Mod+ Platonism & Mathematics [Resources][Platonism]



Like other resources threads, idea here is mostly to provide material for people wishing to investigate the topic.

Some commentary/debate is useful but please, if such discussion seems to be getting long [over 3-5 posts] create a separate thread and link to continue.


While this thread was gobbled up twice, going to try this again, starting with Massimo Piggilucci's essay on why he became open to Mathematical Platonism:

If one ‘goes Platonic’ with math, one has to face several important philosophical consequences, perhaps the major one being that the notion of physicalism goes out the window. Physicalism is the position that the only things that exist are those that have physical extension [ie, take up space] – and last time I checked, the idea of circle, or Fermat’s theorem, did not have physical extension. It is true that physicalism is now a sophisticated doctrine that includes not just material objects and energy, but also, for instance, physical forces and information. But it isn’t immediately obvious to me that mathematical objects neatly fall into even an extended physicalist ontology. And that definitely gives me pause to ponder.
eta: Massimo's not a Platonist anymore, see here.
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Wigner's Unreasonable Effectiveness of Mathematics in the Natural Sciences

Here's the full paper for those so inclined to read it.

Most of what will be said on these questions will not be new; it has probably occurred to most scientists in one form or another. My principal aim is to illuminate it from several sides. The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it. Second, it is just this uncanny usefulness of mathematical concepts that raises the question of the uniqueness of our physical theories. In order to establish the first point, that mathematics plays an unreasonably important role in physics, it will be useful to say a few words on the question, "What is mathematics?", then, "What is physics?", then, how mathematics enters physical theories, and last, why the success of mathematics in its role in physics appears so baffling. Much less will be said on the second point: the uniqueness of the theories of physics. A proper answer to this question would require elaborate experimental and theoretical work which has not been undertaken to date.
Penrose is also a proponent of a Platonic reality, and believes he makes at least a limited case for it in Emperor's New Mind + Shadows Of The Mind. 'Eureka Moments' also hint at this possibility, and of course the Shamanic/Psychedelic Experience (as documented by the likes of Narby, Strassman, Hancock, etc.) lends credence to this notion of a Platonic "Idea Space" where useful information is extracted out of. And of course there is much discussion in mainstream circles (Laughlin, Kauffman, Roy, Bohm, Hiley, etc.) about the idea of space-time, photons, etc. as phase transitions that emerged from "Pre-Space" or "Proto-Space".

Not interested in really debating any of this, just thought I'd mention some ideas relating to it.

Bart V

straw materialist
Massimo talks about this again in the latest episode of his podcast "rationally speaking", and the one before that.
Also in the latest (repeat) episode of the "big picture science" podcast they talk about mathematical platonism.
All of these are interesting, but they only speak of mathematical platonism. In the title of this thread you seem to equate that with a platonic reality.
Do you think they are the same?


Discussions on whether Mathematics is Eternal.

Part 2 here.

Interesting ideas relating beautiful proofs to eroticism. I do sort of see what Chatain's getting at, the mystery of subjective attraction and association of beauty with grace does relate to the feeling of seeing an elegant proof, as does the artistic confrontation with Truth and the new truth that emerges from extant theorems.

I've talked with friends before about how proofs seem to come from a mysterious room in the mind, where one can marvel at a proof coming together in one's own mind. I recall one problem from a class that require proving something or other about permutations. I had no clue how to do it yet when I sat and just let the mind work it seemed to emerge almost miraculously from my own thoughts.

Chaitan discusses Ramunajan in the first part, and how the Indian mathematician thought a Hindu goddess communicated the truths of Math to him. In the second he mentions Cantor trying to use math to get to God, and Euler's explosion of discoveries that seem to defy expectation. Chaitan thinks mathematics is fundamentally irrational, and a communion with something mysterious.

