The best description of math that I have ever encountered of what math is is the book The Mathematical Experience by Davis and Hersch.
Don't be scared by the fact that it is a book. It is very accessible since it is divided into little articles on different areas of math. A lay person can read it and gain a sense of what this math stuff is and how varied it is. A PhD can read the same material and will appreciate it on a much deeper level.
Let me give one data point. I know more mathematicians who became interested in math because of reading The Mathematical Experience than any other book. Seriously.
My father-in-law was a fan of that one as well. I was not as taken with Hardy's "my life is useless" argument, and I find it a delicious irony that advances in cryptography have made Number Theory applied.
Yes, I too found that ironic. But it remains a beautiful and well written insight into the mind of a great mathematician and into why he went into mathematics himself.
Hm. I've not read it, but I have read a rather scathing review of it by Martin Gardner (whose opinions I usually respect). disagreeing with it on fundamental philosophical grounds.
That's interesting. Gardner describes himself as a journalist interested in math and science, but specifically not a mathematician. It's interesting that he disagrees with two mathematicians on the question of what math is.
I find MG fantastic, but sometimes his articles and books lack depth. He is a wonderful writer and populariser, but I wonder whose side I would take if this were a real debate.
But it isn't. There's room for both opinions. I've never met Gardner and am now unlikely to do so, but I know a few people who know him fairly well. I'll ask their opinion when I get the chance.
Here is the primary source I found of Gardner's review. I suggest anyone interested read it themselves, as the review itself is a well-written reflection on Gardner's own mathematical philosophy.
I did not find it scathing, but rather of mixed opinions. He criticizes the sort of mathematical "cultural relativism" that the authors sometimes suggest, but he also praises the book's treatment of specific topics as accessible and inspiring.
Thanks for noting this review, because this book is one I like and often recommend to less-technical people, and I had never encountered it before.
No, I don't think he would. He'd point out that there are things which cannot be placed in the realm of verifiable and undebatable, but that doesn't mean there are no things which can be placed in that realm.
Gödel's intent was to show that there are things that are "true" but not "provable". So to say that math is the study of things that are verifiably and undebatably true is at odds with that.
I think the OP still commits the common error of focusing too much on what math studies rather than how it studies it, to a mild extent. Math, to me, is a style of thinking, and the corpus of math knowledge is this style of thinking applied to particularly well-suited domains.
That style of thinking is characterized by a few things. One, obviously, is rigor: every step is justified by formal rules, there are no appeals to intuition or leaps of logic. Most people's impression of mathematics stops there, that it's a list of rules like you're taught in grade school (because that style of memorization is the only way your teacher got through math in college). There's a lot more to it though.
For example, there's a heavy focus on interpreting a single fact in many ways. There's a lot of bridging knowledge between different sub-disciplines of math, and they work a lot by interpreting a mathematical structure in new ways. There's a lot of treating abstractions as new concrete topics of study. There's a lot of exploration.
Abstractions are well-suited to math because you can make them clean and simple enough to be amenable to complete rigor. This doesn't mean that math-style thinking is limited to worlds populated only by platonic ideals. And I think that's another core disconnect between the potential of math, and its realization in the education system, is that the arbitrary rules of symbol-manipulation are described as applying to illusionary ideals, not real things. There needs to be a bigger focus on motivating the ideals as modeling something useful, because they do, and that connection to the concrete helps a lot at getting a grasp on the topics.
I don't do very much "math" in my day job (other than working with numbers ;P) but that style of thinking is still a great benefit. The real world doesn't often allow math-level rigor, but you can still show exactly where the rigor is missing in your line of reasoning, which lets you identify assumptions, and then challenge them. The focus on abstraction and fundamental principles of systems is also useful, especially because it helps to apply experience from one domain onto another.
Thank you for your post. The notion of appealing to intuition is a very serious one. Intuition does not exist, at least not in the sense that it is commonly understood. There exists simply familiarity and permutation, but most explanations of intuition appeal to some form of ex nihilio proposition generation. What we refer to as intuition is the subconscious process of synthesizing previous experiences and transformations (permutations) of component parts. It is precisely for the rigor of explicit predicates and rules governing the transitions from one statement of fact to another that distinguish the reliability of math from the fallibility of informal thought. As you so deftly described, the ability to entertain varying predicates and explore the possible routes without violating the rules is a stimulating task that is unfortunately absent from most lower math education.
