time and energy, would be capable of calculating any “ computable ” problem. In that paper, “ On Computable Numbers, with an Application to the Entscheidungs Problem, ” Proceedings of the London Mathematical Society, Ser. 2 42 (1937), Turing demonstrates the range of problems that in fact are not computable. The figure of the Turing machine, as a philosophical and machinic hypothesis, stands for the technology of universal computation and for the ultimate limits of computation within mathematics.
- Nicholas Gane and Stephen Sale, “ Interview with Friedrich Kittler and Mark Hansen, ” Theory, Culture, and Society 24 (2007): 323 – 329.
- Lucretius, De Rerum Natura, 1 B.C., in Lucretius on the Nature of the Universe, ed. R. E. Latham (London: Penguin Books, 1951):
We wish thee also well aware of this:
The atoms, as their own weight bears them down
Plumb through the void, at scarce determined times,
In scarce determined places, from their course
Decline a little — call it, so to speak,
Mere changed trend. For were it not their wont
Thuswise to swerve, down would they fall, each one,
Like drops of rain, through the unbottomed void;
And then collisions ne’er could be nor blows
Among the primal elements; and thus
Nature would never have created aught …
… For whatsoever through the waters fall,
Or through thin air, must quicken their descent,
Each after its weight — on this account, because
Both bulk of water and the subtle air
By no means can retard each thing alike.
- The implications continue to play out in contemporary debates from artificial intelligence to the philosophy of physics (e.g., if emergent calculative form is universal, then not only is “ human thought ” no longer a necessary referent model for artificial intelligence, but its own ideas about deterministic rationality do very little to guarantee the regime of computational processing, and the other way around). Computing machines do not differentiate “ rational ” instructions from irrational ones, and we needn ’ t even know what is going on within a computer that makes computation interesting. Computation takes place outside any direct reliance on necessary reason, or “ regimes of knowledge primarily associated with the laws of thought — [it] does things outside of it. ” See Robert Jackson, “ Negarastani, Computing and Knowledge, ” Algorithm and Contingency, December 2, 2012, http://robertjackson.info/index/2012/12/negarestani-computing -and-knowledge/.
- Turing himself began to address the theoretical foundations of biological systems, with “ The Chemical Basis of Morphogenesis, ” published in 1952 in Philosophical Transactions of the Royal Society of London B 237 (641): 37 – 72.
- Once again, I am not qualified to offer even an implied opinion on whether matter is fundamentally discrete or continuous, or related questions. For a general account on behalf of the strong digital physics position, see, for example, Seth Lloyd, Programming the Universe: A Quantum
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