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New Directions in the Philosophy of Mathematics
The traditional debate among philosophers of mathematics is whether there is an external mathematical reality, something out there to be discovered, or whether mathematics is the product of the human mind. This provocative book, now available in a revised and expanded paperback edition, goes beyond foundationalist questions to offer what has been called a "postmodern" assessment of the philosophy of mathematics--one that addresses issues of theoretical importance in terms of mathematical experience. By bringing together essays of leading philosophers, mathematicians, logicians, and computer scientists, Thomas Tymoczko reveals an evolving effort to account for the nature of mathematics in relation to other human activities. These accounts include such topics as the history of mathematics as a field of study, predictions about how computers will influence the future organization of mathematics, and what processes a proof undergoes before it reaches publishable form.
This expanded edition now contains essays by Penelope Maddy, Michael D. Resnik, and William P. Thurston that address the nature of mathematical proofs. The editor has provided a new afterword and a supplemental bibliography of recent work.
This expanded edition now contains essays by Penelope Maddy, Michael D. Resnik, and William P. Thurston that address the nature of mathematical proofs. The editor has provided a new afterword and a supplemental bibliography of recent work.
456 pages, Paperback
First published January 12, 1998
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April 4, 2016
The aim of this book is to present a different side to the philosophy of mathematics other than what is considered to be a foundational approach. To achieve this Tymoczko has put together this anthology to show how foundationalism is limited and should be de-emphasized. Instead, the philosophy of mathematics should include the actual practice and history of mathematics. What is it that mathematicians actually do and think they are doing? How has mathematics changed throughout its history? Does it share aspects with the natural sciences? Is quasi-empiricalism a valid viewpoint in looking at mathematics? Another factor discussed is the social aspects of mathematics. How do mathematicians share their work? How does the working mathematical community affect their practice? How is mathemtics taught? These things are amongst the ideas presented in this book.
There are three major foundationalist approaches, plus the realism issue. The foundations are logicism, formalism, and constructivism.
Logicism is that mathematics can be shown to be nothing but logic. This is often combined with set theory, which provides the objects for logic to work on. The problem with this approach is that it leads to paradoxes that need at least a bit of gerrymandering to get rid of.
Formalism, based on the mathematician David Hilbert, is that mathematics is a set of formal rules operating on certain symbols. One problem with this approach is that these formal systems don't necessarily need to be about numbers or other mathematical objects. A more serious problem is that it runs afoul of Godel's incompleteness theorem, which is that a formal system rich enough to include arithmetics has true statements within it which cannot be proved within that system. A corollary of this theorem is if a system could be shown to be complete it would be inconsistent (there are statements which can be shown to be both true and false).
Constructivism is harder for me to convey. Its basic premise is that for something to be proved about a mathematical object one needs to show how to construct it. How that works I am not sure. What its affect is is that an existence proof is not enough, such as a proof involving infinity, since one cannot actually construct an infinite number or complete an infinite series, although with other approaches these can be completed or summed. Another key aspect of of this approach is the denial of the law of the exclude middle (something is either true or false). With this law not accepted it is not good enough to show that something is not false in order to show that it is true and vice versa. This rules out reductio ad absurdum arguments. It is not enough to show that something leads to a contradiction in order for the opposite to be true.
The realism issue revolves around where mathematical objects preside. Do they inhabit their own indpendent realm or do they live in mathematicians' minds. The first is often referred to as Platonism. Another way to look at it is whether mathematicians discover or invent matematical objects. The realism issue is semi-independent from the other foundational programs. One could be a realist or not in any of these approaches. As is shown in the other parts of the book there are other ways of accepting some form of realism. Once something is shown about a mathemtical object it becomes real in the sense that it lives within the community of mathematicians, and I would imagine anyone accepting the work of mathematicians.
It is interesting that while watching NOVA's The Great Math Mystery Mario Livio put forth the idea that the natural numbers have an independent existence and are, hence, discovered, but the relations that mathematicians show between them are invented. I am not sure if I would buy this explanation or whether it would convince some of the writers in this book.
The rest of the book covers what components might or should go into the making of the philosophy of mathematics. Some of these components are quasi-empiricism, history, practice, social, education, and computers.