I whole-heartedly agree that the capacity to identify your assumptions is a valuable skill that rigid & formal thinking help to cultivate.
That "ex nihilio" implication is why I avoid the word "intuition" in my own writings and even notes. For the unconscious processes you describe I use the word "insight" instead.
One, obviously, is rigor: every step is justified by formal rules, there are no appeals to intuition or leaps of logic.
I am not certain that is entirely true. In areas that are long settled, this has often been made true, but that is often long after the fact, and often done through a process of "monster barring".
That is someone proposes a definition upon which theorems are built, and someone else later will present an object which meets the technical definition but which clearly did not meet the intent of the definition. So the technical definition is revised to exclude that. And the process repeats, narrowing the definition to remove that problem but often in the process making it more difficult to come to an intuitive understanding of that formal definition. One historical example of this was in the definition of polygon.
Areas of active research do tend to appeal to intuition, at least to a degree, until the formalism can be completed. Historically that formalism has often come from other hands than the original researchers and often either in the process of "monster barring" or as the final touch after several iterations of it.
In practice, even in areas where full rigor is absolutely possible, it can be terious beyond any use if there is no debate or challenge. Mathematical papers often do not have complete formalism and even text books on fully settled areas often do not try to justify every single step if there is no reason for it. Perhaps a better definition is that it must be possible in principle to justify every step by formal rules.
You are correct that the definitions are usually motivated by a particular problem or attempt at modeling something. However, if a mathematician arrives at a "bad" result, the reaction is not always to revise the definition.
Sometimes we think it's really awesome that the definition is general enough to capture additional phenomena in the same framework. Sometimes these degenerate ideas lead to interesting areas of research in their own right, like how adding some strange Godel-related axiom to number theory leads to Supernatural Numbers. Sometimes the theorems that are developed are not applicable to their original subject of study, but can be applied to another by re-interpreting their meaning, like non-Euclidean geometry. Sometimes the unintuitive result is simply accepted, like Godel's theorem. Sometimes our intuition is simply wrong, so unintuitive edge cases like Dirac spikes end up being fundamental tools in physics and signal analysis.
Intuition is often used to motivate the definitions or the axioms, but they are clearly labeled as definitions and axioms. What I was trying to say is that the chain from axioms to conclusion must be complete, with every step explained and approved. It's this method of explicit exposing your thought process, to yourself as well as to others, that I've found really useful outside of just the realm of mathematics.
However, if a mathematician arrives at a "bad" result, the reaction is not always to revise the definition.
Certainly not, I meant that as an example, not as a full description of all cases.
What I was trying to say is that the chain from axioms to conclusion must be complete, with every step explained and approved.
But this is what I am trying to say does not always happen, or at least does not always happen right away. Another commenter, mreid cited Thurston's essay "On proof and progress in mathematics" which discusses this in far more detail and far more authoratitively than I possibly could. Hardy also touches on it in his "A Mathematician's Apology".
Mathematics which is long settled, the type of thing generally studied in high school and for most of undergraduate mathematics education, is often (but not always) presented in this format, with axioms and definitions cleanly laid out and tiny steps form one to the other to arrive at the theorem.
Mathematics for working mathematicians does indeed involve this, but it also often involves intuitive leaps, thoerems where the first proof is only vaguely sketched out, and reasoning from diagrams. They normally try to clean this up by the time it is published, but for a variety of reasons even there it is often not perfectly rigorous and hardly ever in tiny steps with every step justified by axiom or previous proof.
There are many things held in common knowledge that are never cited. This is of course fine if the proofs are available, but steps away from that school house rigour immediately. Even beyond that, it is not uncommon for some ideas to only be sketched, and occassionally errors are found in the omitted steps in those sketches years after they have been formally peer reviewed and published.
These sketches are used in proof of absolute formalism often just for the sake of time and space in the journal, but occassionally because no one has ever actually gone through and filled in every detail at all, trusting to the more intuitive logic of the sketch.
The assumption in most cases is that it could be made perfectly rigorous if the need arised, but that is often only actually done if there is some question or debate.
Since everybody has a definition for math, it seems, I'll give it a whirl.
