Plato's Parmenides IX, The One IV

Again, I'll put this past the jump.  Just to remind you, this is a very extended discussion in which Parmenides proposes to talk through both sides of every part of the question. So the traps get run one way: "What if the One is not?" And then the other: "What if the One is?" Problems abound on every side. At this point the problems are going to start looking familiar, because we found them running the traps the one way; you'll see the same problems arising if we make the contrary assumption about the One.

With any luck we'll get through this in one more post after this one. 

"The Disintegration Directive"

I hate Twitter, but sometimes it's the place where important things get said. This thread on how cancel culture resembles less Mao's Cultural Revolution than a practice of the East German government is worth considering. 

As far as I know the government does not employ a Stasi-like secret police to harass ordinary citizens, although the actions against Trump core allies like Roger Stone looked rather similar. Yet there do seem to be cadres in our institutions and cities, which was a major similarity to the Cultural Revolution and its Red Guards. 

Someone else voices my usual rant

 From Hugo Gurdon, Washington Examiner editor-in-chief:

In 1985, singer Bob Geldof was interviewed backstage at London’s Wembley Stadium about the massive Live Aid rock concert he’d organized . . . raising money for famine relief in Ethiopia . . . .

At this moment, the greatest triumph in his already successful career, Geldof remarked with a note of bitter irony that Live Aid involved “the privatization of compassion.” He didn’t mean it as a compliment. “Privatization” had become a dirty word in the left-wing lexicon, as industries previously taken over by socialist governments were released from central control and sold to the public as businesses quoted on stock exchanges.

Geldof’s comment struck me forcefully at the time, for it was the precise opposite of what I took then and still take to be the gem-like truth stated by columnist T.E. Utley that one of the cruelest aspects of socialism is that it delegates compassion to the state. Socialism encourages individuals to think caring for their neighbor is not their responsibility but is, instead, a function of government.

Socialists often suggest that private provision of help for the needy is a failure of the state. [Senator Bernie] Sanders has spoken disdainfully of charity, as have many unappealing politicians elsewhere. They regard the care of others through individual acts of kindness as demeaning the recipient because they believe or at least declare that goods and services received should be taken as a right rather than accepted as a gift. One also suspects that socialists dislike charity because it places a claim on them as individuals, which they’d rather shrug off.

Eudaimonia is Not Therapy

I was going to leave this article alone, even though it addresses something I've been wondering about, until the last paragraph. The subject is the infusion of therapy-speak into everyday life. 

It's from the New Yorker, so 'everyday' means 'everyday upper-middle-class-and-higher life within certain fashionable communities.' Still, I know at least one person who services such a community, and runs a business doing it centered on Facebook and Instagram. She is all the time telling her clients how important it is to "heal" their "trauma." As far as I can tell her clients are at least relatively rich white women. Maybe some of them genuinely have trauma, but it seems to be the kind of stuff the article talks about:

During this exchange, Twitter served me an advertisement that urged me to “understand my trauma” by purchasing a yoga membership. Ridiculous, I thought. I’m not a sexual-assault survivor. I’ve never been to a war zone. But, countered my brain, after four years of Trump and four seasons of covid, are you not hurting? The earth is dying. Your mother issues! Your daddy issues! A clammy wave engulfed me. My cursor hovered over the banner.

So, as I said, I was going to leave it alone. After all, I don't want to beat up on fragile people, and these people are genuinely so fragile that they think they are being traumatized by living the life of a wealthy white woman in the better neighborhoods of the richest country on earth. There's definitely something wrong with them, but it isn't "trauma." Pointing out that their entire worldview is fundamentally unhealthy might seem like I was beating up on the weak.

But then that last paragraph:

Therapy seems to have absorbed not just our language but our idea of the good life; its framework of fulfillment and reciprocity, compassion and care, increasingly drives our vision for society. Writing this piece, I thought especially of the Greek concept of eudaimonia, or human flourishing. Some might call it blessedness. In any case, it seems worth talking about.

No, you thought wrong. 

Eudaimonia is often translated as "flourishing" or simply as "happiness," but the real thing it means is being fully engaged in pursuing excellence with all your vital powers. Now she says she's never been traumatized because she's never been to war; but in fact, war is the closest thing I've seen to eudaimonia.

