Descartes’ self-described aim in the Meditations is to engage in a systematic examination of his belief system so as to be assured that it rests upon solid grounds. In the earlier Discourse on Method (1637), he compared systems of knowledge to buildings, which he said were always better when designed by a single architect and built from scratch than those which developed piecemeal over time. When confronted with the second sort of building, it is better to tear it down and rebuild it than it is to try and repair it, and we should think of our knowledge in the same way. Rather than try to root out all the false and half-baked beliefs we’ve accumulated over the course of our lives, we should start entirely anew.

The buildings which a single architect has planned and executed, are generally more elegant and commodious than those which several have attempted to improve, by making old walls serve for purposes for which they were not originally built…   

In the same way I thought that the sciences contained in books…, composed as they are of the opinions of many different individuals massed together, are farther removed from truth than the simple inferences which a man of good sense using his natural and unprejudiced judgment draws respecting the matters of his experience…

As for the opinions which up to that time I had embraced, I thought that I could not do better than resolve at once to sweep them wholly away, that I might afterwards be in a position to admit either others more correct, or even perhaps the same when they had undergone the scrutiny of reason. 

He begins this task in The Meditations by entertaining a series of skeptical considerations, each one more far-reaching in scope than the last, and by the end, he has managed to cast doubt on every possible variety of belief, whether grounded in sensory experience or in deductive reasoning. 

Whichever of Descartes’ skeptical scenarios we choose and whether we are talking about the justifications we offer for empirically grounded beliefs or for those based in reason, there is a common underlying logic to the critique offered. So let’s examine each in turn.

[1] Belief: It’s raining outside.

[2] Justification: I am looking out of the window, and I can see that it is raining.

In his famous “Dreaming Argument,” Descartes points out that we can have exactly the same experience in a dream that we have while awake and that from within the experience, there is no way to distinguish one from the other. Thus, I can dream that I am looking out a window and seeing rain and that experience can “feel” identical to the experience I have when I am actually looking out the window and seeing rain.

Consequently, [2] is not sufficient as a justification for [1]. Indeed, it only provides such a justification, if I can also justify the following:

[3] Belief: I am not currently dreaming, but really looking out of the window.

The trouble, of course, is that this belief cannot itself be justified by appeal to other sensory experiences – since it is the validity of my sensory experiences that is under question – but it also cannot be deduced or otherwise justified via reason: there is no way to prove that I am not currently dreaming.

Now, let’s turn to knowledge in mathematics. 

[1a] Belief: 2+2=4.

[2a] Justification: Some deductive proof.

In his “Malicious Demon” argument, Descartes maintains that it is possible that our minds are being manipulated by a powerful, malicious being, such that we sometimes – or always – reason incorrectly. Consequently, [2a] is not sufficient as a justification for [1a]. It only provides such a justification, if I can also justify the following:

[3a] Belief: I am not reasoning incorrectly.

The trouble, of course, is that this belief cannot itself be justified by appeal to reasoning – since it is the soundness of my reasoning that is under question – and it also cannot be justified empirically, insofar as mathematical statements are not knowable a posteriori.

When confronting these arguments, many have focused unduly on Descartes’ somewhat exotic examples, but they really are entirely dispensable and only serve an illustrative function. What both arguments rest upon is recognition of the inherent fallibility of our faculties and of the problem of trying to verify the sound exercise of those faculties, by employing the very same faculties.  

Notice something else. To the extent that our common belief in the external world and an objective reality – that is, in a world and reality that exists independently of our minds, thoughts, and experiences – is grounded in beliefs that come from sensory experience and reason, we clearly are engaged in circular reasoning. After all:

[1b] Here is a hand. (Holding out my hand) 

only provides evidence for 

[2b] The external world exists.

if I already presuppose the external world’s existence. If I am a disembodied consciousness, living in a perpetual dream, then seeing a hand is no evidence at all that there is an external world.

Precisely the same argument can be made on the “reason” side of the ledger. And so what we are left with is this – precisely the things that we take as evidence that an objective reality exists only count as evidence if one already presupposes that reality’s existence. Descartes’ solution to these problems, in which he appeals to the Divine and the perfection associated with it and its creation, is unacceptable for obvious reasons. If we can’t even justify the belief that there is a hand, we certainly can’t justify believing there is an invisible, perfect superbeing. My own view is that problems such as these have no solutions and that the value in engaging with them is rather to disabuse us of various notions we may hold about what we know, how well we know it, how good our reasons really are for the things we claim to know, and the like.