That’s an interesting question because most cells in our bodies will die and be replaced a number of times. Many brain cells die but they are not replaced. They are just gone. So what, if anything, remains the same?

One well-known professor of psychology, Susan Blackmore (pictured), argues that there is no continuity between our present selves and our past selves:

Susan says there is an “illusion of continuity”, but what we think is “us” is just a “multiple parallel system” with “multiple parallel things going on”.

So, she says, “the so-called me now is just another reconstruction. There was another one half an hour ago, and there’ll be another one, but they’re not really the same person, they’re just dust happening in the universe.”

However she admits that the sense of self is hard to shake. Intellectually, she thinks the self is an illusion, but emotionally and psychologically she still feels like there’s someone in there.

Eric Hatfield, “Susan Blackmore: there’s no ‘me’!” at Is there a God? (April 29, 2019)

Neurosurgeon Michael Egnor finds that perspective hard to understand:

If Blackmore is not the same person moment to moment, then her publishers are sending the royalty checks from her books to a long line of impostors. When she goes to the bank to cash checks, she’ll be withdrawing from another woman’s account. Of course, if someone called the cops (“Hello 911, there’s a different woman trying to withdraw from Susan Blackmore’s bank account!”), Blackmore could just claim that she (they) were continuously different persons, so the woman attempting the financial fraud against the previous woman isn’t her contemporary self anyway…

Michael Egnor, “Interview with a woman (or women) formerly called Susan Blackmore” at Mind Matters News

In any event, a University of Madrid research team decided to study the problem. The researchers found that “the essence of our being remains largely stable over the years”:

“In our study, we tried to answer the question of whether we are the same person throughout our lives. In conjunction with the previous literature, our results indicate that there is a component that remains stable while another part is more susceptible to change over time,” explained Miguel Rubianes, a researcher at the Department of Psychobiology and Behavioural Sciences Methodology at the UCM and the Centre for Human Evolution and Behaviour (UCM-ISCIII).

The ‘continuity of the self’—the capacity for self-awareness and self-recognition— remains stable whereas other components such as physical aspects, physiological processes and even attitudes, beliefs and values are more liable to change.

Even components such as personality traits tend to change slightly over the years, but “the sense of being oneself is preserved, improving our understanding of human nature,” according to Rubianes.

University of Madrid (UCM), “Are We the Same Person Throughout Our Lives? In Essence, Yes” at Neuroscience News (November 28, 2020)

The study published in Psychophysiology, used electroencephalography (EEG) to follow the brain patterns of 20 volunteers when images were shown of themselves and close friends during various stages of their lives. The researchers concluded that “the neural representation of oneself (i.e., “I am myself”) seems to be stable and also updated across time.”

Incidentally, they found that it took the 20 volunteers’ brains around 250 milliseconds to recognize our own identities as distinct from that of others.

So even though our bodies are almost entirely changed, both in appearance and composition from what they were decades ago, we feel the same and recognize images of ourselves almost instantly. That’s a good argument for the existence of a self that goes beyond mere matter.

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Oxford philosopher Richard Swinburne: Without a soul, there is no self. He presents new philosophical arguments, supported by modern neuroscience, in defense of the soul.

and

An Oxford neuroscientist explains mind vs. brain. Sharon Dirckx explains the fallacies of materialism and the logical and scientific strengths of dualism.