Aristotle says that the goal of ethics is eudaimonia, a state of happy flourishing that you find when all of your vital powers are aligned in rational activity. More, he says, to fully experience this state you need a community that is set up to support it. The military deployed comes much closer to attaining Aristotle's ideal than anything else I've seen in the world. Everyone is working together towards some strategic good. They all have different jobs, but those jobs must align. Thus, there is constant rational communication and consideration of how to align different fires on a target, or different staff sections on a mission. This 'small, close knit' community is also a community that works together toward some goods that they pursue together through rational activity.

War being war, as Clausewitz says, 'everything is simple, and the simplest things are hard.' Thus, one needs all of one's vital powers in alignment to accomplish these goals. It is a very engaging sort of life.

It may well be that the broader society lacks a number of things that these smaller, close-knit and rationally ordered communities offer. Are these goods we can replicate? Certainly: any number of organizations could be set up to pursue goods in this way, although they will not all be as fully engaging of all of one's vital powers absent the extremes of war.

Are they goods that we do replicate? No, not really, not for the most part.

You develop tight knit friendships at war -- and then, if you study philosophy, you notice that Aristotle's ethics ends with a long discussion of the importance of friendship. 

If you want to find eudaimonia, stop ever going to therapy. Stop focusing on your problems, whatever they are. The only thing to do with death -- and whatever you disliked about your childhood -- is to ride off from it. Go join the local volunteer fire department, and work with them putting out fires or saving lives in medical emergencies. Study philosophy and argue about Truth and Justice with friends over beer. Take up an extreme sport with a good community that supports each other. Ride motorcycles. Ride horses. Learn a martial art and practice it intensely. 

Do everything you can except dwelling issues of 'care' and 'sensitivity' and all the 'hurts' and 'trauma' you've suffered. Stay away from anyone who tries to convince you that you're a suffering victim, or who is willing to treat you like one if you ask -- or so you'll pay them to help you dwell on your 'problems.' Dwelling on your problems is in fact the problem. Do great things instead.

Cause of death

The Chauvin trial is at last underway, with excellent daily reporting from PowerLine. The defendant is fortunate enough to have drawn a judge who takes his job seriously, almost alone in a city that appears unified in its desire to make this a show trial against an unjust society. As a result, to my amazement, the opening arguments largely stuck to the issue of what exactly it was that killed Mr. Floyd. After a disgraceful piece in the local rag arguing that the trial was a "vessel into which a splintered society places its rage, anxieties and hopes"--pretty much a recipe for lynch law--the same paper actually ran a fairly reasonable piece this week trying to sort through the conflicting evidence on the cause of death. It's hard to imagine how he got it past his editors.

If this were a civil case, it would be difficult to sort through the relative weight of the various contributing factors to Mr. Floyd's death. The picture is intolerably complicated by a fatal dose of fentanyl. I try to imagine how I'd view the evidence if it were myself being arrested, or someone from my own social circle that I knew and loved. Would I be thinking, "The person you're arresting is in clear medical distress; why are you so cavalier about the risks?" Or would I conclude that, if you resist arrest and act crazy and have to be restrained, then it turns out that you're dying of an OD, then it's just a slow-motion suicide by cop? Looking at it from the opposite perspective, if the defendant in this trial had for some reason forcibly administered the fatal dose of fentanyl to Mr. Floyd, then tried to argue that what really killed him was a knee on the neck, wouldn't I scoff? Wouldn't I reject the argument that the fentanyl wasn't all that dangerous, because Mr. Floyd, an addict, had such a high tolerance?

In a civil trial, the jury might be allowed to allocate percentages to the liability of the various people who created dangerous conditions, which in this fact pattern might well result in a very high percentage being allocated to the decision to swallow the fatal evidence of fentanyl. In a criminal trial, however, the jury has to resolve any reasonable doubt over whether Mr. Chauvin's actions were the effective cause of death. It can't help much that the jury will be allowed and/or instructed to take into account that there can be multiple causes of death, or that a potential fatal treatment can be considered the cause of death even if someone else (including the decedent) had contributed a cause of death that either took effect seconds before the action of the criminal defendant, or would have caused death within seconds in any case.

The only certain conclusion I can draw is that I wouldn't want to live there, and I certainly wouldn't want to serve on a police force in that city. It's a mystery how they keep a police force together at all. Frankly it's hard to see how such a broken city survives.

Heroic Materialism

Raven mentioned this series, and this is the conclusion of it. 

