by Scott Soames
Eudoxus (born 395–390, died 342–337) knew of Plato’s Academy, which opened in 387, as a young man, and, after first establishing his own distinguished school of mathematics elsewhere, he moved it to the Academy, where he is credited with solving the proportion problem mentioned by Heath, (probably) prior to Plato’s death in 347. Thus Plato and the Academy had essentially all the results that would later be systematized in 300 BCE by Euclid’s Elements, the most influential work in ancient mathematics.
In Plato’s Academy philosophy and mathematics were seen not as independent disciplines, but as intimately related inquiries contributing to one another. The first European university, the Academy educated students in a curriculum that proceeded through arithmetic (number theory), geometry, stereometry (the measurement of solid bodies), and astronomy (the discovery of the mathematical properties of the heavens), before culminating in philosophy, or “dialectic” (reasoned philosophical argument). Thus, the words inscribed above its doors are reputed to have been “Let no one destitute of geometry enter my doors.”14 According to Heath, Plato was right to find the genius of the spectacular achievements of ancient Greek mathematics in their connection to philosophy. Speaking of those achievements, Heath asks, “How did this all come about? What special aptitude had the Greeks for mathematics?” He answers,
The answer to this question is that their genius for mathematics was simply one aspect of their genius for philosophy. Their mathematics indeed constituted a large part of their philosophy down to Plato. Both had the same origin.15
Rigor and precision were the origins of Greek mathematics and Platonic philosophy. Nothing is more characteristic of that philosophy than the search for definition. When Socrates asks What is goodness, beauty, truth, knowledge, virtue, piety, or happiness?, he is asking for definitions of the Greek words we would roughly translate as ‘good’, ‘beauty’, ‘truth’, ‘knowledge’, ‘piety’, and ‘happiness’. Let us use the term concepts, as it is often used in philosophy, for the meanings that abstract nouns and noun phrases like these share with their translations in other languages—the realities they are used to talk about.
Confessing not to know the proper definitions of these words/concepts, Socrates asks his interlocutors for help, and they typically offer examples—of good things, pious practices, and virtuous individuals. But Socrates doesn’t want examples; he wants to know what goodness, piety, and virtue really are. To know this is to know the real properties we attribute to something when we call it good, pious, or virtuous. Consider another case. Suppose we ask What is a circle? and someone answers by showing us a single circular figure, saying That’s a circle. We would respond: We don’t want an example of a circle; we want to know what it is for any conceivable thing to be a circle: what conditions it must satisfy to be a circle. The answer we seek is a definition: A circle is the set of all points in a plane equidistant from a single central point.
The definitions Socrates sought were similar—definitions that give necessary and sufficient conditions for any person or thing to be good, pious, or happy, for any statement to be true or known to be true by someone, or for anything to be beautiful. The goal of the joint Socratic and Platonic enterprise was to extend the objectivity and precision of Greek mathematics to the study of all reality. Just as stunning mathematical discoveries required concepts that were precisely defined (like circularity) or rigorously governed by axioms (like point), so the advances in knowledge of the world, and of ourselves, that Socrates and Plato hoped to achieve required precise, well-regulated concepts.
For Plato, this quest for knowledge rested on Platonic forms, which, he believed, were the precisely delineated concepts needed for knowledge in any domain. They include:
(i) the forms goodness, justice, knowledge, virtue, and happiness needed for general laws explaining human behavior, human nature, and human institutions;
(ii) forms for identity and distinctness, forms for different kinds and properties of natural numbers, forms for two-dimensional geometrical shapes (circularity, triangularity, etc.) and their properties (area, circumference, etc.), and forms for three-dimensional geometrical figures (sphere, cone, etc.) and their properties (volume, surface area, etc.), needed to state timeless mathematical truths;
(iii) forms for body, space, velocity, motion, rest, proportion, weight, dimension, etc. needed to describe aspects of the natural environment.
