Thought is not bad, but some thoughts contribute to the good life more than others. Though we wish to think well at all times, there are physical limitations on the quality of the thoughts we produce. As a skill, supervised practice improves the quality of our thoughts.
Clear and exact thought is more likely to guide our actions than vague and inexact thought. Thus clear and exact knowledge whose acquisition is inspired by love is likely to guide us towards the good life.
Clarity and exactness are difficult to develop. Among human activities those that are mathematical are often identified as requiring clarity and exactness in order to be properly performed. Math is also that collection of human activities that few would be happy to perform.
A mathematical thought which is vague or inexact is unlikely to lead to a mathematically relevant action. For example, when performing the elementary mathematical act "add two and three" any thought which does not produce "five" is vague, inexact, or simply wrong. In the case of the acts of addition it is easy to engage in supervised practice. Consequently, most people refer to elementary arithmetic when expressing an opinion on what clear and exact mean.
We are not yet certain as to what is meant by clear and exact thought; there is no universal agreement among experts as to what they might refer to or what qualities characterize them. So, much like all skills, we are most likely to achieve clarity and exactness in our thoughts by practicing things that seem to demand clarity and exactness. Math, and logic, are, for now, those collections of human activities that are most likely to require clarity and exactness in order to be performed.
A mathematician studies math for the mathematics, everyone else benefits from a study of math not for its mathematics but for its development of our sense of clarity and exactness of thought. The focus, in education, which benefits the good life the most is the way in which mathematics achieves its clarity and exactness and how these qualities guide our actions towards desirable solutions.
The taste for clear and exact thought, once acquired, is acute. We need only give each person a single event composed of the pure delight of a clear and exact thought and they will forever seek it out in all that they love.
Some people feel more fear and anxiety than others, and some people are better at working _with_ it rather than _against_ it. Those with a natural skill for working with their anxiety and fear are often misidentified by others as having no fear or no anxiety. In all but the most medically obscure situations this is not the case. Almost everyone experiences fear and anxiety in any situation that is not similar to any previous events in their life.
When solving math problems you are often confronted by events which challenge your skill for identifying similarities between current and past events. This disconnect between what is familiar and what is foreign is the origin of most fear and anxiety. Because it is easy to underestimate the weight of that statement I will make it again. Our inability to identify similarities between events is the cause of most fear and anxiety.
We are lucky that through practice and training we can enhance our skill for identifying similarities between events, and consequently we can reduce the adverse effects of fear and anxiety on our ability to satisfy our desires. The most common desire of a student of mathematics is to solve the problems that are placed before them. Problems are often solved by identifying new similarities with old similarities. That is, a new problem is most easily solved by seeing it as being similar to an old problem whose method of solution can be extended to the current situation.
Surprisingly this skill for solving math problems is similar to the skills a survivalist would use to navigate life threatening situations. It is likely that our power of imagination causes or minds to react as if we are in a life threatening situation even though we are simply struggling to solve a homework problem. I have found this perspective to be useful when brainstorming ways of helping students who are really struggling with their math or physics classes.
So, though I haven't constructed an argument, I believe it valuable to consider teaching math and physics as if they are about survival. As the world becomes more dependent on the tools of science and technology to satisfy our day to day desires, we will find that our mathematical skills will have a greater and more present impact on our ability to survive. We are lucky at the moment that few of us are required to know much about math in order to benefit from the tools of mathematics, but this is unlikely to be the case for too long. Our sense of personal privacy is deeply controlled by the principles of mathematics, and as people desire greater power over what is theirs and what others have access too, they will only find refuge and certainty in special collections of mathematical knowledge.
In my last post I said that it was common knowledge among mathematicians that basic algebra is best described as expression arithmetic. I'm continuing my monologue on these ideas here.
Addition, as an operation of elementary arithmetic, is a procedure that takes a pair of decimals and gives a decimal that represents the sum of the numbers denoted by the taken decimals. It is often frowned upon within some mathematical circles to describe addition in this way, as it requires an overly pedantic distinction between decimals and the numbers that they represent. I can not agree with this perception, nor can I allow it to be propagated, it is not pedantic. In fact, it cuts to the heart of the mathematical method: represent a problem in a form that you can understand and easily manipulate.
The theory of cardinals is rich and deep, it requires a lot of work to prove that the sum of a pair of cardinals is another cardinal, yet in the case of finite cardinals, we need only know that the arithmetic we've developed is an adequate representation of the theoretical backings. If we are curious, and we should be, we will eventually push past the function of arithmetic towards the form of cardinals so that we might see how and why arithmetic is done as it is.
Children, on the other hand, do not need to follow the path of logic through modern mathematical topics. If we can give them tools for thinking about mathematics efficiently then we can leave it to their individual curiosity to look for deeper meaning and form. Not every person will have an impulse to understand the way things work, though they probably could have such an impulse if they were properly educated. Some people will naturally be compelled to follow the facts past what they've been told towards the truth that they are able to demonstrate themselves.
It benefits both types of people to learn efficient methods of dealing with mathematics that might otherwise have deep underpinnings. Both will learn to use math, and the one who is interested in going deeper will have already developed an effective intuition as to what principles and problems are of central importance in the use of mathematics as a tool for thought. Finally, if we are lucky, the one who searches for truth will discover that there is more to mathematics than its utility as a tool for thought, they will be compelled, we hope, to dream of ever more beautiful mathematics that eliminates the illusion that utility and ornament are disjoint.