Math is the act of creating a self-consistent system of symbology and symbolic abstractions that, at some place, exactly mirrors reality (even if that place is only the counting numbers). The belief is that since nature is also self-consistent and ontologically hierarchical that by working inside a parallel symbolic system we can discover aspects of nature we never knew existed. Note that the systems have gotten so complex that in many fields math is entirely theoretical and never "closes the loop" back to observed reality. In fact, some would argue that applied math is more in the realm of physics, whereas "true" math is entirely abstract.
Math seems to me a bit like love, you have to experience it to understand it. Definitions and discussions like the OP might help a little, but don't get at the essence. No-one who reads such a definition will suddenly come away knowing what mathematics is.
I had a few epiphanies in my life that made me understand what mathematics is. One of them was understanding why it is that there are infinitely many primes - I think I learnt more about what mathematics is in the moment that proof clicked than I have from reading hundreds of discussions of "what is mathematics". The experience was, simply, miraculous.
Professor John Stillwell writes, in the preface to his book Numbers and Geometry (New York: Springer-Verlag, 1998):
"What should every aspiring mathematician know? The answer for most of the 20th century has been: calculus. . . . Mathematics today is . . . much more than calculus; and the calculus now taught is, sadly, much less than it used to be. Little by little, calculus has been deprived of the algebra, geometry, and logic it needs to sustain it, until many institutions have had to put it on high-tech life-support systems. A subject struggling to survive is hardly a good introduction to the vigor of real mathematics.
". . . . In the current situation, we need to revive not only calculus, but also algebra, geometry, and the whole idea that mathematics is a rigorous, cumulative discipline in which each mathematician stands on the shoulders of giants.
"The best way to teach real mathematics, I believe, is to start deeper down, with the elementary ideas of number and space. Everyone concedes that these are fundamental, but they have been scandalously neglected, perhaps in the naive belief that anyone learning calculus has outgrown them. In fact, arithmetic, algebra, and geometry can never be outgrown, and the most rewarding path to higher mathematics sustains their development alongside the 'advanced' branches such as calculus. Also, by maintaining ties between these disciplines, it is possible to present a more unified view of mathematics, yet at the same time to include more spice and variety."
Stillwell demonstrates what he means about the interconnectedness and depth of "elementary" topics in the rest of his book, which is a delight to read and full of thought-provoking problems.
Does not quite answer the question, but a true ode to mathematics:
"Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry."
For me, math is all about consequences. I'm extremely poor mathematician and my background is much more about computer science, yet I'm fascinated with this simple fact.
You define few simple structures, and, in consequence, get an enormous amount of structures' properties, more abstract structures and facts about newly created structures.
Thing like an algebraic structure can be understood by a clever seven years old. However, it simply takes stating structure such as an algebraic structure combined with a space, to enable existence of many other structures, more abstract or more specific, with their own properties, and similarities.
This is a very powerful thing. For me, math is all about this 'generative' way of thinking.
I agree. The way most people understand what math is and the way we teach math is horrible. The math curriculum is a disservice to everyone that's been exposed to it. I think mathematicians and scientists alike are realizing this and beginning the long process of fixing it in schools and eventually the culture. Some of the most creative people in the world today are not musicians or artists but rather mathematicians. (and as a corollary computer scientists, engineers, etc.)
I recommend "A Mathematician’s Lament" by Paul Lockhart [1]. He further investigates what math is, how we can better teach it and how it's portrayed in our culture.
35 comments
[ 3.2 ms ] story [ 52.5 ms ] threadDon't be scared by the fact that it is a book. It is very accessible since it is divided into little articles on different areas of math. A lay person can read it and gain a sense of what this math stuff is and how varied it is. A PhD can read the same material and will appreciate it on a much deeper level.
Let me give one data point. I know more mathematicians who became interested in math because of reading The Mathematical Experience than any other book. Seriously.
I find MG fantastic, but sometimes his articles and books lack depth. He is a wonderful writer and populariser, but I wonder whose side I would take if this were a real debate.
But it isn't. There's room for both opinions. I've never met Gardner and am now unlikely to do so, but I know a few people who know him fairly well. I'll ask their opinion when I get the chance.
Do you have a reference for that review?
http://www.cs.nyu.edu/pipermail/fom/1997-November/000128.htm...
http://books.google.com/books?id=oXEaTdstD7gC&pg=PA280&#...
