

A mathematical formalisation of dimensional analysis - mjn
http://terrytao.wordpress.com/2012/12/29/a-mathematical-formalisation-of-dimensional-analysis/

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evincarofautumn
This article appears to be describing a type system without making use of
anything type-theoretical. Maybe it’s just my inexperience with mathematical
language, but the presentation seems to suffer badly as a result. The system
in question is actually rather simple.

“There are several reasons why it is advantageous to retain the limitation to
only perform dimensionally consistent operations. One is that of error
correction: one can often catch (and correct for) errors in one’s calculations
by discovering a dimensional inconsistency, and tracing it back to the first
step where it occurs.”

This is basically describing the “stack trace” you get when typechecking a
program.

“By performing dimensional analysis, one can often identify the form of a
physical law before one has fully derived it.”

This is tantamount to saying that it’s possible to glean an implementation
from a type, which is true—the implementations of all total functions with a
given type are (I think) recursively enumerable.

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banachtarski
It's possible, because you aren't a mathematician, that you aren't aware of
who Terrence Tao is. He is one of the most celebrated mathematicians of our
time, and a prolific one at that. His goal is emphatically _not_ to explain
what dimensional analysis is, but to provide a basis for a mathematical
formalism for it that does not fully exist yet.

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gliese1337
That doesn't seem to be any sort of refutation of the given complaint. Sure,
Terrence Tao is a great mathematician, and he may trying to do something that
type theorists _haven't_ done yet, and we can reasonably give him the benefit
of the doubt and assume that, even though he didn't mention it, he may well be
familiar with type theory and left out reference to it on purpose.

Nevertheless, for some audience, in which I am included, the connection to
type theory seems blatantly obvious and it would've been a better article for
us had it been presented in that light.

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banachtarski
It's not the same as type theory because the relationships between physical
units are actually tied by real mathematical operations. A float and a string
do not combine in any sensible way (aside from type-casting, which is entirely
different). This suggests that type theory as you are familiar with isn't as
connected as you might think.

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evincarofautumn
There are plenty of systems making use of the algebraic operations of product
(tuple) and sum (disjoint union) on types. These are regularly extended with
notions of exponentiation, division, subtraction, complement, union, and
intersection. There is little formal reason why an algebra of types shouldn’t
include even more interesting operations.

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mturmon
I know it's a side point, but I loved the comment:

"...explaining for instance why in any trigonometric identity such as

    
    
        sin(x+y) = sin(x) cos(y) + cos(x) sin(y)
    

the number of odd functions (sine, tangent, cotangent, and their inverses) in
each term has the same parity."

I never thought of it that way. I always converted to exponential notation to
derive them, but you could use this units approach to provide what you needed.

