Product Description: This distinguished little book is a brisk introduction to a series of mathematical concepts, a history of their development, and a concise summary of how today's reader may use them.
A true gem! This is one of those rare works by a true master. The following quote from page 161 definitely applies to the book itself:
"If we understand the preceding ideas, we understand the foundations of modern mathematics".
This is what this book is about. If you're looking precisely for this, as I was, you'll be truly enlightened by its reading.
Excellent for its time I assume that it was very unusual, in the early 20th century, for scholars of Whitehead's stature to write for an educated lay readership, and that IM is one of only a handful of 'popular mathematics' books of its day. As such, IM and Whitehead are to be commended. Aside from an occasional reference to the 'ether', as others have noted, IM is current and compelling in every respect. It is a fine read (though rather curiously organized: it's final two chapters seem to belong much earlier in the book). I give it 4 stars only because others, standing on Whitehead's shoulders, have done a much better job of covering similar ground. I have in mind, especially, Tobias Dantzig's first-rate "Number: The Language of Science" (recently reissued).
A word to the wise: avoid the Barnes and Noble edition of IM. It is rife with typos, not to mention a missing diagram. I don't know whether the errors are B&N's own, or owe to the fact(?) -- it seems -- that this edition corresponds to (though does not photo-reproduce) the original 1911 edition, which (judging from the latest offerings on Amazon), was superseded by a corrected edition in 1948. Whatever the source of these errors, they are many and greatly distracting.
Great Introduction --- Better Adjunct Whitehead's "Introduction to Mathematics" is an illustrative, lucid, and concise discourse on the "three great mathematical ideas of the variable, of algebraic form, and of generalization." As other reviewers have indicated, the author presupposes that the reader have at least *some* experience with elementary algebra, geometry, and trigonometry. The book's greatest strength, however, lie in its ability to supplement the rigor of an undergraduate math education (or something akin) with the "why" of said education. In sum, both math- and non-math oriented folk will benefit from reading this book --- the non-mathematicians may be turned on to the elegance of the discipline whereas the mathematicians may be reminded (gasp!) of its beauty and relevance.
Intro to the PHILOSOPY of mathematics This is a nice, little book: short, clear, and very well written. I confess, though, that I'm not sure who its best audience really is. If you know some math, and have thought and read AT ALL about the philosophy of math, you will not find much new in this book; still, since it will be quick and easy to read, you will probably find it worthwhile, for the occasional new insight or alternative way of looking at things. I found the section on series particularly worth reading, because series were not well covered in my own math education. I also found the comments on the measurement of time to be subtle and thought-provoking.
If you know little or no math, you MIGHT find this a good introduction (as the title implies), but don't expect any detailed exposition on the actual PRACTICE of math. This book is really an introduction to the philosophy of math. It is concerned with WHY we do math, and why math takes the form that it does. Whitehead's goal is to introduce some key concepts, common to all math, such as variables and abstraction. Any actual proofs or expositions in the book are included only as examples of how these concepts play out in seemingly different areas of study.
Perhaps the reader best served by this book would be one who is comfortable with the practice of math at least through the basic high school level (geometry, algebra, trigonometry), and possibly more, but is just starting to think about the underlying philosophy: the "why" of math as opposed to the "how" of it.
For those who don't know, Whitehead was, of course, one of the premier philosophers of math of the early 20th century, co-author with Bertrand Russell of the 3-volume magnum opus "Principia Mathematica". The present book was written around 1911, and is definitely dated in spots - for instance he talks about electro-magnetic vibrations in the "ether" - but that doesn't detract from either its usefulness or readability.
Insightful and Provocative "The study of mathematics is apt to commence in disappointment."
"One of the causes of the apparent triviality of much of elementary algebra is the preoccupation of the textbooks with the solutions of equations."
In discussing Descartes' coordinate geometry, Whitehead states, "Philosophers, when they have possessed a thorough knowledge of mathematics, have been among those who have enriched the science with some of its best ideas. On the other hand, it must be said that, with hardly an exception, all remarks on mathematics made by those philosophers who have possessed but a slight or hasty and late-acquired knowledge of it, are entirely worthless, being either trivial or wrong."
"Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle - they are strictly limited in number, they require fresh horses, and must only be made at decisive moments."
"The really profound changes in human life have all had their ultimate origin in knowledge pursued for its own sake."
Alfred North Whitehead, a remarkable British mathematician and philosopher, enlivens his look at the fundamental ideas underlying mathematics with provocative observations. Nonetheless, Whitehead does not avoid mathematics while trying to explain mathematics. While this book is clearly for the layman, it may occasionally require some effort. An Introduction to Mathematics is delightful, insightful, and intellectually stimulating.
Whitehead argues that mathematics is an abstract science that is primarily concerned with generality, not specificity. In trying to master the techniques and mechanics of arithmetic, algebra, and geometry, many students fail to recognize the fundamental ideas. They become lost in a murky fog of details.
I found myself surprised by Whitehead's insightful explanations of familiar topics like variables, constants, and simple algebraic equations. I know math. But I now recognize that I had not really given sufficient thought to some very basic concepts. Just a few pages into this little book I was actually looking at some familiar concepts from a very different perspective.
Later discussions on mathematical symbolism, imaginary numbers, conic sections, trigonometry, and infinite series move more slowly and may require rereading. But the insights gained will more than offset any additional effort.
Whitehead occasionally digresses to discuss the act of mathematical creation. He agrees with the poet Shelley who compared the discovery of "some great truth" to the slow snowflake by snowflake accumulation that leads to an avalanche. While not discounting the role of genius, Whitehead sees breakthroughs in mathematical thought, often as unexpected as an avalanche, the natural result of the accumulation of knowledge through the centuries.
Whitehead's small book could serve as the basis for a short class or tutorial for high school students (or perhaps even for humanities majors with less than fond feelings for mathematics.) An Introduction to Mathematics is an effective counterbalance to standard textbooks that focus too much on technique, manipulation, and mechanics. Five stars.
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