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Mathematics is hard for mathematicians to understand too

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Re: Mathematics is hard for mathematicians to understand too

#71

Post by youoy »

pathikrit wrote: post #51

I love math but the symbology and notations get in my way. 2 ideas: 1. Can we reinvent notation and symbology? No superscripts or subscripts or greek letters and weird symbols? Just functions with input and output? Verifiable by type systems AND human readable 2. Also, make the symbology hyperlinked i.e. if it uses a theorem or axiom that's not on the paper - hyperlink to its proof and so on..

Notation an symbology comes out of a minmax optimisation. Minimizing complexity maximizing reach. As with every local critical point, it is probably not the only state we could have ended at.

For example, for your point 1: we could probably start there, but once you get familiar with the notation you dont want to keep writing a huge list of parameters, so you would probably come up with a higher level data structure parameter which is more abstract to write it as an input. And then the next generation would complain that the data structure is too abstract/takes too much effort to be comunicated to someone new to the field, because they did not live the problem that made you come with a solution first hand.

And for you point 2: where do you draw the line with your hyperlinks. If you mention the real plane, do you reference the construction of the real numbers? And dimensionl? If you reason a proof by contradiction, do you reference the axioms of logic? If you say "let {xn} be a converging sequence" do you reference convergence, natural numbers and sets? Or just convergence? Its not that simple, so we came up with a minmax solution which is what everybody does now.

Having said this, there are a lot of articles books that are not easy to understand. But that is probably more of an issue of them being written by someone who is bad at communicating, than because of the notation.

Re: Mathematics is hard for mathematicians to understand too

#73

Post by gjulianm »

zwnow wrote: post #52
pathikrit wrote: post #51

I love math but the symbology and notations get in my way. 2 ideas: 1. Can we reinvent notation and symbology? No superscripts or subscripts or greek letters and weird symbols? Just functions with input and output? Verifiable by type systems AND human readable 2. Also, make the symbology hyperlinked i.e. if it uses a theorem or axiom that's not on the paper - hyperlink to its proof and so on..

I'd love getting rid of all the weird symbols in favor of clear text functions or whatever. As someone who never learnt all the weird symbols its really preventing me from getting into math again... It is just not intuitive.

I'm not sure that symbols are the thing actually keeping you away. Clear text functions might not be as clear, as it will be harder to scan and it will still contain names that you might not be familiar with. Those "weird symbols" are not there because people liked to make weird symbols. No one likes them, it's just that it makes things easier to understand.

Re: Mathematics is hard for mathematicians to understand too

#74

Post by sfpotter »

pathikrit wrote: post #51

I love math but the symbology and notations get in my way. 2 ideas: 1. Can we reinvent notation and symbology? No superscripts or subscripts or greek letters and weird symbols? Just functions with input and output? Verifiable by type systems AND human readable 2. Also, make the symbology hyperlinked i.e. if it uses a theorem or axiom that's not on the paper - hyperlink to its proof and so on..

Go ahead. Write a math paper with your proposed new notation with hyperlinks and submit it to a journal somewhere.

Re: Mathematics is hard for mathematicians to understand too

#75

Post by gjulianm »

abraxas wrote: post #64
Jensson wrote: post #61

Earlier quoted context omitted.

Those are used since it makes things easier, if you write everything out basically nobody would manage to learn math, that is how it used to be and then everything got shortened and suddenly average people could learn calculus.

There has to be a happy medium between the tersness of the current notation systems and the verbosity of code-like expressions. We just need to rethink this so more people can learn it. Math still stands a bit like writing did in ancient culture. It's a domain reserved for a few high priests inducted into the craft and completely inaccessible to everyone else.

I wonder why so many people are under the impression that the notation is what is keeping them away and if only the notation was easier then the underlying concepts would be clear. For example, if you don't know what the pullback of a differential form is, it doesn't matter if I write it in clear text or if I write the common notation φ^* ω.

> It's a domain reserved for a few high priests inducted into the craft and completely inaccessible to everyone else.

It's a domain reserved for people who want to learn it, and there's ton of resources to learn it. Expecting to understand it without learning it does not make any sense.

Re: Mathematics is hard for mathematicians to understand too

#76

Post by nyrikki »

abraxas wrote: post #64
Jensson wrote: post #61

Earlier quoted context omitted.

Those are used since it makes things easier, if you write everything out basically nobody would manage to learn math, that is how it used to be and then everything got shortened and suddenly average people could learn calculus.

There has to be a happy medium between the tersness of the current notation systems and the verbosity of code-like expressions. We just need to rethink this so more people can learn it. Math still stands a bit like writing did in ancient culture. It's a domain reserved for a few high priests inducted into the craft and completely inaccessible to everyone else.

The problem is that math is not some universal language, it is a broad field with various sub domains with their own conventions, assumptions, and needs.

Polysemy vs Homonymy vs Context Dependency will always be a problem.

There are lots of areas to improve, but one of the reasons learning math is hard is that in the elementary forms we pretend that there is a singular ubiquitous language, only to change it later.

