Mod+ Platonism & Mathematics [Resources][Platonism]



Plato’s Affinity Argument

In fact, as Michael Pakaluk has pointed out, the affinity argument is not an “argument from analogy” at all, but rather “an argument about the nature of things.” Plato is not saying: “The soul is like the Forms in one way, so there is some significant probability that it is like them in this other way too.” Rather, he is saying something like: “The soul of its nature is X. But as we know from the example of the Forms, which are also X, things that are X are imperishable. So the soul is imperishable.”

What is X? As Lloyd Gerson suggests in his book Knowing Persons: A Study of Plato, what Plato seems to be emphasizing here is that the soul, like the Forms, is immaterial (p. 86). What the argument is (arguably) saying, then, is that whereas the senses pass away just as the things they know pass away, because both are material, the soul by contrast must be as imperishable as the things it knows – namely the Forms – because like the Forms, it is immaterial. In other words, it is, on this interpretation, the immaterial nature of the Forms that makes them imperishable, so that something that shares that nature – as (the argument claims) the soul does – must be equally imperishable. Notice, again, that this is not a probabilistic argument from analogy, but, in effect, an attempt at a proof.
Also, a bit from Nagel's The Last Word:

"Reason, if there is such a thing, can serve as a court of appeal not only against the received opinions and habits of our community but also against the peculiarities of our personal perspective. It is something each individual can find within himself, but at the same time it has universal authority. Reason provides, mysteriously, a way of distancing oneself from common opinion and received practices that is not a mere elevation of individuality--not a determination to express one's idiosyncratic self rather than go along with everyone else. Whoever appeals to reason purports to discover a source of authority within himself that is not merely personal, or societal, but universal--and that should also persuade others who are willing to listen to it.

If this description sounds Cartesian or even Platonic, that is no accident: The topic may be ancient and well-worn, but it is fully alive today, partly because of the prevalence of various forms of what I (but not, usually, its proponents) would call skepticism about reason, either in general or in some of its instances. A vulgar version of this skepticism is epidemic in the weaker regions of our culture, but it receives some serious philosophical support as well. I am prompted to this inquiry partly by the ambient climate of irrationalism but also by not really knowing what more to say after irrationalism has been rejected as incoherent--for there is a real problem about how such a thing as reason is possible.

How is it possible that creatures like ourselves, supplied with the contingent capacities of a biological species whose very existence appears to be radically accidental, should have access to universally valid methods of objective thought?"


A Different Take on Creation: Karl Svozil's Digital Dualism

One obvious explanation for what [physicist Eugene] Wigner calls 'the unreasonable effectiveness of mathematics in the natural sciences,'" suggests Svozil, "seems to be the Pythagorean assumption that numbers are the elements out of which the universe was constructed; and what appears to us as the laws of Nature are just mathematical theorems or computations."


Here's something you don't see every day:

Platonic Atheism

ABSTRACT: Platonic atheism is an affirmative atheism. It affirms modern analytic metaphysics and ethics. The platonic atheist is a metaphysical and moral realist. Reality is lawful. The Law includes the laws of logic, mathematics, actuality, and morality. All things fall under the Law. Gods exist only if the Law permits them to exist. The existence of any god is a scientific question. And if any gods do exist, they are subject to the Law. Hence science decides what the gods can and cannot do.

Any actions of any gods can be evaluated using the moral laws. Platonic atheism allows the soul to be defined as the form of the body. It allows for life after death via lawful resurrection in other universes. Since all persons are equal before the law, platonic atheists are committed to justice. For the platonic atheist, the Law is divine. The projection of any King above the Law is idolatry.

The platonic atheist has a rich system of atheological concepts (piety, impiety, eschatology, soteriology, etc.). Platonic atheism liberates religion from theism.


Does Mathematical Beauty Pose Problems for Naturalism?

