Heraclitus said “You cannot step into the same river twice” (fragment 41). But he also said “We step and do not step into the same rivers; we are and are not” (fragment 81).
The first view has a very large problem, if by it Heraclitus meant that there was no sense in which the river was the same and no sense in which the person stepping in the river was the same. If absolutely everything changes moment to moment, than truth also changes moment to moment, which is to say there is no such thing as truth. Thus the statement “All things change” has something of the form of the Cretan liar paradox: the statement cannot be true, because it denies the possibility of truth.
Fragment 81 provides the common sense view of the world: the river is constantly changing, yet there is something about the river that is the same today as it was yesterday, and there is something about the person stepping into the river that is the same today as yesterday. But what is this sameness? What endures?
I drive to and from work along the Mississippi River. Between the morning and evening commute, I and the river change. The river carries new waters and I have new gray hairs, more dead brain cells, and occasionally a novel thought. But the river is still the Mississippi and I am still Thomas. The one thing that obviously has not changed day to day is the word, the designator. Perhaps Heraclitus meant no more than this by reference to the enduring quality of the logos, but I think not.
There is something more than just the name that remains of the river and something more than just my name that remains of me. The river is an orderly process of change. The Mississippi came into existence after the last ice age through a process and someday it will cease to exist through a process. Similarly, I came into the world through a process of birth and am in the long process of dying. Change is orderly, it follows rules. I believe this is what Heraclitus meant by “logos” which is also what is meant by the “ology” of the various sciences. Science is the study and identification of the “rules” that “legislate” the orderly process of change. Hence the metaphor “laws” of Nature.
Science and spirituality both seek to know the enduring aspects of Nature. In the form of geology, science identifies what is enduring in a river like the Mississippi. In the form of psychology science attempts to identify what is enduring in a human self. But the science of psychology never quite gets at the sense that as “I” drive home at night I am the same person that drove to work in the morning. That sense is concrete and existential.
As Heraclitus implies, the self is like a river. The contents of the self, like the waters of a river, are moving and changing. “Stream of consciousness,” is an apt metaphor for the perceptions, thoughts, imaginings, emotions, sensations and urges that occupy our attention through the waking hours. Yet, there is also a sense of self that is like the banks of the river, a constant that holds this stream. The discipline of meditation (dhyana) teaches us to move our attention from the water to the bank, from the constantly changing to the constant. And that constant of the self seems other than the self; the Hindus call this constant “Atman.” (The word "God" is used by many to indicate one or the other or both constants.)
Heraclitus wrote that “the way up and the way down are one and the same.” The constant within and the constant without are also one and the same, but that is a more difficult thing to recognize.