The red numbers correspond to the Atomic Number on our Periodic Table, with these you can write a formula to manipulate the mana of the element represented. This was the old way of Magic, when the identities of these particular elements and mana were known.

In the current age, magic has become more instinctual, relying on you imagining the manipulation of let's say fire mana with your own mana rather than entrusting it to a formula, since the formulas are mostly gone except for what remains in Artificing/spell circle stuff. So many mages are capable of "silent casting," but for most it's just easier to remember how to cast something by connecting it with a phrase. SImple stuff can be done silently since if one wouldn't have to think too much about it, almost like breathing. The reason why they chant spells out loud and such is to help remember the feeling of manipulating mana into that particular spell, such as perhaps a special technique.

The technique could be done silently or with a word that has no relation to its name other than what the mage gives it, it's really what the mage means with the words that are spoken, what they mean and imagine by saying those words.

Sort of like Initialism, like how FBI is the Federal Bureau of Investigation. Literally, FBI isn't the Federal Bureau of Investigation, it's just three capital letters, but it's meaning is the Federal Bureau of Investigation.