The Master

 

“Let no one ignorant of geometry
enter here.”

~Inscription written
over the door of Plato's Academy

 

Math on Plato

Our introduction to Plato was through his philosophy of mathematics (in the previous lesson). I admit that this is not the conventional way of discussing Plato in philosophy courses. However, mathematics had a profound effect on Plato. He was so taken by the subject that he believed that all knowledge could be like mathematical knowledge. In other words, math literally shaped how Plato understood the world. For Plato, mathematics is a portal into the transcendental, reality as it really is.

I’m not alone in this assessment. Mathematician and historian of mathematics Morris Kline sees that mathematics is central to Plato’s entire conception of knowledge, and he points out that Plato may have even been part of the quasi-religious sect of mathematicians headed by the intellectual descendants of Pythagoras (see Kline 1967: 62-63). There is definitely something to this idea. Notice, for example, the cryptic language that Plato uses when discussing the subject matter—that is, mathematical objects—of mathematicians:

"These very things that they [the mathematicians] model and draw, which also have their own shadows and images in water, they are now using as images in their turn, in an attempt to see those things themselves that one could not see in any other way than by the power of thinking"" (Republic, 510e-511a; emphasis added).

Agreeing that mathematics had a profound impact on Plato, Stewart Shapiro writes on how mathematics influenced Plato's views on knowledge:

“Plato’s fascination with mathematics may also be responsible for his distaste with the hypothetical and fallible Socratic methodology. Mathematics proceeds (or ought to proceed) via proof, not mere trial and error. As Plato matures, Socratic method is gradually supplanted. In the Meno Plato uses geometric knowledge, and geometric demonstration, as the paradigm for all knowledge, including moral knowledge and metaphysics... Plato finds things clear and straightforward when it comes to mathematics and mathematical knowledge, and he tries to extend the findings there to all of knowledge” (Shapiro 2001: 62-63).

Plato’s views on math, in other words, are a good introduction to his philosophical system. As such, let’s recap Plato’s views about mathematical objects before diving into his other views.

Plato on Math

Recall that if one searches for a truthmaker for "2+2=4", then one has a few options. One could side with physicalism, arguing that mathematical objects are just piles of physical stuff. But, unfortunately for the physicalist, physical objects can't account for mathematical concepts like CIRCULARITY and INFINITY, since there are neither perfect circles nor an infinite amount of stuff in the universe. You can also be a conceptualist, arguing that the truthmaker to "2+2=4" is a thought, presumably dependent on the firing of certain neural networks in my brain. But if we all build our mathematical ideas independently and subjectively in our brains, then conceivably this might make mathematical errors impossible. One might say, in other words, that the way they've mentally constructed the number FOUR is as a prime. You can also be a nominalist/fictionliast about mathematical objects, claiming that they are just convenient fictions. But this leaves you having to say that there is no truthmaker for "2+2=4", and that kinda sucks. And so, Plato's idea suddenly becomes appealing. He argued that mathematical objects are abstract objects. That is, mathematical objects are non-physical, unchanging, eternal objects that exist independent of human minds. They exist in another realm, and we typically refer to this other plane of existence where these objects exist as Platonic Heaven (in honor of Plato). It is these objects in Platonic Heaven that serve as truthmakers for "2+2=4".

As a metaphor for how he conceived of the world and the objects within, Plato gave us The Divided Line. What Plato has laid out for us is his hiearchy for the reality of objects. In other words, Plato is organizing types of objects from less fundamental to more fundamental. At the bottom are mere copies of things: reflections in the mirror, paintings, and the like. These are only copies of the real thing; for example, your reflection is a mere copy of the real you. The next level up is the realm of physical objects. This is where you and I live, along with all the physical things that we interact with on a day-to-day basis. We might think that this is the ultimate level of reality, but Plato disagrees. Upon reflection, we might come to think that Plato might have a point. After all, the reality that we see with our senses can't be the ultimate reality. We know that physicists tells us about a world of atoms and smaller subatomic particles that we can't see with the naked eye.

