Quite the contrary. They made a lot of effort to get the numbers right when they sized up enemy armies. They were also usually very specific about the number of their own troops. The problem is that we modern people tend to think of numbers as either true or false: empirical facts in the raw. They are actually a great deal more than that, and especially so in something like Herodotos' Histories.

I'll start with that source and its infamous claims about Persian numbers. Yes, Herodotos says the Persians invaded Greece with a force of over 2,000,000 soldiers. Yes, this is obviously wrong; in the environment of Ancient Greece, such a force would be geographically impossible to fit and logistically impossible to feed. A century ago, Hans Delbrück applied his famous Sachkritik to the problem, using the reported size of the Persian camp and other ground rules to reach a more plausible estimate. Modern authors tend to estimate the size of Xerxes' army somewhere between 60,000 and 200,000 men.

But here's the thing: those figures are not based on any ancient evidence. They are nothing but informed speculation. The passages in which Herodotos reaches his absurd totals, on the other hand, are some of the most "scientific" of his entire work.

When Xerxes' army is brought together in Asia Minor, Herodotos makes his first claim as to its size: 1.7 million (Hdt. 7.60.1). He goes on to explain how the Persians themselves established this figure:

They were counted in this way: ten thousand men were collected in one place, and when they were packed together as closely as could be a line was drawn around them; when this was drawn, the ten thousand were sent away and a wall of stones was built on the line reaching up to a man's navel; when this was done, others were brought into the walled space, until in this way all were numbered.

Modern scholars have rightly questioned this story; apart from anything else, it would have cost Xerxes a tremendous amount of time. But simply to say it didn't happen is to ignore the point of the passage. Herodotos was not satisfied to just give his readers the number. He thought it necessary to explain that it was reached by dividing the army into multiples of ten thousand and acquiring the total through empirical observation. In other words, far from "asspulling numbers", he insisted that his numbers were based on deliberate and painstaking scientific inquiry.

Something similar happens on the eve of Thermopylai, when Herodotos once again gives us the numbers of the Persian force (7.184-186). It is worth quoting this at length just to show how much work Herodotos does to justify his total:

Calculation proves to me that its numbers were still such as I will now show. The ships from Asia were 1,207 in number, and including the entire host of nations involved, there were a total of 241,400 men, 200 being reckoned for each ship. On board all these ships were 30 fighting men of the Persians and Medes and Sakai in addition to the company which each had of native fighters; the number of this added contingent is 36,210.

To this and to the first number I add the crews of the pentekonters, calculating 80 men for each, whether there were actually more or fewer. Now seeing that, as has already been said, 3,000 of these vessels were assembled, the number of men in them must have been 240,000.

These, then, were the ships' companies from Asia, and the total number of them was 517,610. There were 1,700,000 footsoldiers and 80,000 cavalrymen; to these I add the Arabian camel-riders and Libyan charioteers, estimating them to have been 20,000 in number.

The forces of sea and land added together would consist of 2,317,610 men. So far I have spoken of the force which came from Asia itself, without the train of servants which followed it and the crews of the grain ships.

I must, however, also take into account the force brought from Europe, and I will rely on my best judgment in doing so. The Greeks of Thrace and the islands off Thrace furnished 120 ships, and the companies of these ships must then have consisted of 24,000 men. As regards the land army supplied by all the nations—Thracians, Paionians, Eiordoi, Bottiaiai, Chalkidians, Brygoi, Pierians, Macedonians, Perrhaiboi, Enienes, Dolopes, Magnesians, Achaians, dwellers on the coast of Thrace—of all these I suppose the number to have been 300,000.

When these numbers are added to the numbers from Asia, the sum total of fighting men is 2,641,610.

As for the service train which followed them and the crews of the grain ships and all the other vessels besides which came by sea with the force, these I believe to have been not fewer but more than the fighting men. Suppose, however, that they were equal in number, neither more nor fewer. If they were equal to the fighting contingent, they made up as many tens of thousands as the others. The number, then, of those whom Xerxes son of Darius led as far as the Sepiad headland and Thermopylai was 5,283,220.

There are obviously a lot of rough-and-ready estimates here. Nevertheless, the thing to take away from the passage is that Herodotos was not content simply to throw a large number out there. Instead, he meticulously went through each contingent of the combined force, ascertaining the number of its ships and men, hedging his estimates, and using standard multiplication tables where applicable. He spells all this out to forestall any accusation that he was fudging his totals. Anyone who wishes to question Herodotos cannot just laugh at the final number, but must engage with this breakdown and show where the errors are. What is more, the first critic of Herodotos' totals was none other than Herodotos himself (7.187):

I do, however, wonder how there were provisions sufficient for so many tens of thousands, for calculation shows me, that if each man received one choinix of wheat a day and no more, 1,100,340 bushels would be required every day. In this calculation I take no account of the provisions for the women, eunuchs, beasts of burden and dogs.

This is not "asspulling". This is conscientious and critical evaluation of the results of empirical observation and mathematical calculation. This is, in a word, science. Recent work by Rosalind Thomas has shown Herodotos' debt to the medical authors of Ionian natural philosophy; he was raised in the methods of the earliest Greek "scientists" and valued descriptions of reality that were based on verifiable fact.

But then how did he end up with numbers that are incredible and cannot be right?

Here we get beyond what these numbers are and into the arguably more interesting question what these numbers do. For Herodotos, it was not possible to get at the exact figure even for the Greek armies that fought in the Persian Wars; round figures were known for hoplites, but the historian was forced to carry out similar mathematical gymnastics to get at the number of light troops on the Greek side, since these were rarely officially counted. Greater exactitude than that displayed by Herodotos cannot really be expected. However, what he did know when he was writing his work was what the numbers were supposed to reflect. They were an expression of the full might of the Persian Empire - the largest empire the world had ever seen. They indicated the size of a display army raised to justify Xerxes' ascension to the throne; Xerxes had a personal interest in proving that this army was as large as possible. They were also a necessary element of a story in which the Greeks, with the help of the gods, won a victory that no rational observer would have thought possible.

When we go beyond the Persian army numbers as facts and consider them as elements in Herodotos' story, it becomes much easier to understand why they are so stupendously large. As far as Herodotos was concerned, the Persians mobilised the entire empire against Greece, and he therefore set about calculating and estimating the combined forces of every single region within that empire, adding all of it together just before it hits Leonidas at Thermopylai. Both the historical Xerxes and the villain Xerxes from Herodotos' Histories were personally interested in proving the sheer size of their army to the world, and to the Greeks in particular. Meanwhile, to put the decisions of the Greeks in the right perspective - both those who resisted and those who chose to submit - it was necessary for Herodotos to highlight that what they faced was truly an enemy of superlative, incomprehensible strength. Only then would every part of his story - from his extensive survey of the Persian empire, to the character of the king, to the narrative of the resistance and eventual Greek victory - click together. Would any of this have made sense if Herodotos had admitted that the Persian army was probably just 150,000 strong at its height? Would the Persians have launched their invasion with a force barely larger than the Greek alliance that fought them at Plataia, and if so, would any Greek have submitted to that force?

Herodotos' calculations and doubts show that he knew exactly what he was doing when he came up with these numbers. It was as far away from an ass pull as the Greeks could conceive of an army number to be. Herodotos was not making up numbers; he was trapped in a situation where one form of scientific observation and reasoning clashed with another, and he chose the solution that left his narrative intact.