In this essay, I will argue and conclude that Descartes is not successful in showing that there is a real distinction between the mind and body.
Substance Dualism is the position that the mind is a separate substance and exists independently of the body and possesses mental properties through the change of which it persists. For Descartes, the mind/soul is a res cogitans, a thinking thing. A substance is something which can exist independently of other substance, which possess properties and which persists through property change.
Although we now very easily critique Descartes’ arguments, Princess Elisabeth of Bohemia critiqued them during Descartes’ lifetime, and they discussed these problems.
The main issue she raised was the interaction problem.  The interaction problem is the issue of how mental states can have physical effects, so how can the non-physical cause the physical.
The interaction problem is actually a problem because in order for x to cause y, ie something cause something else, it needs to be moved. In order for it to be moved, it needs to be pushed, and this pusher logically needs to be able to be causally extended, and a mental state cannot push and does not have extension.
This objection is the first of four reasons I will be addressing, in order to conclude that Descartes is not at all successful in showing a distinction between mind and body.
Throughout the sixth meditation, Descartes presents a number of arguments for his position. One such argument, the conceivability argument, can be outlined as follows:

I have a clear and distinct idea of myself as something that thinks and isn’t extended,
I have a clear and distinct idea of body as something that is extended and does not think
If I have a clear and distinct idea thought of something, God can create it in a way that corresponds to my thought.
Therefore, God can create mind as something that thinks and isn’t extended and body as something that is extended and does not think.
Therefore, mind and body can exist independently of one another.
Therefore, mind and body are two distinct substances.

We can understand the first two premises as the claim that it is conceivable that the mind and the body are distinct. It is not a logical contradiction to say “the mind and body are distinct”in the same way in which it is a logical contradiction to say “being a bachelor and being an unmarried man are distinct”.
If it is conceivable that the two are distinct, then there must be a possible world in which they are distinct. But if it is possible for the mind and the brain to be distinct then they must really be distinct since it would be impossible to say the mind and brain are identical if they can exist independently in some possible world in the same way as it would be impossible to say a bachelor is not an unmarried man in a possible world.
Conceivable distinctness therefore seems to entail the possibility of distinctness, and the position that the mind and brain can possible be distinct things is “substance dualism”. This argument is therefore an argument for substance dualism.
However, it is actually far from clear that conceivability entails any kind of possibility. In case of an identity there is no possible world in which the M can come apart from B if M = B, There is no possible world in which a bachelor is not an unmarried man. Similarly, if water is H2O there is no possible world in which water could not be H2O. However, although it is not be conceivable to you that there can be bachelors that are not unmarried as long as you understand the meaning of the word bachelor, it might seem conceivable that water could be something else than H2O. This, however, is probably because the identity between water and H20 is an a posteriori identity (and hence the necessity that obtains could be an a posteriori necessity) and not an a priori identity as the identity between bachelor and unmarried man. So although it might seem as if it is conceivable that the mind and the body can exist separately, it doesn’t follow that it is actually possible that they can exist separately if the identity between mind and brain is an a posteriori identity.

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It can be argued that the conceivability argument misapplies Leibniz’ law of indiscernibles and makes an intensional fallacy. Just because I can conceive of two things as separate it doesn’t follow that they are separate – only that I do not know that they are the same thing. You cannot draw ontological conclusions about how the world is and what exists in it from epistemological premises of what you can or cannot doubt or conceive.
A typical issue that faces substance dualism is the problem of other minds. The problem of other minds is the question of how we can realize that there are minds known to man other than our own. We each experience, or introspect, our own minds directly, from ‘inside’ yet our insight into other individuals’ minds is altogether different. We can’t encounter other individuals’ mental states. It appears that all we need to go on is other individuals’ conduct which is communicated through their bodies.

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This is a test for substance dualism in that if minds and bodies are independent of each other, then how might I derive from seeing a body that there is a mind ‘connected’. Different bodies could all be machines, modified to act as they do, yet without any mental states. If there are no other minds, at that point my brain is the special case that exists. This is solipsism and therefore the test to substance dualism is, how would we prove that solipsism is false?
It could be argued that we can infer the existence of minds from behaviours:

This behaviour has a mental cause
That behaviour has a mental cause
The third behaviour has a mental cause
etc.
Therefore, many behaviours have a mental cause
Other people exhibit the same types of behaviour as cited above
Therefore, those behaviours also have mental causes
Therefore, other people have minds.

However, the argument still does not fully refute the objection to substance dualism as it relies on mental causation and thus suffers from the interaction problem. We can also object that the belief that other people have minds is not a hypothesis, nor do we infer, on the basis of evidence, that they have minds. The whole way we think about other minds is mistaken.
Another of Descartes’ arguments to be found in the sixth meditation is the divisibility argument. It can be formalised as follows:

The mind has no parts within itself – it is either there or not.
The body does have parts. You can remove arms, legs etc.
Only that which has parts can be separated.
Therefore the mind cannot be separated.
Therefore the body can be separated.
Therefore, mind and body are entirely distinct types of thing.

This is by far the strongest of Descartes’ arguments for his cause, but there are still some objections to counter and ultimately devalue it.
First of all, we have the objection that not all things that the mental is divisible in some sense. For example, cases of mental illness such as bipolar or schizophrenia could suggest that the mind can be divided. In these cases it seems that some parts of the mind are unable to interact with others. Furthermore, there is a distinction drawn between consciousness and subconsciousness, and these in themselves could be separate “parts”.
Further still, not everything thought of as physical may be divisible, because when you break things down eventually you will get to the smallest possible thing that cannot be broken down any further for example an atom: you cannot have half an atom, or half a proton, neutron, etc.
Overall, these objections severely discredit the divisibility argument, which makes it a useless tool in Descartes’ arsenal.
In conclusion, it does not seem that Descartes is successful in proving that there is a real distinction to be made between mind and body. Based on the strong objection from Princess Elisabeth, the interaction problem, as well as the problem of other minds and the weakness of his conceivability argument, along with the objections to his divisibility argument, it seems logical to conclude that Descartes is unsuccessful in this regard.
References/Bibliography

Descartes, R. (1986). Meditations on First Philosophy. Cambridge [Cambridgeshire]; New York: CAMBRIDGE University Press.
Dualism (Stanford Encyclopedia of Philosophy). (2019). Retrieved from https://plato.stanford.edu/entries/dualism/
Lacewing, M. (2015). Philosophy for A2. [Place of publication not identified]: Routledge.
Lacewing, M. (2019). Substance Dualism. Retrieved from http://s3-euw1-ap-pe-ws4-cws-documents.ri-prod.s3.amazonaws.com/9781138793934/A22014/dualism/Substance%20dualism.pdf
Other Minds (Stanford Encyclopedia of Philosophy). (2019). Retrieved from https://plato.stanford.edu/entries/other-minds/