Abstract
In the present chapter, we shall deal with the general logical–epistemological foundations of the quantum theory, where the stress is on categorisation. This aspect and the physical–ontological ones previously discussed need finally to agree. I first give a general account of category theory. Then, we shall see its applications to QM and especially to quantum information. Then a logical and epistemological assessment of the theory follows.
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Notes
- 1.
Abramsky and Tzevelekos (2011).
- 2.
- 3.
See Chaitin (1998, Sect. 1.1).
- 4.
- 5.
- 6.
Leinster (2014, pp. 71–73).
- 7.
- 8.
Leinster (2014, pp. 79–80).
- 9.
- 10.
I have previously used the symbol \(\longmapsto \) for some mappings between physical systems. In this sense, I have treated them as individual systems, what is in general correct. There is some ambiguity when we deal with information, since in that case we abstract from the particular physical characters of the system. Nevertheless, also in this case, we often deal with a qubit in a well definite state.
- 11.
Spivak (2013, Chap. 3 and Sects. 4.1, 4.2).
- 12.
Geroch (1985, pp. 18–19).
- 13.
The error was found by Russell and deals with the fact that both the definitions of “the set of the sets that are not members of themselves” and of “the set of the sets that are members of themselves” (and similar sets) lead to a contradiction that cannot be resolved within the system that defines the set itself (Russell 1902, 1903) .
- 14.
Leinster (2014, Chap. 5).
- 15.
Spivak (2013, Sect. 5.1).
- 16.
Leinster (2014, Chap. 4).
- 17.
Spivak (2013, Sect. 4.3).
- 18.
Baez and Stay (2011).
- 19.
For what follows see Abramsky and Coecke (2009) .
- 20.
For what follows see Abramsky and Coecke (2009) .
- 21.
I make here a simplification avoiding the complexities arising from compact closed categories.
- 22.
Auletta (2013b).
- 23.
Boole (1854).
- 24.
Auletta (2013c).
- 25.
For a canonical introduction to Boolean algebra see Givant and Halmos (2009).
- 26.
See Boole (1854, p. 33).
- 27.
Spivak (2013, Sect. 3.4).
- 28.
Auletta (2013b, Chap. 1).
- 29.
Auletta (2013b, Chap. 8).
- 30.
Givant and Halmos (2009, p. 45).
- 31.
- 32.
Leibniz (1666).
- 33.
Givant and Halmos (2009, pp. 117 and 127).
- 34.
Givant and Halmos (2009, Chap. 4).
- 35.
Poincaré (1902, p. 49).
- 36.
Poincaré tells us that formal logic is nothing else than the study of the properties that are common to any classification (Poincaré 1909, p. 9).
- 37.
See Givant and Halmos (2009, pp. 149–150).
- 38.
Presented for the first time in Auletta (2015c).
- 39.
For the notion of vector space see Byron and Fuller (1969, I, Sect. 3.1).
- 40.
Byron and Fuller (1969, I, Sect. 3.2).
- 41.
Byron and Fuller (1969, I, Sect. 3.3).
- 42.
Auletta (2015c).
- 43.
Gödel (1931) .
- 44.
Carnap tells us that a class does not consist only of its members (Carnap 1928, Sect. 37).
- 45.
Peirce (1898, p. 247).
- 46.
See e.g. Peirce (1898, pp. 162–163).
- 47.
- 48.
Chellas (1980).
- 49.
Armstrong (1983, p. 82 and ff).
- 50.
- 51.
- 52.
Peirce (1884, pp. 553–554), Peirce (1891, p. 106). See also Auletta (2011a, Chap. 3). Peirce’s view has been also supported in Smolin (2013) , although the main thesis there is quite different from that supported here as far as the author rejects conservation laws and symmetries, which are, at the opposite, central to my approach. I remark that in the Introduction, Smolin quotes interesting statements of Dirac that also support the evolution of laws.
- 53.
Peirce (1887, p. 208).
- 54.
- 55.
It seems that what follows is not far away from the spirit of Epperson and Zafiris (2013). Anyway, they are among the few scholars to take Boolean algebra as fundamental for QM.
- 56.
- 57.
- 58.
As well understood by Ludwig
- 59.
See also Busch et al. (1995, pp. 25–26).
- 60.
Eddington (1939, p. 5) He further noted that physics is more and more based on epistemological principles as well as mathematics on logical ones. He seems however to forget this when he repeatedly affirms (for instance Eddington 1939, p. 89) that in relativity we observe relations and in QM we observe probabilities. See also Eddington (1939, pp. 87–88) .
- 61.
Auletta (2011a, Chap. 8).
- 62.
On this point see Auletta (2016a).
- 63.
- 64.
Auletta (2013a).
- 65.
Uhlen and Ponten (2005).
- 66.
Alberts et al. (1983, Chaps. 2, 7) .
- 67.
- 68.
Auletta (2016b).
- 69.
Conant and Ashby (1970).
- 70.
Auletta and Jeannerod (2013).
- 71.
- 72.
- 73.
- 74.
- 75.
Auletta (2013a).
- 76.
A first, still immature, analysis in Auletta (2008a).
- 77.
