Book review: George P. Lakoff and Raphael E. Nunez, "Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being"
(Dated: October 18, 2007. Note that this review was originally written as a postal correspondence with Professor Dana Scott, co-winner of the 1976 ACM Turing Award. The initial few paragraphs in the correspondence are not directly related to the review of the Lakoff-Nunez book. The book review, proper, starts from the sixth paragraph of the correspondence given below).
(Dated: October 18, 2007. Note that this review was originally written as a postal correspondence with Professor Dana Scott, co-winner of the 1976 ACM Turing Award. The initial few paragraphs in the correspondence are not directly related to the review of the Lakoff-Nunez book. The book review, proper, starts from the sixth paragraph of the correspondence given below).
Urbana-Champaign, IL
October 18, 2007
Thursday
Dear Professor Dana Scott,
As I mentioned during our discussion last Saturday, I do not write on philosophy directly, because it is very tricky. So, I have been approaching the philosophy of mathematics quite indirectly, by first considering history, culture and identity. In view of this indirect approach, it seems to me that the best way to introduce my views on mathematics would be to show you some of my recent writings. I have enclosed two unfinished articles in this mail:
1. An essay titled, "Effective Philanthropy in India: a case study for contributing to the strengthening of the foundations of a developing nation". In this essay, I am making several observations about the relative strengths of the Western society and the Indian society, in their ability to support a life of mathematics. There are also several other discussions centered on mathematics.
2. Line-by-line commentary to the first few pages of Professor Amartya Sen's recent book, "The Argumentative Indian". In these comments, I touch on the mathematical nature of some ancient Indian literature, particularly, the Bhagavad Gita and the Vedas. I also make some observations about famous mathematicians like Andre Weil, Kurt Godel, G. H. Hardy and Srinivasan Ramanujan.
The first article is really a huge project, as you would realize immediately upon reading the abstract. My plan is to write about a hundred pages for this article. I have written six pages (in
12 font size) till now. But the main ideas are already mentioned in the parts I have managed to write so far -- a detailed abstract and the first few pages of an introduction. I have also included
a five page section called Resources, where further indications about some of the directions I intend to proceed are given in bulleted-points format.
The second article is nearly in its final form. There are twenty comments in all. I might come back on a later occasion to add comments to subsequent pages of Professor Amartya Sen's book. But, the twenty comments I have written so far are self-contained, and may be read as they are.
Now, I had originally had the opportunity to speak with you for a few minutes last Thursday (October 11). You had told me then that there are a lot of articles in recent published literature, on the philosophy of mathematics, that criticize Platonism. Moreover, we agreed that the main theme in the recent book, "Where Mathematics Comes From: How the Embodied Mind brings Mathematics into Being" by the Berkeley Linguist, Professor George Lakoff, and the Cognitive Psychologist, Professor Raphael Nunez, was a 'theory of the embodied mind' which had no place for Platonism. In addition, you told me that the Lakoff - Nunez theory needed a lot more empirical / scientific justification before they can claim that their 'conceptual metaphors' explained all of modern mathematics from a cognitive perspective.
Subsequently, on Saturday, I defended Platonism very strongly, and held that the main weakness of the Lakoff - Nunez theory is not that it lacked adequate justification, but that it tried to eliminate Platonism. In view of these discussions, I feel that I should make some points here to clarify what I believe Platonism to be. Again, this is because my view of Platonism is more unambiguous than what it is usually believed to be --
that it posits the existence of an 'ideal' world of ideas which a human being can inhabit, only occasionally, by way of much striving and thought.
1. Most of Plato's writings that have survived to the present day are in the form of dialogues. The participants in these dialogues, other than Socrates, are the young men of Athens who were rather unsophisticated and untrained in philosophical arguments. These dialogues were written in a simple language, meant to be understood by the common man, and are devoid of any nuanced and hairsplitting interpretations of philosophy. In the interest of full disclosure, I interject here to say that my own experience with Plato's dialogues has only been in the reading of Philebus, which Plato wrote as a result of a debate started by Eudoxus, the mathematician.