Plato’s goal was to use these concepts to construct general, exceptionless laws about ourselves and the cosmos, the truth of which would be knowable, yet independent of the knower.
It is telling that the Greek word Plato used for these concepts is translatable as “shape,” as well as “form,” indicating the role in his thinking of geometry, which was well understood in the Academy. Just as there is such a thing as the precisely defined form/concept circularity, which is neither itself circular nor located at any distance from anything else, so there is such a thing as the precisely delimited form/concept beauty, which is neither itself beautiful nor perceptible through the senses. The same is true of the other forms. Although they are real—there are, after all, such things as circularity, truth, and beauty -–to ask where they are in space and how long they have been there seems misplaced. Since Plato took them to be constituents of the objective truths about reality which, with proper study, we may come to know, they had to be independent of our minds. For him, this meant that they were real things existing outside of space and time, yet capable of being recognized by the intellect.
These abstract concepts needed to state general explanatory truths were half of the Platonic equation. Objective knowledge requires not only propositions to be known but also a mind capable of knowing them. What is this mind, or psyche, that it may know itself and the world? Havelock addresses the question, as it confronted Socrates and Plato.16
[T]owards the end of the fifth century before Christ [about the time of Plato’s birth], it became possible for a few Greeks to talk about their ‘souls’ as though they had selves or personalities which were autonomous and not fragments … of a cosmic life force.… Scholarship has tended to connect this discovery with the life and teaching of Socrates and to identify it with a radical change which he introduced into the meaning of the Greek word psyche.… Instead of signifying a man’s ghost or wraith … a thing devoid of sense and self-consciousness, it came to mean “the ghost that thinks,” that is capable both of moral decision and of scientific cognition, and is the seat of moral responsibility, something infinitely precious, an essence unique in the whole realm of nature.17
Plato outlines his conception of the soul in Book IV of the Republic, where he distinguishes three of its aspects or parts—the appetites or desires, willpower or emotive force, and reason. Proper education trains the will to be the ally of reason. Reason then controls desire, the soul is unified, and the agent achieves self-mastery. Plato speaks of this condition of the soul, in which each of its parts plays its proper role, as one of justice between the parts, drawing an analogy with the ideal state in which the philosopher king (reason) makes decisions for the good of all that are enforced by guardians (the will) to ensure proper order among the self-interested citizens (the desires). In Book VII he describes the education of philosophers in which they acquire abstract theoretical knowledge, which requires understanding the concepts that make such knowledge possible. To achieve such understanding, they must turn their attention to the forms, which are innately available to everyone.
The aim of Plato’s Academy was to provide the education that leads to this self-realization. It did this by taking students through a rigorous curriculum in logic, mathematics, and philosophy, designed to enable them “to define the aims of human life in scientific terms and to carry them out in a society which has been reorganized upon scientific lines.”18 In bringing this educational plan to fruition, Plato invented the idea of a liberal education and founded the first institution dedicated to providing it.
Plato’s g
reatest student, Aristotle (384–322/1 BCE), moved to Athens from his home in Stagira (in Thrace) in 368/7, joining the Academy at age 17. He remained there for twenty years, first as Plato’s student and then as his colleague, until Plato died in 347. At that point, Plato’s nephew became the head of the Academy and Aristotle left to found a branch of the Academy in Assos. In 343/42, he relocated to Macedonia, where he took over the education of a thirteen-year-old later known as Alexander the Great. Leaving his post when Alexander ascended the throne in 336/35, Aristotle returned to Athens a year later, where he founded his own “Peripatetic” school, in competition with the Academy.19 There he remained until two years before his death.
