Heraclitus (who was active around 500 BCE) is the "Pre-Socratic" philosopher with probably the most influence today and together with Parmenides (it's not clear which of the two lived first or whether they read each other) is considered the inventor of Western metaphysics. His book, if he even wrote a full book, did not survive antiquity, but he was quoted extensively by other authors, though in some cases (we think) erroneously. In particular, the characterization of him as a philosopher who posited that existence is pure flux (Plato's characterization of him) seems to be bogus.
Eva's emphasis in the book is on the Greek term "logos"--in Latin "ratio"--which has such a rich variety of meanings that she thinks we should really stop trying to translate it and just add "logos" to our lexicon. It's about knowledge, being at the root of all the "-ology" words we have, and rationality, but not just in someone's mind, but in nature itself. To Heraclitus, "listening to the logos" is about trying to suss out the underlying logic in nature, and he was (probably) the first one to try to figure out not just what the underlying element of nature is (as did his proto-physicist predecessors), but what the underlying principle of it is, and he asked the question whether this lawlike appearance of nature is something immanent within nature itself or imposed transcendentally from without. His answer, strangely to our ears, is both. Fragment 32 states: "The wise is one only. It is unwilling and willing to be called by the name of Zeus."
Several of Heraclitus's aphorisms make use of and refer to double-meanings (in Greek, so you don't really get this from just reading one translation). Fragment 93 says "The lord whose is the oracle at Delphoi neither utters nor hides his meaning, but shows it by a sign." So whereas when a person says something ambiguous, you figure he means some one of the possible meanings and the ambiguity is just an epistemic matter of your having to ask him for more information, but, like the notoriously ambiguous oracle, a written sign doesn't have intentions: it just has the multiple meanings. Heraclitus thinks that nature itself is like this. Fragment 61 famously states: "The sea is the purest and the impurest water. Fish can drink it, and it is good for them; to men it is undrinkable and destructive." More famously, fragment 12 states, "You cannot step twice into the same rivers; for fresh waters are flowing in upon you." In both of these cases, our modern mindset is to disambiguate the apparent paradox, and it's true that you can't dismiss Heraclitus as a relativist in the sense that he thinks facts will change depending on your perspective; on the contrary, you need a stable situation, a stable set of facts, in order to have two perspectives on it. But he wants us to expand our minds to hold both perspectives in mind at once, and in particular to get beyond our limited human concerns. Fragment 102 states, "To God all things are fair and good and right, but men hold some things wrong and some right."
According to Eva, "god" and "logos" and "the common" (i.e. the logos is phenomenologically accessible to each of us; it's objective) and, it turns out, "fire" (which does not refer just to visible fire, but to a principle of measured exchange between all elements; Eva posits that Heraclitus came up with a version of the law of conservation of matter) all end up coming down to the same thing: to this vibrant, eternally tense unity of opposing forces. The unity, the bringing together (one of the meanings of "logos"), is not just something we do as knowers, but is there in nature itself. Fragment 50 states, "It is wise to hearken, not to me, but to my Word [logos], and to confess that all things are one." But this "one" is also an irreducible multiplicity governed by strife. Fragment 80 states, "We must know that war is common to all and strife is justice, and that all things come into being and pass away through strife." Eva makes a big deal of the depiction of this as tension, as in the ancient Greek carving of wrestlers (above). Fragment 51 states, "Men do not know how what is at variance agrees with itself. It is an attunement of opposite tension, like that of the bow and the lyre."
Though this may all sound a bit mystical, that's not Eva's take: Heraclitus is appealing to the ability that we all have (but he thinks we don't in general use) to rationally evaluate the world around us, and thought that his logos constituted an explanation of sorts, covering everything from astronomy to politics. Eva's book has a section devoted to Heraclitus's influence on philosophers like Schopenhauer, Nietzsche, and Heidegger, and how parallels can be seen even in James Madison's account of political factions in Federalist Paper 10. He's sometimes called the father of process philosophy. Fragment 125 states, "Even the posset separates if it is not stirred," which is about a certain kind of salad dressing-like sort of drink that is only what it is when it's in motion. One can generalize this to all living beings: we are what we are because we're changing constantly. To be static is to be dead (and of course even the dead start to decompose right away). One could (and some have) generalize this further to physics, but Heraclitus certainly doesn't go into enough detail for us to attribute that to him with certainty. It's a creative act to tease out a whole philosophy from this set of 100some fragments, and Eva's creativity here is impressive.
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The version of the fragments we used for discussion purposes (where the fragment numbers match those in Eva's book) can be found here. Another translation that includes the context of the other author in which the fragment occurred can be read here. Librivox offers a recording of all the fragments read aloud here, and if you want to get a jump on understanding them before our episode is released, listen to episodes 5 and 6 of the Philosophy Without Any Gaps podcast or this discussion featuring three scholars on BBC's "In Our Time" podcast.