If the use of greek letters is a major hurdle to learning maths, I'm guessing that the learner has misunderstood something fundamental about symbols in maths. It shouldn't make any significant difference if the symbols are Latin, Greek or even made up by the lecturer (as long as they are sane).

In theory it shouldn't matter but it does. We brainwash students to think in terms of y being a function of x.

y = f(x)

If you give a problem with x = f(y) it really confuses people. If I label the horizontal axis y and the vertical axis x in the cartesian plane then confusion ensues.

t = b(a) where t is the velocity and a is time

That's confusing. I suppose it's analogous to me saying

"The male, fluffy, white, large sheep."

There is a certain amount of brainwashing that occurs with the symbols that we use. Using symbols other than the ones that have been standardized on is very difficult cognitively. What you say is true in theory but I think not in practice.

If this is enough to "really confuse" someone than there's a serious problem with understanding VS rota learning.

You're totally right... but the previous poster is too. I encounter many students who really freak out at f(t) = y vs f(t) = x vs f(x) = y. Obviously they don't understand very well. I go back and forth between using notation as a crutch (always stay consistent so we can focus on the topic instead of the notation) and trying to really wean students from notation-dependence (let's do f(smily face) = y today! let's rotate all the axes and draw this with the z-axis sticking out!).

When people have very fragile understanding, they may be genuinely unsure if derivative with respect to x is computed the same way as derivative with respect to n. After all, x is always a real-valued variable and n is always a non-negative integer.... right?

This is an enormously serious problem with humans. All humans. Cognitive biases, miscommunication, misinterpreting patterns and data and evidence, inability to differentiate what something looks like from what's inside, inability to believe statistics that contradict personal experience, the list goes on and on...

People who have learned math and are comfortable with it's symbols and abstractions are less susceptible, but all people are susceptible to this confusion.

I'm certain you would be confused by non-standard usage of notation. It's not just symbols and that any symbol can be used. Try reading or speaking English with the adjectives in an order that is not expected. It's confusing as hell but grammatically correct. Our brains get trained to see things in a certain way and when the symbols get jumbled up so do we.

(x1(x2+x3) - x1(x2))/x3

is simply not as understandable to a first year calculus student as

(f(x+h) - f(x))/h

I've been teaching university level mathematics for 25 years and I know mixing up symbols confuses people. Try reading Newton's Principia. It's really hard to know what he's talking about.

EDIT: Try this equation. What famous one does it represent?

(sex(sxe) - sex(xes))/(sxe - xes) = xse sex/xse sxe

Here's something else to consider. There used to be obfuscated C programming contests. If symbols are just symbols and using them in nonstandard ways isn't confusing then shouldn't this mean that there is no such thing as obfuscated C programs? Does the existence of obfuscated C programming contests imply that programmers have a serious problem with understanding vs rote learning?

There's a nuance here too.

in programming, y = f(x) = x^2 is one way; 'y' is a label for a function of one variable.

But in math there is no fixed typing, or distinctions. The symbolic equation represents a relationship.

Hence:

y(x) = x^2 <=> x(y) = (+/-) sqrt(y)

the domain and domain can be switched and the entire equation rearrange so that a 'variable' becomes a 'function'.

Here is an interesting observation on what you wrote:

x(y) = (+/-) sqrt(y)

is not valid notation. On the left hand side you are stating that x is a function of y. But on the right hand side your usage of (+/-) indicates that y can go to two different values. Hence x is not a function of y. To be a function one must have each input going to a distinct output.

Yep, that was on purpose somewhat. A choice of +/- will essentially make a restricted codomain, to positives say.

That means switching the variables around switched domain/codomain.

An example where they are both the same:

y(x) = x + 1 <=> x(y) = y - 1

The learner presumably doesn't know maths or what the symbols in maths are yet, so perhaps they haven't understood yet, rather than they misunderstood something.

Symbols are in fact intimidating to some people. Not after you learn them, but before.

Your argument is abstract and theoretical. It doesn't make any difference to a computer which symbols you use, because it's a lookup table. But to a human, learning new characters takes effort. That effort could be used to learn the concepts, but instead we take time learning new symbols.

Try typing your reply in trinary. It shouldn't make any difference if the symbols are ascii or trinary. You should be able to type just as fast.

In lower-level courses, Greek letters aren't used that much (at least, I don't use them that much), so I wouldn't describe it as a fixation (or a major problem). Examples of use:

College algebra: pi.

Calc 1/2: pi, theta, capital sigma for sums. epsilon and delta to state the definition of a limit (but epsilon-delta proofs have largely disappeared from mainstream calc courses, so those two letters actually don't appear much).

Linear algebra: pi, capital sigma. Some people like to use stuff like phi or psi for linear maps. I don't.

