3 ms·
This is provably untrue and all I need to do is think about how weird stuff gets when increasing dimensionality of "household" shapes like circles and squares.
by dvt 9y ago
This is provably untrue and all I need to do is think about how weird stuff gets when increasing dimensionality of "household" shapes like circles and squares. It literally goes against every intuitive fiber of my body.
If you think that coming up with theorems about, e.g. infinitary algebra, is intuitive, then I guess you're just a lot smarter than I am. Infinities always throw a wrench into our intuitions. Something being a "logical consequence" of something else is an incredibly simplistic way of looking at things, not to mention that I'm not exactly sure what that has to do with intuition anyway.
- sdenton4 9y agoIt /becomes/ intuitive with time, thought, and exposure, as you pick up more tools for understanding the problem space. That 'non-intuitive' aspect of high-dimensional spheres having very small volume relative to a hypercube becomes a point that builds intuition for how those spheres behave, once you're familiar with it. And then you can start throwing out things that seem intuitively wrong, and following intuition towards new ideas.
- dvt 9y agoYou just described the opposite of intuition.
- tashi 9y agoAt this point a mathematician might step in and say that you all need to define your terms. You're basing your arguments on two subtly different definitions of the word "intuition."
- dvt 9y agoThe differences aren't subtle. I'm using the run-of-the-mill definition you might find in a dictionary[1], what other definition could you possibly be using? [1] https://www.merriam-webster.com/dictionary/intuition https://www.merriam-webster.com/dictionary/intuition
- emmelaich 9y agoFrom that I would choose 2.c, with the emphasis on evident > the power or faculty of attaining to direct knowledge or cognition without evident rational thought and inference There is such a thing as developing your intuition. I think your meanings are not qualitatively different.
- nickpsecurity 9y agoIntuition is a part of the mind that makes approximate models based on experience up to that point. The experiences without anything that would help model infinites would make them non-intuitive. Experiences that would help model infinites would make them at least more intuitive. Experience with rational investigation of them going through examples is actually one way to do that. There's been organizations like Marine Corps incorporating intuition into training for a long time where they set out to program it using realistic scenarios in drills that people do over and over until the intuitive processes pick up the patterns/model. It will happen as many people study that topic as well. The trick is that you need good data to feed it that makes the patterns clear. You might have not gotten a lot of that on the topic you had no intuition of. On some topics, there hasn't been much of an attempt to do that since people expect it to be hard with everyone forced to do the methods that are non-intuitive. Lots of traditions are like that. Then, there's the possibility something can't be intuitively handled at all. I don't know if those exist but they might. That many activities of people doing math seem to get an intuitive boost during their day-to-day work makes me lean towards intuitive possibilities for about everything that can be approximated (esp with heuristics).
- naasking 9y agoYou aren't born with all of your intuitions, you learn most of them. This includes mathematical intuitions.
- dorkwood 9y agoA chess master can intuitively tell when mate is three moves away. They weren’t born with that intuition, though. They gained it over time.