Editing Church–Turing thesis (section)

==Philosophical implications==
Philosophers have interpreted the Church–Turing thesis as having implications for the [[philosophy of mind]].<ref>{{cite journal |author-last=Abramson |author-first=Darren |title=Philosophy of mind is (in part) philosophy of computer science. |journal=Minds and Machines |date=2011 |volume=21 |issue=2 |pages=203–219 |doi=10.1007/s11023-011-9236-0 |s2cid=32116031 |url=https://dl.acm.org/doi/abs/10.1007/s11023-011-9236-0}}</ref><ref>{{Cite SEP |author-last=Copeland |author-first=B. Jack |author-link=Jack Copeland |date=2017-11-10 |title=The Church-Turing Thesis |url-id=church-turing}}</ref><ref>For a good place to encounter original papers see {{cite book |editor-first=David J. |editor-last=Chalmers |editor-link=David Chalmers |date=2002 |title=Philosophy of Mind: Classical and Contemporary Readings |publisher=Oxford University Press |location=New York |isbn=978-0-19-514581-6 |oclc=610918145}}</ref> [[Jack Copeland|B. Jack Copeland]] states that it is an open empirical question whether there are actual deterministic physical processes that, in the long run, elude simulation by a Turing machine; furthermore, he states that it is an open empirical question whether any such processes are involved in the working of the human brain.<ref>{{cite book |author-first=B. Jack |author-last=Copeland |author-link=Jack Copeland |section=Computation |editor-first=Luciano |editor-last=Floridi |editor-link=Luciano Floridi |title=The Blackwell guide to the philosophy of computing and information |publisher=Wiley-Blackwell |date=2004 |isbn=978-0-631-22919-3 |page=15}}</ref> There are also some important open questions which cover the relationship between the Church–Turing thesis and physics, and the possibility of [[hypercomputation]]. When applied to physics, the thesis has several possible meanings:

#The universe is equivalent to a Turing machine; thus, computing [[Computable_function|non-recursive functions]] is physically impossible. This has been termed the strong Church–Turing thesis, or [[Church–Turing–Deutsch principle]], and is a foundation of [[digital physics]].
#The universe is not equivalent to a Turing machine (i.e., the laws of physics are not Turing-computable), but incomputable physical events are not "harnessable" for the construction of a [[hypercomputation|hypercomputer]]. For example, a universe in which physics involves random [[real numbers]], as opposed to [[computable number|computable real]]s, would fall into this category.
#The universe is a [[hypercomputation|hypercomputer]], and it is possible to build physical devices to harness this property and calculate non-recursive functions. For example, it is an open question whether all [[quantum mechanics|quantum mechanical]] events are Turing-computable, although it is known that rigorous models such as [[quantum Turing machine]]s are equivalent to deterministic Turing machines. (They are not necessarily efficiently equivalent; see above.) [[John Lucas (philosopher)|John Lucas]] and [[Roger Penrose]] have suggested that the human mind might be the result of some kind of quantum-mechanically enhanced, "non-algorithmic" computation.<ref>cf. {{cite book |author-last=Penrose |author-first=Roger |author-link=Roger Penrose |date=1990 |chapter=Algorithms and Turing machines <!--|section=The Church–Turing Thesis--> |pages=47–49 |title=The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics |publisher=Oxford University Press |location=Oxford |oclc=456785846 |isbn=978-0-19-851973-7 |title-link=The Emperor's New Mind}}</ref><ref>Also the description of "the non-algorithmic nature of mathematical insight", {{cite book |author-last=Penrose |author-first=Roger |author-link=Roger Penrose |date=1990 |chapter=Where lies the physics of mind? |pages=416–418 |title=The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics |publisher=Oxford University Press |location=Oxford |oclc=456785846 |isbn=978-0-19-851973-7 |title-link=The Emperor's New Mind}}</ref>

There are many other technical possibilities which fall outside or between these three categories, but these serve to illustrate the range of the concept.

Philosophical aspects of the thesis, regarding both physical and biological computers, are also discussed in Odifreddi's 1989 textbook on recursion theory.<ref>{{cite book |author=Piergiorgio Odifreddi |author-link=Piergiorgio Odifreddi |title=Classical Recursion Theory |location=Amsterdam, Netherlands |publisher=North Holland |series=Studies in Logic and the Foundations of Mathematics |volume=125 |date=1989}}</ref>{{rp|pages=101–123}}