He says at the end that the collapse of Marxism -- a little over-eager, sadly -- left us with no alternative but "Heroic Materialism." By this I assume he meant that we should endeavor to live boldly and virtuously according to the ancient pattern, but with no belief in any metaphysical beings or realities that might reward such virtues in an afterlife. We should do well because doing well led to physical prosperity; and if it didn't, quite, we should work harder, because working harder might. But ultimately it is heroic because it is a fight without hope; and worse, without faith.

       "Night shall be thrice night over you,
         And heaven an iron cope.
         Do you have joy without a cause,
         Yea, faith without a hope?"

God save us from that; and only a god can. The Greeks built their great temples to divinity, not to materialism. They would not have built them if they had not believed. The Cathedrals of the Middle Ages were built to a God who was believed in because of the best arguments of Athens, and the best poetry of Jerusalem.

Where comes anything without a cause, joy or any other thing? Where comes faith in a thing you don't even hope to be true? 

It is Holy Week. What better time to ask these questions?

Plato's "Hidden Theorem on Distribution of Primes"

While I was trying to remember where Aristotle says the thing about primes that Parmenides said, I came across this paper. It argues that the 5,040 number in Plato's Laws (which you will remember from the long series on the Laws here over the winter) hides a secret theorem. 

I'm not sure about that, but read it for yourself if you like. The full text is there, and it's short.

Against Karl Popper

I've always liked Popper's answer, but hear them out. First, what is the answer?

Popper was struck by Einstein’s prediction. “Now the impressive thing about this case,” he wrote decades later, “is the risk involved in a prediction of this kind.” Had the measurements found Einstein in error, Popper said, the physicist would have been forced to abandon his theory. Popper built his demarcation criterion around the bravado of wagering against refutation: “One can sum up all this by saying that the criterion of the scientific status of a theory is its falsifiability, or refutability, or testability.”

Second, what was the answer meant to contrast?

All demarcation criteria are designed to exclude something. What Popper really wanted to do was to show why psychoanalysis and Marxism were not scientific. Those theories had been widely understood as “scientific” in his Viennese milieu because of a logical empiricist theory called verificationism. According to this view, a theory is scientific if it is verified by empirical data.

From my perspective, successfully excluding Marxism and psychoanalysis from science justifies the project. However, others have disagreed, and not only Marxists and psychoanalysts. 

First, it is difficult to determine whether you have actually falsified a theory....  Not so fast. How do you know that your experimental result was accurate?

That's a terrible objection. Falsification is about whether or not the theory could in principle be falsified. The fact that it may be hard to do is quite different from the objection to Freud's theories, wherein a man who e.g. didn't appear to have any Oedipal concerns would just be said to be suppressing or projecting them or whatever. Marxism, too: Lenin wrote a book about how Marx's theory's failure to materialize was just a further proof of how right he was about the evils of capitalism. These theories aren't falsifiable because they let you just tell another story about how they were right all along. 

Yes, the instrument could be wrong; do it again. But if there's not a fact of the matter that could falsify a theory if known, it's not science.

The second problem with Popper’s proposal has to do with the actual demarcations it gives us. The very minimum we should expect from a demarcation criterion is that it slices the sciences in the right places. We want our criterion to recognize as scientific those theories that are very generally accepted as hallmarks of contemporary science, such as quantum physics, natural selection, and plate tectonics. At the same time, we want our criterion to rule out doctrines such as astrology and dowsing. Popper’s falsifiability standard is not especially helpful in this regard. For starters, it is difficult to present the “historical” natural sciences, such as evolutionary biology, geology, or cosmology—those fields where we cannot “run the tape again” in the laboratory—exclusively in terms of falsifiable claims. Those sciences provide persuasive explanations of nature through the totality of a narrative chain of causal inference rather than a series of empirical yes-no votes. Popper thus inadvertently excludes important domains of contemporary science.

I would argue that this is also a terrible objection. To say that you want to "slice the sciences in the right places" indicates that you think you already know where the slices should be made. Actually, Popper raises an important objection to theories even from these widely recognized fields: they aren't doing quite the same thing as the sciences that can provide replicable testing, and their theories remain rather more theoretical as a consequence. They're not in the same category as Marxism, but maybe that only means that there is Hard Science, and then degrees of distance from it. I'm inclined to score this Popper 2, objections 0 so far.

