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.

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.

You just described the opposite of intuition.

Intuition 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).

A chess master can intuitively tell when mate is three moves away. They weren’t born with that intuition, though. They gained it over time.

You aren't born with all of your intuitions, you learn most of them. This includes mathematical intuitions.

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."

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

From 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.

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 :)

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.

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.