Mathematical notation is ambiguous and therefore unsuited to parsing as a computer syntax. s(x + y) can represent scalar multiplication, or function application. Mathematica only uses s(x + y) for scalar multiplication and s[x + y] for function application to avoid this problem. Sin[alpha], etc. is just a symptom of its consistent naming convention for built-ins. The use of Capitalised names frees you to create your own function names in lowercase without fretting about namespace conflicts.

I've downloaded Mathics onto my Mac, unpacked it, installed it and tested it without issue. The web interface works well too. I'm very impressed with this software and appreciate all their efforts, especially what appear to be well commented sources. To complain that a better notation could have been used is to completely miss the point. This is for people who would like to use Mathematica syntax but can't afford the licence. I've got four books on Mathematica, but can't remotely afford to purchase it.

How about $300 for the home version?

"Mathematicians don't use square brackets for functions"

Not true at all:

1) Operators acting on a distribution: E[X], Var[X], Cov[X,Y]

2) Square brackets are used in nested parentheses to avoid confusion: [1(2 + 3) + 4]/5

3) Functionals: L[f] for lagrangian

The main reason why [] isn't used as frequently as () is because most domains interpret the square brackets to mean something special. For example:

1) [x] is often used to denote floor(x)

2) [i=j] is used to denote the kronecker delta

On point (2) -- In four years of being a math major, I don't think a single professor ever used brackets in that situation, only additional parentheses.

I'm inclined to agree that it's use would be uncommon, I've not encountered it in 30 odd years of reading papers and talking math (or rather I don't recall noticing it's use).

What I must point out is you make a weak case based on myopia- for example; in 15 years of working with tens of thousands of hours of high resolution radiometrics I've never once met and talked to a dendrochronologist who's looked at the radiometric signature of individual bands of Japanese pine trees.

It's my understanding that such beasts do exist.

Another math major here (in my final year). My professors routinely use square brackets after the second or third level of bracket nesting.

I think it must depend on the professor/university.

It's common in engineering. I like it because it makes matching parenthesis much easier.

This is a rather superficial objection. Using square brackets for function application frees up juxtaposition for multiplication (without introducing ambiguity like in standard mathematical notation), so that's just a tradeoff. The use of capitals is consistent and helps distinguish builtin functions from user-defined functions and variables. These particular conventions are really things you get used to quickly, unless you're deliberately trying to be annoyed by them.

It uses Sympy and SAGE under the hood, you'd probably prefer their syntaxes.

Personally, I like that Mathematica uses capitals for functions: it means that the entire namespace of lower-case variables is free - unlike in Sage, MATLAB etc... In fact, this is how you're supposed to name your variables in Mathematica.

For one, Mathematica tries hard to keep its syntax consistent, even if it has to break backward compatibility sometimes. Naming convention in Matlab is often inconsistent (even though Matlab also at times changes things to break backward compatibility).

Also Matlab's documentation is often not complete, frequently requiring Google searches. I have rarely needed that with Mathematica.

My core reasons to use Matlab at times are:

1. More toolboxes, including those from third-parties 2. More team members familiar with Matlab than Mathematica

You can either have (a) consistency or (b) convention. Most people would jump at the opportunity to ditch (b), re-invent the wheel, and establish (a) just for the cleaner semantics.

Any mathematician will agree: mathematics is invariant under change of notation.

But sometimes the right notation can make things simpler and easier to understand. (e.g. Einsteins Four-Vectors)

But mathematics communication is not, hence new fields coming to life when a great notation is introduced.