Though it is common knowledge within the mathematical community that elementary algebra is best described as the arithmetic of basic algebraic expressions, the consequences of this perspective have yet to penetrate into the elementary level. Not every description of a subject impacts its practical use, though it is often better to have multiple descriptions of a familiar object, each of which reveals some feature that might otherwise be obscured in alternate descriptions.
That elementary algebra is an arithmetic of expressions is of great practical importance. It not only gives a sense of what elementary algebra is, but also how one should go about developing its skills. A student of elementary algebra is often mystified by the use of variables, something which continues to be misunderstood by professional and nonprofessional mathematicians alike. By viewing expressions as nothing more than an extension of the numerals used in elementary school, variables become just another thing to play with, not some sort of mystical quasi number or some other such equally vague notion of "unknown".
The fact which is of greatest importance, and most often ignored in our modern presentations of algebra, is that just like the distinction between numbers and numerals, expressions are just special assemblies of marks that we may interpret and operate on without tending to their meaning in a specific context. We do not need to know what number is in order to operate with numerals, and we do not need to know what sentential functions are in order to operate with expressions. In fact, our knowledge of number is informed by our use of numerals, and the same is true of sentential functions.
This wasn’t always the case. People thought they were dealing directly with number when what they were really working with were numerals, some more efficient in their use than others. In general, we tend to work with physical representations of mathematical concepts because, as humans, we have a certain intuition for naive spatial relations and motions. The extent to which there are physical representations of mathematical concepts that faithfully translate the features of the original mathematical concepts often separates good notation from bad notation.
Perhaps the most important reason to reduce algebra to an arithmetic, is that an arithmetic is by its very essence a physical representation of a mathematical concept. Often an arithmetic is a set of procedures performed on a collection of marks. From this perspective we might as well call an arithmetic a set of instructions or algorithms for operating on assemblies of marks.
Furthermore, it is now completely satisfactory to operate as if logic is itself a special algebra. This perspective has been used with great benefit both practically and theoretically. By interpreting logic as algebra, and algebra as arithmetic, we quickly see how it might be possible to place ever more abstract concepts into the hands of unknowing children by properly translating things into an appropriate arithmetic. If a child can learn elementary arithmetic, then they ultimately have all the mathematical tools they will need for the rest of their mathematical lives, what they lack is a certain set of problem solving skills that are not strictly mathematical (though it is often easier to develop problem solving skills in general by tending to mathematical problems whose content are much more precise than the vague problems we are confronted with in everyday life).
I am not only changing the tone of this blog from formal to informal, but I am also going to widen the scope of topics covered here. In my natural habitat I’m friendly, fun, and informal. So, rather than working against myself, I intend on embracing those characteristics of my personality that shine throughout my everyday life.
Math, and science, are, for me, a part of my deep, and seemingly instinctual, love for knowledge. I actually enjoy knowing things, sharing what I know, and learning new things. Rather than sitting in front of a television to watch the latest sitcom or picking up a game controller, I’m likely to spend my free time reading and taking notes on math and physics because it makes me happy to do so. What gives me the greatest joy is understanding something that was once vague and imprecise in terms that are not only simple but absolutely clear. Once I understand something well, then I am often able to explain it to people in a way that is both fun and functional.
Unlike most people who enjoy knowing things, I do not feel that much satisfaction in knowing things without knowing how to share them. That means that when I take notes, or try to figure things out on my own, I’m really trying to figure things out in a way that is blindingly obvious and is connected to relevant features of day to day life. Things that dangle from a long chain of abstraction have got to be connected to something that I can make sense of, something that I can think of in terms of a well defined problem, or something which helps me to better organize my knowledge of the world, otherwise I loose sight of it among the multitude of meaningless abstractions one can build up with just a bit of mathematical imagination.
What does this mean for the blog? I’m not exactly sure yet, but I do know that rather than being a collection of well constructed facts, or a presentation of some new connection between facts, I will be posting more frequently about things that I’m collecting, not just those things which I’ve already organized and subsequently revised.
I honestly think that mathematics is a practical tool for living and working in this world. Most people think this means that I want to use math in order to engineer a better sprocket or create the next internet. This is common because not too many people are familiar with the methods of mathematics, they only know of its products. When mathematicians open up a new problem, they go through a process of identifying similarities with previous mathematical problems as well as any relevant figures or illustrations that draw up the most vivid and inspiring associations. That process of making sense out of the impossible is what really draws me to mathematics and continues to bring me back to the problems of this world.
Most, if not all, of the problems in this world can be remedied with clarity and exactness. Rather than standing in a vague haze of words that connect only to half-facts, we must move ourselves closer towards the vague outlines that are presented to us until we see something that is distinct or clear in form. Without clarity, we tend to argue over nonissues, nonfacts, and nonproblems. The path from vagueness to clarity is rugged, and, like any good mountaineer will tell you, one does not simply walk up to the top of Mount Everest on a whim. We traverse rugged terrain by acclimating ourselves to the environment through successively more difficult outings. The problems of mathematics are something of a rose garden in this way. They are often well organized, well traversed, and the beauty is almost self evident no matter how far from a solution you are (that is if you have a good mathematical gardner keeping watch).
One day I hope to develop the skill needed to write with that proper combination of formal and informal tone. For now, I will allow myself to be satisfied with an informal tone that, I hope, will occasionally dip into something more formal when I’m inspired or when the material demands it.