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Mathematical Expression Editor

This section aims to show how mathematical reasoning is different than ‘typical
reasoning’, as well as showing how what we are doing is mathematical.

Really though, what does any of this have to do with math?

If you are wondering why I haven’t given you anything to memorize yet, I would first
refer you to earlier; learningmemorizing is the antithesis of learningmemorizing. But, if you are wondering what this has to do with “math”, especially since I’ve
been somewhat vague as to what that is, that is a fair question. The answer is... this
has everything to do with math.

Remember, mathematics is not about formulas, numbers and variables. It is
true that mathematics utilizes these things (and we will get to that part
later). Perhaps a better question then is; “what does any of this have to do
with what we’ve done so far?” (my claim, after all, is that we are doing
math right now!). Mathematics is the language of deduction, it is a very
carefully developed language built with the intent of bringing precision and
structure to what we casually refer to as “thinking”.

(Don’t worry I’m not
a megalomaniacal proselytizer and I am not trying to say all thought is
mathematics. Here I mean more that; when we say things like “thinking through a
problem” what we typically mean is trying to deductively work through
a problem from some beginning set of information. This specific type of
“thinking” is really what I’m referring to when I’m talking about math)

Why does “thinking” need to be precise? What does that even mean?

You may already believe that your thinking is precise. Or you may believe that
thinking can’t be made precise. The truth is, thinking itself is pretty nebulous, since
it depends on the individual. What we really mean is that mathematics aims to bring
precision to the process of communicating one’s thoughts. By way of example,
think of some amount of money. Really, think of it. Now think about some
amount of milk... yes really. Chances are, when you thought of the money,
you probably imagined a number, say one million dollars. You could have
pictured a pile of bills, but instead you had a number to represent the value. In
contrast, when you thought of milk, you probably had a “glass” of milk
in mind, or maybe a container. But a “glass” is nonspecific. Is the glass
half full? filled to the point of surface tension? Is it a twelve ounce glass?
sixteen? eight? Keep in mind that if I had said to think of a “glass” of milk
instead of “some” milk, most would say I’m being more precise, but am I
really?

One of the key aspects of mathematics is to bring specifics into play when you are
trying to discuss these things.

Instead of saying “I have some milk”, which of the
following is more helpful in conveying exactly how much milk I have?

I have a
bunch of milk.I have thirteen ounces of milk.I have a container of mulk.I
have a full glass of milk.

Notice that the precision is useful because the person you are talking to may be
picturing a vastly different glass than you are. For example; you could be imagining a
travel mug of milk, whereas the other person may be considering one of those tiny
Styrofoam cups or wax paper cones of milk... or if they are Canadian, a bag of milk.

(Yes, Canadians buy milk by the bag. But in fairness so do a ton of other
countries)

One may notice in our numeric models above that we had to go through a process of
’quantifying our information’. Another way to think of this is that, any problem we
are given to solve will almost inevitably entail nonprecise information or goals. Thus,
one of your jobs as a problem solver (ie modeler) is to bring mathematical precision
to the “thinking” of the one requesting a model. This is typically the hard part
of the “clarifying” steps and then the future “quantifying” steps, but it is
necessary since otherwise your solution may be dismissed as pointlessbe only applicable to a small subsetbe wholly inaccurate due to faulty
assumptions.