Figure 3.1 The Divided Line, from Shapiro (2000)
Figure 3.1
from Shapiro (2000).

As you can see in the diagram, the next level up consists of mathematical objects. This means that all mathematical objects (like numbers), as well as mathematical relations (like equality) and functions (like adding 1, squaring, and all the more complicated functions) exist in this realm, independent of all human thought. In other words, numbers are real and they are more fundamental than the reality we inhabit. This shouldn't sound too strange if you believe that mathematics has the power to help us understand our world: mathematics has this power because it is upon mathematics that our world is ordered. This is why mathematics is always involved in the sciences and in any other enterprise that involves knowing our world in a deeper way.

Above mathematical objects are The Forms on which our reality is based. These are the actual properties of the universe upon which everything we see is based, according to Plato. To see this realm is to have a "god's-eye-view". In other words, it is to understand reality as it really is. Try to imagine combining all the knowledge of all the disciplines and then extending that knowledge until it is complete, where everything that can be known is known. That is what it is to know The Forms. And of course, The Forms are ordered too, just like everything below it. At the top of the hierarchy is The Good.

And so Plato believed that this realm of The Forms could be understood via the realm of mathematical objects. In other words, to come to know The Forms, one must first know mathematics. Put counter-factually, if you don't go through mathematics, then you will not understand reality as it really is. All of our knowledge, then, will ultimately be based on mathematics; mathematics is the medium through which we can understand the basis of all reality (The Forms). I might add here that this is not at all a far-fetched idea. In his introduction, Turchin (2018) reminds us that academic disciplines mature as they are formalized, i.e., as mathematical modeling is incorporated into the discipline. Plato seems to have intuited that mathematics is fundamental.

Sidebar: Contemporary views

Was Plato right? Is there a primacy to mathematics such that it is more fundamental than our physical reality? What we can say with certainty is that many of the thinkers we've covered so far seemed to have thought so. They readily relied on mathematics as the avenue through which to understand the world. This is so much so the case that, in his intellectual history, John Randall made the following point:

“Science was born of faith in the mathematical interpretation of Nature, held long before it had been empirically verified” (Randall 1976).

This is what I've been repeating over and over again in this course: we're jaded. We know science is the best method for answering empirical questions. But thinkers in the past didn't have science's track-record to go on. Science as we know it today was still not in existence. And so, these thinkers from antiquity into the early modern period of philosophy tended towards a quasi-divinical, semi-mystical view of mathematics. They were, in other words, Platonists about mathematics.

I close this section with an interesting report. The cognitive scientist Donald Hoffman (2019) makes arguments, using evolutionary game theory, that imply that Plato was right—at least about mathematics.

 

 

Important Concepts

 

Decoding Plato

 

 

Food for thought

Food for Thought

Before you accept Plato's aristocracy as the best form of government, beware of the authoritarian nature of what Plato appears to be suggesting. First off, Plato begins the training of the armed forces of the state early in their childhood. From when they are babies, they are to be told stories that reinforce their feeling of kinship with members of their city and animosity to those who are alien. Plato also recommends that we tell these future soldiers The Noble Lie. This is a myth that encourages them to believe that they already have all that they need: it was a gift to them from the gods. As such, their priority is to embrace their fate, enthusiastically play their role in the city, and protect its inhabitants with their lives. In a lesson called Teaching Submission (in my course on Critical Thinking and Discourse), I go into more detail. You should be able to tell by the title that it is a dark topic: Plato is recommending that we breed nationalism and subservience to the state in our soldiers.

“We will tell them that they have gold and silver of a divine sort in their souls as a permanent gift from the gods, and have no need of human gold in addition. And we will add that it is impious for them to defile this divine possession by possessing an admixture of mortal gold, because many impious deeds have been done for the sake of the currency of the masses, whereas their sort is pure...

No, they alone among the city’s population are forbidden by divine law to handle or even touch gold and silver. They must not be under the same roof as these metals, wear them as jewelry, or drink from gold or silver goblets. And by behaving in that way, they would save both themselves and the city...

