Diels–Kranz (DK) numbering is the standard system for referencing the works of the ancient Greek pre-Socratic philosophers, based on the collection of quotations from and reports of their work, Die Fragmente der Vorsokratiker (The Fragments of the Pre-Socratics), by Hermann Alexander Diels. The Fragmente was first published in 1903, was later revised and expanded three times by Diels, and was finally revised in a fifth edition (1934–7) by Walther Kranz and again in a sixth edition (1952). In Diels-Kranz, each passage, or item, is assigned a number which is used to uniquely identify the ancient personality with which it is concerned, and the type of item given. Diels-Kranz is used in academia to cite pre-Socratic philosophers, and the system also encompasses Sophists and pre-Homeric poets such as Orpheus.

Stephanus pagination is the comparable system for referring to Plato, and Bekker numbering is the comparable system for referring to Aristotle.

The works of the pre-Socratics have not survived extant to the present day. Our knowledge of them exists only through references in the works of later philosophers (known as doxography) in the form of quotations and paraphrases. For example, our knowledge of Thales of Miletus comes largely from the works of Aristotle, who lived centuries after him. Another interesting example of such a source is Hippolytus of Rome, whose polemic Refutation of All Heresies is a source of many direct quotations of Heraclitus as well as of other philosophers, thereby perpetuating the work of those he was refuting.

These quotations, paraphrases and other references to pre-Socratic philosophers were collected by Diels and Kranz in their book, which became a standard text in modern pre-Socratic education and scholarship. Because of its influence, Diels-Kranz numbering became the standard way of referencing the material: in literature, conferences, and even in conversation.

The number corresponding to an item was made up of three parts:

The above text has a DK number of 11A9, since it refers to Thales who is, as mentioned above, chapter 11's subject. The source is Theaetetus (one of Plato's dialogues), and gives an account of Thales' life, hence it is a testimonium, represented by the letter A. Finally, it is the ninth item in its chapter, giving it the overall number of DK 11A9.

Sometimes, the chapter (personality) number may simply be replaced by the name, which can be helpful in cases where the former is the same as the passage number, to avoid ambiguity. For example:

Rather than "22B22" the above may also instead be referred to as "Heraclitus B22" as it is a direct transmission of the words of Heraclitus (thus, B) and is the 22nd item in the chapter about Heraclitus (whose chapter number is also 22) in the Fragmente.^[6]

The following table gives the Diels-Kranz numbering of Pre-Socratic philosophers.^{[7][8][9][10][1]} Note that the numbering scheme presented is that of the fifth edition of Die Fragmente der Vorsokratiker, the first to be revised by Kranz. The fifth edition's numbering is the scheme which has since gained the most traction in modern Pre-Socratic scholarship, and it is the one used consistently throughout this article. It should not be confused with the numberings given in other versions, which changed frequently depending on the particular edition of the Fragmente.^[2][11]

Most entries (78) are concerned with a single, named individual, while the remaining minority of entries (12) have more complex context. Of these latter, eight (10, 19, 39, 46, 53-56) are each concerned with groups of named personalities, who typically have a clear relationship of some kind to justify their association in each entry. Two entries (58, 79) are devoted not to individuals, but to schools of thought (Pythagoreanism and Sophism), and the last two (89, 90) reproduce contemporaneous anonymous texts. Although "the Seven Sages of Greece" implies a clearly defined set of seven people, historical disagreement renders intractable the problem of exactly who they were, with multiple sources suggesting several different candidates. If one takes the Seven Sages as a group of seven and includes the later Iamblichus, Diels-Kranz encompasses 106 named personalities and two anonymous authors. The chapter on Sophism is concerned with the named sophists who take up most of the rest of the scheme, and per Freeman with regard to the chapter on Pythagoreanism, a catalogue due to Iamblichus lists 218 named men and 17 named women as Pythagoreans, along with other probable, anonymous adherents.^[12]

In several cases, the personalities listed are so obscure that they are merely mentioned by name in other sources, commonly with hints as to their geographical and philosophical associations, and without even surviving paraphrases of any of their ideas, or what they might have written. That is, these more obscure personalities survive in the historical record only as names cited by others, and so came to be included in Diels-Kranz for the sake of scholarly completeness.