Timeline for Mathematical habits of thought and action which would be of use to non-mathematicians
Current License: CC BY-SA 3.0
18 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 2, 2019 at 20:58 | comment | added | LSpice | I think the thing that is very clear about this habit is that we must always make arguments by first considering the most extreme cases, or else we must never do so. | |
| Jun 5, 2016 at 6:28 | comment | added | bubba | Another way of saying this, ... mathematicians typically get pre-occupied with "corner" cases, as opposed to mainstream cases. They are not interested in things that are usually/typically true, only in things that are always true. I find this attitude annoying and misleading, both in mathematics and in everyday life. | |
| Jul 4, 2014 at 3:19 | comment | added | goblin GONE | "In my experience, mathematicians will frequently argue (in general, not just in mathematics) by passing to an extreme case at the beginning." Yes, but in real life, we're interested in the what's probable or typical, not the extreme cases. I think arguing from extreme cases is basically a hallmark of not having done enough science and/or humanities, and is really not something to be celebrated. | |
| Jun 27, 2013 at 22:05 | comment | added | Pait | I agree that arguments to the extreme are useful when applicable, but I wanted to point out that often they are misused. | |
| Jun 26, 2013 at 21:04 | comment | added | Emerton | ... e.g. imagining that there is just one company listed on the stock exchange, or imagining that the population consists of just two classes, one whose annual income is $x$ and the other whose annual income is $1000 x$. Of course that's unrealistic, but it can help clarify ones thoughts about possible phenomena or mechanisms, without getting distracted by all the messy details of a real-world situation. Hopefully, one can then modify the ideas coming from the ideal model so that they say something about reality. (Of course, that's always a challenge ... .) Regards, | |
| Jun 26, 2013 at 21:02 | comment | added | Emerton | @Pait: Dear Pait, I think you misunderstood what I am describing. As an example, when I used to try to explain physics to non-scientist friends, and would explain that one first begins to analyze the case without air resistance or friction, and then (perhaps) later adds those parameters back in, they found it hard to appreciate: they couldn't get a sense of how the ideal context could provide a good starting point with few complications, and that later one could then try to perturb it to a more realistic situation. I find the same kind of analysis useful sometimes in thinking about economics: | |
| Jun 26, 2013 at 15:28 | comment | added | Pait | I find arguments to the extreme a very bad habit in complex situations, for example politics. Something along the lines of "if you favor more individual freedom then you must be against traffic lights and that would be a disaster" or "if you favor regulation of business then you must be against private property and see what happened with the Soviet Union". I hope mathematics training prevents people from using arguments to the extreme when they are not applicable. | |
| Sep 10, 2011 at 5:49 | comment | added | Terry Tao | The Monty Hall problem is a good litmus test in this regard. Resolving that problem by using an extreme case (1,000,000 doors!) to illustrate the main issue tends to be convincing if and only if one has had exposure to mathematical thinking. | |
| Sep 9, 2011 at 18:17 | comment | added | Dylan Wilson | +(a lot). As everyone else says, I do this all the time and it really does seem to anger other people... I pretty much sympathize word for word with what Spencer said as well... | |
| Sep 8, 2011 at 20:51 | comment | added | Gil Kalai | Dear Matthew, I definitely agree with that... | |
| Sep 8, 2011 at 13:27 | comment | added | Emerton | Dear Gil, I agree that it not always the best way to analyze a situation. But, as I wrote, I think it can sometimes be valuable. Best wishes, Matthew | |
| Sep 8, 2011 at 6:21 | comment | added | Gil Kalai | Actually,In areas outside mathematics, passing to extreme cases first and considering unrealistic situations is a habit of thought and action which is more often damaging than useful. | |
| Sep 7, 2011 at 14:33 | comment | added | Spencer | +1. Yes!; well-articulated. I do this all the time when talking with my girlfriend and she nearly always argues that the extremity of my example makes it invalid and irrelevant to the discussion, whereas I see the extremity as potentially highlighting the salient points of the general case. | |
| Sep 7, 2011 at 8:10 | comment | added | Dirk | I guess that this technique is widely known in management under the name ""scenario technique"... | |
| Sep 7, 2011 at 6:05 | comment | added | Mariano Suárez-Álvarez | Going to extremes also puts relevant features of the situation to the fore---as well as angers others. | |
| Sep 7, 2011 at 5:27 | comment | added | David Roberts♦ | I've definitely done this in my non-academic job, and it surprises me that no one else thought to consider such situations (and it wasn't necessarily an extreme hypothetical). | |
| Sep 7, 2011 at 4:41 | comment | added | James D. Taylor | I think this is an extension of being able to recognize definitions, or lack thereof. Going to extremes sets the parameters of the definitions in discussion. | |
| Sep 7, 2011 at 4:14 | history | answered | Emerton | CC BY-SA 3.0 |