The human brain and the human mind are both Turing Machines.

I think Alan Turing would agree with you on this.

I'm also sure that the proponents of Strong AI would agree with you as well. These are the people who believe that it's just a matter of time before we have robots that can do anything that humans can do. Robots and computers can already do some things better than humans, such as performing calculations at lightning speeds with no mistakes, and safe inexpensive self-driving cars may only be just a few years away. But will we ever have robots that are conscious of their own existence, enjoy beautiful music, have free will, and be able to make up a really good joke?

The proponents of Strong AI certainly feel that is possible, and that it will all come when the algorithms controlling the robot become long enough and complicated enough, so that these abilities will somehow emerge just from the sheer complexity.

However, Artificial Intelligence is only possible because someone wrote a program that tells the computer or robot what to do. In other words, AI is just algorithms. They may be very long and complicated algorithms, but nevertheless, they're still just algorithms. In his book The Emperor's New Mind (Oxford University Press, 1989), Roger Penrose explains how he cannot imagine how self-consciousness, intuition, insight, etc., can emerge from an algorithm, regardless of how complicated the algorithm can be.

I think the most convincing argument comes from Gödel's Incompleteness Theorem.

Using the symbolic notation that you can find in the book Principia Mathematica by Russell and Whitehead, Kurt Gödel constructed the statement "This statement cannot be proved." When you think about it, you realize that there is no way that you could either prove or disprove this statement starting with the Axioms and rules of inference found in Principia Mathematica, but that also means that the statement is true!

In 1959, Hao Wang wrote a computer program which, starting with the Axioms and rules of inference found in Principia Mathematica, it proved hundreds of theorems in Principia Mathematica, and it did it in only 9 minutes.

Of course, Hao Wang's computer program, or any other computer program, could never come up with the statement that Kurt Gödel came up with, and also realize that it's true. In fact, a human, strictly following the rules of some algorithm, would not be able to come up with Kurt Gödel's statement either.

Furthermore, it wasn't just Kurt Gödel who constructed a statement beyond the reach of any algorithm, and found it interesting. Any sane human of reasonable intelligence, after thinking briefly about Kurt Gödel's statement, also understood its significance and found it interesting. Could there ever be a computer that could find Kurt Gödel's statement interesting?

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All that being said, everyone admits that there have been unbelievable strides in Artificial Intelligence, particularly Weak AI. Just a generation ago, who could have imagined that today we would use smart phones, GPS, download all kinds of cool apps to our computers, etc. We can't even imagine what AI will bring to the next generation.

AI has the potential to surprise us and impress us in unimaginable ways for generations to come! I'm sure it will!