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.


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