Abstract: This essay takes up a question posed by the physicist Eugene Wigner in 1960, who was amazed at the success of mathematics in predicting the physical world. What accounts for this success? Wigner ultimately dubbed it a "miracle," and R. C. Hamming, after an attempt to fill in some gaps left by Wigner, declared mathematics to be "unreasonably effective." This paper looks at some evolutionary accounts to explain cognition, and expands some ideas of Mark Steiner. Specifically, it explores the role of aesthetics in theory formation and suggests that theism may well offer a better explanation of mathematical success than naturalism.


The great mystery of mathematics is its lack of mystery

There are statements that happen to have been proved and that we can therefore agree are true, as we might agree with our accountant that 532+193=725. But if a statement hasn’t yet been proved or disproved then, in Motl’s view, there’s no way even to guess, better than chance, which way it will turn out. There are no valid analogies to any previously proven statements, no broad patterns, just one damn lemma after another.

To put it mildly, this hasn’t been my experience, or the experience of anyone else I know who works in any part of math. Yes, sometimes people are surprised; surprises are part of the thrill of what we do. But the surprises are surprises only because of their rarity, because of all the other times when things worked out pretty much the way the experts expected them to. And yet the rarity of surprises is itself a surprise, a genuine mystery. A priori, math could have been like Motl said it was, with the statements we care about lacking any humanly comprehensible reasons for being true or false. By and large, though, it isn’t that way. Why?
A priori, Fermat’s Last Theorem, the Poincaré Conjecture, and pretty much every other statement of mathematical interest could have been neither provable nor disprovable: if true, then totally disconnected from all the other interesting truths, an island unto itself, with the only question (a question of taste!) being whether we should add it on as a new axiom. But it didn’t turn out like that. Instead of millions of islands, mathematicians discovered a supercontinent, with just a few islands here and there off the coast – and many of the islands, when explored further, ended up being connected to the mainland after all.

So again: why? One possible answer is a selection effect. Sure, there are plenty of pattern-less parts of math, but for precisely that reason, those parts aren’t interesting to humans. The parts that we teach students, put in textbooks, pontificate about in essays such as this one, etc, are the parts that ended up being interconnected and elegant. Likewise, no one wonders why the subjects of biopics so often turn out to have lived riveting lives; if they didn’t, there wouldn’t be biopics about them.

This strikes me as clearly part of the answer. But it can’t be the whole answer, because it doesn’t account for something all mathematicians have experienced: namely, the frequency with which there turn out to be striking patterns and connections between seemingly unrelated concepts, even when no one had thought to expect them beforehand, even when no one had charted out the territory and assured the latecomers that such patterns were there to be found.

A second possible answer is that even the parts of math that look far removed from physics are indirectly inspired by our experience with the physical world – and that they are coherent because the physical world is. This answer would push the mystery of math’s comprehensibility and elegance back to a different kind of mystery, what the Hungarian-born mathematician Eugene Wigner called the ‘unreasonable effectiveness’ of math in the physical sciences. A third sort of answer might focus on the peculiarities of the human brain, on its ability (which begs yet another ‘why?’) to zero in on mathematical questions that will turn out to be answerable and concepts that will turn out to be interestingly interrelated, even when the brain has no idea that it’s doing so.

I don’t know which of these answers is closest to the truth, or whether it’s something else entirely. Still, I feel confident in saying that, yes, there is something mysterious about math – but the main thing that’s mysterious is why there isn’t even more mystery than there is.


Plato's stave: academic cracks philosopher's musical code

"Historian claims Plato's manuscripts are mathematically ordered according to 12-note scale"

Kennedy's breakthrough, published in the journal Apeiron this week, is based on stichometry: the measure of ancient texts by standard line lengths. Kennedy used a computer to restore the most accurate contemporary versions of Plato's manuscripts to their original form, which would consist of lines of 35 characters, with no spaces or punctuation. What he found was that within a margin of error of just one or two percent, many of Plato's dialogues had line lengths based on round multiples of twelve hundred.

The Apology has 1,200 lines; the Protagoras, Cratylus, Philebus and Symposium each have 2,400 lines; the Gorgias 3,600; the Republic 12,200; and the Laws 14,400.

Kennedy argues that this is no accident. "We know that scribes were paid by the number of lines, library catalogues had the total number of lines, so everyone was counting lines," he said. He believes that Plato was organising his texts according to a 12-note musical scale, attributed to Pythagoras, which he certainly knew about.

