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Ad Infinitum: The Ghost in Turing's Machine: Taking God Out of Mathematics and Putting the Body Back In

by Brian Rotman

This book is not for those who are looking for an easy read into the inner workings of mathematics. Brian Rotman explores the inner workings of the number system. He explores the philosophical properties of numbers and that of the infinite. Many of the arguments used are very complex.

The book begins by exploring mathematics as a language. It discusses what qualifies a system as a language and whether or not mathematics qualifies in this certain area. Many believe mathematics to be a Platonist science. This means that there has always been he “natural’ numbers and man have simply worked with those numbers which have been provided to discover the rest of the mathematical knowledge that we know today. The book talks about mathematics as a language and mentions how many today believe it to be a language. The book backs up its claim by describing how “we have no choice but to translate in and out of mathematical expressions and terminology on their way to other interests and forms of engagement; and certainly not scientists” (p. 16). Many times in life there is a language of mathematics expressing ideas, and being able to be recorded and understood by other individuals throughout the world. Language is based on correspondence. Properties correspond to objects and statements in the real world. These statements are either true or false depending on their existence in the real world. Rotman explains that part of mathematical language is describing objects that pre-exist. The Language of mathematics simply makes sense of the objects which were previously not described by any other language. A difference between mathematics and written languages is that writing is the written oral word, as where mathematics holds no such equivalent. We have adapted mathematics into oral phrases which represent mathematical writing. Although there are different verbal forms of each mathematical form, its absolute meaning is completely internal. There is an argument that the true pinnacle of Mathematics is that of mathematical thought. The writing is simply the representation of the mathematics which individuals have conceived in their minds. Rotman argues that this cannot be true because of the way that notational systems are so important to document what previous individuals had thought about.

Rotman continues on in his book to discuss infinity and the immense concept that accompanies it. Many aspects of mathematics are dependent on the infinite such as calculus. In mathematics infinity is seen as a legitimate object for use and investigation. The simplest infinity to think about is that of the counting numbers. If one could conceivably count forever, then they could continue counting with no end. Constructivists believe that infinity is not “endless” because there is no possible creature or thing which is capable of such a feat. There is much philosophical thought which goes into infinity and of some attempting to overthrow the idea of an infinite. Rotman suggests the “ad infinitum” principle. This is what allows for there to always be “one more time” this allows for every x there be an (x+1). Going back to the previous argument about no object can past forever and therefore it is unknown whether or not it is possible to continue counting. This is contrasted by asking “why not one-hundred lifetimes” of counting? This is not a valid argument and therefore can be ignored when attempting to dispute the idea of infinity. One thought about infinity is that of an unchanging “heap” of something. If an individual has a “heap” of sand grains, and that individual were to remove a single grain, a “heap” would still remain. This means that as long as the individual removes a single grain of sand, then there will always be a heap remaining. This is a simplified way of contemplating the infinite.

Much of mathematics takes place because of experimental thought. Much in the history of science has been possible because of the use of experimental thought. Galileo used this idea extensively to create his physical laws. The beginning of all mathematical thought begins with that which takes place in the mind. Although the entirety of mathematics does not take place here, the ideas are born here and then they are converted to the paper. A model is built in the mind which allows for certain things to be executed and go beyond that previously thought. Many times, not just in mathematics, the individual who carries out the experiment wishes to find the idea behind the proof. They search for principles or concepts that will help them to form a unified argument. This can in turn be changed in order to create other proofs which explain other relationships. The manipulation and writing of mathematical signs goes hand in hand with the thought experiments of mathematics. It would not be possible to perform any sort of successful thought experiment without writing mathematical signs and performing computations. Conversely one would not be able to manipulate and write mathematics without the thought experiments which occur at all times.

Limits are important aspects of mathematics and in fact are what make all of calculus possible. Leibniz and Newton both were very close to calculus but both were missing a small piece of the puzzle. Limits were what made calculus possible. Physics relies heavily on this type of mathematics. Only through a limit is it possible to picture the mechanical in relation to the infinitesimal.

Rotman also deals with non-Euclidean geometry. He posits the question of whether it is possible to use the existence of non-Euclidean geometry to show that it is possible to show the existence of a non-Euclidean number. Is it possible that all of mathematics has slipped into a paradigm that it is unable to escape? All of the axioms are derived from the everyday experience of the mathematician. The idea of arranging and grouping objects led to the use of positive whole numbers. This is similar to that used in geometry. There is a difference between the local and the global. Such an example is the globe, from a local view it appears flat, but when zoomed out it is easily seen as round.

Rotman writes a very interesting, but also very difficult book. This is not a simple book one would pick up for comfort on a rainy day. He does not waste time, and does not wait for the reader to catch up. There are many very interesting aspects to the book however. His ideas of the infinite are fascinating and he does a wonderful job of explaining all sides of every argument. This is very beneficial to the reader, because they are not forced to believe what the author tells them, but instead is free to choose which seems most logical. The author goes deep within that which most readers had taken for granted. Rotman discusses the natural numbers and whether or not they are truly natural or were invented by man, I agree with many mathematicians and believe that they are there and were always there. I believe that Leopold Kronecker made a very good statement when he said “God made the integers, all the rest is the work of Man.” Many of the ideas which were discussed it the book are not easily answered. It will require many more hours and also a second reading of the book before it would be possible to come to any conclusion about the topics discussed in this book.

There is a large aspect of philosophy associated with this book. There are many deep philosophical arguments and ideas that truly expand the thoughts of the reader. It requires deep thought and challenges the beliefs of the individual reader. If an individual would like to look deeper into the most elementary building blocks of all of mathematics, this book would be an excellent start. It takes the ideas that we encounter every single day and dissects them and breaks them down. When these ideas are broken into their most basic parts it is possible to examine and understand the meaning and more about the process of the individual who utilizes them. This book was an incredible way to expand my mind and think more deeply about mathematics and all of the mathematics that I consider to be basic. I look at these “basic” concepts and will see more of what is hidden behind the basic computational properties of the numbers.