I did not find it scathing, but rather of mixed opinions. He criticizes the sort of mathematical "cultural relativism" that the authors sometimes suggest, but he also praises the book's treatment of specific topics as accessible and inspiring.
Thanks for noting this review, because this book is one I like and often recommend to less-technical people, and I had never encountered it before.
A certain Mr. Gödel would disagree with you there.
don't hurt me ...
no more ...
I think the OP still commits the common error of focusing too much on what math studies rather than how it studies it, to a mild extent. Math, to me, is a style of thinking, and the corpus of math knowledge is this style of thinking applied to particularly well-suited domains.
That style of thinking is characterized by a few things. One, obviously, is rigor: every step is justified by formal rules, there are no appeals to intuition or leaps of logic. Most people's impression of mathematics stops there, that it's a list of rules like you're taught in grade school (because that style of memorization is the only way your teacher got through math in college). There's a lot more to it though.
For example, there's a heavy focus on interpreting a single fact in many ways. There's a lot of bridging knowledge between different sub-disciplines of math, and they work a lot by interpreting a mathematical structure in new ways. There's a lot of treating abstractions as new concrete topics of study. There's a lot of exploration.
Abstractions are well-suited to math because you can make them clean and simple enough to be amenable to complete rigor. This doesn't mean that math-style thinking is limited to worlds populated only by platonic ideals. And I think that's another core disconnect between the potential of math, and its realization in the education system, is that the arbitrary rules of symbol-manipulation are described as applying to illusionary ideals, not real things. There needs to be a bigger focus on motivating the ideals as modeling something useful, because they do, and that connection to the concrete helps a lot at getting a grasp on the topics.
I don't do very much "math" in my day job (other than working with numbers ;P) but that style of thinking is still a great benefit. The real world doesn't often allow math-level rigor, but you can still show exactly where the rigor is missing in your line of reasoning, which lets you identify assumptions, and then challenge them. The focus on abstraction and fundamental principles of systems is also useful, especially because it helps to apply experience from one domain onto another.
If you haven't read it already I highly recommend Thurston's essay "On proofs and progress in mathematics" - http://arxiv.org/abs/math/9404236 .
I whole-heartedly agree that the capacity to identify your assumptions is a valuable skill that rigid & formal thinking help to cultivate.
I am not certain that is entirely true. In areas that are long settled, this has often been made true, but that is often long after the fact, and often done through a process of "monster barring".
That is someone proposes a definition upon which theorems are built, and someone else later will present an object which meets the technical definition but which clearly did not meet the intent of the definition. So the technical definition is revised to exclude that. And the process repeats, narrowing the definition to remove that problem but often in the process making it more difficult to come to an intuitive understanding of that formal definition. One historical example of this was in the definition of polygon.
Areas of active research do tend to appeal to intuition, at least to a degree, until the formalism can be completed. Historically that formalism has often come from other hands than the original researchers and often either in the process of "monster barring" or as the final touch after several iterations of it.
In practice, even in areas where full rigor is absolutely possible, it can be terious beyond any use if there is no debate or challenge. Mathematical papers often do not have complete formalism and even text books on fully settled areas often do not try to justify every single step if there is no reason for it. Perhaps a better definition is that it must be possible in principle to justify every step by formal rules.
Sometimes we think it's really awesome that the definition is general enough to capture additional phenomena in the same framework. Sometimes these degenerate ideas lead to interesting areas of research in their own right, like how adding some strange Godel-related axiom to number theory leads to Supernatural Numbers. Sometimes the theorems that are developed are not applicable to their original subject of study, but can be applied to another by re-interpreting their meaning, like non-Euclidean geometry. Sometimes the unintuitive result is simply accepted, like Godel's theorem. Sometimes our intuition is simply wrong, so unintuitive edge cases like Dirac spikes end up being fundamental tools in physics and signal analysis.
Intuition is often used to motivate the definitions or the axioms, but they are clearly labeled as definitions and axioms. What I was trying to say is that the chain from axioms to conclusion must be complete, with every step explained and approved. It's this method of explicit exposing your thought process, to yourself as well as to others, that I've found really useful outside of just the realm of mathematics.
Certainly not, I meant that as an example, not as a full description of all cases.
What I was trying to say is that the chain from axioms to conclusion must be complete, with every step explained and approved.