2. In view of the simplicity of the language of the dialogues, all of Plato's views on the philosophy of mathematics cannot be explained unambiguously by a narrow interpretation like the 'ideal world of ideas', mentioned in italics above. Interpretations of that type only became common in the Western world much later, as a result of religious theologies, which engaged in much nuanced interpretations of the original texts of the religions. Plato never really intended his dialogues to become theological doctrines, but rather, he intended them to provide basic material for common, every day discussions about philosophical matters. However, an unambiguous interpretation of Plato's views, sophisticated to any degree one wants, can be inferred quite accurately by relying on three factors -- (i) the spirit in which Plato wrote his dialogues and the principles he adhered by, (ii) the legacies he left behind, and (iii) the historical usefulness of views ascribed to Plato, in making progress on long-standing problems in mathematics -- all three of which we discuss below.
3. An important legacy of Plato for the modern world has been the university as a place of higher education and research. This is particularly relevant for the pure mathematician, because the modern university has been the main source of employment for pure mathematicians, whereas applied mathematicians, engineers, scientists and computer professionals have successfully found career employment elsewhere. To be sure, research in mathematics has also benefited from the efforts of technical societies, national academies, multi-disciplinary centers, institutes of advanced study, R & D labs, private corporations and software companies. In fact, during the middle-ages, royal patronage of scientific academies in Europe was a major reason that made new mathematical discoveries possible. But, for sheer longevity, Plato's legacy is unrivaled. The academy that Plato set up survived for nearly nine hundred years, before it was closed down by the Roman Emperor Justinian I in 529 AD, as a possible threat to Christianity. Much of the modern university system is modeled after Plato's academy (except that his academy did not issue academic degrees). However, I have to mention a disclaimer here. It would be a big mistake to ascribe everything that happens in the modern university to the legacy of Plato. The university system was heavily subjected to later influences like the Church and the Roman Empire. So, the functioning of the modern university is determined, not just by a quest for knowledge and wisdom, as Plato would have liked, but also by subtle influences of power and money.
4. Another important legacy of Plato to the modern mathematician has been the thesis advisor - graduate student relationship. The sense of respect and admiration in which Plato held Socrates in all his dialogues, written over the course of over forty years, has been the enduring source of inspiration for the way research is to be conducted by the advisor - student duo. Again, this is particularly important to the student of pure mathematics. For doctoral students in other disciplines, the necessity for a physical infrastructure - e.g. computers, laboratories - and attending conferences and seminars to keep abreast of latest developments may be equally important as interacting with the thesis advisor.
5. The particular relevance of Plato's views to the working mathematician of today can be traced back to Sir Isaac Newton's writing of the Principia Mathematica. Just before Newton's time, mathematics was dominated by Rene Descartes. In the Cartesian world view, theorems and proofs (in the spirit of Euclid's high school geometry) did not matter when it came to advanced studies. What mattered were ideas and perceptions. It was with this view that Descartes introduced the Cartesian axes. In one stroke, he had managed to bring together numbers, pictures, motion and measurements -- all the main concerns of mathematics then. This view could be characterized as cognitive existentialism (Cogito Ergo Sum), and it placed much emphasis on geometric intuition (via the Cartesian axes) and empirical evidence (via the Scientific method). Newton in his early years was often engaged in claims and counter-claims with other scholars of the day who were active in the same areas he worked in. However, the publication of the Principia Mathematica in 1686 settled, once and for all, the supreme dominance of Newton's intellect. Every one of his strongest critics, from all over Europe, conceded that Newton's mind existed in a different world -- the Platonic world of ideas. The level of clarity in the exposition, and the complete mastery of the mathematical principles in applying them to physical problems were unprecedented. The Principia Mathematica installed Platonism as the highest ideal of mathematical investigation, sidelining the Cartesian world view, as far as mathematics was concerned, for the next 300 years. We mention here that Newton was also an early pioneer of the school of thought named Formalism, and he was perhaps the greatest intuitionist the world has ever known.
6. The narrow interpretation of Platonism mentioned in italics above completely overlooks some other important aspects of Platonism. The sense of respectful adoration that society bestows on its great thinkers is in the belief that the thinkers are roaming in the Platonic world with no care or worry about the physical world. Also, associated with Platonism is a strong sense of romance and fascination with the subject that is being investigated. Platonism shares such romantic traits with Intuitionism, though the emphasis in Platonism is less on the human element and more on the objects of study than in Intuitionism. Moreover, in Platonism, there is no sense of urgency, as clearly exists in Formalism, for example. Progress happens at its own pace. It is beauty, clarity, depth and finality that one is looking for. For this reason, when a Platonist finally manages to establish a result after a lot of hard work, it comes as a blessing. In fact, under Platonism, engaging in the work itself is therapeutic in a way that the other approaches could not be. In the words of Karl Weierstrass, "The mathematician who is not also something of a poet will never be a complete mathematician".
