Does anyone here have a knowledge of economics or taxation in any detail? If so, could they give me an economic reason why progressivity of a taxation schedule is desirable, or if not, why the vast majority of the world’s tax systems adopt such a schedule?
No economics training at all, but I could render a guess. The vast majority of the world is poor. Including you and me, and, oh, 95% of the entire earth’s population. So we get jealous of the rich, decide to tax them more, call a progressive system “justified” so that we don’t feel guilty, and live happily in peace. Once we’re all truly brainwashed that inequitable (“progressive”) taxes are good, they can come after us. Well, they already did. I’m not lower middle class, and I’m sure that to them, I look “rich.” So in their minds, a progressive tax is surely fair, right? After all, I have more, so can afford it.
So, really, the “desireabilty” is driven by the poorer. This is why we should give IQ tests to potential voters, and weight votes according to your declared (taxable) income. 
This would appear to be more suited to GD.
That being said, I would first request a cite for your contention that the vast majority of tax schemes are progressive. That is untrue, even in the United States. While the Unites States federal income tax is progressive, most other taxes are not progressive, including: sales tax, state tax (usually a fixed percentage of income), corporate tax, excise taxes, sin taxes, capital gains taxes. While I am not sure of specifics, many countries have a flat tax on the income of employees and many countries also employ an extra sales tax call a V.A.T.
With respect to the idea behind a progressive individual income tax, the most often stated basis for such a tax is that individuals need a certain amount of money to survive and sustain themselves. Therefore, is someone is making a small amount of money, then they should not have to use up an overly-burdensome amount of their income to support the common welfare (i.e., they shouldn’t have to sacrifice their own welfare). On the other hand, those individuals who make a larger amount of money can sustain their own health and security and still be able to pay a larger portion of their income to promote the common welfare of society.
Again, whether such a system is “fair” is more of a question for GD. This posts merely responds to your factual request for the main basis for such a system.
db
When duck hunting, go where the ducks are.
When raising revenue for any purpose, follow the money. The rich have it, the poor don’t.
I have more tax training than economics training but here goes:
There is an economic concept of diminishing marginal utility. Imagine, if you are wandering alone in the desert and dying of thirst. You find one of those nice water coolers. As you pull one of the paper cups and fill it with water you are absolutely beside yourself with joy. You drink the cool water and are feeling great. The next cup is nice but not the same. By the time you are half way through the bottle, you are washing your hands. At the end you are using the water to dampen your comb to make yourself tidy.
Essentially, there is the belief that the poor have less money and will be harmed more by paying more. If you make $10,000, you will miss $1,000 much more than someone that makes $1,000,000 will miss $100,000. The hope is that the progressivity of tax rates will cause the least anguish in the population.
Aside from that, any income tax system will necessarily go after the rich because that’s where the money is.
Just as an aside - the US corporate income tax system is progressive. In 2004, the marginal rates went from 15% to 35% with spikes to 39% and 38%.
From an economic perspective, it’s a easy way to redistribute wealth to government priorities. I don’t mean redistribute wealth in terms of steal from the rich and give to the poor, but in the more general sense of tax the rich more so that spending on all government programs may be increased by whatever amount, no matter whether that spending is for defense, foreign aid, welfare, education, or whathaveyou.
Since the poor don’t have much wealth to tax, maintaining a high rate of taxation on the poor – or even the middle classes – would basically eat into savings and investments before it would eat into consumption. That is equally true for the wealthy, of course, but the potential cost of eliminating savings and investment from lower classes could be the risk that they would be more dependent on government spending in their golden years, which is not really as much a risk for the wealthy.
However, the practical question of why progressive tax schemes are popular has more to do with political decisions than economic ones.