But this is what I am trying to say does not always happen, or at least does not always happen right away. Another commenter, mreid cited Thurston's essay "On proof and progress in mathematics" which discusses this in far more detail and far more authoratitively than I possibly could. Hardy also touches on it in his "A Mathematician's Apology".
Mathematics which is long settled, the type of thing generally studied in high school and for most of undergraduate mathematics education, is often (but not always) presented in this format, with axioms and definitions cleanly laid out and tiny steps form one to the other to arrive at the theorem.
Mathematics for working mathematicians does indeed involve this, but it also often involves intuitive leaps, thoerems where the first proof is only vaguely sketched out, and reasoning from diagrams. They normally try to clean this up by the time it is published, but for a variety of reasons even there it is often not perfectly rigorous and hardly ever in tiny steps with every step justified by axiom or previous proof.
There are many things held in common knowledge that are never cited. This is of course fine if the proofs are available, but steps away from that school house rigour immediately. Even beyond that, it is not uncommon for some ideas to only be sketched, and occassionally errors are found in the omitted steps in those sketches years after they have been formally peer reviewed and published.
These sketches are used in proof of absolute formalism often just for the sake of time and space in the journal, but occassionally because no one has ever actually gone through and filled in every detail at all, trusting to the more intuitive logic of the sketch.
The assumption in most cases is that it could be made perfectly rigorous if the need arised, but that is often only actually done if there is some question or debate.
Math is the act of creating a self-consistent system of symbology and symbolic abstractions that, at some place, exactly mirrors reality (even if that place is only the counting numbers). The belief is that since nature is also self-consistent and ontologically hierarchical that by working inside a parallel symbolic system we can discover aspects of nature we never knew existed. Note that the systems have gotten so complex that in many fields math is entirely theoretical and never "closes the loop" back to observed reality. In fact, some would argue that applied math is more in the realm of physics, whereas "true" math is entirely abstract.
Is that anywhere close?
I had a few epiphanies in my life that made me understand what mathematics is. One of them was understanding why it is that there are infinitely many primes - I think I learnt more about what mathematics is in the moment that proof clicked than I have from reading hundreds of discussions of "what is mathematics". The experience was, simply, miraculous.
"Mathematics is the minimal environment to preserve ideas."
"What should every aspiring mathematician know? The answer for most of the 20th century has been: calculus. . . . Mathematics today is . . . much more than calculus; and the calculus now taught is, sadly, much less than it used to be. Little by little, calculus has been deprived of the algebra, geometry, and logic it needs to sustain it, until many institutions have had to put it on high-tech life-support systems. A subject struggling to survive is hardly a good introduction to the vigor of real mathematics.
". . . . In the current situation, we need to revive not only calculus, but also algebra, geometry, and the whole idea that mathematics is a rigorous, cumulative discipline in which each mathematician stands on the shoulders of giants.
"The best way to teach real mathematics, I believe, is to start deeper down, with the elementary ideas of number and space. Everyone concedes that these are fundamental, but they have been scandalously neglected, perhaps in the naive belief that anyone learning calculus has outgrown them. In fact, arithmetic, algebra, and geometry can never be outgrown, and the most rewarding path to higher mathematics sustains their development alongside the 'advanced' branches such as calculus. Also, by maintaining ties between these disciplines, it is possible to present a more unified view of mathematics, yet at the same time to include more spice and variety."
Stillwell demonstrates what he means about the interconnectedness and depth of "elementary" topics in the rest of his book, which is a delight to read and full of thought-provoking problems.
http://www.amazon.com/gp/product/0387982892/
"Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry."
Betrand Russell: "Mysticism and Logic and Other Essays". [http://www.gutenberg.org/etext/25447]
Some very eloquent responses in there. I was moved.
You define few simple structures, and, in consequence, get an enormous amount of structures' properties, more abstract structures and facts about newly created structures.
Thing like an algebraic structure can be understood by a clever seven years old. However, it simply takes stating structure such as an algebraic structure combined with a space, to enable existence of many other structures, more abstract or more specific, with their own properties, and similarities.
This is a very powerful thing. For me, math is all about this 'generative' way of thinking.
I recommend "A Mathematician’s Lament" by Paul Lockhart [1]. He further investigates what math is, how we can better teach it and how it's portrayed in our culture.
[1] http://www.maa.org/devlin/LockhartsLament.pdf
EDIT: I also posted this in the linked article. But wanted to provide this great paper to the HN crowd too.