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/or variables. It is true that mathematics utilizes these things (and we will get to that part later), but saying math is about formulas/numbers/variables because those things are used to write mathematics, is like saying that poetry is just about vocabulary because it uses words and grammar - it’s true that one uses their vocabulary to write poetry, but that is missing the point - and really the heart of what poetry is.

Perhaps a better question 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!). The answer to that question is that 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”.

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, spend a second and come up with a specific amount in your mind. Now think about some amount of milk... yes really, picture a specific amount of milk in your mind. 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? Most people would say “a glass of milk” is more precise than “some” milk... but if I can’t see the exact picture in your mind of what you mean by “a glass of milk” then... is it really?

One of the key aspects of mathematics is developing a framework that requires specifics for when you are trying to discuss these things.

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. This leaves room for misunderstanding and mistakes that can get out of control with surprising speed and consequences (ok, maybe not for milk... but when discussing money or strength of a bridge or building?)

One may notice in our previous numeric model 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 begin with nonprecise information or goals. Thus, one of your jobs as a problem solver is to bring mathematical precision to the “thinking” of the person posing the problem. This is typically the hard part of the “clarifying” step and the “quantifying” step, but it is absolutely necessary, since otherwise your solution may be dismissed as pointlessbe only applicable to a small subsetbe wholly inaccurate due to faulty assumptions .