On the one hand, in writing my course notes I have moved away from Greek letters somewhat in courses like abstract algebra. On the other hand, there is at least one good reason to use them: It helps you distinguish different kinds of objects by using different kinds of symbols. So, for example, I can write "f(x)" for a group map f applied to a group element x. If you follow the rule of picking letters from disparate parts of the alphabet for different kinds of things, that's okay. But I think some people prefer "phi(x)" exactly because the function phi "stands out" from the argument x.

Also, if I need 4 functions "in parallel", I could use alpha-beta-gamma-delta. If I try to do this using Roman letters, I use f, g, h, ... now what? ("i" is sort of reserved for indexing; "e" might mean the constant, but in any case is not often used to name a function.) You can subscript (f_1, f_2, f_3, f_4) but that introduces another kind of notational density and possibility of confusion. There are tradeoffs both ways.

There are places where I'm definitely in sympathy with the your general point. I've sometimes read something and wondered why the writer didn't just use a Roman letter (particularly when I don't recognize the Greek letter - is that a capital Greek whatsis or an emoji?). Sometimes, people use calligraphic script caps (e.g. for ideals), not realizing that they will be a pain for handwriting or TeXing notes. And in some subfields of math, people seem to like Fraktur or Hebrew characters which are a real nuisance (it took me a few tries to learn how to handwrite a passable aleph), particuarly when they're handwritten and the person writing has shitty penmanship so you can't tell what letter they wrote.

I agree with the general point, though: Greek letters should be used like seasoning.

Symbols are symbols, are symbols.

Instead of trying to compress everything into a very limited set of "daily language" symbols we should be trying to expand our ideas, including our language(s).

I'm not sure what that means. Are you saying the use of Greek letters is a good thing, and that we need more of them, not less? Or do you mean that all symbols are equally easy to learn for beginners, whether they're English or Greek or Chinese or abstract? Or do you mean that we should do away with symbols that don't mean anything and use real language to describe math instead?

The symbols we use in math aren't to be confused with the ideas, they are shorthand. Ideas and symbols are independent, and we still have to explain what Greek letters do using English anyway. We use Greek letters for convenience, not for expressive power. Writing "∫ x ∂x" is only faster and more compact than writing "The integral of the equation x over x", it is not an improvement of the concept, right?

Use of symbols is the act of trying to compress everything into a smaller set, not the other way around.

For the modern audience, the use of Greek letters may hamper learning because the symbols are unfamiliar and foreign looking. It takes cognitive effort to learn the Greek letters in addition to the cognitive effort to learn the mathematical ideas. What if we could remove the cognitive effort on letters, and focus on the pure math ideas? What if the Greek letters are a speed bump to learning? That we use ∑ for sum and ∆ for change are arbitrary - canonical, but completely arbitrary. History could have decided that a circle was the integral symbol, or the capital letter I. Maybe math would be easier to learn if we switch to capital letter I.

> The symbols we use in math aren't to be confused with the ideas, they are shorthand. Ideas and symbols are independent, and we still have to explain what Greek letters do using English anyway. We use Greek letters for convenience, not for expressive power. Writing "∫ x ∂x" is only faster and more compact than writing "The integral of the equation x over x", it is not an improvement of the concept, right?

It is not, strictly, an improvement on the concept. It's the same concept expressed two ways.

But you cannot symbolically manipulate that prose text, where as you can stick that integral statement into an equation and manipulate it there along with the other expressions of the equation. The ability to perform those manipulations is an improvement.

Very true, good point, it's much easier and more convenient to do algebra on the symbolic expression. There is a reason math has evolved the way it has. Not using any symbols would suck, no question.

But, nothing about writing math in text prevents manipulating the text, right? You can do it, it would just be laborious and exhausting for people who are familiar with the use of symbols.

It wouldn't just be laborious for those of us familiar with the symbolic notation, it'd be laborious for the new students as well.

Try a "solve for x" problem with the equation described in prose (still using short variable names and numbers though):

  x added to 4 all divided by 3 is equal to 16.

What's the next step in solving this? First, because we're abandoning the conventions of symbolic notation the groupings of equation parts (terms and expressions) becomes a bit hairier. I've tried to be clear, and I think in this instance it is, that "divided by 3" applies to "x added to 4". So we know that we should multiply both "sides" by 3. I put sides in quotes because, well, now there are no sides. An equally valid prose description is:

  one third of x plus four is the same as sixteen

If we lack conventions for expressing these statements we could probably come up with an absurd number of equally valid prose presentations of this one expression. This makes reasoning about the next steps much, much harder, as we first have to parse a near arbitrary expression and then discern the author's intent (and seek clarification for more complex ones).

  the result of dividing four and x by three is sixteen.

By the time you settle on a subset of english for prose math specification you'll be down to such a small or specialized portion of the language that you might as well add back in the symbols and at least lose the English dependency for your work so you can collaborate with people around the world.

