Habits of highly mathematical people

Concerning example 2: If you want some 'common sense' property to hold, why don't you specifiy it then?

>I'm not a mathematician and was miserable at math in school, but I apply these habits in the business world every day. They help me cut through a lot of crap that comes from other people's sloppy/lazy thinking.

That's possible. The OP does not claim that the traits he listed apply if and only if you are a mathematician. It's certainly possible that you hone these skills without being mathematically inclined.

I do contest, however, that these habits apply to all "highly" rational people. I know plenty of people who are rational (or so are they deemed to be by their friends and colleagues) yet decidedly do not possess some of the listed habits, especially "scaling the ladder of abstraction" and "Being wrong often and admitting it".

On "being wrong often and admitting it," based on my experience, it's not that you become modest as a result of studying math. By studying math, you realize how prone your mind is to arriving at a faulty, unproven conclusion, and you learn to be more skeptical of your own thinking and become more at ease with being "wrong", allowing yourself to let your intellectual guard down.

-- Here, I think you've distilled the basic assumption of the op and I would claim someone with enough exposure to mathematics will know this claim is wrong.

Informal arguments may indeed be carefully and even exactly argued but they nearly always involve implicit emotional appeals, appeals to unstated common assumptions and definitions and so-forth.

Mathematics is different - the arguments are absolute, given the assumptions is can be demonstrated that the conclusions follow, even to the point of the proofs being mechanically verifiable. The reason someone making mathematically assumptions often has to admit their wrong is the arguments have follow a determined series of steps whereas someone with a strongly held, strongly articulated belief-system about the ordinary world can have a plausible answer to every objection but with some portion of those answers being appeals to commonly held beliefs in society.

- "if and only if", versus mere implication

- TFAE, as in, the following are equivalent ways of establishing some property

- Defining a relation, say R, between entities, and establishing properties about R, like transitivity.

- Defining notations, labels and symbols for certain quantities or relationships within a problem.

You typically first learn these idioms when doing proofs or reading lots of math and trying to summarize the ideas in it. Some of them are also aligned with the more abstract side of CS (unsurprisingly).

Someone who has had a little experience with these habits can make a real contribution to clear thinking. How many times have you stood at a whiteboard discussion where people repeatedly talk about "the properties that this object inherits from its parent" (or similar) -- but without just inventing a notation for that thing?

Mathematical definitions are usually abstractions of concrete observations, but they aren't the observations themselves. This is why coming up with a mathematical definition often requires more work than coming up with any other kind of definition.

> What is "1"? The idea of "1" can only be understood in terms of experience

1 is very rigorously defined: As a natural number, it's the successor of 0. As an integer, rational, real or complex number, it's the result of mapping the natural number 1 into Z, Q, R or C in a suitable way (compatible with the semiring structure of N).