7. Some scholars of the Renaissance period might view Newton's contribution to the revival of Platonism, as only one part of a mass awakening, spanning several centuries during the Renaissance enlightenment, to the teachings of Plato. We recall here that it was Aristotle who was the dominant philosopher throughout the first millennium. In any case, after Newton, the standards of mathematical scholarship were predominantly focused on Platonic ideals. This state of affairs was further sanctioned by the mathematical developments in the late Renaissance period -- e.g., the discovery of the imaginary number, the proof of the unsolvability of the general quintic polynomial by radicals, and the development of projective geometry. Humanity had gone through generations and generations of thinking about mathematics. Whole civilizations had developed, decayed and disappeared over the course of millenniums. But, in all that time, nobody had thought of the imaginary number. This was the clearest indication that there was indeed something called a Platonic world that was quite different from the physical world. The development of projective geometry is important here for two reasons. Firstly, Blaise Pascal, one of its founders, was the prime example of the human striving for higher knowledge. He made his mark equally in religious theology as well as in mathematics, his Pensees being considered 'the most eloquent book in French prose' by Will Durant. Secondly, the Renaissance painters and artists with their lifestyles revolving around the laid-back, romantic side of Platonism, were intimately involved in using projective geometry in their works. Next, the string of unsolvability results established in this period -- e.g., the impossibility of trisecting an angle, doubling the cube and squaring the circle using only compass and straight-edge, the Abel-Ruffini theorem on polynomials of degree five or higher -- were developments that would be hard to explain solely through a theory of embodied mind for mathematics.
8. In 1854, Bernhard Riemann gave his Habilitation lecture, "On the Hypotheses which lie at the Foundations of Geometry". The topic of the lecture was chosen by Carl Gustav Gauss. In this celebrated address, Riemann stated, "the propositions of geometry cannot be derived from general notions of magnitude, but the properties which distinguish space from other conceivable triply extended magnitudes are only to be deduced from experience". This was wisdom gained from centuries of effort by mathematicians on the independence of the parallel postulate from the first four axioms of Euclid. Clearly, the symbolic manipulation skills of the Formalist or the cognitive skills of the Intuitionist do not account for the techniques and the approaches that were needed for tackling problems of such scale. Platonism alone gave a credible account of the nature and the kind of mathematical reality that mathematicians needed to be concerned with, in order to be able to make progress with such problems that remained unsolved for millennia. Throughout the nineteenth century, pre-occupation with the shape of physical space led frequently to one of the central debates among mathematicians -- whether mathematical reality existed outside of us, or inside. After a lot of deliberations, debates and discussions, a consensus emerged in the twentieth century. By 1940, G. H. Hardy was able to say in his famous essay, 'A Mathematician's Apology', "I believe mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations', are simply our notes of our observations."
I hope that the eight observations given above show convincingly that Platonism will always be an indispensable philosophical disposition among mathematicians. However, it is true that Formalism began to occupy center-stage among mathematicians in the twentieth century, under the influence of David Hilbert. Swearing a poetic sensibility in the spirit of Weierstrass today would do little to advance the cause of a young mathematician towards his academic tenure. But, it would be a mistake to think modern mathematics has left Platonism behind. In fact, Formalism can be seen as an adaptation of Platonism for the busy, competitive, cosmopolitan lifestyle.
At present, I am trying to understand how the pre-occupation with the building of an empire among the social classes of the late nineteenth century Europe could have motivated a Formalist world view among mathematicians. The socio-political problems that occupied the intellectuals in Europe during the nineteenth century, were very much the concerns of an empire. The second Nobel Prize in Literature was awarded in 1902 to Theodore Mommsen, "the greatest living master of the art of historical writing, with special reference to his monumental work, A history of Rome". Of course, Formalism is founded on the age old debate on form versus function, and hence may be a natural human instinct. This particular project of mine, towards understanding the roots of Formalism, is still work in progress, and is far from final form.