"My claim," says Kennedy, "is that Plato used that technology of line counting to keep track of where he was in his text and to embed symbolic passages at regular intervals." Knowing how he did so "unlocks the gate to the labyrinth of symbolic messages in Plato".

Believing that this pattern corresponds to the 12-note musical scale widely used by Pythagoreans, Kennedy divided the texts into equal 12ths and found that "significant concepts and narrative turns" within the dialogues are generally located at their junctures. Positive concepts are lodged at the harmonious third, fourth, sixth, eight and ninth "notes", which were considered to be most harmonious with the 12th; while negative concepts are found at the more dissonant fifth, seventh, 10th and 11th.

Kennedy has also found that the enigmatic "divided line" simile in the Republic, in which Plato describes a line divided by an unstated ratio, falls 61.7% of the way through the dialogue. It has been thought that the line refers to the golden mean, which expressed as a percentage is 61.8%.

Copies of the paper have been circulating among senior scholars, who believe Kennedy's argument should be taken seriously.

Professor Andrew Barker, a leading authority on ancient Greek music, said that "the results he's come up with look too neat to be accidental" and that if scholars confirm them, "he will have shown something quite startling about Plato's methods of composition".

Kennedy believes his findings restore what was the standard, mainstream view which held for 2,000 years "from the first generation of Plato's followers, up through the renaissance". This held that "he wrote symbolically and that if you worked hard and became wise you could understand the symbols and penetrate his text to his underlying philosophy." Only in the last few hundred years has an emphasis on the literal meanings of texts led to a neglect of their figurative meanings.

It also explains why it is that Aristotle, Plato's pupil, emphatically claimed that Plato was a follower of Pythagoras, to the bafflement of most contemporary scholars.

The secrecy was because Plato's was "a dangerous idea", claims Kennedy. "It meant that mathematical law governed the universe and not Zeus." Given that Plato's teacher, Socrates, had been executed for sowing impiety among the youth he would have been "very cautious abut revealing doctrines that threaten the gods of Olympus".

For once, Alfred North Whitehead's description of western philosophy as "a series of footnotes to Plato" looks like being an understatement. "We've got some 2,000 pages of Plato," says Kennedy. "We now know that underneath all of those genuine dialogues there's another layer of symbolic meaning. This is the beginning of a big debate. It will take years to make sense of all this."


The Deep Mystery of the Prime Numbers (page 8)

When we speak of prime numbers, or indeed other mathematical entities, such as pi, e or the square root of 2, what are we speaking about? Are these entities merely human inventions that describe how the world works? Or are they abstract identities which have a mysterious existence of their own outside of human minds? The question is an ancient one, with many strong arguments to support each point of view. Many philosophers and mathematicians believe we invent mathematics, and that consequently mathematical entities have no existence outside of human minds. But many others are not so convinced. They are of the opinion that these entities appear to have an abstract existence of their own, independent of human minds. For instance, the Higgs boson was recently discovered to exist. In 1964 the British physicist Peter Higgs predicted that such subatomic particles must exist merely because of a mathematical equation that implied that they exist. In 2013 the Higgs boson was discovered. If mathematics is a human invention how or why has it this power to predict the way nature behaves? On the other hand, if mathematics has a reality outside of human minds, perhaps the laws of nature follow mathematical reality. The discovery of the Higgs boson would seem to support this viewpoint. One other recent discovery is worth noting. It has been conjectured (and there is some evidence to support the conjecture) that the physics of the atomic nucleus is apparently connected to the distribution of the prime numbers. If this turns out to be true, it will strengthen the argument that mathematical entities, including the primes, exist “out there” in the world and are not confined to human minds. Perhaps all reality, in a profound way, is mathematical beyond description

Paul C. Anagnostopoulos

Nap, interrupted.
I have no idea what anyone means by numbers existing "out there" versus "in our minds." This pile of coins here has 6 coins. Really. I can count them and so can you. Most people (but not all) can count them immediately upon seeing the group, without having to count one by one. Does that 6 exist out there or just in my mind?


