Читать книгу Applied Univariate, Bivariate, and Multivariate Statistics - Daniel J. Denis - Страница 18

Ian Stewart (1995) said it best when he wrote that the mathematician is not a juggler of numbers, he is a juggler of concepts. The greatest ambivalence to learning statistical modeling experienced by students outside (and even inside, I suppose) the mathematical sciences is that of the presumed mathematical complexity involved in such pursuits. Who wants to learn a mathematically-based subject such as statistics when one has “never been good at math?”

The first step in this pursuit is to critically examine assumptions and prior learned beliefs that have become implicit. One way to help “demystify” mathematics and statistics is to challenge your perception of what mathematics and statistics actually are in the first place. It is of great curiosity that so many students claim to dislike mathematics and statistics, yet at the same time cannot verbalize just what mathematics and statistics actually are, and then even worse, proceed to engage in real‐life activities that utilize very much the same analytical cognitive capacities as would be demanded from doing mathematics and statistics!

More than likely, the “dislike” of these subjects has more to do with the perceptions one has learned to associate with these subjects than with an inherent ontological disdain for them. Human beings are creatures of psychological association. Any dislike of anything without knowing what that thing is in the first place is almost akin to disliking a restaurant dish you have never tried. You cannot dislike something until you at least know something about it and open your mind to new possibilities of what it might be that you are forming opinions about. Not to sound overly “Jamesian,” (the analogy isn't perfect, but it's close) but perhaps you are afraid of mathematics because of your fear of it rather than the mathematics itself. That is, you run, not because of the mathematics, but because of the fear. If you accept that you are yet unsure of what mathematics is, and will not judge it until you are knowledgeable of it, it may delay derogatory opinion about it. It is only when we assume we know something (to some extent, at least) that we usually feel free to judge and evaluate it. Keep your perceptions open to revision, and what you may find is that what was disliked yesterday curiously becomes likable today, simply because you have now learned more about what that something actually is. But to learn more about it, you need to first drop, or at minimum suspend, previously held beliefs about it. Have an open mind from the outset, and refresh that mindset each time you open a book or attend a lecture in a course.

The first point is that statistics is not mathematics. Statistics is a discipline unto itself that uses mathematics, the way physics uses mathematics, and the way that virtually all of the natural and social sciences use mathematics. Mathematics is the tool statisticians use to express their statistical ideas, and statistics is the tool that scientists use to help make sense of their research findings. The field of theoretical or mathematical statistics is heavily steeped in theorem‐building and proofs. Applied statistics, of the kind featured in this book, is definitely not. Thus, any fear of real mathematics can be laid to rest, because you will find no such mathematics in this book. Upon browsing this book, if you are of the opinion that it contains “lots of math,” then quite simply, you do not know what “lots of math” looks like. Rest assured, the mathematics in this book is simply used as a vehicle for understanding statistics.

Mathematics and statistics are not things “mysterious” that can only be grasped by those with higher mental faculties. A useful working definition might be that mathematics is a set of well‐defined and ever‐expanding rules and consequences about symbolic abstract objects based on fundamental assumptions called axioms. The axioms of mathematics are typically assumed to be true without needing to be proved. Theorems and other results built on such axioms usually require proof. What is a proof? It is an analytical argument for why a proposition should be considered true. Any given proof usually relies on other theorems that have already been proven to be true. Make no mistake, mathematics is a very deep field of intellectual endeavor and activity. However, expecting something to be deeper than it is can also lead you to just as well not understand it. Sometimes, if you are not understanding something, it may very well be that you are looking far beyond what there is to be understood. If you retreat in your expectations slightly of what there is to see, it sometimes begins to make more sense. Thinking “too deep” where such depth is not required or encouraged, is a peril. Many “bright” students have this “gift” of critical analysis, and to understand a concept, need to actually retreat somewhat in their depth of inquiry (at least for the moment).

For a general overview of the nature of mathematics, the reader is encouraged to consult Courant, Robbins, and Stewart (1996), and for an excellent introduction to basic mathematical analysis, Labarre (1961). Hamming (1985) is another good introduction to the field of mathematics, as well as Aleksandrov, Kolmogorov, and Lavrent’ev (1999). For more philosophical treatments, the reader should consult Dunham (1994) and Stewart (1995). For an in‐depth and very readable history of mathematics, consult Boyer and Merzbach (1991).