4 ms·

> It always saddens me how mathematicians seem to look down on “intuition”. They do this for good reason. I had to take, as part of my philosophy concentration

by dvt 9y ago

> It always saddens me how mathematicians seem to look down on “intuition”. They do this for good reason. I had to take, as part of my philosophy concentration, a bunch of mathematical logic classes. First Order and Second Order logic, as you might imagine, are pretty simple. The rules make sense in a very intuitive way. I'd get 100 on exams just because I'm good at programming. But when I had to study metalogic, model theory, Henkin Proofs, and Godel's Theorems, that intuition quite literally flies out the window. I guess my point is that most interesting stuff is rarely intuitive.

goldenkey 9y ago
That is ludicrous. The intuition is just harder to find, harder to grasp. But it is still there because these theorems are logical consequences of the base axioms. And the base axioms are logical consequences of our desire for sensible assumptions.
- lostpathfinder 9y ago
  Working mathematician does derive his ideas from axioms, it is ridiculous perception of mathematical process. And axioms are not logical consequences of sensible assumptions, it is try and fail process, long historical process. Intuition in mathematics has nothing to do with common sense intuition. Respectfully.
  - goldenkey 9y ago
    In abstract mathematics, one can choose any axioms one wants. To derive all kinds of consequences/theorems. But respectfully, ZFC came about because of logical paradoxes that couldn't be accepted as consistent. You can create a new number system and derive all kinds of consequences but the truth is, most mathematicians care more entirely about prime number theory on the naturals that are entirely based on counting. Most of modern math is based on the natural numbers. You can't remove all intuition. The thread that holds our love for math is also the same one that tells us we are exploring consequences that tell us a dearth about our universe.
- 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 ago
    It /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 ago
      You just described the opposite of intuition.
      - tashi 9y ago
        At 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 ago
        The 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
- eftychis 9y ago
  It depends how you define intuition. Maybe we could define intuition as your personal "model" for making guesses on how things work. Let's claim that writing a proof is similar to writing a program. You start from a state, your premise, and have your assumptions, input, which you process through a series of steps, maybe taking cases in between (branching) and return an output (hopefully True). Then considering this correspondence, it will be fine to say that you, as a programmer, can guess the output/behavior/semantics of any program without running it. Similarly, a mathematician would be able to reach the conclusion that the theorem holds, without going through all the details of the proof. Sure, in most cases you can, and as you get more and more experience and knowledge as a programmer you can say you improve your guesses. But this ends up backfiring with a good probability, when our model skipped a step, e.g. the interaction of two concurrent data structures. Thus I don't think it is right to say "as you are able to know all axioms and rules of inference, you should be able to achieve perfect intuition". You can know the fundamentals of each layer on your stack and you would find it insane if one asked you to guess the output of any non-trivial terminating program. Note also that the way of inference (logical consequences) can and does change (e.g. ZF vs ZFC). () I may be misusing the word model here -- it is usually defined (at least to things I work with) as the interpretation of a theory that satisfies all theorems (inferences). () constructive proofs yes. No flame wars please :)

by dvt 9y ago

goldenkey 9y ago

That is ludicrous. The intuition is just harder to find, harder to grasp. But it is still there because these theorems are logical consequences of the base axioms. And the base axioms are logical consequences of our desire for sensible assumptions.

lostpathfinder 9y ago
Working mathematician does derive his ideas from axioms, it is ridiculous perception of mathematical process. And axioms are not logical consequences of sensible assumptions, it is try and fail process, long historical process. Intuition in mathematics has nothing to do with common sense intuition. Respectfully.
- goldenkey 9y ago
  In abstract mathematics, one can choose any axioms one wants. To derive all kinds of consequences/theorems. But respectfully, ZFC came about because of logical paradoxes that couldn't be accepted as consistent. You can create a new number system and derive all kinds of consequences but the truth is, most mathematicians care more entirely about prime number theory on the naturals that are entirely based on counting. Most of modern math is based on the natural numbers. You can't remove all intuition. The thread that holds our love for math is also the same one that tells us we are exploring consequences that tell us a dearth about our universe.
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 ago
  It /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 ago
    You just described the opposite of intuition.
    - tashi 9y ago
      At 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 ago
        The 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
eftychis 9y ago
It depends how you define intuition. Maybe we could define intuition as your personal "model" for making guesses on how things work. Let's claim that writing a proof is similar to writing a program. You start from a state, your premise, and have your assumptions, input, which you process through a series of steps, maybe taking cases in between (branching) and return an output (hopefully True). Then considering this correspondence, it will be fine to say that you, as a programmer, can guess the output/behavior/semantics of any program without running it. Similarly, a mathematician would be able to reach the conclusion that the theorem holds, without going through all the details of the proof. Sure, in most cases you can, and as you get more and more experience and knowledge as a programmer you can say you improve your guesses. But this ends up backfiring with a good probability, when our model skipped a step, e.g. the interaction of two concurrent data structures. Thus I don't think it is right to say "as you are able to know all axioms and rules of inference, you should be able to achieve perfect intuition". You can know the fundamentals of each layer on your stack and you would find it insane if one asked you to guess the output of any non-trivial terminating program. Note also that the way of inference (logical consequences) can and does change (e.g. ZF vs ZFC). () I may be misusing the word model here -- it is usually defined (at least to things I work with) as the interpretation of a theory that satisfies all theorems (inferences). () constructive proofs yes. No flame wars please :)