~~ Paul


I have no idea what anyone means by numbers existing "out there" versus "in our minds." This pile of coins here has 6 coins. Really. I can count them and so can you. Most people (but not all) can count them immediately upon seeing the group, without having to count one by one. Does that 6 exist out there or just in my mind?


~~ Paul
The way I understand it is that mathematics is the underlying reality - I think the first post in the thread, with Massimo's arguments for and then against Mathematical Platonism, provides a good introduction to both sides of the argument.

Personally I'm wary of the idea that "stuff" of any kind - Platonic Forms or swirling atoms - can have an intrinsic, preferred meaning - which seems to be what Platonism requires - but at the same time I leave the door open as mathematics deals with universal truths reached by reason.

Might be worth a new discussion thread...


From Gödel to Mathematical Neo-Platonism?

Mathematical Platonism (MP), also known as "Mathematical Realism", is a well-known position from the philosophy of mathematics. It states, roughly, that mathematical objects have a real existence, outside of space and time, independent of what humans think and know about math. Mathematical objects would in that way be analogous to the Platonic Forms, which are likewise supposed to have an ideal, non-spatiotemporal existence. In fact, it seems that Plato himself counted mathematical objects to be among the Forms that give intelligible order to the physical flux, so in that sense MP can be said to have originated with Platonism (P) itself.

Now the central question I would like to explore in this post and the following posts is this: If MP is a possibly true position, then perhaps Mathematical Neo-Platonism (MNP) is possible as well? Neo-Platonism (NP), of course, is the school of philosophy founded by Plotinus in the third century CE and developed further by later Neo-Platonists, though it is fair to say that none of them reached the originality, scope and profundity of the Plotinian system. Now what I am suggesting is that there could be a theory about the nature of mathematics that is analogous – or perhaps even equivalent – to NP. Why? Well, as NP systematized P by deriving all the Forms from a single transcendent source called "the One" (anticipated, admittedly by Plato's "Good beyond being"), so MNP might systematize MP by deriving mathematical reality as a whole from a single transcendent source. Thus MNP would stand to MP as NP stood to P. In this post and the following posts I want to explore the possibility of MNP. Does it make sense? Are there any arguments for it? What are its consequences?


Free book on Neoplatonism.

...The Alexandrian Jew named Philo, who lived between 30 BCE and 50 BCE, taught that people, by freeing themselves from matter and receiving illumination from the divine, may reach a mystical, ecstatic, or prophetical state in which they become absorbed into Divinity. But the most systematic attempt at formulating a philosophical system of a mystical character was that by the Neoplatonic School of Alexandria, especially that of Plotinus, arguably the greatest philosopher-mystic the world has ever known, who lived between 205 and 270 CE.

In his Enneads, Plotinus sets out a system which has as its central idea the concept that there exists a process of ceaseless emanation and out-flowing from the One, the Absolute. He illustrates this concept using metaphors such as the radiation of heat from fire, of cold from snow, fragrance from a flower, or light from the Sun. This theme leads him to the maxim that “good diffuses itself” (bonum diffusivum sui). He concludes that entities that have achieved perfection of their own being do not keep that perfection to themselves, but spread it out by generating an external image of their internal activity. The ultimate goal of human life and of philosophy is to realize the mystical return of the soul to the Divine. Freeing itself from the sensuous world by purification, the human soul ascends by successive steps through the various degrees of the metaphysical order, until it unites itself in communion with the One...


Seems like a good introduction to the idea of Mathematical Platonism, and to some degree Platonism in general:



Why I am a Platonist by David Mumford

...Probably most mathematicians get a gut feeling that math is “out there” from their personal experiences struggling to understand some mathematical situation, to prove or disprove some theorem. But this is such a slippery subjective argument that I want to take a somewhat different tack. I want to say why studying the History of Mathematics makes mathematics seem to me to be universal and unchanging, invariant across time and space. Historians are disposed to dismiss amateurs like me as being naïve by imposing their modern point of view on ancient writings and not understanding the cultural infl uences, the proper historical context in which the work was done. I would counter: is a metallurgist imposing modern biases when he/she analyzes the metallic content of an ancient weapon, using the periodic table? It really all depends on whether you accept the Platonic universal view of mathematical truth or not. If you accept this view, using modern mathematics to analyze writings from other times and places is no different from the metallurgist’s using modern knowledge of metals. So let me present my reading of several historical writings which seem to me to shout out that all mathematicians are working on one and the same body of truths...