While subjective experience can be deceptive I do encourage people to try to do some proofs here and there and just feel what it's like. It might feel tedious....but just maybe you might feel like either some other is communicating with you, or you're experiencing Anamensis - the gnostic Loss of Forgetfulness that made Plato think the soul was immortal and all learning merely remembering.



Brian D. Josephson & Tethys Carpenter

We present an account of the phenomenon of music based upon the hypothesis that there is a close parallel between the mechanics of
life and the mechanics of mind, a key factor in the correspondence proposed being the existence of close parallels between the concepts of gene and musical idea. The hypothesis accounts for the specificity, complexity, functionality and apparent arbitrariness of musical structures. An implication of the model is that music should be seen as a phenomenon of transcendental character, involving aspects of mind as yet unstudied by conventional science.


Karl Popper's the Three Worlds of Knowledge

Karl Popper theorizes that there are three worlds of knowledge:
  • World 1 is the physical universe. It consists of the actual truth and reality that we try to represent, such as energy, physics, and chemistry. While we exist in this world, we do not always perceive it and then represent it correctly.
  • World 2 is the world of our subjective personal perceptions, experiences, and cognition. It is what we think about the world as we try to map, represent, and anticipate or hypothesis in order to maintain our existence in an every changing place. Personal knowledge and memory form this world, which are based on self-regulation, cognition, consciousness, dispositions, and processes. Note that Polanyi's theory of tacit and explicit knowledge is based entirely within this world.
  • World 3 is the sum total of the objective abstract products of the human mind. It consists of such artifacts as books, tools, theories, models, libraries, computers, and networks. It is quite a diverse mixture. While knowledge may be created and produced by World 2 activities, its artifacts are stored in World 3, for example a claw-hammer, Maslow's hierarchy of needs, and Godel's proof of the incompleteness of arithmetic. Popper also includes genetic heredity (if you think about it, genes are really nothing more than a biological artifact of instructions


The mathematical world -> Some philosophers think maths exists in a mysterious other realm. They’re wrong. Look around: you can see it.

Still, despite its clean lines and long history, Platonism cannot be right either. Since the time of Plato himself, nominalists have been urging very convincing objections. Here’s one: if abstracta float somewhere outside our own universe of space and time, it’s hard to imagine how can we see them or have any other perceptual contact with them. So how do we know they’re there? Some contemporary Platonists claim that we infer them, much as we infer the existence of atoms to explain the results of chemistry experiments. But that seems not to be how we know about numbers. Five-year-olds learning to count don’t perform sophisticated inferences about abstractions; their contact with the numerical aspect of reality is somehow more perceptual and direct. Even animals can count, up to a point.
Aristotelian realism stands in a difficult relationship with naturalism, the project of showing that all of the world and human knowledge can be explained in terms of physics, biology and neuroscience. If mathematical properties are realised in the physical world and capable of being perceived, then mathematics can seem no more inexplicable than colour perception, which surely can be explained in naturalist terms. On the other hand, Aristotelians agree with Platonists that the mathematical grasp of necessities is mysterious. What is necessary is true in all possible worlds, but how can perception see into other possible worlds? The scholastics, the Aristotelian Catholic philosophers of the Middle Ages, were so impressed with the mind’s grasp of necessary truths as to conclude that the intellect was immaterial and immortal. If today’s naturalists do not wish to agree with that, there is a challenge for them. ‘Don’t tell me, show me’: build an artificial intelligence system that imitates genuine mathematical insight. There seem to be no promising plans on the drawing board.


If It’s Possible, It Happened -> ‘Our Mathematical Universe’: A Case for Alternate Realities

...Our reality, in other words, is not just described by mathematics, it is mathematics.

The vision of a purely mathematical universe, one that can be understood solely through rigorous mathematical reasoning, is far from new. As far back as the sixth century B.C., the Pythagoreans declared that “all is number,” and in the 17th century A.D. Descartes tried to deduce the universe from first principles. Other 17th-century rationalists, including Hobbes and Leibniz, offered their own versions of the mathematical universe.

