What makes you the same person over time may not be that there is some essence that you have at each time, but rather that your different 'slices' are related to each other in the right way. Think of climbing rope. Modern synthetic ropes do have a single filament running their entire length, but the old-fashioned ropes are made of of many relatively short fibers woven together. So there is no fiber that runs the entire length, but still different 'slices' of the rope are all part of the same rope, because of the way they are connected to each other. Admittedly, just what 'related to each other in the right way' comes to in the case of people is a more difficult question. If you want to pursue the topic of personal identity, a good place to start is John Perry's A Dialogue on Personal Identity and Immortality.
This is, indeed a matter of great controversy, and one that has generated a vast literature. There are those who argue that it simply wrong to say that you are REALLY the same person you were before, because there are so many differences; others who argue that because it is correct to say that you are that there must be SOMETHING about you, or the continuum that constitutes you that undergirds that fact. Others suggest that you identity is a kind of very useful fiction or construct, and so that while it is true that you are the same person now as you were when you were 2, that is true in the same sense that it is true that Ahab was captain of the Pequod, that is because we say so, and this is the kind of thing that can be made true by say-so, as opposed to by discovery. You might enjoy reading, from the Buddhist literature, the QUESTIONS OF KING MELINDA, relating a lovely dialogue between a king and a Buddhist philospher regarding this point, or David Hume's discussion in A TREATISE OF HUMAN NATURE, or for more recent treatments, Derek Parfit's REASONS AND PERSONS or Mark Siderits' PERSONAL IDENTITY.
In Plato’s Cratylus, the character Socrates makes thefollowing comment about Heraclitus: “Heraclitus is supposed to say thatall things are in motion and nothing at rest; he compares them to thestream of a river, and says that you cannot go into the same rivertwice" (402a). Ever since Plato, the view that we can’t step twice intothe same river has been attributed to Heraclitus.
However,let’s consider the following two fragments about rivers that manyancient scholars regard as Heraclitus’ own words (in translation):
"On those who enter the same rivers, ever different waters flow– and souls are exhaled from the moist things" [B 12].
"We step and do not step into the same rivers, we are and we are not" [B 49a].
Inthe first fragment, Heraclitus suggests that we do step into the samerivers, even though the water in these rivers changes. The secondfragment raises interpretative problems of its own, but here tooHeraclitus speaks of the same rivers.
So how can we choosebetween the interpretation that Plato seems to give of Heraclitus, thateverything is always changing in every respect (a river is not even ariver from one moment to the next) and a less radical interpretationaccording to which Heraclitus is saying that things are always changingin at least some respect, but, for all that change, may well remainstable in at least some other respect?
Well, first it’s notclear that the position that Plato attributes to Heraclitus is evencoherent. But more importantly, it’s hard to reconcile Plato’sinterpretation with other Heraclitean fragments. Consider, for example:
"Theworld, the same for all, neither any god nor any man made; but it wasalways and is and will be, fire ever-living, kindling in measures andbeing extinguished in measures" [B 30].
Fire isclearly very volatile and is in a constant state of change, butnonetheless Heraclitus is suggesting here, it doesn’t change in respectof being fire. In fact, Heraclitus was reported by Theophrastus assuggesting that the change that an object undergoes in one respect canaccount for its stability in another respect:
"Things which have this movement by nature are preserved and staytogether because of it– if indeed, as Heraclitus says, the barley-drink separates if it is not moving" [B 125] (Theophrastus, On Vertigo 9).
Abarley drink cannot continue to be a barley drink over time, unlessit’s in constant movement. Fire can’t remain fire unless it’s inconstant movement. A river can’t remain a river, unless it’s watercontinues to flow. Or as Heraclitus says:
"Changing, it rests" [B84a].