"There's a secret world out there. A hidden parallel universe of beauty and elegance, intricately intertwined with ours. And it's invisible to most of us."

Imagine that you had to take an art class in which they taught you only how to paint a fence or a wall, but never showed you the paintings of the great masters. Alas, this is how math is taught, and so for most of us it becomes the intellectual equivalent of watching paint dry. Edward Frenkel wants to open this secret world to all of us because it can teach us so much about the mysteries of the Universe. In this talk, he weaves the discovery of math with his personal journey, addressing the existential questions of finding out who we are; of truth, courage, and passion.

Edward Frenkel is a professor of mathematics at the University of California, Berkeley, which he joined in 1997 after being on the faculty at Harvard University. He is a member of the American Academy of Arts and Sciences, a Fellow of the American Mathematical Society, and the winner of the Hermann Weyl Prize in mathematical physics. Frenkel has authored three books and over eighty scholarly articles in academic journals, and he has lectured on his work around the world. His YouTube videos have garnered over 3 million views combined.

Frenkel’s latest book "Love and Math" was a New York Times bestseller and has been named one of the Best Books of 2013 by both Amazon and iBooks. It is being translated into 14 languages. Frenkel has also co-produced, co-directed and played the lead in the film "Rites of Love and Math" (2010).


Infinity and Beyond

The standard conception of the infinite is that which is endless, unlimited, unsurveyable, immeasurable. Ever since people have been able to reflect, they have treated the infinite with a curious combination of perplexity, suspicion, fascination and respect. On the one hand, they have wondered whether we can even make sense of the infinite: mustn’t it, by its very nature, elude our finite grasp? On the other hand, they have been reluctant, indeed unable, to ignore it altogether.

In the fourth century BCE, Aristotle responded to this dilemma by drawing a distinction. He believed that there is one kind of infinity that really can’t be made sense of, and another that is a familiar and fundamental feature of reality. To the former he gave the label ‘actual’. To the latter he gave the label ‘potential’. An ‘actual’ infinity is one that is located at some point in time. A ‘potential’ infinity is one that is spread over time. Thus an infinitely big physical object, if there were such a thing, would be an example of an actual infinity. Its infinite bulk would be there all at once. An endlessly ticking clock, on the other hand, would be an example of a potential infinity. Its infinite ticking would be forever incomplete: however long the clock had been ticking, there would always be more ticks to come. Aristotle thought that there was something deeply problematic, if not incoherent, about an actual infinity. But he thought that potential infinities were there to be acknowledged in any process that will never end, such as the process of counting, or the process of dividing an object into smaller and smaller parts, or the passage of time itself.
The conception of sets involved here is, as I’ve already said, relatively intuitive. But isn’t it also strikingly Aristotelian? There is a temporal metaphor sustaining it. Sets are depicted as coming into existence ‘after’ their members, in such a way that there are ‘always’ more to come. Their collective infinity, as opposed to the infinity of any one of them, is potential, not actual: its existence is spread ‘over time’ rather than being located at any one point ‘in time’. Moreover, it is this collective infinity that arguably has the best claim to the title. For recall the concepts that I listed earlier as characterising the standard conception of the infinite: endlessness, unlimitedness, unsurveyability, immeasurability. These concepts more properly apply to the full range of sets than to any one of them. This in turn is because of the very success that Cantor enjoyed in subjecting individual sets to rigorous mathematical scrutiny. He showed, for example, that the set of numbers is limited in size. It is limited in size because it doesn’t have as many members as the set of sets of numbers. He also showed (although I didn’t go into the details of this) that its size can be given a precise mathematical measure. Isn’t there a sense, therefore, in which he established that the set of numbers is ‘really’ finite and that what is ‘really’ infinite is something of an altogether different kind? Didn’t his work serve, in the end, to corroborate the Aristotelian orthodoxy that ‘real’ infinity can never be actual, but must always be potential?


