Walsh's interesting paper considers six arguments against the singularity.

The fast thinking dog argument
Computers are fast. I agree that it is not fundamental for achieving our goal. Intelligence is more than considering many possibilities as fast as possible. If one handles them badly, one can waste a lot of time. However, it can be very useful.

The anthropocentric argument
Many suppose that human intelligence is something special, and they assume that it is enough to design a system which could reach the singularity. Here again, I completely agree with Walsh: our intelligence is only a particular form of intelligence, which evolution allows us to have. Why could this state allow us to realize systems very much clever than ourselves? And even if we create them, it will perhaps be not enough to reach the singularity.

The meta-intelligence argument
The capacity to accomplish a task must not be confused with the capacity to improve the capacity for accomplishing tasks. With present methods, excellent results have been obtained in several domains; however, the systems have always been realised by teams of many experts; it is not an AI system that solves the problem. Therefore, if a system is learning to play Go, it does not learn to write better game playing programs. An improvement at the basic level, solving a particular problem, does not lead to an improvement at the meta-level, solving large families of problems.
However, there are exceptions: CAIA uses the same methods for solving a problem than for solving some meta-problems. For instance, it finds symmetries in the formulation of a particular problem. Finding the symmetries of a problem (which is a meta-problem) will improve CAIA's performances for solving this problem. In this case, it is bootstrapping.
Unfortunately, this situation happens rarely. The reason is that most of the meta-problems are not defined as the problems solved by AI systems, which have a well-defined goal. Usually, the goal of a meta-problem is vague: can we tell that the monitoring of the search for a solution is perfect? We are glad to have solved it: we feel that we have not wasted too much time, but is it possible to do it better? Their goals cannot be defined as well as checkmate in chess. For achieving a bootstrap successfully, one must solve many meta-problems, where one is interested in the way problems are solved.  They are often very different from the problems for which AI researchers have developed efficient methods. However, learning to monitor the search for a solution would be useful for many problems, including this meta-problem itself: a virtuous circle would be closed. This is a part of the singularity.

The diminishing return argument
It often happens that we have very good results when we begin the study of a family of problems. This explains the hyper-optimistic predictions made in the beginning of AI: we did not see that forto progressing just a little more, a huge amount of work is necessary. Here, I do not completely agree: it may happen that discontinuities suddenly entail an impressive progress. For instance, the appearance of the reflexive consciousness brought an enormous discontinuity of the intelligence for the living beings. It is one of the main reasons of the existing gap between the intelligence of the smartest animals and that of the man. Other kinds of discontinuities may exist, which can also lead to an extraordinary increase of the performances. It is difficult to predict when it is going to arrive, no more than a dog can understand our reflexive consciousness.
Self-consciousness is precisely a domain where we can predict a discontinuity in the performances of AI systems, without any idea of when it is going to occur. Indeed, for us, it is a wonderful tool, but it is very limited: the largest part of what takes place in our brain is unconsciously made. Moreover, we have difficulty observing what is conscious because we do not manage to store it. Yet, we can give to our AI systems many possibilities in this domain: CAIA can study all of its knowledge, it can observe all the steps of its reasoning that it could want to, it can store any event. Naturally, it is impossible to observe constantly everything, but it is possible to choose anything among what happens. The difficulty is that I do not know how CAIA could use these capacities efficiently: I have no model because humans cannot do this. Therefore, I am only using them for debugging. Super-consciousness is an example of what could someday be given in the future AI systems; for the present time, the instructions for use are still missing. This is one of the improvements that could lead to AI systems with behavior as incomprehensible for us as ours is incomprehensible for dogs.

The limits of intelligence argument.
The intelligence of living and artificial beings have limits. This is well known since the limitations theorems such as Gödel incompleteness: some sentences are true, and there does not exist a proof showing that it is a theorem. It is possible that it is the case with a sentence as simple as Goldbach conjecture. However, this does not mean that it is impossible to go considerably further than what we achieve now.

The computational complexity argument
For some problems, even very much faster computers would never be able to solve them with the combinatorial method: there are too many branches. This is true, but it is possible that these problems could be solved by a non combinatorial method. Let us consider the magic squares NxN, with N odd. When N is very large, we cannot use the combinatorial method: there are 2N+2 constraints, each of them has N+1 unknowns, which can take any value among N² possible values. If N=100,001, there are 200,003 constraints, each of them with 100,002 unknowns with 10,000,200,001 possible values. This is a very hard problem, even if we are using heuristics for reducing the size of the tree.
Nevertheless, by 1700, a Belgium canon discovered a non combinatorial method that directly generated the values for all the unknowns. I wrote, a small C program (only 26 lines) that generated a solution in 333 seconds. Therefore, is it impossible that, for many problems apparently insoluble with the combinatorial approach, a super-intelligent system would discover a method for finding solutions without any combinatorial search? Complexity is related to an algorithm, but one may solve this problem without using a combinatorial algorithm.