Lovelace realized that even though a computer was, at its heart, a mathematical machine, it wasn't restricted to doing mathematics. She realized that a computer could be used to process other types of 'information' by having numbers represent anything else. She realized that a computer could handle text, or music, or practically anything.
That's Lovelace's Leap.
Speaking to the BBC about his book The Information, author James Gleick says:
She understood even better than Babbage did what the potential of this machinery was. Babbage was always thinking in terms of just numbers. Ada thought it might not just be numbers. You could have words or music or anything that could be expressed in the form of symbols and these machines could operate on them.Writing in Computer Resurrection Issue 53, Doron Swade says:
It was Lovelace who appears to have made the essential transition in understanding. Babbage saw the Analytical Engine as a sophisticated programmable computing machine capable of executing any sequence of arithmetical operations under program control, but still operating only on number. Ada saw the potential of computational machines to manipulate symbols of which number was but one example. She speculates that if the rules of harmony and composition were appropriately represented then the Analytical Engine “might compose elaborate and scientific pieces of music of any degree of complexity or extent”. She also wrote of the machine operating on numbers directly representing entities other than quantity “as if they were letters or any other general symbols” and that it might produce outputs in notational form. Nowhere, at least in his published writings, does Babbage write in this way nor does he speculate about his machines in any context other than mathematical.The quotations there are taken from the notes attached, by Lovelace, to Sketch of the Analytical Engine published in 1842.
In this Ada saw what none of her contemporaries, Babbage included, appear to have seen. Ascribing this essential and historic understanding to her is not a backwards projection from our own age. She is explicit. This is Ada, banging the table, saying it is this that is significant and revolutionary about automatic computation. So Lovelace is rightly celebrated but not for the reasons commonly cited.
Supposing, for instance, that the fundamental relations of pitched sounds in the science of harmony and of musical composition were susceptible of such expression and adaptations, the engine might compose elaborate and scientific pieces of music of any degree of complexity or extent.and then
Many persons who are not conversant with mathematical studies, imagine that because the business of the engine is to give its results in numerical notation, the nature of its processes must consequently be arithmetical and numerical, rather than algebraical and analytical. This is an error. The engine can arrange and combine its numerical quantities exactly as if they were letters or any other general symbols; and in fact it might bring out its results in algebraical notation, were provisions made accordingly. It might develope three sets of results simultaneously, viz. symbolic results (as already alluded to in Notes A. and B.), numerical results (its chief and primary object); and algebraical results in literal notation. This latter however has not been deemed a necessary or desirable addition to its powers, partly because the necessary arrangements for effecting it would increase the complexity and extent of the mechanism to a degree that would not be commensurate with the advantages, where the main object of the invention is to translate into numerical language general formulæ of analysis already known to us, or whose laws of formation are known to us. But it would be a mistake to suppose that because its results are given in the notation of a more restricted science, its processes are therefore restricted to those of that science.What Lovelace appears to be referring to is the type of machines we all have today. Underneath the hood only numbers abound, but numbers can be used to represent almost anything and computation on numbers gives us word processors, streaming video, MP3 players and more.
Calling Lovelace the first programmer has always seemed a bit silly because surely Babbage would have written some programs for his machine. But recognizing her for "Lovelace's Leap" seems far more realistic.
Why be the first programmer when you could be the prophet of the information age?