During our discussions last Saturday, a gentleman named Dr. Michael Kelly joined us. He had made two comments. The first was that when I mentioned that the Lakoff - Nunez team can explain the imaginary number I, using their conceptual metaphors, Dr. Kelly asked what
about 2*I, 3*I, and so on. I suppose his point is that there are an infinite number of Platonic objects in mathematics, and they would have to explain each one cognitively. This is really quite
a nuanced interpretation of Platonism for which I would refer the gentleman to my point (1) above on Platonism. The second comment was made when I mentioned that since the Lakoff - Nunez book went against Platonism, it is going to run into serious trouble with a lot of mathematicians. Dr. Kelly said something to the effect that there are whole departments working on the meaning of Platonism, and if there was a consensus against it, surely their book could go against it. My answer is to refer the gentleman to point (2) above on Platonism.
I have several other comments to make about the Lakoff - Nunez book. Last Thursday, you had mentioned that Professor George Lakoff's work has become quite political, the title of his latest book being, 'Don't think of the elephant:...' Well, mathematics is in a crisis now. In the nineteenth century, mathematics occupied a pre-eminent position among the scientists and the thinkers. To mention just one example, fifty years before Albert Einstein developed his special theory of relativity, Riemann had already built the mathematical infrastructure required for it, by defining the Riemannian manifold. However, over the course of the twentieth century, an interesting combination of psychology, linguistics, anthropology, evolution, genetics, medicine and ecology (all seen from a computational perspective) has taken over the intellectual center-stage of the modern society. In fact, the most important discoveries are happening elsewhere, and mathematics is simply trying to imitate the exciting developments in these other areas. I am sure that the Lakoff - Nunez team sees this clearly. According to them, the game is already over. They don't believe they need to make any more justifications. They have nothing to learn from mathematicians. They say explicitly in their book:
"But mathematics by itself does not and cannot empirically study human ideas; human cognition is simply not its subject matter. It is up to cognitive science and the neurosciences to do what mathematics itself cannot do -- namely, apply the science of mind to human mathematical ideas. That is the purpose of this book."
Professor Lakoff has moved onto politics, perhaps hoping to make a bigger impact there. It appears that the Lakoff - Nunez team see themselves as philosophers, as intellectual heirs of Rene Descartes. Since philosophers have always concerned themselves with the political situation of the day, they probably are not embarrassed by becoming politically active.
On the other hand, the Cartesian world view, building on the Aristotlean world view, has continued to be the dominant tradition in the biological sciences and the experimental sciences. Descriptive prose and pictures are the major medium of communication. In these disciplines, the level of certainty provided by experimental verification allows the researchers to forego any historical concerns, if and when necessary. In fact, the same situation also applies in some other natural sciences, like Physics and Chemistry. As a result, existing theories in these disciplines have been replaced many times by newer and simpler theories that explain the experimental
results better. Examples are (i) Newton's work on chemistry which was later relegated to alchemy, (ii) the theory of the ancient Greeks that matter is made of fire, water, earth, air and ether would not have any subscribers today.
The Lakoff - Nunez team's strategy of quickly explaining the cognitive foundations of a given discipline and then moving on to another discipline would work well in such areas where history can be consigned to the trash bin without much trouble. However, in mathematics that strategy would not work. History matters quite a lot here, since history is usually the best guarantee against the false proofs and missteps. And, it is Platonism that has often shown the way forward during crucial periods in the history of mathematics.
The Lakoff - Nunez book also criticizes modern mathematicians that they use highly abstract language which is socially exclusive, and that mathematicians greatly emphasize formal manipulation of symbols and ignore the perception of new ideas. Rene Descartes said that in transcendental matters, one must be transcendentally clear. To their credit, the Lakoff - Nunez team has followed that example quite sincerely in the writing of the book. The book is really clear and is easy to read.
What makes the Lakoff - Nunez book tick is that most of the material has been worked out thoroughly over many centuries. Aristotle, Rene Descartes, Charles Darwin, Sigmund Freud and Noam Chomsky have all left their deep impressions on the material dealt with in this book. But, one would not get that impression upon reading this book. One would think that the Lakoff - Nunez team invented it all by themselves. In fact, the material in the first few chapters explains how humans understand arithmetic by extending on their biological capabilities for counting numbers. This material is nearly the same as found in Tobias Dantzig's book, which Albert Einstein recommended highly, and was written more than seventy five years ago.
