What natural symbolic representations could be used for Mathematical constants?

Question

We know mathematics is a language by itself.

But to evoke any constants or any arbitrary values as such to solve anything, prior knowledge of a particular symbol and its usage must be understood.

For example, for Pi, we use the well known symbol. But hypothetically, think of a child who did his mathematics with this number as a ratio called 'gobbledegook', he lives his life fine without ever assigning it a symbol, hence avoiding convinience.

My question is rather than assigning an arbitrary symbol without a meaning, and associating a meaning with it, how can one approach designing or linguistically portraying a value that in plain words- represents the idea directly (only speaking for mathematics). Like for Pi, could we say a circle with a line beneath it? (For this, the symbolism of division is a precursor you see, how to avoid stuff like that?)

Answer 1

Strictly speaking, a name like "gobbledegook" is a symbol. That's a reasonable systematic, and important to note as a derived grapheme might strive for similarity with the name. Thus it becomes apparent that you are basically asking for how to name things, equivalent to the question of how does language work. In the same vein we see that maths does have aspects of language. That's true all the more if you are asking for a logical system of naming, if logic is a basic tenant of maths, though one might argue that it's foundational and maths is what's building on top. Naming, eh? The distinction is not very important, actually.

I will just focus on high-school maths for a quick answer.

The name Pi for the ratio of square area to area of a circle where the radius equals the square's side-length--to stay with your example--was an abbreviation of the Greek word perimeter. We see that the definition doesn't need to rely on the perimeter.

The choice of Greek letters might stem chiefly from the need to choose symbols that type foundries and print shops had already in stock, and it might have been traditional already prior to printing, but the decision for a name from natural language is well descriptive. All letters of the various common alphabets are already reserved for common constants and variables, not even uniquely, rather overloaded with meaning depending on subfield, or assigned arbitrarily by any individual author. On the other hand many function names have longer abrevs. That's the important point, in my humble opinion, as the trend in professional programming, probably the biggest field in applied maths, goes to speaking variable names. A multiletter grapheme like gobbledegook is still a symbol. There are arguments to be made in favor of compact symbols the size of a single character, but that's a matter of fine arts or industrial design more than linguistics.

Relatedly one could ask for the optimal size of a languages character inventory and compare e.g. the mess that is the Chinese logogram-syllabary potpourri.

Corollary: If the name is arbitrary, then the symbol may be, too. Thus there can't be one correct answer to your question.

In many ways tradition trumps ergonomics; At least one might argue that linguistics is more adapt at describing what-is, not inventing what-could-be.

You might want to take a look at APL, the obfuscating programming language, for a heavy use of single-character-width identifiers and operators. It still shows numbers written out. That's the most common depiction of numbers we know. In essence, we describe numbers by the computation that derives them, so any computable number can be identified by a function that produces the number.

The lowest decimal numbers are a different matter. The arabic numerals have been developed over millenniums for optimal legibility. There is no need to reinvent them. The smallest ones are even homologic (or ideographic), 1 is one stroke, 2 is two and so on. And zero was for a long time not written at all. This reminds of the integration of counting into the visual cortex. Few other things are integrated as deep, that would require mathematical symbolism and could make use of it at the same time. Audible symbolism may rely to a degree on the auditory portions of the brain. Letters that describe the phonetics of the word somewhat marry both ideas. You'd be hard pressed to find other graphemes that imply the sound of pie. Since maths is for better or worse still taught mainly auditory, or discovered in non-symbolic ways, the need for intricate symbol design is just not there, I guess.

Answer 2

The number 4 is represented "naturally" in a number of languages as 四 ٤ Ⰳ ४ ௪ IV Δ ד or 4. They fall into 3 groups: 4 sticks (the most common system), the 4th letter, and the obscure system the Romans used which seems to be derived from a stick-stacking system. As you can see, the actual letters for the same thing have diverged massively over the millenia that they have been used. It is trivial to devise a symbolic written system that "represents the idea directly", applied to numerals – if you have a theory of what the basic numerals are. It is virtually impossible to force people to write "8" as "||||||||".

Beyond that, there is a potload of mathematical consonants, all of which are given rather modern and arbitrary symbolization. The Ramanujan–Forsyth series is not natural, so you would not expect there to be a natural symbol for it. Indeed, the only natural constants are the numerals.

The other way that constants are symbolized is with words: in that usage, there is absolutely no hope of devising a natural, iconic representation of "4". the number 4 alone has thousands of pronunciations, depending on what language you are using.