But if they acquire private land, houses, and money themselves, they will be household managers and farmers instead of guardians... For all these reasons, let’s declare that that is how the guardians must be provided with housing and the rest, and establish it as a law. Or don’t you agree?” (Republic, 416a-417b).

The next concerning aspect of Plato's kallipolis is all the censorship that appears to be integral for the well-being of society. Not only should the children's stories be safeguarded from anything that will make them question the Noble Lie, but we must ensure that our future soldiers are brave. So we must censor the poets and storytellers so that their stories are only of the type that make men brave. With regards to the children, Plato intends to "mold their souls" so that they are ideal citizens (see 377b-c). In fact, even the truth is to be regulated. Authorities need complete control of the facts that are going out to the population, otherwise stability might be jeapordized.

“So it’s appropriate for the rulers of the city, if for anyone at all, to lie for the benefit of the city as far as either enemies or citizens are concerned, but for everyone else, such a thing is not to be touched” (Republic, 389b-c).

Plato's beautiful city also requires a strict division of labor. In order for the city to be just, all citizens must do what they are best at. The merchants engage in their trade. Soldiers engage in the duties of a soldier. And the rulers rule with absolute power.

"Listen, then, and see whether there is anything in what I say. You see, what we laid down at the beginning when we were founding our city, about what should be done throughout it—that, I think, or some form of that, is justice. And surely what we laid down and often repeated, if you remember, is that each person must practice one of the pursuits in the city, the one for which he is naturally best suited... Moreover, we have heard many people say, and have often said ourselves, that justice is doing one’s own work and not meddling with what is not one’s own" (433a-b).

Even Plato's theory of the Forms has some authoritarian elements. The historian Charles Freeman discusses how the very existence of the Forms downgrades some humans and their gods. How so? Well, the pantheon of Greek gods was at least intelligible to regular people through the myths and the cults. That is, the gods were knowable to basically everybody. But it took time and effort to come to know Plato's Forms. You had to be part of an intellectual elite. And so, if you're not part of this intellectual elite, you probably couldn't help but feel inferior when faced with the timeless, non-physical, hierarchically-organized Forms. Thus, Plato categorizes human beings into two categories: the elite minority that understands the Forms and the masses who need to be guided by the minority.

Many are shocked by Plato's principles of statecraft. For example, deliberately sanitizing a nation's history and teaching lies to its young people in order to "mold their souls" and breed nationalism seems morally abhorrent. However, when some people hear this, they think to themselves, "That's exactly what nations do!" Similarly, Plato's concerns about democracy almost seem prophetic given the global rise of populism and factionalism in the 2010s.

 

 

Why Plato?

Why cover Plato at this juncture? Well, of course, Plato is integral to intellectual history. No introductory course to philosophy would be complete without some Plato. Here are some reasons for why it's important to know Plato:

  • Plato's views are an essential part of the history of Christianity and for that reason and essential part of Western history (see chapter 10 of Freeman 2007). By the middle of the second century CE, Christians began making use of Greek philosophy and “disentangling” the Christian teachings within it. Essentially, Christian intellectuals were selecting aspects of Greek philosophy that could be Christianized and discarding the rest. Clement of Alexandria, for example, claimed that God gave philosophy to the Greeks as a schoolmaster in order to pave the way for the coming of the Lord. Justin Martyr, a Platonist by training, believed could draw from both scripture and Greek philosophy to argue for its positions.

    As it turns out, Platonism was ideally suited for providing the intellectual backbone for Christianity. Platonists were used to dealing with an immaterial world in which the Good is at the top of a hierarchy and where the material world is inferior to the immaterial world. This became the Christian views that God is the creator of all and that the material world is full of sin. Platonists also developed the idea of a soul that was independent of the body—an idea that Christians readily accepted into their doctrine. Importantly, Platonists taught that only a few could glimpse and come to understand the reality of the immaterial world, and this gave backing to the church’s hierarchy, where bishops understood God’s message and the rest had to rely purely on faith—something Protestants would take issue with centuries later.