A prodigious worker, Aristotle produced an enormous volume of work that began in the Academy, continued in Assos, and reached its zenith at his Peripatetic School. His writings extended nearly every domain of human learning. They are organized around the following major topics: Logic and Language (including definitions of truth and falsity, their bearers, the nature of judgment, predication, generality, patterns of logically valid argument, and fallacies), Epistemology (including proof, knowledge, and deductive and inductive reasoning), Metaphysics (including substance, essence, accident, existence, and God), Physics (the natural world and the cosmos), Biology (including the history, generation, life, and death of animals), Psychology (including perception, memory, reasoning, sleeping, and dreaming), and Ethics, Politics, and Aesthetics (including rhetoric and poetics).
A close follower of Plato in his early days, Aristotle then believed in the immortality of the soul, its preexistence before birth when it was acquainted with Platonic forms, and the need in later life to recollect or rediscover the forms through philosophical argument.20 In time, he gave up those views and modified the theory of the forms in a far-reaching way. He also produced the first systematic codification of principles of logically valid inference and developed theories of ethics and politics that were more realistic and widely applicable than Plato’s. Finally, he began to make good on the implicit Platonic promise of advancing empirical knowledge of the physical world and our place in it. While it was Plato who, more than anyone else, provided the inspiration, conceptual foundations, and institutional framework to advance human knowledge, it was Aristotle who, more than anyone else, gave us the systematic beginnings of logic, physics, biology, and social science (including psychology and political science).
None of the steps that Aristotle took away from his teacher on fundamental philosophical matters was more important than his modification of Plato’s theory of the forms. To paraphrase what I noted earlier, it is plausible to suppose that just as there is such a thing as the form/concept humanity, which is neither itself human nor something of any height or weight, so there is such a thing as the form/concept redness, which is neither itself red nor any other color. Although it is plausible that humanity and redness are real, it seems strange to ask where they are in space and how long they have been there. However, Plato’s conclusion—that the forms exist outside of space and time, and so are eternal and unchanging, despite being accessible to the mind—isn’t the only way to think about them. Perhaps it sounds strange to ask when and where they are because they don’t exist at any single place or time, but do exist at many places and times—namely, at all and only the places and times at which humans and red things do. If so, they are contingent existents of our world just as we are. That is how Aristotle thought of them.21
This modification of Platonic doctrine brought with it a new metaphysics of form, matter, substance, essence, and accident, which Aristotle used in studying physical change. Consider an individual man, Socrates, and a particular mountain, Mount Vesuvius. Both came into being at a certain time, endured through many changes, and at some point ceased, or will cease, existing. As a baby Socrates was small, and couldn’t walk, talk, or survive on his own. In time, he grew larger and learned to do these things, while acquiring many new properties and losing others. One property he never lost was being human, which was essential to his nature. In contrast to this property, which is an essential property of everything that has it, the properties Socrates once lacked but later acquired, as well as those he once had but later lost, were inessential, or “accidental.” A similar story could be told about Mount Vesuvius. It, too, has essential properties, including being a mountain, that it can’t exist without having, as well as accidental properties that can be acquired or lost without affecting its continued existence.
Aristotle called things like Socrates and Mount Vesuvius, which endure through changes of the sort just illustrated, “substances.” But, since these substances themselves come into existence and pass away, there must also be changes in, or of, substance. How could there be? Aristotle’s model presupposes that in order for there to be change, something must change. What can it be, if no substance endures the change? His answer is that what changes is prime, or first, matter, which is the stuff out of which substances like Socrates and Mount Vesuvius are constituted. And what is that? The answer, unfortunately, is that prime matter is undifferentiated stuff—an all-purpose we-know-not-what in which properties, like humanity or mountainhood, somehow inhere. It is no good to ask what the essential properties of prime matter are. To think of a bit of prime matter as having such properties—in addition to accidental properties like making up Socrates, which it had for a time and then lost—would be to take it to be a substance, which it can’t be, if, as Aristotle demands, prime matter is to be part of the analysis of substance. For him, every natural substance—i.e., every physically enduring entity E—is some bit of prime matter M shaped by a substantial form F into the kind of thing that E essentially is by F’s inhering in M.