I hope that you enjoyed reading the arguments I have made in this letter. I have printed out the pages of this correspondence and mailed them to you at my own cost. Moreover, I drafted this correspondence outside of office work. This correspondence does not involve my current employer, (-employer's name edited out-) in any way, except that (i) I met you in the (-employer's name edited out-) Conference last week, and (ii) I store my writings in my office computer. I request that this letter, and the enclosed articles, be treated as a private correspondence that does not involve (-employer's name edited out-).
I would really appreciate an opportunity to work with you on this subject of identifying future directions for doing research in mathematics. Please let me know about your opinions. Thank you very much, Sir.
Sincerely,
T. V. Selvakumaran
October 18, 2007
Thursday
Dear Professor Dana Scott,
As I mentioned during our discussion last Saturday, I do not write on philosophy directly, because it is very tricky. So, I have been approaching the philosophy of mathematics quite indirectly, by first considering history, culture and identity. In view of this indirect approach, it seems to me that the best way to introduce my views on mathematics would be to show you some of my recent writings. I have enclosed two unfinished articles in this mail:
1. An essay titled, "Effective Philanthropy in India: a case study for contributing to the strengthening of the foundations of a developing nation". In this essay, I am making several observations about the relative strengths of the Western society and the Indian society, in their ability to support a life of mathematics. There are also several other discussions centered on mathematics.
2. Line-by-line commentary to the first few pages of Professor Amartya Sen's recent book, "The Argumentative Indian". In these comments, I touch on the mathematical nature of some ancient Indian literature, particularly, the Bhagavad Gita and the Vedas. I also make some observations about famous mathematicians like Andre Weil, Kurt Godel, G. H. Hardy and Srinivasan Ramanujan.
The first article is really a huge project, as you would realize immediately upon reading the abstract. My plan is to write about a hundred pages for this article. I have written six pages (in
12 font size) till now. But the main ideas are already mentioned in the parts I have managed to write so far -- a detailed abstract and the first few pages of an introduction. I have also included
a five page section called Resources, where further indications about some of the directions I intend to proceed are given in bulleted-points format.
The second article is nearly in its final form. There are twenty comments in all. I might come back on a later occasion to add comments to subsequent pages of Professor Amartya Sen's book. But, the twenty comments I have written so far are self-contained, and may be read as they are.
Now, I had originally had the opportunity to speak with you for a few minutes last Thursday (October 11). You had told me then that there are a lot of articles in recent published literature, on the philosophy of mathematics, that criticize Platonism. Moreover, we agreed that the main theme in the recent book, "Where Mathematics Comes From: How the Embodied Mind brings Mathematics into Being" by the Berkeley Linguist, Professor George Lakoff, and the Cognitive Psychologist, Professor Raphael Nunez, was a 'theory of the embodied mind' which had no place for Platonism. In addition, you told me that the Lakoff - Nunez theory needed a lot more empirical / scientific justification before they can claim that their 'conceptual metaphors' explained all of modern mathematics from a cognitive perspective.
Subsequently, on Saturday, I defended Platonism very strongly, and held that the main weakness of the Lakoff - Nunez theory is not that it lacked adequate justification, but that it tried to eliminate Platonism. In view of these discussions, I feel that I should make some points here to clarify what I believe Platonism to be. Again, this is because my view of Platonism is more unambiguous than what it is usually believed to be --
that it posits the existence of an 'ideal' world of ideas which a human being can inhabit, only occasionally, by way of much striving and thought.
1. Most of Plato's writings that have survived to the present day are in the form of dialogues. The participants in these dialogues, other than Socrates, are the young men of Athens who were rather unsophisticated and untrained in philosophical arguments. These dialogues were written in a simple language, meant to be understood by the common man, and are devoid of any nuanced and hairsplitting interpretations of philosophy. In the interest of full disclosure, I interject here to say that my own experience with Plato's dialogues has only been in the reading of Philebus, which Plato wrote as a result of a debate started by Eudoxus, the mathematician.
2. In view of the simplicity of the language of the dialogues, all of Plato's views on the philosophy of mathematics cannot be explained unambiguously by a narrow interpretation like the 'ideal world of ideas', mentioned in italics above. Interpretations of that type only became common in the Western world much later, as a result of religious theologies, which engaged in much nuanced interpretations of the original texts of the religions. Plato never really intended his dialogues to become theological doctrines, but rather, he intended them to provide basic material for common, every day discussions about philosophical matters. However, an unambiguous interpretation of Plato's views, sophisticated to any degree one wants, can be inferred quite accurately by relying on three factors -- (i) the spirit in which Plato wrote his dialogues and the principles he adhered by, (ii) the legacies he left behind, and (iii) the historical usefulness of views ascribed to Plato, in making progress on long-standing problems in mathematics -- all three of which we discuss below.
