Communications of the ACM
What Came First, Math or Computing?
One of the most fundamental conundrums in the philosophy of mathematics is the question of whether mathematics was discovered by humans or invented by them. On one hand, it seems hard to argue that highly sophisticated mathematical objects, such as inaccessible cardinals, were discovered. On the other hand, as Albert Einstein asked, "How can it be that mathematics, being after all a product of human thought, which is independent of experience, is so admirably appropriate to the objects of reality?" The 19th century mathematician Leopold Kronecker offered a compromise, saying "God created the integers, all else is the work of man."
So let us consider the natural numbers. The Lebombo Bone is a bone tool made of a baboon fibula with incised markings, discovered in a cave in the Lebombo Mountains in Africa. More than 40,000 years old, the bone is conjectured to be a tally stick, its 29 notches counting, perhaps, the days of the lunar phase. It has referred to as the oldest mathematical artifact. But should we not call it the oldest computing artifact? Counting is, after all, the most basic form of computing.
Comments
Martin Wheatman
Dear Moshe,
You forgot to mention one important machine in your list of early computers: The Manchester Baby, the first electronic stored program computer, which ran on the 21st June, 1948. But why is this so important? It demonstrates Turings notion of Universality: the property that both program and data can be stored in memory, not merely programmed using switches on the front of a device.
With this omission, however, comes an important thread missing from your piece. Mathematics would be difficult without the invention of the clay tablet about 3500 BCE. This supported the logistics to sustain the city of Sumer in Mesopotamia. This developed into paper, the printing press and other machines which can use the printed word.
But where were we before the clay tablet? Were there cultures before writing? Presumably, the example of a 40,000 year old tally stick shows there were. What Im asking you to consider is, can an aural culture compute? If so, then perhaps speech is the Universal Machine.
But this is the thing, this is self-evident, if I can conditionally state this conjecture, then Im already reasoning with ordinary language. John Austin, in his William James Lecture at Harvard in 1955, laid out the idea of conditionality in speech, in his idea of felicity or the happiness of the outcome of a statement. Conditionality is one of the main ideas behind Turing Completeness. So if I can say, if so, , or if not, , I am reasoning in speech.
Similarly with loops, another idea behind Turing Completeness. If you can say, Do X with this, and Do X with these, you can also say, Do X with the first of these, and Do X with the rest of these, and you are processing a listcreating a loop, essentially what Kurt Godel was doing with recursive functions.
If youve made it this far, I hope we can agree that speech precedes writing, and that math is (at best) limited to simple arithmetic without writing. However, by successfully conveying an idea, I am essentially programming your mind, and speech is computational. Therefore, computation precedes math.
Many thanks for your thought provoking piece,
Martin
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P.S. If you need a demonstration of this: of how speech contains both data and action, and is therefore a Universal, this can be demonstrated by the software at bitbucket.org/martinwheatman/enguage
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