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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.

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.

lmkg 17y ago
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.
- timwiseman 17y ago
  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.

lmkg 17y ago

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.

timwiseman 17y ago
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.