4 ms·

(I have a math degree) 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.

by lmkg 17y ago

(I have a math degree) 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.

mreid 17y ago
Great points, especially the one on multiple perspectives. 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 http://arxiv.org/abs/math/9404236 .
elblanco 17y ago
If the population as a whole learned just the principle of intellectual rigor, I truly believe the world would be a better place.
- yters 17y ago
  Yes, copy academia's intellectual rigor mortis.
aaronblohowiak 17y ago
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.
- billswift 17y ago
  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.
timwiseman 17y ago
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.

by lmkg 17y ago

mreid 17y ago

Great points, especially the one on multiple perspectives. 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 http://arxiv.org/abs/math/9404236 .

elblanco 17y ago

If the population as a whole learned just the principle of intellectual rigor, I truly believe the world would be a better place.

yters 17y ago
Yes, copy academia's intellectual rigor mortis.

aaronblohowiak 17y ago

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.

billswift 17y ago
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.

timwiseman 17y ago

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.