3. An important legacy of Plato for the modern world has been the university as a place of higher education and research. This is particularly relevant for the pure mathematician, because the modern university has been the main source of employment for pure mathematicians, whereas applied mathematicians, engineers, scientists and computer professionals have successfully found career employment elsewhere. To be sure, research in mathematics has also benefited from the efforts of technical societies, national academies, multi-disciplinary centers, institutes of advanced study, R & D labs, private corporations and software companies. In fact, during the middle-ages, royal patronage of scientific academies in Europe was a major reason that made new mathematical discoveries possible. But, for sheer longevity, Plato's legacy is unrivaled. The academy that Plato set up survived for nearly nine hundred years, before it was closed down by the Roman Emperor Justinian I in 529 AD, as a possible threat to Christianity. Much of the modern university system is modeled after Plato's academy (except that his academy did not issue academic degrees). However, I have to mention a disclaimer here. It would be a big mistake to ascribe everything that happens in the modern university to the legacy of Plato. The university system was heavily subjected to later influences like the Church and the Roman Empire. So, the functioning of the modern university is determined, not just by a quest for knowledge and wisdom, as Plato would have liked, but also by subtle influences of power and money.
4. Another important legacy of Plato to the modern mathematician has been the thesis advisor - graduate student relationship. The sense of respect and admiration in which Plato held Socrates in all his dialogues, written over the course of over forty years, has been the enduring source of inspiration for the way research is to be conducted by the advisor - student duo. Again, this is particularly important to the student of pure mathematics. For doctoral students in other disciplines, the necessity for a physical infrastructure - e.g. computers, laboratories - and attending conferences and seminars to keep abreast of latest developments may be equally important as interacting with the thesis advisor.
5. The particular relevance of Plato's views to the working mathematician of today can be traced back to Sir Isaac Newton's writing of the Principia Mathematica. Just before Newton's time, mathematics was dominated by Rene Descartes. In the Cartesian world view, theorems and proofs (in the spirit of Euclid's high school geometry) did not matter when it came to advanced studies. What mattered were ideas and perceptions. It was with this view that Descartes introduced the Cartesian axes. In one stroke, he had managed to bring together numbers, pictures, motion and measurements -- all the main concerns of mathematics then. This view could be characterized as cognitive existentialism (Cogito Ergo Sum), and it placed much emphasis on geometric intuition (via the Cartesian axes) and empirical evidence (via the Scientific method). Newton in his early years was often engaged in claims and counter-claims with other scholars of the day who were active in the same areas he worked in. However, the publication of the Principia Mathematica in 1686 settled, once and for all, the supreme dominance of Newton's intellect. Every one of his strongest critics, from all over Europe, conceded that Newton's mind existed in a different world -- the Platonic world of ideas. The level of clarity in the exposition, and the complete mastery of the mathematical principles in applying them to physical problems were unprecedented. The Principia Mathematica installed Platonism as the highest ideal of mathematical investigation, sidelining the Cartesian world view, as far as mathematics was concerned, for the next 300 years. We mention here that Newton was also an early pioneer of the school of thought named Formalism, and he was perhaps the greatest intuitionist the world has ever known.
6. The narrow interpretation of Platonism mentioned in italics above completely overlooks some other important aspects of Platonism. The sense of respectful adoration that society bestows on its great thinkers is in the belief that the thinkers are roaming in the Platonic world with no care or worry about the physical world. Also, associated with Platonism is a strong sense of romance and fascination with the subject that is being investigated. Platonism shares such romantic traits with Intuitionism, though the emphasis in Platonism is less on the human element and more on the objects of study than in Intuitionism. Moreover, in Platonism, there is no sense of urgency, as clearly exists in Formalism, for example. Progress happens at its own pace. It is beauty, clarity, depth and finality that one is looking for. For this reason, when a Platonist finally manages to establish a result after a lot of hard work, it comes as a blessing. In fact, under Platonism, engaging in the work itself is therapeutic in a way that the other approaches could not be. In the words of Karl Weierstrass, "The mathematician who is not also something of a poet will never be a complete mathematician".
