If you're suggesting that the 753 BC date is archaeological, you're incorrect. 753 BC is the legendary date. There's archaeological evidence of habitation on the hills of Rome earlier than that.
When do you have a handful of sand? If you take 1 grain of sand and put it in your hand, that's not a handful. Same with 2, 3. How many grains of sand constitutes a handful?
In the example of a handful: the average "handful" is around 100-50mL. Say we define a "handful" as 50mL +/- 1 order of magnitude. On the high end, you have Andre the Giant, who could easily fit a 12 oz beer in his hand no problem (354 mL.) On the low end, you have children; even kids with small hands can hold 5mL.
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Sure, there's some ambiguity, and of course there's not a hard number on it. But that doesn't mean you can't tell if something is a city. Nobody would argue that, for example, NYC isn't a city, or London or Amsterdam or whatnot.
As the world population grew, so did the definition of "city". 12,000 years ago, there were somewhere around ten million people in the entire world, so a definition of somewhere in the neighborhood of 0.1% of the world population (10,000 people) for a city seems reasonable, plus or minus one order of magnitude. I'll call it Owen's Law :) That holds for today: 0.1% of the world population is 7.8 million, plus or minus 1 order of magnitude - I'd say that's a fine definition for a city. In any case, it works very well today: at the large end you have cities like Chongqing with 30+ million in the municipality, and at the small end you have cities like Rotterdam with around 700k people in the city proper.
To check if my rule of thumb worked: At the year 1, assuming 200 million people in the world, that's setting a city around 200,000 people, plus or minus 1 order of magnitude. The largest city in the world was Rome, between 1MM-400k people; all the top 10 largest cities fell within the rule.
- It holds at the year 1000 (world pop. ~300MM, largest city Córdoba with 400k, smaller cities like Nimes with 56k.)
- It holds at the year 1500 (world pop. 500MM, largest city Constantinople 250k, smaller cities like Palermo, Florence, Madrid, Moscow with ~70k.)
- It holds at 1800 (world pop. ~1 billion, largest cities Vienna and London ~1MM, smaller cities like Berlin, Lyon, Venice 150k.)
- It holds at 1900 (world pop. 1.6 billion, largest city London 5.5MM, smaller cities Naples, Madrid, Amsterdam ~500k.)
The fallacy lies in the question, that a collection of sand is either individual grains aggregated, or a heap. One response is that "heap" describes not quantity but behaviour, or even more crucially, useful mental models.
I'd argue that the transition between "grains" and "heap" occurs somewhere between 16 and 128 grains, the 6th to 8th squares (2^5 to 2^7). Sixteen grains is still, mostly, individual grains. 128 grains is almost certainly a heap.
In urban geography, urbanisations are virtually always classed by the log, usually base ten, of their population. This gives settlements of class 0 (one inhabitant) to about class 7.6 (roughly 40 millions of souls, greater Tokyo). Distribution nearly perfectly follows a power distribution -- frequency distribution is linear with the log size.
Going back through human history, the first settlements we might have termed "cities" may well have been only a few hundreds of inhabitants, but they would have dominated their regions. Anthropologists typically identify Uruk as the first settlement recognised as a city. "Many ancient cities had only modest populations, often under 5,000 persons."