Quasi-empircalism is the view that mathematics operates along similar, though no exact lines, as the natural sciences, in particular physics. It is said that mathematicians experiment with numbers, which then can lead them to investigate various properties of these numbers, leading to a proof of some theorem involving these numbers and/or properties.
The history of mathematics is consider important because it shows what mathematics was like in the past and how it has developed up to the present. One writer sees similarities with Thomas Kuhn's view of scientific change. Others have taken different approaches, such as, looking how the method of proof has change over the years, focusing on the nineteenth century.
With the practices the writers move into what mathematicians do. Some see most mathematicians as providing informal proofs. These proofs do not rely on a set of axioms and rules for proving theorems. Often a mathematician sees the truth of some piece of mathematics, and then goes and shares it with his contemporaries, only later presenting it at a seminar or conference, and then finally writing a paper. None of this involves formal proof working with axioms. It is said that the majority of theorems are never given any formal proof. This is said to show the inappropriateness of foundational mathematics to most of mathematical activity.
The social atmosphere of mathematicians' lives they find themselves in is said to matter a lot. Mathematicians practice in a community with other mathematicians. These communities share a particular topic, practice, and interaction.
How mathematics is taught is also believed by some of the writers to matter. After all, what gets transmitted to a new generation of mathematicians appears to be of importance. How mathematics is taught to non-mathematicians can also play a role in how mathematics is viewed.
Finally, work with computers is look at. The four-color theorem is look at by the editer using it to dissect what exactly is a mathematical proof, and that provides the mathematician. In other words, how do mathematicians see the truth of a given proof. Chaitin provides a look at computational theory, and how it relates to Godel's theorem.
An interesting bit for me was in Wang's presentation of what it means to be able to multiply. There he gives an inductive proof as such: “1 is small; if 1 is small, n + 1 is small; therefore, every number is small.” An inductive proof I can actually follow, not given that kind of mathemtical skill. But, it occurred to me that this proof, like maybe all proofs, certainly many, is in need of interpretation. For this proof to mean anything to me I had to compare every number is small with infinity. It seems to me that every number is small in relation to infinity. I even thought that maybe you could prove that every number greater than 1 is large. This could be in relation to all real numbers less than 1, or possibly in comparison with the negative numbers. If there is any proof of these, I will leave it to a competent person.
I found to the book to be an eye opener to how much more broad the philosophy of mathematics could be from what I had known before, which had been limited to pretty much the foundations, realism, and the discovery/invention controversy. While I can't claim to understand everything in the book, especially those parts that actually use mathematics in what I would call a technical manner, I think I have comprehended the main components of this book. Not being very interested in the technical side of mathematics, this was okay with me. I believe that the book lives up to its stated purpose including presenting the ideas in it to a broader audience than just mathematicians and philosophers. For the most part I found the book enjoyable, especialy those parts that I felt I could ask pertinent questions of the various writers as I tend to do in most books that I read.
I recommend this book for those interested in the philosophy of mathematics, and who would like to explore different approaches to it other than foundationalism. It would be good if the reader is already familiar with basic mathematical concepts and areas of study, but I do not think that is absolutely nesessary.
There are three major foundationalist approaches, plus the realism issue. The foundations are logicism, formalism, and constructivism.
Logicism is that mathematics can be shown to be nothing but logic. This is often combined with set theory, which provides the objects for logic to work on. The problem with this approach is that it leads to paradoxes that need at least a bit of gerrymandering to get rid of.
Formalism, based on the mathematician David Hilbert, is that mathematics is a set of formal rules operating on certain symbols. One problem with this approach is that these formal systems don't necessarily need to be about numbers or other mathematical objects. A more serious problem is that it runs afoul of Godel's incompleteness theorem, which is that a formal system rich enough to include arithmetics has true statements within it which cannot be proved within that system. A corollary of this theorem is if a system could be shown to be complete it would be inconsistent (there are statements which can be shown to be both true and false).
Constructivism is harder for me to convey. Its basic premise is that for something to be proved about a mathematical object one needs to show how to construct it. How that works I am not sure. What its affect is is that an existence proof is not enough, such as a proof involving infinity, since one cannot actually construct an infinite number or complete an infinite series, although with other approaches these can be completed or summed. Another key aspect of of this approach is the denial of the law of the exclude middle (something is either true or false). With this law not accepted it is not good enough to show that something is not false in order to show that it is true and vice versa. This rules out reductio ad absurdum arguments. It is not enough to show that something leads to a contradiction in order for the opposite to be true.
