>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.

Basically, it's a basic vs. applied science critique; basic science is quite brutal, and you will be wrong very often. Usually there's little at stake regarding which conclusion you come to, so long as you're investigating something interesting. Someone trying to create effective medicine is in a very different situation.

> there’s very little fame outside of the immediate group of people you’re talking to

This is very true in most basic science fields. There are about 4 or 5 other labs in the world that are familiar with the details of my sub-field, and this is true for most of my colleagues. Mathematics is certainly not privileged in this sense.

The fact that you jumped to 'other people's sloppy thinking' as your first example is somewhat telling. The point is that your whole perspective can shift from thinking that you're smarter than other people and need to correct their mistakes, to understanding how to collaborate with other smart people over difficult problems while everyone is making and correcting mistakes.

You'll have to take my word for it but I'm under no illusion as to my smartness relative to others. More often than not I'm the dumbest person in the room. I learned the hard way that I have plenty of blind spots and I'm the first person I should be checking for 'sloppy thinking'.

Getting meaning across via words is fundamentally the same process.

-- 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.

It's all a difference in context and environment. When talking through the specification of a product your building and prerequirements for it, nitpicking the definitions will save you pain and money.

On the other hand, nitpicking whether someones analogy in covers exact 100% cases of the issue while ignoring the point of the conversation will just make you look like an arsehole in front of the group of people. And the latter is still painfully common on HN as well. As the article says - understanding when some detail is important for the current conversation is an important social skill.

And if you have ever worked with customers to elicit requirements, then you will know that the human race has far more fuzzy thinkers than it does logical pedants. (And of course some of the worst nitpickers are fuzzy thinkers too).

Maths in school is not really like the maths that mathematicians do.

There's a huge difference between reasoning about abstract statements and arithmetic calculation, even when it comes to abstract statements about arithmetic. I'm not saying there's no correlation, but the conditioned probability in the direction of bad at arithmetic => bad at mathematical reasoning is probably much lower than you'd expect.

At the very least, I'd encourage those that are bad at arithmetic to not assume they will be bad at mathematics or data science. If I had, I'd be much worse off and less happy than I am now with math in my life :)

It's similar to the Grothendeick story -- a mathematician making a simple mistake -- that get exaggerated to "mathematicians can't do arithmetic." https://en.wikipedia.org/wiki/57_(number)

Another common myth was that Einstein was bad at math, because he (and others) had trouble in school because the material was too easy and he quarreled with teachers, and "failed" tests for early college admissions administered in a foreign language. https://www.washingtonpost.com/news/answer-sheet/wp/2016/02/...

When a mathematician confesses being bad at calculation, they don't mean "couldn't pass high school", they mean compared to their elite peers.

There's a wide margin between "couldn't pass high school" and "had difficulty with math in high school". And I've known plenty who confessed that were quite explicit that was not what they meant (e.g. by comparing themselves unfavorably with their children).

[1]: http://math.stackexchange.com/questions/551074/are-all-mathe...

But because I was specialized towards mathematics, I had been ... "inconvenienced" in favor of "easier" courses for the non-specialists. I think there's a severely unexamined premise in that state of affairs - but I have to ask - would others be as math-phobic if they really knew how it worked?

The head of the math/sci department literally said "Calculus keeps the dumb people outta med school" - and he was only partly kidding.

The question is - is rigor, even partial, informal rigor ( if that even makes any sense ) of value to everyone or not? It's obviously painful. But it really helps with goal of not being a sucker.

University maths is much more logical than the school voodoo I had.

Some of the most blatantly off-putting characteristics of school math, like mindless repetition, are antagonical to real math.

I think a broader term would be, habits of highly "analytical" people.

For example, many of the moral dilemmas that get tossed around as interesting become trivial to answer once you give a mathematically rigorous definition to words like "good", "moral", "utilitarian" or whatever.

Yes, but the while the conclusions become trivial, actually coming up with those rigorous definitions is not.

