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> the sentence "a monad is a monoid in the category of endofunctors" is pretty much impossible to represent in a script like egyptian heiroglyphics or this one

单子是自函子范畴中的幺半群

It bears remembering that spoken language existed long before written language, and written language developed as a form of encoding spoken language. Purely pictorial communication utilises a small number of large symbols that make it clear what is being conveyed from pictures alone, but the language depicted is too complex and abstracted to be purely pictorial; it uses a great number of small symbols, and you cannot understand what it is trying to convey merely by looking at it as a series of pictures. For a reader to understand what is written there would require understanding the relation of symbols to spoken language.

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Somehow it bothers me that mon- has been translated as "alone" in 单子 ("monad") but as "one" in 幺半群 ("monoid").
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The very first letter in your example sentence started as an ox 𓃾 then via 𐤀 α etc. turned into a https://en.wikipedia.org/wiki/A#:~:text=1-,History,a – but even in the hieroglyphics it was used to represent the sound (the sound of the start of the word for ox), not literal oxes. The road towards more abstract / less pictorial systems was created for speed and ease of writing, not in order to represent more abstract thought. (Note also how alphabets are changed by the implements used to write them, e.g. runes for knife-on-wood, wedge incisions for clay tablets, square capitals on stone – this too was a pressure from ease of writing, nothing to do with representing more complex ideas.)
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Egyptian hieroglyphs are mostly phonetic. You can represent any abstract concept in it that you could in the spoken version.
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This is a line of thinking that's very common amongst people who only speak languages that use alphabets but it's not remotely true. Egypt became one of the greatest empires ever with hieroglyphs and those evolved into a phonetic writing system. Chinese and Japanese of course function and they evolved from pictographs. A pictograph is only limiting if a character that resembles a dog can only carry the meaning of a dog and nothing else. But that's not the case in any language. They all evolved to use the symbol of a dog, or any given character, to carry other meanings.
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But that symbol can also begin referring to many other things rapidly, especially when combined with other symbols e.g. doglike things such as wolves and dogfish, canine teeth, even doglike characteristics of humans such as aggression and ravenousness.
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Egyptian hierghlyps had phonetic values too so it definitely would be able to express everything spoken Egyptian could.

The main issue with hieroglyphics is that it was a very convoluted system, where symbols meant different things depending on context. A symbol of a bird could be a literal bird, some abstract concept related to that bird or part of the sounds of the word for that bird.

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I am sorry but this like saying that "chinese cannot represent abstract notions" because "picture" .

In middle egyptian (the language you probably assume) "pictures" are just syllables. They are phonetic, not semantic, in the same way letter of modern language correspond to sounds, not meanings.

Egyptians had no problem expressinyg conplex concepts and they also had cursive writing, which is much easier to write.

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Nit pick: the Egyptian writing system never had any symbols for syllables.

The phonetic symbols included in the Egyptian writing system represented 1 consonant or 2 consonants or 3 consonants, not syllables. Any syllables or short syllable sequences with the same consonants were written with the same symbol.

This makes the Egyptian writing system an exception, as all other writing systems that have developed completely independently, instead of being inspired by an existing system, have used phonetic symbols for syllables.

This is the very reason why the Egyptian writing system has generated the ancient Semitic alphabet with 29 consonannts, from which all later Semitic consonantic alphabets have been derived, then the Greek alphabet and other European alphabets, and the Indian writing systems and other Asian writing systems derived from them.

Since the beginning, the Egyptian writing system had two variants, depending on the writing instruments: hieroglyphic for inscriptions carved in stone and hieratic for texts written with a reed brush on papyrus. The latter is what you mean by "cursive". "Cursive" is not really appropriate, as hieratic was still a very complex script, difficult to write, even if it was simplified in comparison with hieroglyphic. Millennia later, a more cursive form of hieratic developed into the demotic script.

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That is correct, but my knowledge of middle egypytian is limited to a single introductory book, and didnt want to muddy the waters with details.

The point stands still: the writing was not as clean as modern alphabets but was capable of expressing abstract concepts, it is completly orthogonal to concepts expressed in writing.

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As a language matures, it moves away from concrete things towards abstract things. Eg cave paintings -> pictorial scripts -> modern languages which are very detached from pictorial/phonetic meaning (even modern chinese). These days we have programming languages which do not have any phonetic or pictorial representation. And this trend will keep going on. I think I still stand by my point that this script isn't as refined as a modern language. Just like the great pyramids aren't as refined as burj khalifa.
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> Just like the great pyramids aren't as refined as burj khalifa.

I'd wager the Great Pyramids will still be around in 1000 years and the Burj Khalifa will not, if anyone wants to take bets.

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I dont think this chnages anything. The writing was less convenient than modern alphabets but still capable of expressing all things necessary.

The closest analogy might be: modern alebraic notation is compact and clean but this doesnt mean algebra didnt exist much, much earlier.

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To make this claim, wouldn't you need to understand the meaning of the script? It's probably not about monads, but you don't know for sure.
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The mental image of an ancient Mesoamerican civilization writing about monads thousands of years before the rest of us, only for it to go unappreciated because we can't comprehend their script, is a great one.
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While the proto-writing systems are based on pictograms, both the Sumenrian cuneiforms and the Egyptian hieroglyphic (used for inscriptions on stone) and hieratic (use for writing with a reed brush on papyrus) have made the transition towards having phonetic symbols, used together with ideograms.

Using the phonetic sign subsets of the Egyptian and Sumerian scripts, it was possible to write any sentence that could be spoken in their languages.

This was the most important advance in writing and both in Mesopotamia and in Egypt there is evidence about this transition from an earlier writing system that could write only a subset of the words of a language, so it could not be used to write arbitrary sentences, but only things like lists of objects with their amounts and owners, like needed for accounting, to a writing system that added phonetic symbols for writing any words that did not have their own symbol.

I cannot read the paywalled article, but it seems that now there is evidence that also the Proto-Elamite writing system has also passed around the same time through this transition from having only symbols for certain words to having phonetic symbols too, e.g. for syllables, which can be used to write arbitrary words and sentences.

Before phonetic symbols began to be used, we cannot know the language spoken by the users of a proto-writing system.

While in Egypt there is little doubt that the first users of writing spoke some kind of Old Egyptian, in Mesopotamia there is doubt the users of the first proto-cuneiform writing system spoke Sumerian. However, by the time when phonetic cuneiform signs were introduced, the language of the writers was Sumerian.

In the territory later known as Elam (in the West of present Iran), it is not known what language was spoken by the users of the Proto-Elamite writing system. It could have been an ancestor of the Elamite language spoken a millennium later, or it could have been a completely different language. Elamite is not related to the Indo-European languages that spread much later in that territory, like Old Persian.

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It does look complex. These are not mere pictographs but a refined and codified system of representing ideas.

By the way, Chinese uses a modern example of such a script and succeeds in representing such concepts.

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