This is the way I see it happening. As the transmissions from the early days of radio propagate into space in an ever expanding sphere, outposts on distant planetary systems will begin to detect those transmissions and send us back carefully engineered ‘commercials’ that depict themselves as everything that we desire of an alien intelligence; that they are benevolent, wise, and deeply concerned for our welfare. They will then give us instructions on how to build a machine that will cure all the problems of the world and make us all happy. When the machine is complete, it will ‘disinfect’ the planet of competing life forms and begin to scan the skies from earth in search of further nascent planets.
But the concept of numbers runs much deeper than a simple one, two, three. Before you can even think of counting, you have to understand the concept of countable things. Whether you are counting people, coins, or pebbles on a beach, you must decide on the set of things you want to count, and what requirements you should establish to qualify for inclusion in that set. How big can a pebble be before it counts as a rock instead of a pebble, and how small before it counts as a grain of gravel or sand? The answer depends on why you are counting them, that requires a fine mathematical judgment. Counting requires that all the members of the set can be considered equivalent, interchangable, they each count for exactly one. This is so natural and obvious an assumption (to us humans) that we don’t even realize that we are making it. But it is an artifact of the human mind, not of the natural world, that things come in identical countable units. In reality pebbles are not at all identical, each one has its own unique size and shape and color, and the distinctions between rock and pebble and gravel are overlapping and indefinite. The fact that we can make ourselves see a set of pebbles as identical units is itself a conceptual feat that is a prerequisite to learning to count them.
The schema is the framework that gives meaning to the math. It is what poses the question in the first place, and what interprets the meaning of the results at the end. That is the real mathematical thinking beyond mere arithmetic, the framing of the problem to be solved, the selection of the algorithm to be used to solve it, and understanding the significance of the results after the counting is done. The counting itself is the simplest part of the problem – so simple that even a stupid computer can do it. But conceptualizing the situation and seeing the schema in it, is in fact the real mathematical part of the task. The rest is merely arithmetic.
And yet the digital computer is totally incapable of even the most primitive mathematical thought. The computer is completely incapable of conceptualizing a schema, or understanding the significance of the quantities that it calculates. It seems that there are two starkly contrasting aspects of math, one which is thoroughly understood, which can be performed by stupid machines even better than by humans, the other that we all do unconsciously and instinctively virtually every waking moment, but have no idea how we do it or even what it is we are doing.