The realism issue revolves around where mathematical objects preside. Do they inhabit their own indpendent realm or do they live in mathematicians' minds. The first is often referred to as Platonism. Another way to look at it is whether mathematicians discover or invent matematical objects. The realism issue is semi-independent from the other foundational programs. One could be a realist or not in any of these approaches. As is shown in the other parts of the book there are other ways of accepting some form of realism. Once something is shown about a mathemtical object it becomes real in the sense that it lives within the community of mathematicians, and I would imagine anyone accepting the work of mathematicians.
It is interesting that while watching NOVA's The Great Math Mystery Mario Livio put forth the idea that the natural numbers have an independent existence and are, hence, discovered, but the relations that mathematicians show between them are invented. I am not sure if I would buy this explanation or whether it would convince some of the writers in this book.
The rest of the book covers what components might or should go into the making of the philosophy of mathematics. Some of these components are quasi-empiricism, history, practice, social, education, and computers.
Quasi-empircalism is the view that mathematics operates along similar, though no exact lines, as the natural sciences, in particular physics. It is said that mathematicians experiment with numbers, which then can lead them to investigate various properties of these numbers, leading to a proof of some theorem involving these numbers and/or properties.
The history of mathematics is consider important because it shows what mathematics was like in the past and how it has developed up to the present. One writer sees similarities with Thomas Kuhn's view of scientific change. Others have taken different approaches, such as, looking how the method of proof has change over the years, focusing on the nineteenth century.
With the practices the writers move into what mathematicians do. Some see most mathematicians as providing informal proofs. These proofs do not rely on a set of axioms and rules for proving theorems. Often a mathematician sees the truth of some piece of mathematics, and then goes and shares it with his contemporaries, only later presenting it at a seminar or conference, and then finally writing a paper. None of this involves formal proof working with axioms. It is said that the majority of theorems are never given any formal proof. This is said to show the inappropriateness of foundational mathematics to most of mathematical activity.
The social atmosphere of mathematicians' lives they find themselves in is said to matter a lot. Mathematicians practice in a community with other mathematicians. These communities share a particular topic, practice, and interaction.
How mathematics is taught is also believed by some of the writers to matter. After all, what gets transmitted to a new generation of mathematicians appears to be of importance. How mathematics is taught to non-mathematicians can also play a role in how mathematics is viewed.
Finally, work with computers is look at. The four-color theorem is look at by the editer using it to dissect what exactly is a mathematical proof, and that provides the mathematician. In other words, how do mathematicians see the truth of a given proof. Chaitin provides a look at computational theory, and how it relates to Godel's theorem.
An interesting bit for me was in Wang's presentation of what it means to be able to multiply. There he gives an inductive proof as such: “1 is small; if 1 is small, n + 1 is small; therefore, every number is small.” An inductive proof I can actually follow, not given that kind of mathemtical skill. But, it occurred to me that this proof, like maybe all proofs, certainly many, is in need of interpretation. For this proof to mean anything to me I had to compare every number is small with infinity. It seems to me that every number is small in relation to infinity. I even thought that maybe you could prove that every number greater than 1 is large. This could be in relation to all real numbers less than 1, or possibly in comparison with the negative numbers. If there is any proof of these, I will leave it to a competent person.
I found to the book to be an eye opener to how much more broad the philosophy of mathematics could be from what I had known before, which had been limited to pretty much the foundations, realism, and the discovery/invention controversy. While I can't claim to understand everything in the book, especially those parts that actually use mathematics in what I would call a technical manner, I think I have comprehended the main components of this book. Not being very interested in the technical side of mathematics, this was okay with me. I believe that the book lives up to its stated purpose including presenting the ideas in it to a broader audience than just mathematicians and philosophers. For the most part I found the book enjoyable, especialy those parts that I felt I could ask pertinent questions of the various writers as I tend to do in most books that I read.
I recommend this book for those interested in the philosophy of mathematics, and who would like to explore different approaches to it other than foundationalism. It would be good if the reader is already familiar with basic mathematical concepts and areas of study, but I do not think that is absolutely nesessary.
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