7. Some scholars of the Renaissance period might view Newton's contribution to the revival of Platonism, as only one part of a mass awakening, spanning several centuries during the Renaissance enlightenment, to the teachings of Plato. We recall here that it was Aristotle who was the dominant philosopher throughout the first millennium. In any case, after Newton, the standards of mathematical scholarship were predominantly focused on Platonic ideals. This state of affairs was further sanctioned by the mathematical developments in the late Renaissance period -- e.g., the discovery of the imaginary number, the proof of the unsolvability of the general quintic polynomial by radicals, and the development of projective geometry. Humanity had gone through generations and generations of thinking about mathematics. Whole civilizations had developed, decayed and disappeared over the course of millenniums. But, in all that time, nobody had thought of the imaginary number. This was the clearest indication that there was indeed something called a Platonic world that was quite different from the physical world. The development of projective geometry is important here for two reasons. Firstly, Blaise Pascal, one of its founders, was the prime example of the human striving for higher knowledge. He made his mark equally in religious theology as well as in mathematics, his Pensees being considered 'the most eloquent book in French prose' by Will Durant. Secondly, the Renaissance painters and artists with their lifestyles revolving around the laid-back, romantic side of Platonism, were intimately involved in using projective geometry in their works. Next, the string of unsolvability results established in this period -- e.g., the impossibility of trisecting an angle, doubling the cube and squaring the circle using only compass and straight-edge, the Abel-Ruffini theorem on polynomials of degree five or higher -- were developments that would be hard to explain solely through a theory of embodied mind for mathematics.
8. In 1854, Bernhard Riemann gave his Habilitation lecture, "On the Hypotheses which lie at the Foundations of Geometry". The topic of the lecture was chosen by Carl Gustav Gauss. In this celebrated address, Riemann stated, "the propositions of geometry cannot be derived from general notions of magnitude, but the properties which distinguish space from other conceivable triply extended magnitudes are only to be deduced from experience". This was wisdom gained from centuries of effort by mathematicians on the independence of the parallel postulate from the first four axioms of Euclid. Clearly, the symbolic manipulation skills of the Formalist or the cognitive skills of the Intuitionist do not account for the techniques and the approaches that were needed for tackling problems of such scale. Platonism alone gave a credible account of the nature and the kind of mathematical reality that mathematicians needed to be concerned with, in order to be able to make progress with such problems that remained unsolved for millennia. Throughout the nineteenth century, pre-occupation with the shape of physical space led frequently to one of the central debates among mathematicians -- whether mathematical reality existed outside of us, or inside. After a lot of deliberations, debates and discussions, a consensus emerged in the twentieth century. By 1940, G. H. Hardy was able to say in his famous essay, 'A Mathematician's Apology', "I believe mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our 'creations', are simply our notes of our observations."
I hope that the eight observations given above show convincingly that Platonism will always be an indispensable philosophical disposition among mathematicians. However, it is true that Formalism began to occupy center-stage among mathematicians in the twentieth century, under the influence of David Hilbert. Swearing a poetic sensibility in the spirit of Weierstrass today would do little to advance the cause of a young mathematician towards his academic tenure. But, it would be a mistake to think modern mathematics has left Platonism behind. In fact, Formalism can be seen as an adaptation of Platonism for the busy, competitive, cosmopolitan lifestyle.
At present, I am trying to understand how the pre-occupation with the building of an empire among the social classes of the late nineteenth century Europe could have motivated a Formalist world view among mathematicians. The socio-political problems that occupied the intellectuals in Europe during the nineteenth century, were very much the concerns of an empire. The second Nobel Prize in Literature was awarded in 1902 to Theodore Mommsen, "the greatest living master of the art of historical writing, with special reference to his monumental work, A history of Rome". Of course, Formalism is founded on the age old debate on form versus function, and hence may be a natural human instinct. This particular project of mine, towards understanding the roots of Formalism, is still work in progress, and is far from final form.
During our discussions last Saturday, a gentleman named Dr. Michael Kelly joined us. He had made two comments. The first was that when I mentioned that the Lakoff - Nunez team can explain the imaginary number I, using their conceptual metaphors, Dr. Kelly asked what
about 2*I, 3*I, and so on. I suppose his point is that there are an infinite number of Platonic objects in mathematics, and they would have to explain each one cognitively. This is really quite
a nuanced interpretation of Platonism for which I would refer the gentleman to my point (1) above on Platonism. The second comment was made when I mentioned that since the Lakoff - Nunez book went against Platonism, it is going to run into serious trouble with a lot of mathematicians. Dr. Kelly said something to the effect that there are whole departments working on the meaning of Platonism, and if there was a consensus against it, surely their book could go against it. My answer is to refer the gentleman to point (2) above on Platonism.
