculum that strictly emphasize this.  While computation is an integral component to math, it is merely the mechanism by which genuine math is done.  Math is the epitome of science:  it simply consists of making assumptions and proving what follows (computation is the process generating the proof).  Of course one hopes the assumptions made do not contradict eachother–otherwise anything is provable (at least in traditional binary logic).  In this sense, any experiment is math, with assumptions being hypotheses and results being true if they are consistent with the hypotheses.  Even the practice of law is math, with assumptions being “people follow laws for all laws”, and true/false therefore coinciding with legal and illegal behaviors.  Everyday decision making of individuals (and hence social science) is math:  the assumptions being “biological predispositions and existence of a well-ordering of a set of possible decisions under given circumstances” and true decisions coinciding with those that are optimal (in the sense of the ordering).

3.  Humans don’t have to be mathematical.
This is actually a contradiction.  The human body in fact can be thought of as nothing more than a computer whose hardware is biology (even though our biology is soft/flimsy material–hopefully due for a big upgrade in the upcoming centuries) and whose software is a bundle of cognitive schemas.  Mathematically, we simply have that the assumptions are the hardware and arbitrary software installed at a later point, and the true consequences are just the perceptions that are consistent relative to the hardware and software (versus the false ones which are not).  For what we conventionally call a computer, the software is installed by humans.  For humans, the software is installed by the environment.  Sometimes the software itself may contradict the hardware–or other software for that matter.  Statements (computations) that follow which are in contradiction to the assumptions (hardware/other software) can lead to run on and halting problems (or in the case of humans, perceptions/schemas contradicting other schemas or hardware, possibly leading to psychological disorders).

4.  Any statement is either true or false.
This statement is (naively)false and undecidable.  The (meta)assumptions we have made so far are that true statements are simply those that follow from (direct)assumptions, and false statements are those whose negation follows from the (direct)assumptions.  Given those (meta)assumptions, the statement is (naively)false since the lack of (direct)assumptions (naively)contradicts the (meta)assumptions.  If a statement or its negation do not follow from a set of assumptions, we say it is undecidable/independent with respect to that set of assumptions.  Since we made no (direct)assumptions regarding statement 4, it is undecidable.

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.