Not at all verifiable, at the very least they were rounded up, at worst they are cited from some hagiography (the source of much history at the time) and massively inflated. The citation on the Wikipedia page with such authority is from a massive history of the entire world in 40 volumes based on secondary sources by diodorus 300 years later, so no, it's not likely to be accurate or useful in comparisons with other times. History as a discipline based on verified facts is a relatively modern invention.
That's a reasonable army for the time period. I don't know how historians make their estimates but often these are just estimates. Many armies in the time period have 30-50-80K men. In Medieval times getting an army of 20K was a huge deal, most armies were 10K.
Different economics, populations, fighting styles, time periods. They probably estimate army size by time period sometimes.
Why was it easier to field large armies in Alexander's time than in the medieval period? Does it have anything to do with the feudal system? Or was it that weapons, armor, and equipment was more expensive in the medieval times. Last guess: training was more valuable in the medieval times than it was earlier, so it was more effective to field smaller armies of better trained and equipped men than a horde of untrained peasants with archaic weapons (because that's all they could afford.) ?
In Medieval Europe there was less surplus, a crappier economy, a population closer to the edge. Rome linked prosperous lands in Italy, Egypt, Gaul, North Africa into one empire. Romans overfarmed Italy at their peak and depended on Egypt for grain. After 500AD these luxuries could not be sustained. Economies became local and if you mismanaged your farmland you had to live with it. Italy's recovered by ~1000AD. Much of Europe faced similar issues. Cities declined all over around 500 because farmers could not produce the same surplus locally.
Most fighting inside Europe was done with 5-20K armies during 400-1500. It started with the Romans. As they faced rebel "Romans" ever more frequently, they faced similarly equipped and trained men so results of big battles were unpredictable. They would favor skirmishes over big decisive battles. And if your fighting is indecisive, why field large armies at great expense?
The well trained, well armed foot soldiers, "man of arms" was equivalent to a Roman legionnaire and formed the basis of these armies. At the time a full suit of mail armor cost as much as a Ferrari today. Now this is an issue of economics because there were few complaints about the price of similar kit from Roman times.
Well trained and equipped armies could face numerically superior "peasant uprisings" with predictable results, just like in Roman times. Romans faced a lot of poorly armed opponents in Europe.
BUT it's hard to say what these trends were driven by. There was a lot of traditionalism around warfare.
Macedonian armies were actually poorly equipped early on. They used long pikes instead of reasonable spears like the Greeks. Macedonians wore rags for armor, while some Greeks they faced wore 70 lb almost full body armor. And similarly in Europe, states started fielding cheap pikemen combined with crossbowmen, archers, muskets and a few well trained melee fighters.
Such armies could stand up to more expensive Ferrari suited, trained all their lives, born into a higher class men. It was probably the noise of firearms that changed military traditionalism. But handheld firearms of the period were mostly noise and smoke. And as Macedonians demonstrate not really necessary to face better armed and trained opponents.
But I don't think any historian could untangle all the factors without a Matrix level simulator, complex systems and all that.
In Medieval times there were more local conflicts than large scale invasions - it was a time of fragmented states (that's what the feudal system is about).
Plus, in Medieval times, people were paid to wage war, therefore keeping a large army was extremely expensive, and often you did not keep an army on your own but hired mercenaries to wage war with/for you. In Antiquity, I believe the reward system was different.
It pays here to look at a map of how small an average principality would be in medieval times versus how large Alexander's empire was.
The population densities were, it should be noted, not that different in the two periods. It was not until the mid-1700s that we started to see sustained significant improvements in the carrying capacity of the land thanks to agricultural improvements in England. Until then the carrying capacity would vary, higher during a warm period in the 1200s, falling significantly when things cooled down again in the 1300s.
In Medieval Europe the King was not that important figure and some Kings had to make agreements so that the nobles with their armies would not leave the battle if they did not like it. A very good analysis in a book by Barbara Tuchman of the "Guns of August" fame is in another of her books. http://en.wikipedia.org/wiki/A_Distant_Mirror
This number is vastly exaggerated. Porus was small king of a minor principality( modern day equivalent of a district).
References:
Ancient India : By R C Majumdar,
Wonder That Was India : By A.L.Basham.
I have only hard copy of these books. I don't know if e-book is available or not.
Very typical number for the great battles of the era. This was Alexander's great rampaging army after all, I'm not sure why you think his army shouldn't be on par with Hannibal's.
These number are not verifiable but they are not they are not the army you would think it is. Most of Alexander's army was disbanded after he took the 4 main persian cities and was later using local armies with Macedonian Genarals.
How verifiable are these numbers? This seems like an awfully large army, given the context.
To put this in some perspective, that's about the size of the army Hannibal entered Gaul with in the second punic war [1].
[1] http://en.wikipedia.org/wiki/Second_Punic_War