So the man Socrates was a bit of prime matter shaped by the form humanity. Since that is true of every human being, there must have been something else essential to Socrates that differentiated him from everyone else.22 It is not entirely clear, from Aristotelian texts, what this was. But there is reason to take it to be his individual essence, a form unique to him that Aristotle calls the immaterial soul inhering in and giving life to his body. Something similar is said about every living thing. Naturally, human souls are different from those of other living things. For Aristotle the souls of plants are responsible for their nutritive functions; those of animals and humans have a nutritive component plus a sensitive component responsible for sensation, movement, and memory, while human souls also have a purely rational component called “the active intellect.”
To appreciate this one must remember that for Aristotle, (i) contingently existing substances are combinations of form and matter, (ii) their forms don’t exist independently of the substances they inform, and thus, (iii) when Socrates ceased to exist any form unique to him, i.e., his soul, also ceased to exist. It follows that the soul of Socrates, which was really the form, or principle of organization, of his particular body neither pre- nor post-existed him. There was simply no room for human immortality in Aristotle’s world.23 He did believe in an eternal God of pure thought—which was the uncaused teleological cause, or reason, for the existence, of everything else. But his God didn’t intervene in the world, wasn’t the object of prayer or worship, and wasn’t a being that could either love or be loved.24 This contributed to the impression that Aristotle the philosopher and Aristotle the budding scientist were more or less one and the same. For him, the greatest human goal was not to conquer the fear of death, to find consolation in the face of life’s tragedies and disappointments, or to discover ultimate purpose in a universe impervious to our concerns; it was to understand the universe and everything in it. Although he believed this was the highest human good, he didn’t take it to be the only human good. He was well aware that other, less contemplative but more practical, forms of the good life were possible for a wider range of people.25
If, in looking at all this, we ask What, in sum, did ancient Greek philosophy contribute to the world?, we must include the following.
(i) It playe
d a vital role in transforming an oral narrative tradition based on myth and poetry into a more critically reflective culture in which the chief means of expression was the written word.
(ii) In so doing, it demonstrated the superiority of basing beliefs on evidence, argument, and rational examination, rather than on authority.
(iii) It offered a naturalistic worldview in which observed facts are to be explained not by interventions of deities, but by interactions of fundamental elements according to universal laws.
(iv) It asked the questions and provided the concepts—truth, proof, definition, matter, mind, motion, causation, generation, etc.—that made it possible to think scientifically about the world and ourselves, and, in so doing, laid the basis of what we now know as logic, physics, biology, psychology, and political science.
(v) It founded the first university in the western world, giving birth to the idea of a liberal education, blending mathematical, scientific, and humanistic investigations in a curriculum designed to produce not only technical proficiency, but wisdom.
(vi) It transformed our idea of god from an anthropomorphic one in Homer’s time to Aristotle’s perfectly good, perfectly rational cause of the universe, uniting Plato’s form of the Good with Aristotle’s Active Intellect.
(vii) It elevated our idea of the soul from a ghostly breath of life to the subject of conscious experience, the source of decision and action, and the seat of moral responsibility.
Though long and impressive, the list is still incomplete. In addition to providing the foundations for rational thought and the pursuit of theoretical knowledge in virtually every domain, the Greek philosophers, especially Socrates and Plato, imbued the search for theoretical truth with the urgency of a personal quest for meaning. The special genius that made them so compelling was in placing the idea that objective knowledge in any domain requires precisely defined concepts at the center of their vision of the good life. In order to live such a life, they thought, one should strive to know the essential nature of human goodness and happiness. Although Socrates didn’t claim to have such knowledge, he did claim to know that it could be acquired only by rigorous reasoning. He also thought that to know the good is to be sufficiently motivated to do it. Thus, he reasoned, one’s best hope of living a good life was in acquiring as much knowledge of oneself, of one’s nature, and of goodness as one could.