Rule Britannia said:
Diamonds are as worth as much people think they're worth, it's just carbon. Diamond is an infinite resource, we can synthesize it and have it look exactly the same as a normal diamond. Gold and silver are both finite resources, logically, gold and silver should be worth more.
Your logic fails you, as you are not comparing like items. Gold
is worth more than carbon, even in the form of a diamond. You can buy rough diamonds for dirt cheap on ebay. The problem with your assessment(and the biggest flaw with the OP's argument) is that you are only considering objective measurements when determining worth, yet you are comparing an element with a crafted good.
A diamond's value is determined chiefly by four factors: clarity, carat(size), color, and cut. Clarity and carat are the only ones that can be directly compared to gold, and even then it isn't a perfect match. Gold may be the rarer mineral, but you can melt gold down, extract impurities, combine small quantities to make a larger mass, and after all that, the resulting material will be worth the same(or more) than what you started with. A diamond is what it is. You can't refine an included diamond to make it flawless, and flawless diamonds of any size are rarer than gold(manufactured diamonds can be determined to be such by a professional). Beyond that, color can also affect the rarity(and therefore value) of a diamond.
But the most important distinction that you are ignoring is that diamonds are graded on how well they are cut. A talented and experienced gemcutter can cut a diamond to best show off its natural characteristics, which would increase its value far beyond the ones churned out to most jewelry stores. You can't even really compare it to a well crafted gold ring, as you don't get second chances with gemcutting. If you screw up, you're either stuck with the result, or you lose material(and therefore value).
Someone said it before, but a diamond is a work of art. And like all art, there are people who will appreciate it and people who will not(obviously the OP falls into this category). The reason diamonds can be(but aren't necessarily) expensive is because they are an artform crafted on to a rare(for high quality) and unforgiving canvas.
As for me... well I designed my wife's and my wedding rings(no engagement ring), and my wife's does have diamonds set into platinum, though none were terribly expensive(the center stone was $50 because it apparently has a black speck in it, but neither my wife nor I can see it). My own ring is platinum with a gold sunburst surrounding a topaz. I'm happy with the aesthetic and enjoy knowing that our rings are truly unique.