Vershun’s Brain Dump

I’m not quite sure how or when I exalted mathematics to a point beyond the scope of scrutiny, but when I mentioned yesterday to Doug that I was questioning the infallibility of math he responded with “Yeah… so?”
It’s strange to me that I never questioned mathematics at its fundamental level when I seem to question everything else.  It is extremely easy to throw out criticisms of it when everywhere you look and everything we have is a byproduct of math and science.  I viewed opposition of mathematics as iconoclastic in a way; merely a method to spark argument for argument’s sake (like critiquing the literal translation of parts in the Bible to get a rise out of people).  I regret not listening more.

It started when I was reading “Zen and the Art of Motorcycle Maintenance,” but it could have just as easily started in an introduction to philosophy class, wondering why we thought the Earth was flat (and why we now think it’s round), or any other sort of thought dealing with facts, or truth.  The question is “What is the nature of truth?”
To put it kindly, my knowledge of philosophy is horrible.  I found myself thinking about the question fairly extensively and I came up with truth is subjective and is only valid when it yields some sort of perceived results or convincing indirect results.  The results that can’t be perceived that are regarded as fact are due to an extension of what Stanley Fish calls interpretive communities, or trusted groups of people that tell us results occur and are believed by us.  This view puts truth within the individual and leaves room for religion and superstition to be treated as truths but also opens up a plethora of unsolvable discrepancies between facts.  This quick-n-dirty theory is essentially just a poorly thought out Pragmatic Theory of Truth, but it seems to work in my mind so I’ll regard it as true, for now.
So on to math.  Mathematics seemed like an easy place to start with truth since I believed it was the poster child of pure logical abstraction.  However, the more I thought about math on a conceptual level the less satisfied I became with my prior disposition.

Math as a Language
Parallels Between Mathematics, Linguistics, and Rhetoric

It was George Orwell that introduced (to me, that is) the idea that our thoughts are constrained by our language.  In his book, 1984, Orwell’s dystopia enforces a language called Newspeak which consists of only the most basic elements of language.  This was done in order to make thoughts unapproved by the aristocracy “unthinkable.”  His idea was limit the language, limit the thoughts.
If stripping a language of its superfluous vocabulary results in a more narrow scope of thought, it would seem that extending the vocabulary of a language would result in deeper and more meaningful thought.  The problem I have with this is that all new words introduced into a language has to be defined by the basic elements of the existing language (excluding nouns of external objects which can be defined by interaction and sensation).  A language is built from fundamental experiences that are universally shared and from these simple building blocks we construct more complicated symbols that are used to more accurately describe experience.

Using the basic building blocks of language we can begin to construct logical arguments.  In rhetoric, an enthymene is a starting place for a chain of logic in which you leave off the first step because it’s self-evident or universally agreed upon.  It creates an implicit premise which allows a starting point for the syllogism to be constructed upon.

Mathematics is built entirely from axioms and postulates.  Axioms, in particular a subset called logical axioms, are statements that are taken to be universally true (self-evident) and therefore create a starting point in which a system is built.  Extremely complex and useful systems arise from these simple axioms, but at the very base of any theorem is generally just a few fairly simple axioms; they’re the Newspeak of mathematics.

I’ve heard all my life mathematics described as “the universal language.”  It wasn’t until very recently that I started to see just how much of a language math really is.  Its basic building blocks are its core vocabulary, its rules are its linguistic syntactical structures, its theorems are its rhetoric’s syllogisms.  English, French, Spanish, and math.  Ambiguity comes with language; I think it would be hard to find someone who would argue that any language is perfect (by what measure would you define perfection in the first place?).

Math Abstractions
Abstracting the Real as a Way of Modeling our World

The real power of mathematics is how we can model fairly complex ideas through a more simple abstraction or model.  Nothing is a perfect cube, but l^3 will give us a great idea of the volume of a cube-like object.  We have done this for nearly everything in the macroscopic world.  Modeling from simple stationary objects  to fairly complex equations modeling speed, acceleration, waves, vector fields, population changes over fixed resources, and a myriad of other models we have abstracted from the “real” to put into our ideal frameworks.
I should say right now that I am in no way dismissing or critiquing the usefulness of these techniques and my argument against it wouldn’t even be considered weak.  Inane, useless, and vacuous come to mind.
It doesn’t FEEL right.  Something doesn’t sit well having to use an irrational number to define something as simplistic as a circle.  Doesn’t it seem odd that a circle, a fundamental shape, is IMPOSSIBLE to precisely describe in the language we created to specifically deal with such calculations?  Pi is the easiest example, but what’s probably used even more than pi is Euler’s number; seen everywhere from stuff as simple as compounding interests and population densities up to complex numbers and probability theory.  Even the Golden Ratio is irrational!  Though these are abstractions, which makes irrational numbers legitimate in mathematics, it feels as though these are warning signs that our original axioms in which these abstractions are derived from do not describe the natural world correctly.

A classic example of abstraction weirdness:
1/3 = .333…
2/3 = .666…
3/3 = .999…
?

The last statement is actually correct, since .999… is precisely equal to 1.

Abstracting Too Far
Leaving the Physical World

Anyone who has taken even rudimentary number theory knows that mathematics can stray far from the physical world and create its own little universe.  Amazingly, within this universe of pure abstraction, we can sometimes “get back” to the physical world through either manipulation of unreal systems or through interpreting the systems differently (for this think of control theory or Nyquist).  While these methods have proven to be very useful, they go outside of the realm of what is natural which nature, by definition, doesn’t (we’re still trying to model nature right?).
The main example that I think of when I talk about going outside nature while still staying within our artificial constructs is imaginary numbers and the complex plane.  These simply do not exist in nature and are a pure abstracted result of the system in which we created (those pesky negative roots).  They’re incredibly useful though, and manipulation can yield “natural” models and results.  I believe there should be a way of circumventing this abstraction to stay within the realm of nature while still arriving at the same results.  If nature does it naturally (duh o.O), then why can’t we create a system of describing it that stays within the confines of nature as well?

Problem from the Start
The Fallacy of  Axioms

To tie this rant up, I think the problem we have in describing nature using our current system of mathematics lies in the a posteriori nature of the original axioms.  Math, which has been described as a purely logical system, is based on postulates created by “self-evident truths,” or more simply, universal observations.  With observation you bring bias, and I think there’s the possibility that we’re running into so many mathematical problems describing quantum mechanics  partly due to fundamental axioms which were created based on observations of how we THOUGHT the world worked.  The macroscopic implicit truths don’t translate well at the atomic level and perhaps we need to heavily modify or even  abandon our language in which we describe these natural occurrences for a more concise one.
So…?

I know I don’t offer any sort of alternative here.  Hell, I don’t even know what one would look like.  I’m just questioning things I’ve never really asked myself and nearly everyone who I’ve talked to about this has intrigued and interested me more.  I’m also aware that this isn’t set up as a logical argument at all and any attempt at organization was self-serving (my thoughts meander way too much).

I guess I can summarize everything fairly well with one question:

Is mathematics truly the best possible system for precisely describing the natural world?