While he was on the Moon, transmitting a laconic description of his surroundings as "different, but it's very pretty out here", Neil Armstrong mused: "I suppose they are going to make a big deal of all this."

Richard Nixon, addressing the Apollo XI crew four days after Armstrong and Buzz Aldrin became the first men to walk on the Moon, lived up to expectations, flatly declaring: "This is the greatest week in the history of the world since the Creation."

Armstrong never did want to make a big deal of it, and left NASA and the space programme to teach aeronautics in Cincinnati, Ohio, where he later retired to a farm. It inevitably brings to mind the man for whom that city was indirectly named.

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Lucius Quinctius Cincinnatus was hauled away from his plough and made Dictator of Rome in order to defend the city against an invasion by the Aequi. Having sorted them out, he gave up absolute power and public life, and went back to his farm and obscurity 16 days later. For good measure, if Cicero is to be believed, he did almost exactly the same thing again some 20 years later.

This combination of outstanding achievement and personal modesty was probably always an unusual combination – as the fact that Cincinnatus's name is still remembered for it indicates – but in Armstrong's case it is particularly remarkable. Despite the grandiosity of President Nixon's statement, the primacy of the Moon landings as a human achievement is a difficult one to dispute.

Indeed, it's much more surprising that so many people seem indifferent to the achievements of astronauts (and cosmonauts). The satirical website The Onion once produced a spoof newspaper front page for July 21 1969, with a headline which read: "Holy ****! Man Walks On the ****** Moon!" This still strikes me as by far the most natural response; Walter Cronkite, the American news anchorman covering the landings, later claimed that all he could think to say as the Eagle touched down was: " 'Wow! Jeez!' Not exactly immortal." (He actually said "Whew! Boy!" – which isn't notably more Periclean.)

Yet those born after Armstrong became the first human being to set foot on another planetary body are strangely blasé. Even as Curiosity, the Mars lander, sends pictures of the surface of that planet back to Earth, many people seem oddly incurious. Some years ago, The Independent went as far as to run a story with the headline "Life found on Mars" – at the foot of a page. Surely such a report is either untrue (or at least premature), or the biggest news in the history of humanity, worthy of a special edition with headlines in 92 point-type.

One of Armstrong's very few public pronouncements in later life was to express his dismay at President Obama's downgrading of the space programme and the retirement of the space shuttle.

Armstrong was right, because space exploration is not fanciful, unrealistic, or a wasteful use of money, resources and human ingenuity. Those are, rather, the charges which will eventually be brought against its opponents.

Moving off this planet is, in the long run, not optional. If humanity is interested in survival – which seems to be the one common attribute of all life – then after keeping ourselves alive from day to day and trying not to exterminate each other, our priority must be to get off this lump of rock and into the rest of the universe.

One thing which the scientific study of our own planet's history and of the heavens has made apparent is that extinction is the certain end both of our species and, eventually, the Earth itself. Against this certainty, the sole advantage human beings have (apparently uniquely in all Creation) is our ability to reason, to imagine, and to plan for the future.

In this, Neil Armstrong was right to see his part, as one man, as "a small step" which represented a giant leap for mankind. He lived as if he were not a particularly remarkable man; indeed, in his quiet Midwestern life, avoiding the hoopla of the public round in favour of the golf course and the same restaurant for lunch, he achieved an ordinary existence to a degree which is itself utterly remarkable when one considers that his name is destined to be remembered for as long as people hold on to any notion of history at all.

But he was right that it was not his achievement alone, but that of hundreds of thousands of people involved in the space programme. And their work built upon the achievements of millions of people; on every human advance in mathematics, in physics, in engineering; in short, in every field of technological endeavour since our species began to reason and to make tools.

Armstrong's famous assessment of the significance of his own achievement (leaving aside the dispute about whether he said "a man", as he certainly intended, or fluffed the line by omitting the article) thus seems, in the light of his own undemonstrative personality, an extraordinarily wise judgment.

It has been claimed that the line is an echo of one in The Hobbit ("not a great leap for a man, but a leap in the dark"), a supposition which may be backed up by the fact that Armstrong named his farm Rivendell, after the Elven valley in Tolkien's book. Whether that is true or not, it illustrates an aspect as important as the history of technological progress – the importance of myth and imagination as a spur to those achievements.

It is not only the human race's scientific work which made Armstrong's step on to the surface of the Moon possible. Above all, it was rooted in the human capacity to speculate about the future and to attempt the apparently impossible – in our ability to dream.

It is human beings' capacity for wonder which enables us to do wonders, and our ability to think which allows us to achieve what seems at first unthinkable. It took Neil Armstrong, an apparently ordinary man, to that first step into the extraordinary, the extraterrestrial, the future. To where, if we are to survive, we must all eventually follow him.