Such efforts to comprehend the world entirely through mathematical reasoning certainly had their triumphs, none more so than when Einstein upended our universe using nothing more than careful reasoning and mathematical calculations.

But the overall record for such attempts is decidedly mixed: The Pythagoreans, who insisted that everything was made of whole numbers and their ratios, foundered on the discovery of irrationals, and Descartes concluded that matter was simple extension in space, that a vacuum was impossible and that the planets swirled in vortexes of ether.

It is difficult to say whether Dr. Tegmark’s mathematical universe will ultimately be deemed an Einsteinian triumph or a Cartesian dead end. His conclusions are simply too far removed from the frontiers of today’s mainstream science, and there is little hope that conclusive evidence will emerge anytime soon.


Put up some stuff from Massimo's critique of reductionism and mechanism in the relevant thread, but wanted to make note of his plug for Platonism*:

'One last parting shot, about a topic that the astute reader may have noticed I have bypassed so far: if every thing is gone and we only have mathematical structures and relations, what is the ontological status of mathematical objects themselves? Here are the only relevant quotes from Ladyman and Ross that I could find:

" OSR as we develop it is in principle friendly to a naturalized version of Platonism. ... One distinct, and very interesting, possibility is that as we become truly used to thinking of the stuff of the physical universe as being patterns rather than little things, the traditional gulf between Platonistic realism about mathematics and naturalistic realism about physics will shrink or even vanish. ... [Bertrand Russell] was first and foremost a Platonist. But as we pointed out there are versions of Platonism that are compatible with naturalism; and Russell’s Platonism was motivated by facts about mathematics and its relationship to science, so was PNC [Principle of Naturalistic Closure] -compatible."
Wild stuff, no? Now I don’t feel too badly about having written in sympathetic terms about mathematical Platonism...'

*I note the similarity to Feser's conception of the soul here.

eta: Massimo's not a Platonist anymore, see here.
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I chuckled when I read this JC. I am convinced that the quality of teaching is a factor here.
Definitely. I started off as a terrible math student, almost held back in 2nd grade.

Years later I graduated with honors in the subject, though in the course of work life I've forgotten a good deal.

Who my teachers were made all the difference.

I really do think everyone should take the time to learn some basic math proofs. There's just something incredible about looking an exercise, thinking you'll never have an answer, and then feel it arise from your mind.

Plato himself was incredibly fascinated by this process, as he notes in the Dialogue entitled Meno this inspired him - via arguments elucidated by Feser - to believe mind transcends brain and set the the course for a good deal of Western Philosophy and Christian/Muslim/Gnostic thinking. While one doesn't have to agree with Plato's conclusions regarding reincarnation, the World of Forms, and all that jazz the feeling of solving mathematical proofs does give one a greater appreciation for why mathematics inspired mysticism.

eta: The wiki entry on Meno.


Feser's [blog] goes into this subject as well.

World 1 is the world of physical entities and states – tables, chairs, rocks, trees, fundamental particles and forces, human bodies and behavior, and so forth. World 2 is the world of thoughts, sensations, and mental phenomena in general. World 3 is the world of scientific and philosophical theories, arguments, stories, social institutions, works of art and the like.
Popper’s World 3 is often compared to Plato’s realm of the Forms, and Popper himself acknowledges that there are similarities. But he also emphasizes the significant differences between his view and Plato’s, not the least of which is that he takes World 3, despite its objectivity or autonomy, to be something “man-made,” its objects in the strict sense being what the human mind “abstracts” from their World 1 embodiment. Though Popper does not take note of the fact or develop the theme in much detail, this is clearly reminiscent of an Aristotelian or “moderate realist” approach to the traditional problem of universals, as distinct from the “extreme realism” of Platonism.
Note the similarity of World 3 to aspects of Morphic Resonance, specifically Sheldrake's account morphic fields and cultural institutions.
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