Because the topic of hypercomputation has a strong philosophical component, I will focus on the logical argumentation contained in this book, rather than going through the particular chapters (although I will provide specific examples and quotations).

First, I have to acknowledge the editorial quality of the book--or the lack thereof; this is more a critique of the publisher than the author, considering that Syropoulos might not be a native English speaker. The first typographical error appears on line six of the preface.

According to the book, Syropoulos’ arguments in favor of hypercomputation are roughly as follows: Searle’s Chinese room is a way to immediately acknowledge hypercomputation--Syropoulos presents no real refutations, although there are many and they are well founded; the book subscribes to Lucas’ Gödelian argument, without providing any refutation; models such as DNA computing and quantum computing, among others, are already steps toward hypercomputation; a hypercomputer is more feasible than a quantum computer; there is experimental evidence of a continuous space and time, but none for a granulated space and time, so the author assumes that space and time are continuous--in spite of quantum mechanics; and the fact that “space probes landing on other planets rarely survive more than their expected ‘life’ span” is an indication of the flaws of the Turing machine model and a serious argument in favor of Lucas’ Gödelian argument (this is actually in the chapter devoted to Lucas’ Gödelian argument).

I found an unfortunate comment in Syropoulos’ opposition to the Turing computable view of the mind/universe: “According to these views we are tiny Turing machines.” According to the book, the mind is regarded as a Turing machine by most researchers currently studying the human mind; even though I would have considered such a statement harmless in any other context, I found it an unfortunate oversimplification of the matter. This claim does not make the subtle difference between saying that the human mind might be a Turing machine and saying that the mind might compute the same set of functions as a universal Turing machine. This might be intentional, as the book is willing to mix one thing with another, in order to finally present as obvious the fact that a Turing machine cannot model many aspects of physical phenomena, not even the mind, and therefore concludes the necessity of hypercomputation. Having these kinds of oversimplifications, meant to disqualify opposing arguments instead of coming up with elaborate, thoughtful arguments, are quite unfortunate. But even if the human mind were able to compute just as much as a universal Turing machine, that does not imply a trivial simplification of the human mind. Turing machines have great power, and yet they have limits imposed by the difficulty of problems such as the halting problem, intractability issues, and other irreducibility constraints of various degrees that make Turing machines fascinating and rich systems, with as many dynamical properties as any other dynamical system we have studied thus far.

There are also many mistakes in the Church-Turing thesis chapter. According to the author, the Church-Turing thesis is more of a definition than a thesis in computability theory; later, Syropoulos claims it is the cornerstone of computability, rather than the consequence of the convergence of the definitions of computation. Syropoulos states that David Deutsch reformulated the thesis; he actually extended it in physical terms, so it is a different thesis. (The book continues to present arguments in an oversimplified way, in order to make it easy to refute them.) The book overlooks the attempts of researchers to axiomatize the theory of computability providing a formal framework to Church’s thesis. Finally, the author claims that the Turing-centered view has no experimental evidence and there is no evidence for the validity of the Church-Turing thesis; in fact, it is the other way around: there is no evidence that the thesis may be wrong, but a lot of evidence that it is correct.

According to the book, the field of hypercomputation hasn’t gained acceptance “because no one actually believed that one could compute the incomputable.” I find this argument wrong. It is the Turing model that proved to be feasible, and ever since we have not been able to show a single noncomputable function. The Turing model has gained so many adepts because, based on these ideas, we were able to build a machine that has changed the world. Furthermore, science has been unveiling the Turing computational properties of the universe ever since, by finding and compressing physical phenomena into formulas and theories, with which we have been able to make accurate predictions. All those evaluations of formulas and theories in science are Turing-computable computations.

Researchers have tried to find feasible models of hypercomputation since the birth of computability theory; there was a revival in the 1990s, when models with more of a computational complexity motivation were created, rather than proposals of actual feasible devices. However, according to the book, some of these models are feasible, albeit exotic. Someone can only claim something is feasible if it has some evidence of its feasibility; otherwise, the statement is a subjective claim. This should have been a seminal part of the discussion throughout the book, but it is posited at the beginning, clearly showing the spirit of the book from then on: limiting and discouraging any further discussion on the topic of the feasibility of hypercomputational models, which is the real subject. The rest of the chapters are, therefore, just descriptions of models developed by several researchers.

Perhaps that’s why, later in the book, chapters on interaction machines and persistent machines are introduced as if they are threats to the Turing model in computability terms, or even hypercomputational models by themselves; in reality, they are not that at all. No serious author thinks that Turing machines are the right model to describe the mind, the universe, or even desktop computers, other than to describe their computational power. But the author overlooks this fact and continues his exposition in the exact same way in which other texts presented the subject--in the form of a list of disconnected models, with no further discussion.

The book can be used as a reference text to the subject, although, as it is acknowledged in the book itself, several models are missing and those included are only introductions. Consequently, the book is more a dictionary of models of hypercomputation that Syropoulos chose to include.

Being not that skeptical of some of the hypercomputational models, I was quite surprised to find that this book is not very much help to the field.