*To the Editors*:

I was pleased and honored by Freeman Dyson’s gracious review of my book, *The Fabric of the Cosmos* [*NYR*, May 13]. Nevertheless, statements attributed to me in Part 3 of that review are incorrect and suggest a viewpoint that I do not hold.

Specifically, Professor Dyson suggests that in a public debate we had at the 2001 World Economic Forum in Davos, I claimed: “When we know the fundamental equations of physics, everything else, chemistry, biology, neurology, psychology, and so on, can be reduced to physics and explained by using the equations.”

I can’t imagine making such a statement as it runs thoroughly counter to my long-held beliefs. While there is apparently no transcript available, my views on this issue were expressed in *The Elegant Universe* (1999), page 17, where I write that finding the fundamental equations of physics “would in* no way* mean that psychology, biology, geology, chemistry, or even physics had been solved or in some sense subsumed.” (Emphasis added.)

Professor Dyson also recalls my saying that “once we have the fundamental equations, we are done…. All that will be left for scientists to do will be applied science…. The end of fundamental science is near.” However, referring again to *The Elegant Universe*, page 17, I write: “The universe is such a rich and wonderful place that the discovery of the final theory would not spell the end of science. Its discovery would be a beginning not an end. ”

What I did say in Davos is that the search for the elementary ingredients making up the universe and the deepest laws governing their interactions may be a search that one day draws to a close. The deeper we look, the simpler and more unified the laws become and there may well be a limit to this process. However, achieving this goal would only mean that we were “done” with one fantastically interesting but limited chapter in human exploration—the search for the basic constituents and underlying laws.

Professor Dyson does note that he is reconstructing my remarks from memory and so may be exaggerating the claims that I made. The conclusion he reaches from this reconstruction, though, is erroneous. In particular, Professor Dyson concludes that I believe that only research which “reduces complicated phenomena to their simpler component parts,” what he calls analytic science, “is worthy of the name of science.” This is not my view. Instead, I believe that research seeking to “build up complicated structures from their simpler parts,” what Professor Dyson calls synthetic science, is of the utmost importance and is critical to the success of our ongoing and likely endless scientific journey to understand ourselves and the universe ever more fully.

Brian Greene

Professor of Physics and Mathematics

Columbia University

New York City

*To the Editors*:

In the final section of his lucid review of Brian Greene’s *The Fabric of the Cosmos*, Freeman Dyson recalls a meeting of the World Economic Forum where he and Greene were asked to debate the question “When will we know it all?”: Greene took the position “Soon” and Dyson the position “Never.” Greene argued that the search for the fundamental equations of physics (like superstring theory) would lead to a “complete theory of everything” from the most basic particles through all animate and inanimate matter to the entire cosmos. Dyson argued that the reduction of the other sciences to physics does not—and indeed cannot—work, and thus that science is inexhaustible, a “fact” in which he rejoiced. One of the main reasons Dyson gave that a complete theory of everything *cannot* work is Kurt Gödel’s incompleteness theorem for axiomatic systems of mathematics that are included in any basic physical theory. While I agree with the first part of his argument, I think he claims more for the significance of Gödel’s theorem than it can bear in this context.

Dyson’s formulation of Gödel’s theorem is that “given any finite set of rules for doing mathematics, there are undecidable statements, mathematical statements that cannot be proved or disproved by using the normal rules of logic and arithmetic.” More precisely, the theorem applies only to formally specified systems of axioms—for some part or other of mathematics—that are both consistent and suffice to derive the Peano axioms for the arithmetic of the natural numbers 1, 2, 3, …, where the means of inference are those given by what Dyson calls “the normal rules of logic.” Here “arithmetic” is understood in the sense of higher mathematics in which one investigates among other things whether there exist natural numbers with a certain property and if so, how many such numbers there are; for example whether there are infinitely many *n* such that both *n* and *n* + 2 are prime (the answer to which is not known to this day). Dyson says that Gödel’s theorem implies that “pure mathematics is inexhaustible” and that because of this, “physics is inexhaustible too” since the laws of physics are “a finite set of rules, and include the rules for doing mathematics.” Indeed, if the laws of physics are formulated in an axiomatic system S which includes the notions and axioms of arithmetic as well as physical notions such as time, space, mass, charge, velocity, etc., and if S is consistent, then there are propositions of higher arithmetic which are undecidable by S. But this tells us nothing about the specifically physical laws encapsulated in S, which could conceivably be complete as such.

All this is highly theoretical and speculative. In practice, a much different picture emerges. Beyond basic arithmetic calculations, the mathematics that is applied in physics rarely calls on higher arithmetic but depends instead mainly on substantial parts of mathematical analysis and higher algebra and geometry. All of the mathematics that underlies these applications can be formalized in the currently widely accepted system for the foundation of mathematics known as Zermelo-Fraenkel set theory, and there is not the least shred of evidence that anything stronger than that system would be needed. In fact, it has long been recognized that much weaker systems than that suffice for scientific applications, as discussed in the last chapter of my book *In the Light of Logic* (Oxford University Press, 1998). Note well that the issue here only concerns *applied* mathematics. It is entirely another matter whether, and in what sense, pure mathematics needs new axioms beyond those of the Zermelo-Fraenkel system; that has been a matter of some controversy among logicians.

Whether or not the kind of inexhaustibility of mathematics discovered by Gödel is relevant to the applications of mathematics in science, there is a different kind of inexhaustibility which is much more significant for practice: no matter which axiomatic system S is taken to underlay one’s work at any given stage in the development of mathematics and science, there is a potential infinity of propositions that can be demonstrated in S, and at any moment, only a finite number of them have been established. Experience shows that significant progress at each such point depends to an enormous extent on creative ingenuity in the exploitation of accepted principles rather than essentially new principles. One can join Freeman Dyson in rejoicing in that kind of inexhaustibility as well.

Solomon Feferman

Professor of Mathematics and Philosophy

Stanford University

Palo Alto, California

###### Freeman Dyson replies:

Each time I publish a book review in *The New York Review of Books*, I receive a bimodal set of responses. First come the responses from nonexpert readers who write to tell me how much they like the review. Second come the responses from expert readers who write to correct my mistakes. I am grateful for both categories of response, but I learn much more from the second category. It is inevitable that I make mistakes when writing about fields in which I am not an expert, and I rely on the experts to set the record straight. I am especially grateful to Brian Greene for correcting my misrepresentation of his views. I apologize to him for the misrepresentation, and for not checking with him before publishing the review. I am grateful to Solomon Feferman for explaining why we do not need Gödel’s theorem to convince us that science is inexhaustible. I am grateful to many other readers who have written to me privately to correct other mistakes.

This Issue

July 15, 2004