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Comment by maroonblazer | Hacker News Reader

Couldn't this also be titled "Habits of high rational people", with mathematics simply being one application?

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.

>Anyone who has gone through an undergraduate math education has known a person (or been that person) to regularly point out that X statement is not precisely true in the very special case of Y that nobody intended to include as part of the discussion in the first place. It takes a lot of social maturity beyond the bare mathematical discourse to understand when this is appropriate and when it’s just annoying.

I don't disagree but would argue that the far more common problem is people - not just mathematicians, mind you - not considering definitions enough which ultimately leads to confusion and/or misunderstandings and consequently additional, unnecessary, cycles spent in discussion about "what do you really mean?"

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(I studied math and computer science. Walked out of a math Ph.D. program before starting one)

>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.

My small issue is that he attacks empirical science at its periphery (statistical muckery, mostly going on in politicized debates, pharma, etc.) rather than it's core. My experience in biology is quite similar to what he describes with people more-or-less dispassionately seeking the truth. It's a 'messier' field, so perhaps there's more room for personal pride, but I have my doubts about that. Anyone who holds hypotheses dear is a poor mathematician or biologist, just the same.

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.

I find it ironic how OP's claim is easily ruled out by basic mathematical argument.

I do think there is something unique about training in the 'hard' analytic disciplines (math, computer science, physics, etc.). In particular, one learns to be wrong often, and that the result of one's hard-fought intellectual effort is often objectively incorrect. At the very least, math accelerates the process of discovering this, and so is a very good teacher of intellectual humility.

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.

The "hard analytical disciplines" require understanding the fundamentals of whatever they work with, in order to process those fundamentals together into a useful effect.

Getting meaning across via words is fundamentally the same process.

"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.

Having done graduate work in both computer science and philosophy (and having a healthy respect for both), I agree with this assessment. It's part of what I was trying to articulate. Although I think some of the philosophers (particularly the historians of philosophy) that I've met have been some of the most intelligent, rational, and well-spoken people I've met, I think there's a keen difference between the critical thinking abilities of the computer science undergraduates vs. the philosophy undergraduates that I've seen graduate from a university/college, and I chalk a lot of this up to learning to be wrong very quickly in formal disciplines.

That in my experience is a huge differentiating factor between analytical people and everyone else. Most people look on admitting failure as a great horribleness to be avoided at all costs. at my current corporate job I've often been looked at as insane for being open about such things.

>The fact that you jumped to 'other people's sloppy thinking' as your first example is somewhat telling.

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'.

> I don't disagree but would argue that the far more common problem is people - not just mathematicians, mind you - not considering definitions enough which ultimately leads to confusion and/or misunderstandings and consequently additional, unnecessary, cycles spent in discussion about "what do you really mean?"

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.

Everyone on this thread, from the original article to you seems to agree that sometimes nitpicking is good and sometimes it isn't. But for every minute that has been wasted ignoring and even responding to pedants, days have been lost by people undertaking projects that they never bothered to define.

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).

> I'm not a mathematician and was miserable at math in school

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

True, but if you're not good at school math, I believe there is little hope you can be good at the style of math that mathematicians do...

I'm not sure about this one. I studied CS and was kinda forced to learn university-level maths and I found that it differs both on the pedagogical level (different ways to motivate students, I preferred the ones in the university) and on the logical level, when for the first time in my life I was actually EXPLAINED the terms we operated on.

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

I didn't quite have this experience in my engineering math courses. Admittedly, the other math courses were much better about this.

I wouldn't bet on that, they are really different.

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

Orthogonal?

Sorry. I meant antagonistic.

High School math (often) is all about memorizing a few tricks from the analysis of real numbers and maybe elementary number theory/combinatorics. I think there'd be more mathematicians if people got to see what math is like in actuality. I fell in love with math when I discovered that it's an actual body of knowledge complete with definitions, concepts and proofs just like any other discipline. Compared to that high school math (loosely) could be said to be a bunch of party tricks with no rhyme or reason to them.

This is not true at all. I know many people in mathematics that are below average when it comes to arithmetic, some to the point of suffering from dyscalcula. I myself am probably in at least the bottom quartile of arithmetic ability, and there have been quite a few historic examples of top mathematicians with a similar "problem", Hilbert being a common example.

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 :)

Hilbert's supposed dyscalcula is a myth..unless you have a source we haven't seen?

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.

You're right, it looks like that is a myth (I remember seeing it on this math stackexchange question[1], which is actually a great example of some mathematicians' opinions on the matter).

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...

I suspect the relationship between school maths and what mathematicians do is a little bit like the relationship between touch typing and programming, in the sense they're separate skills but being good at one of those skills reduces friction with the other.

Math is taught so poorly that I am rather skeptical that this generalizes. I was fairly terrible at math ( although I scored well on tests and was able to struggle through it ) until I was in a junior year course that was used for discrete math, which went into proofs, and from that point on, it was simply no longer a problem.

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.

Unless you have natural intuition or memorizing skills that make high school math easy for you, it's for many kids a negative experience due to the combination of a bad curriculum and teachers who are not skilled pedagogically. School math is the number one reason why so many kids learn to dislike it, and programming is the field that made me love and appreciate math for what it really is, while I didn't like it too much in school. With better teachers and curriculum I'm sure I would have appreciated it much earlier.

Nice spot on. The described qualities would also apply to philosophers, logicians, etc.

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

Philosophers? Using precise definitions? Maybe some philosophers do... But not most.

I took parent's comment to mean it's about philosophers who deal with logic. You may recall that logic as a tool free of opinion originated with Aristotle, and the first thing that comes to my mind in this context is Principia Mathematica.

"If you wish to converse with me, define your terms." -Voltaire

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.

Styles vary between schools and the one called "analytic philosphy" is big.

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.

The much larger problem is that philosophers can't agree on one (or at least few) axiomatic base(s) they want to base their argumentations on. Thus lots of philosophical discussions are rather of the kind "we have different axioms and thus come to different conclusions".

Oh, they use them. The problem is that each philosopher has a different precise definition that they feel is natural and matches what we "really mean" when we ask deep questions. Since most arguments and conclusions are phrased fuzzily, you end up with the following situation:

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.

I strongly disagree with the claim that philosophers fit this description. As a mathematician, it often seems to me that 90% of philosophy is spent arguing about things with extremely loose definitions. As a result, you can argue from the same starting point and come to completely different conclusions (as separate philosophers often do), because the starting point was already self-contradictory for some interpretations of the wording. As soon as you clearly define what you are talking about the problem usually becomes trivial.

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.

Define "loose". Edit: Jokes aside, nothing is really "strictly-defined" as we would like it to be since definitions (even the most rudimentary ones) are based on -what we might call- statistical sets of observations. We come to the conclusion that an apple is a round object because we've seen an apple from multiple angles. Math accepts axioms, but those axioms are based on accepting definitions that are rudimentary observations. What is "1"? The idea of "1" can only be understood in terms of experience - hardly "rigorously defined".

> definitions (even the most rudimentary ones) are based on -what we might call- statistical sets of observations

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).

> 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. ... 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 successor" of "0" means nothing because "successor" isn't defined

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.

One of the problems of philosophical discourse, imo, is that it seems to be impossible to give mathematically rigorous definitions of concepts like 'good' or 'moral' or even 'knowledge' that somebody won't be able to disagree with by counter example.

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.

> 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. 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.

In a sense you do. Often when we are giving a definition we are intending to capture a particular idea. Sometimes the definition we give captures too little, or too much. If you find that out early enough then you can change your definition to better match what you intend.

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 }

So the net result is that we have the two definitions: "connected" and "pathwise connected."

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.