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Mathematical Notation Is Awful

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Re: Mathematical Notation Is Awful

#31

Post by paxcoder »

chilie wrote: post #13

These are just issues that someone unfamiliar with a field would face. None of them are problems for those of us in the field. First, the expectation thing. He's using a special case, E(X), and complaining that the more general case doesn't follow the general case. It's like saying "Well the plural of mouse is mice but the plural of house isn't hice!". The general definition of expectation (for a discrete probability…

So why are years of studying math insufficient for one to be "familiar" with it?

Re: Mathematical Notation Is Awful

#33

Post by catnaroek »

bjornsing wrote: post #29
catnaroek wrote: post #27

Earlier quoted context omitted.

LaTeX... now that is unreadable. Traditional mathematical notation is just fine.

Have you seen x86 machine code? ;P

Long ago, when I took a computer architecture course, I learnt the 8086's instruction format. I don't remember a single thing about it.

Re: Mathematical Notation Is Awful

#34

Post by jamessb »

I think the fundamental source of confusion might be due to not understanding the concept of a random variable. In his example, X is a random variable; the expectation E[X] is an functional applied to its probability mass function. Given this, it should not be surprising if it seems to behave differently to "any other function, in math".

If you understand this, I think the notation is natural:

We have a random variable X, which takes value x_i with probability p(x_i). Thus, the random variable X^2 will take value (x_i)^2 with probability p(x_i).

Given that the expection for the random variable X with p.m.f p(x_i) is defined as E[X] = \sum x_i p(x_i), it should be clear that to obtain the expectation of any random variable we must sum over the product of (value) and (probability of that value). It should also be clear that this gives E[X^2] = \sum (x_i)^2 p(x_i)

I'm confused by his comment that: p(xi) isn't, because it doesn't make any sense in the first place. It should really be just PXi or something, because it's a discrete value, not a function!

The probability mass function is a function: for a given value, it gives the probability that the the discrete random variable takes that value. To calculate the expectation we use the values obtained by evaluating the function at discrete points, but what else could we do?

Re: Mathematical Notation Is Awful

#35

Post by posterboy »

chilie wrote: post #13

These are just issues that someone unfamiliar with a field would face. None of them are problems for those of us in the field. First, the expectation thing. He's using a special case, E(X), and complaining that the more general case doesn't follow the general case. It's like saying "Well the plural of mouse is mice but the plural of house isn't hice!". The general definition of expectation (for a discrete probability…

Unfamiliar is a weasel word. It's a No-True-Scottsman. And I don't mean, that it is entirely wrong, I am saying you are unjustly putting a limit to whom you deem worthy for the field. There is no need for imprecision other than speed breaking things while you go.

> "Well the plural of mouse is mice but the plural of house isn't hice!"

Are irregular word forms necessary or essential? Probably, but I doubt you could explain why. Hence you are not qualified to ridicule anyone. It's a perfectly valid complaint, IMHO, but probably only loosely related to the math example, which I can't be bothered to follow at the moment.

Re: Mathematical Notation Is Awful

#36

Post by lifthrasiir »

chilie wrote: post #13

These are just issues that someone unfamiliar with a field would face. None of them are problems for those of us in the field. First, the expectation thing. He's using a special case, E(X), and complaining that the more general case doesn't follow the general case. It's like saying "Well the plural of mouse is mice but the plural of house isn't hice!". The general definition of expectation (for a discrete probability…

In the other words, E(...) has a hidden lambda there, and the fully consistent usage would be E(X -> X^2) (instead of E(X^2)) and so on. The covariance thing would be E((X,Y) -> X*Y) with an implied domain being a Cartesian product of that of X and Y. Of course we humans can easily infer the domains, and writing explicit domains every time is not efficient.

Re: Mathematical Notation Is Awful

#37

Post by [deleted] »

k__ wrote: post #14
klibertp wrote: post #8

Earlier quoted context omitted.

"We call something a Ring to avoid saying" Why a "Ring" and not an "asda2+"?

the problems I had were mostly with good names. So names that are 1. already used in everyday life AND 2. really fit the concept. These words have definitions in my head that I use often and now math uses them differently. Like 'Menge' (german for Set) which has 'almost' the same definition in math and everyday use. I like the physics approach more, gluons, quarks, protons, etc. these are all words that never had any…

[deleted]

Re: Mathematical Notation Is Awful

#38

Post by [deleted] »

k__ wrote: post #14
klibertp wrote: post #8

Earlier quoted context omitted.

"We call something a Ring to avoid saying" Why a "Ring" and not an "asda2+"?

the problems I had were mostly with good names. So names that are 1. already used in everyday life AND 2. really fit the concept. These words have definitions in my head that I use often and now math uses them differently. Like 'Menge' (german for Set) which has 'almost' the same definition in math and everyday use. I like the physics approach more, gluons, quarks, protons, etc. these are all words that never had any…

[deleted]

Re: Mathematical Notation Is Awful

#39

Post by adrianratnapala »

jamessb wrote: post #34

I think the fundamental source of confusion might be due to not understanding the concept of a random variable. In his example, X is a random variable; the expectation E[X] is an functional applied to its probability mass function. Given this, it should not be surprising if it seems to behave differently to "any other function, in math". If you understand this, I think the notation is natural: We have a random variab…

You are right, and the deeper source of the authers beef seems to be that he wants mathematics to be about textual rules. Now that's a fine thing in a computer language -- and there are philosophers who say mathematical truth is no more than computation. But mathematical notation is for humans to talk about things that have meanings that humans can understand.

The fact is that E[X^2] is natural way of an important concept. Whereas "\sum \sum (x_i)^2 p(x_i^2)", need mean nothing at all (especially is $p$ is not defined over all the $x_i^2$).

Re: Mathematical Notation Is Awful

#40

Post by pwdisswordfish »

lifthrasiir wrote: post #36
chilie wrote: post #13

These are just issues that someone unfamiliar with a field would face. None of them are problems for those of us in the field. First, the expectation thing. He's using a special case, E(X), and complaining that the more general case doesn't follow the general case. It's like saying "Well the plural of mouse is mice but the plural of house isn't hice!". The general definition of expectation (for a discrete probability…

In the other words, E(...) has a hidden lambda there, and the fully consistent usage would be E(X -> X^2) (instead of E(X^2)) and so on. The covariance thing would be E((X,Y) -> X*Y) with an implied domain being a Cartesian product of that of X and Y. Of course we humans can easily infer the domains, and writing explicit domains every time is not efficient.

Well, formally a random variable is already a function which assigns values to elements of the probability space (outcomes). The expected value is just another name for an integral over this space. When the probability space is discrete, "the integral over a probability space" is just another name for a weighted sum. The domain is always the same: it's the probability space.

The real bad notation (which is employed here, actually) is f² meaning (x ↦ f(x)²) while at the same time f⁻¹ means the inverse function of f instead of (x ↦ f(x)⁻¹).

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