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Why math is painful to read

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Re: Why math is painful to read

#71

Post by Xcelerate »

nn2 wrote: post #12

Really good point. That always annoys me with most formulas. I have to dig back several paragraphs (with no scope/indexing tool) to figure out what the individual variables mean. And a lot of papers do not even bother to explain some variable but just assume the reader already knows it. One of the earlier commenters made this mistake very prominently enthusing over Maxwell's equations. Yes of course the formula looks…

I agree with you. A book that aims to TEACH should provide the most common form of the equation and provide detailed annotations of what everything is. (Wikipedia should be like this as well since I know no one is using it as a reference).

For a REFERENCE, you can just list the equations in their most commonly used form.

Re: Why math is painful to read

#72

Post by gfodor »

pash wrote: post #64

No doubt about it, mathematical notation can be difficult to read. It's terse, conceptually dense, operators are commonly overloaded, etc., etc. But that's not why it's hard to read. Math is hard to read, I think, because math is hard: (1) Some core mathematical concepts are fundamentally difficult. Notions like infinity (and different kinds of infinity!) are entirely foreign to our daily experience. (2) In addition,…

To your point on naming, the core of Hungarian Notation's thesis is that you can never get a name right, due to the creeping in of unfounded assumptions due to past experiences with that name, so you're doing more damage by trying. So, create a unique 3/4 letter nonsense tag (and define it), and get on with your life. Once you do this, it's crystal clear where the gaps in your understanding are, since if it looks like gibberish, there's a gap. And when you understand it, believe it or not, it doesn't look like gibberish. (Hungarian as originally defined, ie Apps Hungarian, not the horrible Systems Hungarian that came out of Programming Windows.)

It's a powerful yet under-appreciated idea. Of course the proof of the pudding is in the eating and so it does seem Hungarian takes it too far due to it's lack of success stories. But I can't help but think if that thread of an idea were not squished so resolutely that we'd be in a better place than we are now with the problem of naming.

Re: Why math is painful to read

#73

Post by coolSCV »

aroberge wrote: post #56
forrestthewoods wrote: post #44

Earlier quoted context omitted.

I couldn't possibly disagree more with your post. :) The language of math requires exact precision which makes extracting what the series of abstract symbols "mean" incredibly difficult. Trying to learn new mathematical concepts from wikipedia is damn near impossible. Here are two examples. b-spline: http://en.wikipedia.org/wiki/B-spline spherical harmonics: http://en.wikipedia.org/wiki/Spherical_harmonics Trying to…

Wikipedia is not designed to be a pedagogical repository; arguing that the mathematical language sucks because a terse summary in wikipedia is not enough to understand the material is missing the point. Would you be able to calculate the probability of finding an electron within a certain distance from your pretty pictures? I would be able to do so fairly easily from those equations. What about computing special func…

> Wikipedia is not designed to be a pedagogical repository; arguing that the mathematical language sucks because a terse summary in wikipedia is not enough to understand the material is missing the point.

Nicely worded. I think the real issue is that many people want mathematics to be easy to read and understand. Unfortunately, much of mathematics is not easy to read nor is it easy to understand regardless of how it is written. Mathematics, like many other things, often requires a lot of intellectual scaffolding to be built from the bottom up. There is no shortcut to the top.

Re: Why math is painful to read

#74

Post by bane »

Someone wrote: post #47
forrestthewoods wrote: post #44

Earlier quoted context omitted.

I couldn't possibly disagree more with your post. :) The language of math requires exact precision which makes extracting what the series of abstract symbols "mean" incredibly difficult. Trying to learn new mathematical concepts from wikipedia is damn near impossible. Here are two examples. b-spline: http://en.wikipedia.org/wiki/B-spline spherical harmonics: http://en.wikipedia.org/wiki/Spherical_harmonics Trying to…

For the sake of an argument, let's assume you are right, and the language of math sucks. Then, we have had centuries of mathematicians inventing ever changing notations and ending up with a notation that sucks. If so, those mathematicians must be extremely stupid. Surely persons of average intelligence must be able to come up with better notations, and find proofs for stuff that is out of reach of those morons? I do…

I agree, it's a fun exercise to try and come up with a "better" form of notation that works in the general case. I think that most people will find that modern mathematics notation is really good stuff.

Re: Why math is painful to read

#75

Post by amplitwist »

Consider the solution to the quadratic equation

x = (-b ± sqrt(b^2-4ac))/(2a)

What is x? What is a? The wonderful thing about math is that it doesn't matter. x could be length, dollars, area, volume, mass, potatoes, lines of code, chickens, or electron-volts. But in the general case, it represents a number. Why would anyone consider it an improvement to write

Given that number1 answer^2 + number2 answer + number3 = 0, answer = (-number2 ± sqrt(number2^2-4 number1 number3))/(2 number1).

The extra characters convey no additional semantics. Even when presented with an equation where things have specific units, you can usually mentally figure out the units of everything else, due to conventions, previous definitions, and mental dimensional analysis.

In my experience, the notation isn't so much of an issue as knowing definitions, closely followed by being able to translate those definitions into intuition. You must be able to remember them exactly, otherwise nothing makes any sense, and you must have an intuition for what they mean, otherwise you'll never get anything done. Consider some terms from a recent talk I attended that was outside my immediate areas of expertise: "solvable Lie group", "left-invariant metric", "upstairs", "double cover". I was able to understand the main idea of the talk, but my full understanding of the talk was sunk by not knowing the definition of left-invariance [1].