The situation with inclusion is even worse. The difficulty was sharply expressed by philosopher of science Larry Laudan in an influential article from 1983. Popper’s criterion, he wrote,

has the untoward consequence of countenancing as “scientific” every crank claim that makes ascertainably false assertions. Thus flat Earthers, biblical creationists, proponents of laetrile or orgone boxes, Uri Geller devotees, Bermuda Triangulators, circle squarers, Lysenkoists, charioteers of the gods, perpetuum mobile builders, Big Foot searchers, Loch Nessians, faith healers, polywater dabblers, Rosicrucians, the-world-is-about-to-enders, primal screamers, water diviners, magicians, and astrologers all turn out to be scientific on Popper’s criterion—just so long as they are prepared to indicate some observation, however improbable, which (if it came to pass) would cause them to change their minds.

OK, that one's better. Popper's criterion may not be sufficient to exclude fields that are easily falsifiable because they are crazy. We might need to address that objection to have a proper definition of hard science. That seems to be the evolving position, and it's not that hostile to Popper: he said something important, but he may have mistaken that something for everything.

Other approaches might prove more successful. Philosopher and former professor of biology Massimo Pigliucci, for example, has suggested that the problem with falsicationism is its one-dimensionality, not its effort to establish clear criteria. Perhaps we could add more dimensions that correspond to the heterogeneity of scientific practice. 

You Don't Say

Theroux gets that readers might perceive him as cranky, but he thinks the problem might be with the readers. 

Paul Theroux gets older.

His new novel tells the story of Joe Sharkey, an aging North Shore surfer... Sharkey feels acutely that he is being overtaken by younger surfers with big endorsements. For him, surfing was a way of life, an existence centered on catching waves, a commitment to the ocean.

Theroux sees surfing as a metaphor for his own life.... like the surfer past his prime, he is not immune to feeling forgotten, to the sense that the world has become hostile to the pure joy of the waves. There’s a fear of being overlooked, unread.

“I was once a hot shot, I was once the punk,” Theroux said. “And anyone who has once been a punk, eventually you’re older, and you see the turning of the years as it is."

Well, he has had more attention than most; and more wealth and success, too. But there is still a kind of universality to the experience of growing older.  

This looks like fun

"Conservatism" is No Longer Enough

A provocative and worthy piece at The American Mind, by Glenn Ellmers.

Fewer are willing to take the next step and accept that most people living in the United States today—certainly more than half—are not Americans in any meaningful sense of the term.

I don’t just mean the millions of illegal immigrants. Obviously, those foreigners who have bypassed the regular process for entering our country, and probably will never assimilate to our language and culture, are—politically as well as legally—aliens. I’m really referring to the many native-born people—some of whose families have been here since the Mayflower—who may technically be citizens of the United States but are no longer (if they ever were) Americans. They do not believe in, live by, or even like the principles, traditions, and ideals that until recently defined America as a nation and as a people. It is not obvious what we should call these citizen-aliens, these non-American Americans; but they are something else.

That's a very grave problem statement. Is it just? I wonder about the percentage -- "certainly more than half" may be overstating the numbers -- but it is certainly true of some percentage. Percentages aside, too, it seems to be true especially of the part that occupies all the seats of power: in government, both elected and the bureaucracy; in corporations; in the media, to include news, social media, and entertainment; and, at the moment at least, even in top positions of the military. 

Sad times for MIT

I kept reading aloud Townhall excerpts from this extraordinary MIT paper to my husband, who kept insisting that it was a hoax. It does sound as though someone were auditioning for the Bee, and sailing very close to the wind.

[C]alling for increased media literacy can often backfire: the instruction to ‘question more’ can lead to a weaponization of critical thinking and increased distrust of media and government institutions . . . . [C]alls for media literacy can often frame problems like fake news as ones of personal responsibility rather than a crisis of collective action. . . . [Anti-maskers] value unmediated access to information and privilege personal research and direct reading over "expert" interpretations. . . . As a subculture, anti-masking amplifies anti-establishment currents pervasive in U.S. political culture. . . . Data literacy, for anti-maskers, exemplifies distinctly American ideals of intellectual self-reliance, which historically takes the form of rejecting experts and other elites. The counter-visualizations that they produce and circulate not only challenge scientific consensus, but they also assert the value of independence in a society that they believe promotes an overall de-skilling and dumbing-down of the population for the sake of more effective social control. As they see it, to counter-visualize is to engage in an act of resistance against the stifling influence of central government, big business, and liberal academia . . . . Most fundamentally, the groups we studied believe that science is a process, and not an institution.

