Descartes Exists
It is now some years since I detected how many were the false beliefs that I had from my earliest youth admitted as true, and how doubtful was everything I had since constructed on this basis; and from that time I was convinced that I must once for all seriously undertake to rid myself of all the opinions which I had formerly accepted, and commence to build anew from the foundation, if I wanted to establish any firm and permanent structure in the sciences.
Meditations on First Philosophy, in which the existence of God and the immortality of the soul are demonstrated was written by the French philosopher René Descartes and first published in 1641. In Meditations, Descartes begins with the premise that he wanted to find true factual "indubitable" knowledge and that the way to find this was by first doubting everything, and then retaining only what could be defended by rigorous logic.
The first thing Descartes was able to assert was his own existence. First, there seems to be no reason to doubt our own existence. Even the act of doubting seems to reinforce it. Second, no point in trying to ascertain the nature of reality if we ourselves do not exist. Claiming to prove our own existence by assuming our doubt is actually begging the question, but assuming our existence is a necessary assumption for everything that follows. So the first axiom is that the self exists: If you are reading this, then you can safely assume that you exist, not because that can be proven, but because trying to ascertain any truth while doubting your own existence is futile. The second axiom allows for other things to exist, given that they can act upon something else that exists. More concretely, if something that exists is being acted upon, then something must be doing the acting.
Assume that our senses exist, and we perceive an apple. We can deduce that something that can cause us to perceive this apple, exists. Not that the apple exists - this can be an illusion, a dream, a facade - but something that can make us perceive the apple, nonetheless, must exist. Assume a measuring instrument exists, and during its natural course as a measuring instrument, it happens to obtain some measurements. We can deduce that something caused the measurements to be obtained. Again, we can't know for sure what that something is, but it is definitely not nothing.
Let us now try to put on our critical hat and ask ourselves a hard question: Does a Soul exist? Can it act upon something that exists? Can it affect something? Can we measure it in some way? Can we measure its effect on something else? If it were to suddenly disappear, would we be able to tell the difference? The answer is largely dependent on the definition of Soul, and I will leave it to the reader to draw their own conclusions.