Since math is inherently abstract, it is hard, and there is no substitute for the hard work necessary to acquire an intuition for it. When doing high-level math, it is necessary to have a rigorous intuition for the subject, where you are able to intuitively see a path to a proof, and then are able to translate that intuition into a suitably rigorous argument.

[1] To illustrate my point that knowing definitions precisely is one of the keys to understanding math, here's the definition of a left-invariant metric. Let G be a Lie group with metric , L_g be left multiplication, and L_g^* denote the pullback of L_g. The metric is said to be left-invariant if L_g^* = for all u,v in G. It makes no sense unless you know what metrics, Lie groups, left multiplication, and pullbacks are, and you'll only shoot yourself in the foot if you can't define them precisely.

Re: Why math is painful to read

#76

Post by michaelochurch »

davegauer wrote: post #34
michaelochurch wrote: post #10

Disagree strongly. Math is often the easiest and simplest thing that works for the class of problems it exists to solve. Imagine trying to explain singular value decomposition without the notion of a matrix. No group theory, no well-formed concept of a linear transformation or function . You wouldn't be able to do it. No one even had such ideas before generations of mathematical machinery had been built. I've come to…

I think you make a really good point about the power of mathematical notation and conventions. However, I completely agree with the article. Do mathematical formulas have to use Greek letters rather than useful variable names like "distance" or "speed"? Does C's syntax actually allow you to express something you can't in Python? Or is it more terse for historical reasons? (And never mind that it's a good idea to use…

In math, you have a trade-off between entropy (in the information-theoretic sense), rigor, and size of the document (e.g. proof). Entropy loses, so you get something extremely terse.

This is one of the hardest things about reading math. It's not the notation. You can pick that up pretty quickly. It's that every character counts: math is high in entropy. We're not used to that. In fcat, if you pmterue the inenr ltertes of tyicpal wtterin psore at rnoadm, msot pleope can read it at csloe to nmroal seepd. However, in math, one has to pay attention to every jot, because y-hat (ŷ) means something different from y.

Re: Why math is painful to read

#77

Post by SatvikBeri »

Negitivefrags wrote: post #52
Someone wrote: post #47

Earlier quoted context omitted.

For the sake of an argument, let's assume you are right, and the language of math sucks. Then, we have had centuries of mathematicians inventing ever changing notations and ending up with a notation that sucks. If so, those mathematicians must be extremely stupid. Surely persons of average intelligence must be able to come up with better notations, and find proofs for stuff that is out of reach of those morons? I do…

It's not that mathematicians are stupid, they just that they have backwards compatibility problems. The notation of mathematics is centuries old. It's very hard to take notation away from the set that must be learned, but easy to invent more. Eventually old constructs can fall out of favor, but that is a long and slow process.

This is hardly true. Mathematical notation changes all the time-in fact, inconsistency is a significant problem, because Mathematicians love to experiment with notation, definitions, etc.

Seriously, just look at a Mathematical logic book from 50 years ago vs. one today. You won't even recognize half the words, let alone the symbols.

Re: Why math is painful to read

#78

Post by j2kun »

So this article is just the author complaining that he has no practice in reading mathematics. Would you be surprised if someone who never written a computer program before couldn't understand a complicated dynamic programming algorithm? Of course not, they have no practice or background!

And for the record, computer programming is much more precise than mathematics. Most people here don't do mathematics, so they don't realize how many implicit identifications we make between objects which are very very different. In programs, correct types are tantamount to the success of a program. In mathematics, we will readily identify rings with topological spaces and never think twice. Moreover, we need to readily be able to change the rules by which we consider two different things to be identical on the fly. It is simply a different way of thinking that programmers aren't used to.

Re: Why math is painful to read

#79

Post by j2kun »

dxbydt wrote: post #23

Frankly, there are 2 kinds of programmers. People who do this: val result = directProduct(cyclicGroupOfDegree3, finiteAbelianGroupOfDegree7) and the second kind, people like me, who do this: // Compute the direct product of 2 cyclic groups val z = dP( cg1, cg2 ) You can easily guess that the 2nd kind are math majors. If my math professor started writing everything out in plain English like the first example, he'd nev…

I guess by degree you mean order? And by finiteAbelianGroupOfDegree7 you mean cyclicGroupOfDegree7. Of course, you should always use theorems to keep your notation consistent. :)

Re: Why math is painful to read

#80

Post by j2kun »

davegauer wrote: post #34
michaelochurch wrote: post #10

Disagree strongly. Math is often the easiest and simplest thing that works for the class of problems it exists to solve. Imagine trying to explain singular value decomposition without the notion of a matrix. No group theory, no well-formed concept of a linear transformation or function . You wouldn't be able to do it. No one even had such ideas before generations of mathematical machinery had been built. I've come to…

I think you make a really good point about the power of mathematical notation and conventions. However, I completely agree with the article. Do mathematical formulas have to use Greek letters rather than useful variable names like "distance" or "speed"? Does C's syntax actually allow you to express something you can't in Python? Or is it more terse for historical reasons? (And never mind that it's a good idea to use…

Mathematical formulas with real-world interpretations can use those variable names. The problem occurs when the thing you're trying to express has no analogue in the real world. What variable name should I give to a reduced Noetherian subscheme of an arbitrary scheme? If you think "reducedNoethereianSubscheme" is a better variable name than X, then you'll love writing the notation for a function on it:

f: reducedNoethereianSubscheme -> reducedNoethereianSubscheme

It's surprising how everyone thinks they can make math better, but nobody even knows about what it's like to begin with.

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