There's a reason a good portion of elementary mathematics was spent on "word problems", where we reduced problems like this down to their essential equations (a process of abstraction, though not described to us as such). Usually you'd see the above as something like:

  Andrew has some number of apples. Billy gives him 4
  more. He splits the apples evenly between himself,
  Billy, and Charlie. Everyone has 16 apples, how many
  did Andrew have to start with?

All that other crap is noise. Just like in the prose version of the equation "the result of ..." has no value in trying to solve the problem.

I completely agree it would be more laborius, including the beginner.

Maybe I didn't make my point clearly. I'm not arguing against the use of notation or symbols. I'm asking whether Greek specifically is slowing learning down or intimidating people from starting. I'm asking whether learning the symbols with the concepts is the most efficient way to learn math, not whether it's the most efficient way to do math.

Maybe the laborious way is better if more people are comfortable starting that way, and easing more slowly into symbolic syntax.

Maybe there's a more compelling story about math that will get more people interested in it than teaching what symbols do.

If Greek is intimidating enough that many people never start, it doesn't matter how actually laborious the symbols are or not, the battle is lost.

> unfamiliar and foreign looking

The whole point of education is to approach and embrace unfamiliar concepts.

By the way, arabic numbers are... arabic.

Agreed. But learning multiple unfamiliar concepts at once takes longer than learning one unfamiliar concept at once.

It's a fact that some people are intimidated by Greek letters and highly symbolic math, I'm not sure why that's being pushed back on. It's an open question, not a statement on my part, whether using more familiar symbols would speed up learning. Are you absolutely certain it would not? Why? Have you tried?

Lots of people on HN believe music education would be improved by updating the notation to be more "modern", and complain that music is more difficult to learn than it could be.

I'm saying when you learn something new, you are learning new things, including the "language" that "thing" is expressed in in order for others to understand you.

What you seem to be saying is Japanese is too hard to learn, so let's just discard Kanji and use the english alphabet instead. You can do that, but at some point you are not teaching Japanese any more.

In higher level math symbols are definitely used for their expressive power.

I agree if we are talking about syntactically expressive power.

I was talking about conceptually expressive power. Symbols have no conceptually expressive power at all, we assign meaning to them by describing what they do using familiar language, and previously defined symbols. The transitive definition of any symbol is completely and only defined from first principles via familiar language.

∆ - delta - d as in difference

∑ - sigma - s as in sum

∫ - long s - s as in sum

It's not totally arbitrary.

I could use alpha - a as in add.

I could use sigma - s as in subtract.

I could use iota - I as in integrate.

Of course it's arbitrary.

You claimed the symbols were arbitrary, they are (often) not. What you demonstrate here is not that the symbols are arbitrary, but that the concept (sound as abbreviation) maps to more than one potential meaning and the choice of which one is standard or convention is arbitrary.

That is correct, but at this point it is standard to see sigma as summation (or in some other contexts as set or space, but it's also clear in those contexts which meaning is appropriate, certainly an algebra student seeing sigma notation will not also be seeing sigma used for "set" or "space" in that same course).

You're conflating choice with canon/convention. The choice was made arbitrarily by someone. There was no convention at the time the arbitrary choice was made. The choice to use alliteration was as arbitrary as the choice of which Greek letters to use. There are other equally arbitrary possibilities.

Also don't confuse arbitrary with random. Having a reason doesn't mean a choice isn't arbitrary.

Bigger picture, maybe I'm misinterpreting you, but I feel like the logical conclusion of your argument is that nothing can change now, because canon. And yet basic math notations have changed throughout history.

It would take a lot of effort to change the summation symbol from sigma to something else. But if something else made learning math faster, the canon is exactly the part that needs to go, not an excuse to stop improving.

Let's add Chinese, Amharic and Arabic characters. Why not, expanding is expanding.

In some branches of engineering (traditional engineering, not software), there are sometimes too few symbols in the Latin and Greek alphabets, so they have to rely heavily on subscripts. Expanding the set of symbols would be useful here, but I guess it's hard to get people to accept change unless they absolutely have to.

A part of the problem is character entry. If it were easy to enter in all these symbols APL may have taken off and we wouldn't have gotten ASCII variants of it like J. So it's somewhat easier to just take our latin alphabet and turn a single symbol into a pair of letters or a letter with a subscript.

The symbols in math are great for writing by hand, but can become cumbersome when typing.

Arabic numerals are already widely used in mathematics...

Hebrew is already quite widely used in certain fields (specifically set theory).

I doubt that has much to do with it.

As I went through school I saw many students who were bilingual either do poorly or exceptionally well in math.

These were students who spoke Spanish, Tagalog, Chinese, Burmese, etc. at home for the most part and then spoke English at school. Some learned well and some didn't.