- 78.
- 79.
- 80.
- 81.
- 82.
- 83.
As pointed out in Peirce (1903a, p. 167). Note that here and in the following, I use the term invalidating feedback for denoting a signal that contradicts the expectations or indicates that things do not proceed in the right way. Sometimes scholars, included myself, lacking a generally acknowledged term, use negative feedback to this purpose, although the latter term has a different technical use.
- 84.
See Schrödinger (1958, Chap. 2).
- 85.
See Auletta et al. (2013).
- 86.
Also Deutsch seems to agree on the relevance of control (Deutsch 2011, Chap. 3).
- 87.
Mach (1905).
- 88.
See Nielsen and Chuang (2000, p. 249).
- 89.
On this point see Auletta (2011a, Sect. 4.1) and references therein.
- 90.
- 91.
- 92.
- 93.
Auletta (2015a).
- 94.
Crile (1941, p. 211).
- 95.
“Les vérités ne sont fécondes que si elles sont enchaînées les unes aux autres” (Poincaré 1897b).
- 96.
- 97.
- 98.
Marshall-Pescini and Whiten (2008).
- 99.
- 100.
- 101.
Gibson (1979).
- 102.
Hauser (1996).
- 103.
Auletta (2011a, 3rd part).
- 104.
- 105.
Auletta (2011a, Chap. 19).
- 106.
Peirce (1866, pp. 405–406).
- 107.
- 108.
Peirce (1868b, p. 214), Here, not by chance, he says that perceptions represent premises for our further reasoning. It is also true that he did acknowledge that to perceive represents also a kind of discontinuity (Peirce 1903b, p. 191–94), in accordance with our interpretation of information selection. See also Margenau (1950, Sect. 4.1).
- 109.
- 110.
- 111.
Aristotle An. post. (2019a, 75a38-75b2 and 76b13-16).
- 112.
Auletta (2013d).
- 113.
- 114.
Peirce (1865, pp. 187–189 and 284–286).
- 115.
Peirce (1868a, p. 83)
- 116.
Peirce (1878)
- 117.
See Margenau (1950, Sect. 5.6).
- 118.
Auletta (2009).
- 119.
Einstein (1934, pp. 164–165).
- 120.
Peirce (1865, p. 179).
- 121.
Peirce (1865, pp. 187–189 and 284–286).
- 122.
Peirce (1865, p. 292).
- 123.
Peirce (1866, pp. 458–471).
- 124.
Peirce (1866, p. 452).
- 125.
Auletta (2017).
- 126.
- 127.
- 128.
- 129.
- 130.
- 131.
Kuhn (1962).
- 132.
Auletta (2017).
- 133.
- 134.
- 135.
This is the main thesis of Auletta (2011a).
- 136.
Poincaré (1897a).
- 137.
- 138.
Popper (1934, pp. 16–17). See also Deutsch (1997, p. 62). At the opposite, C. Peirce, although having much more clarity about the logical and inferential background of science, showed still some incertitude about the non-positive (non-instructive) character of experience. On the issue of induction in Peirce see also Shimony (1970, pp. 231–235).
- 139.
- 140.
Auletta (2013b).
- 141.
Auletta (2017).
- 142.
- 143.
- 144.
- 145.
To a certain extent, this stage corresponds to the so-called auxiliary hypothesis that does not change the core of the theory, as proposed in Lakatos (1976).
- 146.
Poincaré (1902, Chap. 10).
- 147.
- 148.
Auletta (2011b, Sect. 5.4).
- 149.
Einstein (1936, p. 96).
- 150.
A first hint in Peirce (1898, p. 198).
- 151.
D’Ariano et al. (2017, Sects. 3.1, 3.2 and 4.2).
- 152.
As analysed in Ludwig (1978, pp. 8–9).
- 153.
Auletta and Torcal (2011).
- 154.
D’Ariano et al. (2017, Sect. 6.1).
- 155.
D’Ariano et al. (2017, Chap. 5) .
- 156.
I quote here Bell (1981).
- 157.
Only in this specific sense I could agree with the dictum recalled in the Introduction, when the Nobel Prize winner Feynman says that QM cannot be explained or understood (Feynman et al. 1965, III, 1–1). Otherwise it would imply a renouncement to scientific research.
- 158.
- 159.
As stressed in Auletta (2011a, Chap. 12), where further references can also be found.
- 160.
The philosophy of the postmodern world dominated by simulacra has been introduced in Baudrillard (1981).
- 161.
Hume (1739, p. 253).
- 162.
As pointed out in Auletta (2011a, Sect. 6.1).
- 163.
This has been the basic insight of Auletta (2011a).
- 164.
- 165.
- 166.
Examination and literature in Auletta (2011a, Chaps. 4 and 13).
- 167.
It is not by chance that Poincaré said that organisms immobile (like plants) could not build a geometry, because they do not move and act in the space (Poincaré 1905, pp. 68–69).
- 168.
- 169.
Wheeler (1988).
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Auletta, G. (2019). Category Theory and Quantum Mechanics. In: The Quantum Mechanics Conundrum. Springer, Cham. https://doi.org/10.1007/978-3-030-16649-6_8
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