The solution?

Convincing anti-maskers to support public health measures in the age of COVID-19 will require more than ‘better’ visualizations, data literacy campaigns, or increased public access to data. Rather, it requires a sustained engagement with the social world of visualizations and the people who make or interpret them.

"Social," posited as an alternative to data literacy and explained as a "sustained engagement with the social world," remains a little fuzzy to me. Shaming? Censorship? Police raids? How I hope this turns out to be a Sokal hoax.

Holy Week

If any of you are Jews, Passover was I understand yesterday; I hope your holiday was satisfying. For Christians, of course today is Palm Sunday and the start of Holy Week. (If we have any Muslim readers, I note with surprise that Ramadan will also fall at nearly the same time this year, beginning on the 12th of April. A sacred spring for us all, this year.)

This is the occasion of one of my favorite stories from scripture. Probably I am guilty, like many, of focusing too much on the parts that appeal to me personally. I love the story of Jesus riding an unbroken colt, though. This is Luke 19:28-36:

28 After Jesus had said this, he went on ahead, going up to Jerusalem. 29 As he approached Bethphage and Bethany at the hill called the Mount of Olives, he sent two of his disciples, saying to them, 30 “Go to the village ahead of you, and as you enter it, you will find a colt tied there, which no one has ever ridden. Untie it and bring it here. 31 If anyone asks you, ‘Why are you untying it?’ say, ‘The Lord needs it.’ ” 32 Those who were sent ahead went and found it just as he had told them. 33 As they were untying the colt, its owners asked them, “Why are you untying the colt?” 34 They replied, “The Lord needs it.” 35 They brought it to Jesus, threw their cloaks on the colt and put Jesus on it. 36 As he went along, people spread their cloaks on the road. 

You try riding an unbroken colt sometime, and see if it's as easy as that for you.  

A Good Day for Strongmen

At the Central Georgia’s Strongest Man competition today, resumed after COVID, a man named Shannon Willits finished his courses and then perished. The best of us can hope for no better. He died in competition after finishing a set of heavy farmer’s walks carries and an immediate heavy bag carry. Most people couldn’t do either, given all the time in the world. 

A fit and worthy death. God send us each such when our time comes. 

Plato's Parmenides VIII, The One III: Time

This is a long section, so I'm going to put it beyond a jump. It's important to note at the beginning that this dialogue was written before almost any of the philosophy of time. Aristotle wrote one of the most important pieces on time in the ancient world, Plotinus another, and St. Augustine wrote another one. None of that thinking was available to Plato or to Parmenides. We saw in the preface to this series that Zeno appears to have believed that time, like space, was atomic in nature; Aristotle rejected both, but his rejection has not yet appeared. 

There are a lot of philosophical problems about time. One of them is the status of things (like ourselves) that seem to be extended in time. I can remember existing yesterday. Does that me still exist somewhere? For Zeno, the answer might be yes: there might be a 'time atom' (or rather a whole series of them) in which yesterday me still exists, and permanently (and changelessly). Or it might be 'no,' just as easily: that only the present time atom exists, neither one before nor one after. Both views admit of his apparent theory that change is impossible: the first view for the reasons we've already seen, as about the arrow that must be frozen in time because at any atom of time it is frozen; the second because, if only the current time atom actually exists, there is nowhere to move from or to move towards. 

Lacking a built-out philosophy of time this bit is quite difficult to follow. It is at odds with the way that we tend to think about time. The basic problem is that the one cannot be many, but if it exists through time there will be a part of it that is younger than itself (i.e., the part that existed yesterday) and a part that is older (i.e., the part that will exist tomorrow). The One cannot, then, be in time. This is another reason it can neither 'come to be' nor perish.

You Don’t Say

NCO: Most soldiers thought extremist training was a waste of time. 

Sweden

Preliminary data from EU statistics agency Eurostat compiled by Reuters showed Sweden had 7.7% more deaths in 2020 than its average for the preceding four years. Countries that opted for several periods of strict lockdowns, such as Spain and Belgium, had so-called excess mortality of 18.1% and 16.2% respectively.

UPDATE: Sorry about that; I posted this from my phone, and the mobile version of the Hall didn't show me the formatting error.  

