A
philosopher and
mathematician (c. 417 - 369 BCE) that lived in ancient
Athens. According to the Suda
Lexicon,
1 there are two references to an
ancient scholar that both have the
name of Theaeteus, and experts
debate over whether or not they are the same
individual. Most of the knowledge of Theaetetus comes from Plato's writing about him.
Plato wrote two
dialogues in which he is a central
participant, the Theaetetus and the
Sophist.
Theaetetus made significant contributions to the field of mathematics. None of his writing has survived to the present day, but it is thought that Books X and XIII of Euclid's work Elements are descriptions of Theaetetus' work. From Pappus' commentary on Book X:
"The aim of Book X of Euclid's treatise on the "Elements" is to investigate the commensurable and the incommensurable, the rational and irrational continuous quantities. This science has its origin in the school of Pythagoras, but underwent an important development in the hands of the Athenian, Theaetetus, who is justly admired for his natural aptitude in this as in other branches of mathematics. One of the most gifted of men, he patiently pursued the investigation of truth contained in these branches of science ... and was in my opinion the chief means of establishing exact distinctions and irrefutable proofs with respect to the above mentioned quantities." 2
Theaetetus was heavily influenced in his mathematical work by his teacher Theodorus who is also a participant in the Socratic dialogue Theaetetus. From the Elements:
" ... was considerably developed by Theaetetus the Athenian, who gave proof, in this part of mathematics as in others, of ability which has been justly admired. ... As for the exact distinctions of the above-named magnitudes and the rigorous demonstrations of the propositions to which this theory gives rise, I believe that they were chiefly established by this mathematician. For Theaetetus had distinguished square roots commensurable in length from those which are incommensurable, and who divided the more generally known irrational lines according to the different means, assigning the medial line to geometry, the binomial to arithmetic and the apotome to harmony, as stated by Eudemus ... " 2
Theatetus is also said to have been the first to work with the dodecahedron and the octahedron, two of the five "Platonic" figures, the other three of which were initially worked with by Pythagoras. He is also often credited with developing the Theory of Proportion which appears in the work of Exodus.
Theaetetus is also the name of a Socratic dialogue written by Plato. Partcipants in the dialogue include:
The central particpants are Socrates, Theaetetus and Theodorus. Knowledge is the most important issue explored in the dialogue, as Plato inititally presents the idea that knowledge is comprised of seeing and doing, as in learning a craft. Plato's theory of knowledge is looked at in a more advanced manner in The Republic and the Meno, but here it is first understood as being the mastery of a skill 3. This definition is also looked at in The Apology. 4
At the beginning of the Theaetetus, Socrates prompts Theodorus to talk about his young student, Theaetetus:
"I speak without any qualms; and I assure you that among all the people I have ever met - and I have got to know a good many in my time - I have never yet seen anyone so amazingly gifted. Along with a quickness beyond the capacity of most people he has an unusually gentle temper ... People as acute and keen and retentive as he is are apt to be very unbalanced. They get swept along with a rush, like ships without ballast ... But this boy approaches his studies in a smooth, sure, effective way ... " 5
Socrates goes on to ask Theaetetus to defend this praise that Theodorus has given him by describing his work and what he has studied. Theaetetus does in the following manner:
"We defined under the term 'length' any line which produces in square an equilateral plane number; while any line which produces in square an oblong number we defined under the term 'power', for the reason that although it is incommensurable with the former in length, it is commensurable in the plane figures which they respectively have the power to produce." 6
Later in the dialogue, Socrates asks Theaetetus to give a definition of "knowledge". Theatetus attempts to give three explanations:
- Knowledge is Perception 7 Socrates dismisses this definition because he thinks knowledge presupposes being and truth, which perception cannot access. We perceive, he thinks, not by sensory organs, but via the soul.
- Knowledge is True Judgment 8 Socrates rejects this explanation using the example of Athenian jurors. In their case, they can have true judgment without knowledge as lawyers can pursuade them without teaching them anything (causing them to actually know anything). Instead, they are convinced to hold a certain belief - whichever belief the lawyer wants.
- Knowledge is True Judgment with an Account 9 No definitive conclusion is reached by Socrates on this account, as he attempts to define what giving an "account" entails, and their conclusion is that more inquiry is needed when they gather to discuss the matter again the next day. 10 Socrates is pleased that Theaetetus has become more adept at philosophical inquiry and criticism.
1 The Suda Project's website is at: http://www.stoa.org/sol/
2 http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Theaetetus.html
3 gk. technê
4 Plato. The Apology. tr. G. M. A. Grube. Stephanus pp. 19 e - 20 c.
5 Plato. Theaetetus. tr. M. J. Levett, rev. Myles Burnyeat. Stephanus pp. 144 a-b.
6 Stephanus p. 148 b.
7 gk. aisthêsis. Stephanus p. 151 e.
8 Stephanus p. 187 b.
9 gk. logos. Stephanus p. 201 d.
10 The Theaetetus leads directly into the next dialogue, the Sophist, where similar issues are discussed at length.