I'm not going to provide examples because they are likely to start a flamewar. But if you search for words like "normative" or "core principle" in my comment history, you can see how difficult it is to actually come up with those rigorous definitions. The problem is that when you do, you either come up with unsatisfying arbitrary definitions, or else you wind up implying peripherally related conclusions you probably dislike.

Yes, I am definitively guilty of being "highly mathematical" (and also highly economical and philosophical), and it pushes me to recognize incoherence in mainstream "thought".

> I usually take the presence of any proofs in my code as the cue to question my definitions.

http://stackoverflow.com/a/13241158/46571

What you sometimes see among philosophers is that even definitions that are the result of many iterations are still treated more like rules of thumb than precise definitions. Or they at least acknowledge that their definition is probably flawed, and proceed carefully.

Some proceed as if they are doing math.

I really don't understand this. You give a counterexample to a theorem; you don't give one to a definition. If someone disagrees with the definition and wants to use a different one, then they are talking about a different thing (even if they want to use the same English word to refer to each of them) and their conclusions can not be compared in any meaningful way.

Imo, you can't "proceed carefully" with a definition that is open to interpretation. If you could do so safely, then you know enough about how people could interpret your definition to form a more rigorous definition.

  >> that somebody won't be able to
  >> disagree with by counter example

  > I really don't understand this.
  > You give a counterexample to a
  > theorem; you don't give one to
  > a definition.

The classic example is "connected" from topology. Speaking very loosely, a set is "disconnected" is there is a "disconnection", which is a separation of the set into two pieces which are contained in disjoint open sets. Basically, the set is disconnected if it's made up of two (or more) pieces that can be divided by a "surface". A set is "connected" if there is no disconnection.

The problem is that there are sets we want to think of as not connected, and yet which satisfy the definition of connected as given above. Here's an example:

  X = { (x,sin(1/x)) : x in R, x>0 } u { (0,y) : -1 < y < 1 }

If that example had been thought of earlier, it's possible that the definition of "connected" might have been fixed earlier. So in a sense, X is a counter-example to the definition of connected.

Further reading:

http://planning.cs.uiuc.edu/node140.html

https://en.wikipedia.org/wiki/Connected_space

https://en.wikipedia.org/wiki/Connected_space#Path_connected...

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).

That's all nice and dandy for mathematicians, but such is still meaningless (from an overall perspective) because ultimately you end up with recursive definitions. "The successor" of "0" means nothing because "successor" isn't defined, unless you define it via dimensions (or something else). Once you try to define dimension or otherwise, you have to define dimension, which leads to defining counting, which leads back to defining 1. Appealing to experience is the only way to end the cycle.

That said, it's perfectly fine to talk about a system within itself, but it's applicability to anything real (which is what I'm concerned with - my apologies if that's not where you're going) is limited to whatever can be described in terms of reality.

The existence of a successor function is postulated, because it's an axiom of the theory of natural numbers. It's up to individual models of the theory to define this function concretely.

Fuzzy premise -> precise logic -> fuzzy conclusion

where each of those arrows leaves so much room for interpretation that you can build literally an entire subfield by arguing back and forth about what the most reasonable mapping from fuzzy to precise and back again might be, even if everyone agrees that the manipulations in between are rigorously correct.

But in general when I read philosophers disucssing social matters, I feel they are being too rigorous. Trying to cover all bases when the fuzzy nature of the subject is always going to make their abstractions leak.

They have to do this because they compete with other philosophers who will pick the nits.

Perhaps it depends on what you mean by "precise", but philosophical arguments do tend to start by defining one's terms or disputing the opponent's definitions, expressed or implied. When continental philosophers talk, that's the only part I can follow.

It's why economists are terrible hedge fund managers and not mathematicians. It's why all these script kiddie programmers at silicon valley are terrible.

Tell me ... when you are mathematical, you cannot be wrong in all worlds.