That is why books that try to be rigorous tend to dedicate so much room at the start to definitions.

Abstract algebra is what finally help it click for me, but it is rare for people to be exposed to it.

Re: Mathematics is hard for mathematicians to understand too

#77

Post by zozbot234 »

kragen wrote: post #65
pathikrit wrote: post #51

I love math but the symbology and notations get in my way. 2 ideas: 1. Can we reinvent notation and symbology? No superscripts or subscripts or greek letters and weird symbols? Just functions with input and output? Verifiable by type systems AND human readable 2. Also, make the symbology hyperlinked i.e. if it uses a theorem or axiom that's not on the paper - hyperlink to its proof and so on..

Probably not. The conventional math notation has three major advantages over the "[n]o superscripts or subscripts or [G]reek letters and weird symbols" you're proposing: 1. It's more human-readable. The superscripts and subscripts and weird symbols permit preattentive processing of formula structures, accelerating pattern recognition. 2. It's familiar. Novel math notations face the same problem as alternative English…

> ... It's optimized for rewriting a formula many times.

It's not just "rewriting" arbitrarily either, but rewriting according to well-known rules of expression manipulation such as associativity, commutativity, distributivity of various operations, the properties of equality and order relations, etc. It's precisely when you have such strong identifiable properties that you tend to resort to operator-like notation in any formalism (including a programming language) - not least because that's where a notion of "rewriting some expression" will be at its most effective.

(This is generally true in reverse too; it's why e.g. text-like operators such as fadd() and fmul() are far better suited to the actual low-level properties of floating-point computation than FORTRAN-like symbolic expressions, which are sometimes overly misleading.)

Re: Mathematics is hard for mathematicians to understand too

#78

Post by worthless-trash »

nkrisc wrote: post #43
worthless-trash wrote: post #31

Earlier quoted context omitted.

> You don't have "the manual" of programming languages. " Well, we kinda do when you can say "this python program" the problem with a lot of math is that you can't even tell which manual to look up.

Someone not educated in programming would not know that a given text is Python source code.

Same problem, but unlike math notation, it is MUCH clearer and even my 10 year old newphew can tell python from javascript and C.

Re: Mathematics is hard for mathematicians to understand too

#79

Post by nkrisc »

worthless-trash wrote: post #78
nkrisc wrote: post #43

Earlier quoted context omitted.

Someone not educated in programming would not know that a given text is Python source code.

Same problem, but unlike math notation, it is MUCH clearer and even my 10 year old newphew can tell python from javascript and C.

I would wager that a vast majority of people on the planet could not distinguish Python from JavaScript from C.

Re: Mathematics is hard for mathematicians to understand too

#80

Post by kragen »

zozbot234 wrote: post #77
kragen wrote: post #65

Earlier quoted context omitted.

Probably not. The conventional math notation has three major advantages over the "[n]o superscripts or subscripts or [G]reek letters and weird symbols" you're proposing: 1. It's more human-readable. The superscripts and subscripts and weird symbols permit preattentive processing of formula structures, accelerating pattern recognition. 2. It's familiar. Novel math notations face the same problem as alternative English…

> ... It's optimized for rewriting a formula many times. It's not just "rewriting" arbitrarily either, but rewriting according to well-known rules of expression manipulation such as associativity, commutativity, distributivity of various operations, the properties of equality and order relations, etc. It's precisely when you have such strong identifiable properties that you tend to resort to operator-like notation in…

Hmm, I'm not sure whether operator-like notation has any special advantage for commutativity and distributivity other than brevity. a + b and add(a, b) are equally easy to rewrite as b + a and add(b, a).

Maybe there is an advantage for associativity, in that rewriting add(a, add(b, c)) as add(add(a, b), c) is harder than rewriting a + b + c as a + b + c. Most of the time you would have just written add(a, b, c) in the first place. That doesn't handle a + b - c (add(a, sub(b, c)) vs. sub(add(a, b), c)) but the operator syntax stops helping in that case when your expression is a - b + c instead, which is not a - (b + c) but a - (b - c).

Presumably the notorious non-associativity of floating-point addition is what you're referring to with respect to fadd() and fmul()?

I guess floating-point multiplication isn't quite commutative either, but the simplest example I could come up with was 0.0 * 603367941593515.0 * 2.9794309755910265e+293, which can be either 0 or NaN depending on how you associate it. There are also examples where you lose bits of precision to gradual underflow, like 8.329957634267304e-06 * 2.2853928075274668e-304 * 6.1924494876619e+16. But I feel like these edge cases matter fairly rarely?

On my third try I got 3.0 * 61.0 * 147659004176083.0, which isn't an edge case at all, and rounds differently depending on the order you do the multiplications in. But it's an error of about one part in 10⁻¹⁶, and I'd think that algorithms that would be broken by such a small amount of rounding error are mostly broken in floating point anyway?

I am pretty sure that both operators are commutative.

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