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In his The Road to Reality, Roger Penrose espouses three distinct realities - the physical, mental and mathematical.

The physical and mental are basically good old dualism, although he is an atheist and the mental aspect is accordingly shorn of angels dancing on pinheads. The mathematical is in essence Plato's realm of ideals.

Although my own ideas differ in much detail from his, this eternal triptych or whatever it is called appeals to me. However I remain unconvinced that the worlds of Plato and of experience are necessarily distinct. It might be that every quale of experience is attached to some ideal, and that every ideal has a quale attached to it, such that quale and idea are merely aspects of a single entity. It seem to me that information theory, especially in its philosophical aspect of semantics, must have a strong role to play in resolving this question. But I need to better understand the reasons for considering these worlds distinct.

Dualism has been flogged to death in philosophical circles, but has this kind of triple-ism received significant attention?

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    It's similar to Popper's three worlds, though Popper did not restrict world 3 to math as Penrose does. Don't know what kind of analysis this received from others though. If you're interested in the concept of associating qualia with mathematics or information structures and removing the notion of a distinct physical realm, you might check out Chalmers' "double-aspect principle" at informationphilosopher.com/solutions/philosophers/chalmers and I discussed a version of this at philosophy.stackexchange.com/a/72799/10780 – Hypnosifl Dec 8 '20 at 20:42
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    @sand1 - Logic is about propositions, the "three worlds" is a metaphysical idea which doesn't involve any denial that every well-defined proposition is either true or false. – Hypnosifl Dec 8 '20 at 21:13
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    @sand1 Classical logic is binary, three-valued logic with something like undecided for the third value is arguably implicit in human reasoning, and was explicitly pondered at least as far back as Ockham. Fichte's dialectic triad (often misattributed to Hegel) also comes to mind. Not that this has any non-metaphorical import, even under binary logic one is free to make triple or multiple classifications: Agrippa's trilemma/pentalemma in epistemology, trichotomy (less, more, equal) in classical mathematics, etc. Peirce also constructed trialistic metaphysics (different from Popper's). – Conifold Dec 8 '20 at 21:27
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    @sand1 - Logic doesn't say anything about not being able to "consider meaningfully more than two things at once", and you can certainly have individual propositions in logic that concern more than two objects or properties or sub-propositions. – Hypnosifl Dec 8 '20 at 23:21
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    @sand1 Can't we? Parmenides thought that we can think only One, and even not-One is unthinkable. And yet we do conceive change easily. Already Plato resolved it into a triad, being/non-being/becoming. Hegel suggested that reason can rise above “thinking that belongs to the understanding alone” and “the mere logic of the understanding”. I think many people are with Plato and Hegel on this. Peirce was in his semiotic, his Thirdness is the way of the mind. Secondness is the way of action, we do need a yes/no to act, but that is a pragmatic necessity, not an intellectual one. – Conifold Dec 9 '20 at 1:12
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Roger Penrose, The Road to Reality (Vintage, p.1029)

each of the three worlds - platonic mathematical, physical, mental - has its own kind of reality and each is founded in the one that precedes (the worlds being taken cyclically)

There are monists, dualists, pluralists, etc.

Dualism has been flogged to death in philosophical circles, but has this kind of triple-ism received significant attention?

Obviously it has not; usually it is reduced to some dualism and next it can be flogged to death. Apparently the main problem comes from a kind of prejudice that every ordering is hierarchical while cyclical arrangements created by non-transitive relations are brutally ignored. Popper's proposal is that for any two elements of his construct there is one that is more fundamental but there is not a most fundamental one among the three. German idealism has proposed mind, Physicalism insisted on matter and Platonism upheld mathematics.

'Rock, paper, scissor' game is a well known example and I have always wondered how the Ancient Greeks have not discovered something similar. The earliest mention seems to be from the 18th c. about three chess players among who no one is the best. Kenneth Arrow made popular the idea in mid-20th. c. Penrose exposed his ideas in the Tanner Lectures of 1994/5 and they were published with comments[1]; Tegmark and al. [2] discussed it in 2006 after The Road to reality appeared and their paper contains enough details; a Fqxi essay from 2017 also commented the idea[3].

Refs. [1] Penrose R., The Big, the Smalland the Brain (1997), includes comments by A. Shimony, N. Cartwright, and S. Hawking: see Fig 3.3; earlier it appears in his Shadows of the Mind (1994) [2] Tegmark M., et al On Math, Matter and Mind arXiv:physics/0510188v2 [3] Losev A., A Fundamental Loop

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    Comments have so far led me to Plato, Frege and Popper as triple-ists. All were significant philosophers. So I find the "obviously not" opener to this answer somewhat unjustified. – Guy Inchbald Dec 9 '20 at 11:46
  • This is a good answer, but it can be improved. It would be useful to add reference to Popper's Tanner Lecture tannerlectures.utah.edu/_documents/a-to-z/p/popper80.pdf and Frege as prior major thinkers. This recent triplest essay also cites Paul Davies positively: steelpillow.com/blocki/philosophy/threelevels.html – Dcleve Jan 9 at 21:44
  • @Dcleve That "recent triplist essay" is my own work in hand! Towards the end, it poses the question I am asking here. Davies' main exposition of his thesis appears to be his book The Demon in the Machine (the titular resemblance to Arthur Koestler's The Ghost in the Machine must be no coincidence), which I evidently need to get a copy of. – Guy Inchbald Jan 10 at 8:03
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    @GuyInchbald -- About half of philosophers ascribe to lower p platonism of one type or another (math, and ethics are the two most popular areas to admit to realism for). Platonism plus emergent consciousness gets one to triplism, and that is how Popper got there, despite his initial physicalist inclinations. Popper's book length treatment of this is The Self and Its Brain. – Dcleve Jan 11 at 7:37
  • Information is one subset of the world of ideas. Another is logic. Neither create consciousness nor quales -- they are as different from quales as water is from fear. Popper noted that consciousness has an intrinsic time element, while ideas are timeless. The dramatic failure of the 3/4 century plus long effort of AI to somehow have algorithms create consciousness, has been a pretty definitive demonstration that consciousness and math/logic/algorithms/information are simply two distinct and unrelated areas. – Dcleve Jan 11 at 7:45

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