  • As I've been stressing, it appears that Plato's views on the power of mathematics have been influential throughout intellectual history. Even if he was anti-positivist, it could be the case that Plato's emphasis on mathematics did pay dividends further down the line. In chapter 14 of his Worldviews, DeWitt reports that it is likely that Copernicus was inspired to work out his complex model of the solar system due to his Neo-Platonist leanings.

  • In Shenefelt and White (2003, chapter 2), the authors make the case that Aristotle's work on logic—i.e., the study of validity—took off for reasons other than the brilliance of the author. In particular, the social conditions were ripe for a study of that kind. Why was Aristotle’s audience so receptive to the study of validity? Because 5th century BCE Athenians made two grave errors that 4th century BCE Athenians never forgave: 1. They lost the Second Peloponnesian War (431-404 BCE); and 2. They fell for “deceptive public speaking.” Aristotle's teacher, Plato, blamed both of those errors on a group of professional teachers of philosophy and persuasive speaking referred to as sophists. These first teachers were more pervasive in Athens than elsewhere due to the opulence of the city. But once public opinion turned on the sophists, things got violent. One person accused of sophistry (Socrates) was even put to death. And so the populace was ready to restore the distinction between merely persuasive arguments and truly rational ones. The study of the elenchus (persuasive argumentation) was divided into rhetoric and rational argumentation largely thanks to Plato. It was one step from here to standardize rational argumentation and study its logical form, which is what Aristotle did. As we will learn later, logic was a key ingredient in the digital revolution that we are currently experiencing.

But, to be honest, this is not the main reason why I bring up Plato here. I bring him up because Plato defended the view that we have an immaterial soul that existed before we were born. This is where he fits into our story. We are still trying to solve the Problem of Evil, and it would be helpful to overcome the Problem of Free Will. And so, we will attempt to defend the existence of souls. After all, souls are non-physical, and so they cannot be affected by the laws of nature. As such, determinism (or quantum indeterminacy) have no bearing on our souls. Our souls are free. That is, if we have souls, then they are free. So, here it is... DILEMMA #8: Do we have souls?

 

 

 

Executive Summary

  • Although Plato's work ranges across many fields of inquiry, in this lesson we enter Plato's philosophical system through his views on mathematics.

  • With regards to mathematics, Plato believed that mathematical objects are more fundamental than physical reality itself. This means that mathematical objects are not mere ideas (conceptualism) and they are not merely physical objects (physicalism). Instead, mathematical objects exist independetly of humans; they are eternal and unchanging. Mathematical objects, in fact, are the basis on which physical reality is ordered. There is one level of existence higher than that of mathematical objects: the third realm, the realm of The Forms. The study of mathematics is the gateway by which we understand reality as it really is. This is called Platonism about mathematics.

  • Although Platonism about mathematics might sound strange to some, many of the thinkers we've covered so far seemed to have implicitly believed this.

  • With regardsd to politics, Plato had an interesting cyclical theory. He believed that one form of government gives way to another and that this cycle repeats. He believed the best form of government is an aristocracy, where there is a rigorous training process through which the best candidates for political power are discovered. Aristocracies, however perfect, eventually devolve into a timocracy, rule by those that are martially minded. Timocracies devolve into an oligarchy, rule by those who value wealth above all else. These eventually devolve into a democracy, where all desires are valued equally—something which Plato thought was a recipe for disaster. Lastly, democracies devolve into a tyranny, or rule by a tyrant. Then the cycle begins again.

FYI

Suggested Reading: Plato, Book VIII of The Republic

TL;DR:

 

Supplemental Material—

Related Material—

 

Footnote

1. The sophists were probably more scholarly and less mercenary than Plato makes them out to be. For instance, sophists generally preferred natural explanations over supernatural explanations (i.e., positivism) and this preference might’ve been an early impetus for the development of what would eventually be science. Nonetheless, sophists would often argue that matters of right or wrong are simply custom (nomos). Although this view—which is called subjectivism—is a respectable view in ethics and aesthetics today, the sophists posited it in a somewhat crude way.