An Introduction to Mathematics by Alfred North Whitehead (PDF)

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CONTENTSCHAP. PAGEI THE ABSTRACT NATURE OF MATHEMATICS 1II VARIABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7III METHODS OF APPLICATION. . . . . . . . . . . . . . . . . . . 15IV DYNAMICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29V THE SYMBOLISM OF MATHEMATICS. . . . . . . . . . 43VI GENERALIZATIONS OF NUMBER . . . . . . . . . . . . . . 54VII IMAGINARY NUMBERS . . . . . . . . . . . . . . . . . . . . . . . . . 67VIII IMAGINARY NUMBERS (CONTINUED) . . . . . . . . . 80IX COORDINATE GEOMETRY . . . . . . . . . . . . . . . . . . . . . 90X CONIC SECTIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103XI FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118XII PERIODICITY IN NATURE . . . . . . . . . . . . . . . . . . . . . . 134XIII TRIGONOMETRY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141XIV SERIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159XV THE DIFFERENTIAL CALCULUS . . . . . . . . . . . . . . . 179XVI GEOMETRY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194XVII QUANTITY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202NOTES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Even though I’m a mathematical hobbyist who had four semesters of calculus at a university, I found “An Introduction to Mathematics” to have some tough reading in it – especially, some of the geometric drawings that Whitehead used to illustrate some of his ideas.According to my understanding, Whitehead wrote this book after the advent of Einstein’s development of Special Relativity, in which the idea of an ether permeating space seemed to have been defeated. Yet, Whitehead says the following (page 158):“Again, the motion of vibration of a violin string is submitted to a similar harmonic analysis, and so are the vibrations of the ether and the air….”I wonder what was meant by the use of the term “ether”. Of course, it is possible that Whitehead simply rejected Einstein’s repudiation of a “luminiferous ether”. Einstein’s theory of the relativity of all non-accelerated motion had no use for the theory of an ether as a medium through which light was transmitted.I like what Whitehead says in the last paragraph of Chap. XVI (page 201):“(2) Space-perception accompanies our sensations, perhaps all of them, certainly many; but it does not seem to be a necessary quality of things that they should all exist in one space or in any space.”The idea (very prevalent among materialists) that in order for something to exist, it must exist in a portion of “physical” space, is an idea that is, I am convinced, gravely erroneous, resulting in a most unfortunate devaluing of mind (consciousness), as well as postulating the impossibility of the objective existence of “transcendent spirit”. First of all, if one seeks to sustain such a view as held by many materialists, then what is one to do with mathematical “objects”? If they have any genuine existence, then that existence is OUTSIDE of space-time. If they have no actual existence, then most mathematicians probably operate within the framework of mathematical “superstitions” – being deluded into the conviction that the numbers they study and manipulate really exist and are not mere convenient fictions created in their minds (with minds also being convenient fictions). In such an outlook, reality becomes a desert filled with desiccated and inert entities in which hapless humanity finds itself lost in the dilemma of being forced to create innumerable fictions just to cope and thrive in this desert. I believe that this outlook is deserving of ridicule.Suffice it to say that, given Alfred North Whitehead’s brilliance as a philosopher of the highest rank, his book about mathematics was a somewhat rewarding read for me. Mere mathematical hobbyists, such as myself, might find some portions of “An Introduction to Mathematics” somewhat tough going. It is worth the effort, though.

⭐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.

⭐I neglected my mathematical education. I thought this would be an interesting way to learn more. To be fair, I did learn some things, particularly at the beginning. As he noted in the post script, the actual math can get in the way of most people learning. I enjoyed the historical and theoretical points, which is really what I wanted, but after a while the equations got to me and I realize I just do not have the time, or inclination, to concentrate hard enough to learn it. I do not know how you do it without actually treating it as a course. So, I sped through, glomming the easy parts. I really do think there are ways to generally explain math simply to people with mild interest only in the history, objective and general outline of the discipline without going through complex equations and diagrams. This wasn’t it.I did see a mistake or so early on in the diagrams (e.g., an “x” that came out “y”), but that may be the printing and not his fault. Perhaps the book is better than I know and it is my shortcomings that keep me from appreciating it. And perhaps if I have more time and patience some day, I will try again.

⭐Striking review of fundamental concepts from a distinguished teacher. This is enlightening + engaging because you get the concepts plus its importance within the body of knowledge. Not to mention the historical context. For instance, Whitehead remember us that Archimedes was killed by a Roman soldier while contemplating a mathematical diagram and no Roman ever died in such conditions. This book is full of insights; go for it if you want to refresh your view of Mathematics having lots of fun.

⭐I bought this book because I’m taking a course in mathematics, I haven’t read it yet but I’m sure when I do I will enjoy it.

⭐I can’t stress how much this book concise.It is an amazing overview of the core mathematical ideas explained in such a simple (simple, not trivial) way.It is also the best introduction not only to mathematics but to the best of the human mind – the beauty of an abstract thinking.

⭐Assim termina o primeiro parágrafo do Capítulo I desse notável grande pequeno livro. Logo adiante, Whitehead adverte, cotejando as facilidades técnicas com as idéias fundamentais da mestra de todas as ciências: “In this sense there is no royal road to learning. But is equally an error to confine attention to technical processes, excluding consideration of general ideas. Her lies the road to pedantry.” Um petardo contra os (não poucos !!!) maus autores/professores de Matemática mas que, ‘pari passu’ com seus desafortunados e sofridos alunos, podem usufruir, no decorrer da obra, do estilo humano e acolhedor dos ensinamentos filosóficos, não obstante firme e acessivelmente técnicos, do notável parceiro de Russell no monumental “Principia Mathematica’. Cabe, entretanto, observar que a Editora DOVER, notabilizada por reedições fiéis de clássicos (notadamente em Matemática e Ciências) suprimiu rodapés e algumas notas do autor disponíveis na edição de base, pela Henry Holt & Co. Tradicional na publicação de versões não-abreviadas, como menciona ser o caso da presente obra, é de se surpreender e de lamentar.

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