Neighborhood Ordinance: No Whining

Austin immigrants versus “toxic masculinity,” plus an awesome Caddy.

In honor of our heroes, some Tejano music you can blare as loudly as you wish. 

9th Circuit: "What Do You Mean, 'Bear' Arms?"

Hawaii has a law that permits you to carry firearms, provided you obtain a license from the state that requires you to establish that you have an "urgent need" for such a firearm. A man named George Young applied for such a license, but the state was not satisfied that he proved the urgency of his need to carry one. He just wanted to carry a gun in case he needed to defend himself, but had established no reason to believe that it was likely he would need to do so. They thus turned him down, and he sued. 

The 9th Circuit Court of Appeals has upheld them in doing so, but gone rather farther. They assert that Americans have no right to carry weapons in public at all. 

“There is no right to carry arms openly in public; nor is any such right within the scope of the Second Amendment,” the court ruled in an “en banc” decision that involved all the panel’s judges.

“We can find no general right to carry arms into the public square for self-defense,” the majority wrote, claiming that the Second Amendment applies to the “defense of hearth and home.”

“The power of the government to regulate carrying arms in the public square does not infringe in any way on the right of an individual to defend his home or business,” the judges wrote.

This seems to be the evolving position on the Second Amendment from the statists in response to Heller. 'Ok, maybe you have a right to own a gun that we can't completely reject,' they say, 'but only at home.' The District of Columbia will now let you own a gun, to be kept at home, registered with the police; you may not even possess ammunition for it without that permit. You may also not carry it anywhere without an additional license that they are under no obligation to provide you. (They also have a list of handguns you're allowed to buy if approved to buy them, which is called the "List of Handguns Not Determined to be Unsafe," and a list of guns you're definitely not, which they call an "Assault Weapons" ban though it appears to have been constructed largely based on scary appearances.)

So they are responding to the "right to keep and bear arms" somewhat like Ms. Clinton asking whether she was being asked if she had wiped her server with a cloth. "Yes, you can keep it at home; but bear it? What on earth could that mean? Like ride around on a bear with it? That sounds horribly unsafe."

Addendum to Part VII

Though I'm going to leave the discussion of whether our approach to relations is better than the Greeks for another day, I would like to say some things about how we handle it in logic.

There are different ways that contemporary symbolic logic tends to handle relations like the kinds Parmenides was discussing in the previous post. I'll walk through just one. Let's say that we wanted to express a likeness relation such as "All crows are black." It would look like this:

∀x(Cx⊃Bx)

That is read, "For every x, if x is a crow then x is black."

Notice that this relationship does not include any actual crows, only a variable x. It is a statement that describes every object in the universe, most of which will fail to be crows. Those things that satisfy the crow condition will, if the statement is true, also satisfy the black condition.

It is possible to talk about an actual crow. Let's say object a is a crow. (Variables are from the end of the alphabet, whereas letters taken from the beginning of the alphabet are constants. That is to say that x or y could be anything, but a is a particular something.) Now you can test the proposition, because if Ca is true and Ba is false, then the proposition is false. It only takes one counterexample to falsify a universally quantified statement like the one above (the upside-down A is the universal quantifier).

So one way we can talk about relations between objects is to use the capital letter to indicate a class of things. All of those things are automatically related to each other by being members of the class. 

That doesn't actually solve the problem Parmenides is raising, though, because you still need two things at work to express the relation. Let us say that a is not a crow, but the One or really any Form that is a unitary idea. Being a pure unity, it is just a and not Fa. A genuinely pure unity cannot admit of likeness in this way either.

It also, it turns out, can't admit of unlikeness on this model because you'd have to say that a was not a crow, and that requires three concepts working together: the constant, the class, and the negative operator. (¬ Ca). The pure unity does not admit of either the ¬  or the C, because if it did, it would no longer be just one idea but multiple ideas.

It actually seems like the Form of a Crow would be C, though, not a. After all, it is x's participation in C that makes it a crow. Now you might say that the One is "O," and any x might participate in it without changing it. So you could have Oa and Ob, where a and b each participate in the One without being the One. This addresses Parmenides' concerns somewhat, because whether any x does or doesn't participate in O, O remains singular and unchanged by the participation or lack thereof. 

Viewed that way, contemporary symbolic logic depends upon Platonic forms. So too does mathematical logic, and therefore math itself.