I have several other comments to make about the Lakoff - Nunez book. Last Thursday, you had mentioned that Professor George Lakoff's work has become quite political, the title of his latest book being, 'Don't think of the elephant:...' Well, mathematics is in a crisis now. In the nineteenth century, mathematics occupied a pre-eminent position among the scientists and the thinkers. To mention just one example, fifty years before Albert Einstein developed his special theory of relativity, Riemann had already built the mathematical infrastructure required for it, by defining the Riemannian manifold. However, over the course of the twentieth century, an interesting combination of psychology, linguistics, anthropology, evolution, genetics, medicine and ecology (all seen from a computational perspective) has taken over the intellectual center-stage of the modern society. In fact, the most important discoveries are happening elsewhere, and mathematics is simply trying to imitate the exciting developments in these other areas. I am sure that the Lakoff - Nunez team sees this clearly. According to them, the game is already over. They don't believe they need to make any more justifications. They have nothing to learn from mathematicians. They say explicitly in their book:
"But mathematics by itself does not and cannot empirically study human ideas; human cognition is simply not its subject matter. It is up to cognitive science and the neurosciences to do what mathematics itself cannot do -- namely, apply the science of mind to human mathematical ideas. That is the purpose of this book."
Professor Lakoff has moved onto politics, perhaps hoping to make a bigger impact there. It appears that the Lakoff - Nunez team see themselves as philosophers, as intellectual heirs of Rene Descartes. Since philosophers have always concerned themselves with the political situation of the day, they probably are not embarrassed by becoming politically active.
On the other hand, the Cartesian world view, building on the Aristotlean world view, has continued to be the dominant tradition in the biological sciences and the experimental sciences. Descriptive prose and pictures are the major medium of communication. In these disciplines, the level of certainty provided by experimental verification allows the researchers to forego any historical concerns, if and when necessary. In fact, the same situation also applies in some other natural sciences, like Physics and Chemistry. As a result, existing theories in these disciplines have been replaced many times by newer and simpler theories that explain the experimental
results better. Examples are (i) Newton's work on chemistry which was later relegated to alchemy, (ii) the theory of the ancient Greeks that matter is made of fire, water, earth, air and ether would not have any subscribers today.
The Lakoff - Nunez team's strategy of quickly explaining the cognitive foundations of a given discipline and then moving on to another discipline would work well in such areas where history can be consigned to the trash bin without much trouble. However, in mathematics that strategy would not work. History matters quite a lot here, since history is usually the best guarantee against the false proofs and missteps. And, it is Platonism that has often shown the way forward during crucial periods in the history of mathematics.
The Lakoff - Nunez book also criticizes modern mathematicians that they use highly abstract language which is socially exclusive, and that mathematicians greatly emphasize formal manipulation of symbols and ignore the perception of new ideas. Rene Descartes said that in transcendental matters, one must be transcendentally clear. To their credit, the Lakoff - Nunez team has followed that example quite sincerely in the writing of the book. The book is really clear and is easy to read.
What makes the Lakoff - Nunez book tick is that most of the material has been worked out thoroughly over many centuries. Aristotle, Rene Descartes, Charles Darwin, Sigmund Freud and Noam Chomsky have all left their deep impressions on the material dealt with in this book. But, one would not get that impression upon reading this book. One would think that the Lakoff - Nunez team invented it all by themselves. In fact, the material in the first few chapters explains how humans understand arithmetic by extending on their biological capabilities for counting numbers. This material is nearly the same as found in Tobias Dantzig's book, which Albert Einstein recommended highly, and was written more than seventy five years ago.
I hope that you enjoyed reading the arguments I have made in this letter. I have printed out the pages of this correspondence and mailed them to you at my own cost. Moreover, I drafted this correspondence outside of office work. This correspondence does not involve my current employer, (-
I would really appreciate an opportunity to work with you on this subject of identifying future directions for doing research in mathematics. Please let me know about your opinions. Thank you very much, Sir.
Sincerely,
T. V. Selvakumaran
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