An anti-Platonic article, but inline with the general idea of this Resources thread:

How to play mathematics

But can we say that sea slugs and corals know hyperbolic geometry? I want to argue here that in some sense they do. Absent the apparatus of rationalisation and without the capacity to form mental representations, I’d like to postulate that these humble organisms are skilled geometers whose example has powerful resonances for what it means for us humans to know maths – and also profound implications for teaching this legendarily abstruse field.

I’m not the first person to have considered the mathematical capacities of non-sentient things. Towards the end of Richard Feynman’s life, the Nobel Prize-winning physicist is said to have become fascinated by the question of whether atoms are ‘thinking’. Feynman was drawn to this deliberation by considering what electrons do as they orbit the nucleus of an atom. In the earliest days of atomic science, atoms were conceived as little solar systems with the electrons orbiting in simple paths around their nuclei much as a planet revolves around its sun. Yet in the 1920s, it became evident that something much more mathematically complex was going on; in fact, as an electron buzzes around its nucleus, the shape it makes is like a diffused cloud. The simplest electron clouds are spherical, others have dumbbell and toroidal shapes. The form of each cloud is described by what’s called a Schrödinger equation, which gives you a map of where it’s possible for the electron to be in space.

Schrödinger equations (after the pioneering quantum theorist Erwin Schrödinger and his hypothetical cat), are so complicated that, when Feynman was alive, the best supercomputers could barely simulate even the simplest orbits. So how could a brainless electron be effortlessly doing what it was doing? Feynman wondered if an electron was calculating its Schrödinger equation. And what might it mean to say that a subatomic particle is calculating?
Since at least the time of Pythagoras and Plato, there’s been a great deal of discussion in Western philosophy about how we can understand the fact that many physical systems have mathematical representations: the segmented arrangements in sunflowers, pine cones and pineapples (Fibonacci numbers); the curve of nautilus shells, elephant tusks and rams horns (logarithmic spiral); music (harmonic ratios and Fourier transforms); atoms, stars and galaxies, which all now have powerful mathematical descriptors; even the cosmos as a whole, now represented by the equations of general relativity. The physicist Eugene Wigner has termed this startling fact ‘the unreasonable effectiveness of mathematics’. Why does the real world actualise maths at all? And so much of it? Even arcane parts of mathematics, such as abstract algebras and obscure bits of topology often turn out to be manifest somewhere in nature. Most physicists still explain this by some form of philosophical Platonism, which in its oldest form says that the universe is moulded by mathematical relationships which precede the material world. To Platonists, matter is literally in-formed, and guided by, a pre-existing set of mathematical ideals.

In the Platonic way of seeing, matter (the stuff of everything) is rendered inert, stripped of power and subordinated to ethereal mathematical laws. These laws are given ontological primacy with matter being effectively a sideline to the ‘true reality’ of the equations. Over the past half-century, this vision has been updated somewhat because now matter, or subatomic particles, have themselves been enfolded into the equations. Matter has been replaced by fields – as in electric and magnetic fields – and now it’s the fields that follow the laws. Still, it’s the laws that retain primacy and power; hence the obsession with finding an ultimate law, a so-called ‘theory of everything’.

Platonism has always bothered me as a philosophy in part because it’s a veiled form of theology – mathematics replaces God as the transcendent, a priori power – so if we want to articulate an alternative, we need new ways of interpreting mathematics itself that don’t also slip into deistic modes. Thinking about maths as performative points a way forward, while also offering a powerful pedagogic model.
You don’t have to be a symbol-expert to appreciate this terrain. Just as humans are endowed with an ability to dance and play music (even if education too often crushes this out of us), so we have innate form-making and pattern-playing proclivities. Sea slugs, sound waves and falcons do mathematics; Islamic mosaicists and African architects do it too. So can you.