CHARLES SIMONYI
Computer Scientist; Founder, Intentional Software; formerly Chief Architect, Microsoft

I believe that we are writing software the wrong way. There are sound evolutionary reasons for why we are doing what we are doing—that we can call the "programming the problem in a computer language" paradigm, but the incredible success of Moore's law blinded us to being stuck in what is probably an evolutionary backwater.

There are many warning signs. Computers are demonstrably ten thousand times better than not so long ago. Yet we are not seeing their services improving at the same rate (with some exceptions—for example games and internet searches.) On an absolute scale, a business or administration problem that would take maybe one hundred pages to describe precisely, will take millions of dollars to program for a computer and often the program will not work. Recently a smaller airline came to a standstill due to a problem in crew scheduling software—raising the ire of Congress, not to mention their customers.

My laptop could store 200 pages of text (1/2 megabytes) for each and every crew member at this airline just in its fast memory and hundred times more (a veritable encyclopedia of 20,000 pages) for each person on its hard disk. Of course for a schedule we would need maybe one or two—or at most ten pages per person. Even with all the rules—the laws, the union contracts, the local, state, federal taxes, the duty time limitations, the FAA regulations on crew certification; is there anyone who believes that the problem is not simple in terms of computing? We need to store and process at the maximum 10 pages per person where we have capacity for two thousand times more in one cheap laptop! Of course the problem is complex in terms of the problem domain—but not shockingly so. I would estimate that all the rules possibly relevant to aircraft crew scheduling are expressible in less than a thousand pages—or 1/2 of one percent of the fast memory.

Software is surely the bottleneck on the high-tech horn of plenty. The scheduling program for the airline takes many thousand times more memory than what I believe it should be. Hence the software represents complexity that is many thousand times greater than what I believe the problem is—no wonder that some planes are assigned three pilots by the software while the others can't fly because the copilot is not scheduled. Note that the cost of the memory is not the issue—we could afford that waste. But the use of so much memory for software is an indication of some complexity inflation that occurs during programming that is the real bottleneck.

What is going on? I like to use cryptography as the metaphor. As we know, in cryptography we take a message and we combine it with a key using a difficult-to-invert function to get the code. Programmers using today's paradigm start from a problem statement, for example that a Boeing 767 requires a pilot, a copilot, and seven cabin crew with various certification requirements for each—and combine this with their knowledge of computer science and software engineering—that is how this rule can be encoded in computer language and turned into an algorithm. This act of combining is the programming process, the result of which is called the source code. Now, programming is well known to be a difficult-to-invert function, perhaps not to cryptography's standards, but one can joke about the possibility of the airline being able to keep their proprietary scheduling rules secret by publishing the source code for the implementation since no one could figure out what the rules were—or really whether the code had to do with scheduling or spare parts inventory—by studying the source code, it can be that obscure.

The amazing thing is that today it is the source code—that is the encrypted problem—which is the artifact all of software engineering is focusing on. To add insult to the injury, the "encryption", that is programming, is done manually which means high costs, low throughput and high error rates. In contrast with software maintenance, when the General realizes that he is about to send a wrong encrypted message, no one would think of editing the code after the encryption or "fixing the code"; instead the clear text would be first edited and then this improved message would be re-encrypted at computer speeds and computer accuracy. In other words the message may be wrong, but it won't be wrong because of the encryption and it is easily fixed.

We see that the complexity inflation comes from encoding. The problem statement above is obviously oversimplified, but remember that we used just two lines from our realistic budget of a thousand pages and we haven't even used the aviation jargon which can make these statements even more compact and more precise. But once these statements are viewed through the funhouse mirror of software coding, it becomes all but unrecognizable: thousand times fatter, disjointed, foreign. And as any manual product, it will have many flaws—beyond the errors in the rules themselves.

What can be done? Follow the metaphor. First, refocus on recording the problem statement—the "cleartext" in our metaphor. This is not a program in any sense of the word—it is just a straightforward recording of the subject matter experts' contributions using their own terms-of-art, their jargon, their own notations. Next, empower the programmers to program not the problem itself, but to express their software engineering expertise and decisions as a computer code for the encoder that takes the recorded problem statement and generates the code from it. This is called generative programming and I believe it is the future of software.

CHRIS W. ANDERSON
Editor-In-Chief, Wired


The Intelligent Design movement has opened my eyes. I realize that although I believe that evolution explains why the living world is the way it is, I can't actually prove it. At least not to the satisfaction of the ID folk, who seem to require that every example of extraordinary complexity and clever plumbing in nature be fully traced back (not just traceable back) along an evolutionary tree to prove that it wasn't directed by an invisible hand. If the scientific community won't do that, then the arguments goes that they must accept a large red "theory" stamp placed on the evolution textbooks and that alternative theories, such as "guided" evolution and creationism, be taught alongside.

So, by this standard, virtually everything I believe in must now fall under the shadow of unproveability. Most importantly, this includes the belief that democracy, capitalism and other market-driven systems (including evolution!) are better than their alternatives. Indeed, I suppose I should now refer to them as the "theory of democracy" and the "theory of capitalism", to join the theory of evolution, and accept the teaching of living Marxism and fascism as alternatives in high schools.

VERENA HUBER-DYSON
Mathematician, Emeritus Professor, Dept of Philosophy, University of Calgary; Author, Gödel's Theorems

Most of what I believe I cannot prove, simply for lack of time and energy; truths that I'd claim to know because they have been proved by others. That is how inextricably our beliefs are tied up with labors accomplished by fellow beings. And then there are mathematical truths that we now know are not provable. These phenomena have become favorites with the media but can only be made sense of by a serious scrutiny of the idea of mathematical truth and a specific articulation of a proof-concept,

But running across Esther's contribution I came up with a catchy response:

I believe in the creative power of boredom.

Or, to put it into the form suggested by the Edge question:

I believe that, no matter how relentlessly we overfeed our young with prepackaged interactive entertainments, before long they will break out and invent their own amusements. I know from experience; boredom drove me into mathematics during my preteens. But I cannot prove it, till it actually happens. Probably in less than a generation kids will be amusing themselves and each other in ways that we never dreamt of.

Such is my belief in human nature, in the resilience of its good sense.

Here is an observation from mathematical practice. By now the concept of an algorithm, well- defined, is widely hailed as the way to solve problems, more precisely sequences of problems labeled by a numerical parameter. The implementation of a specific algorithm may be boring, a task best left to a machine, while the construction of the algorithm together with a rigorous proof that it works is a creative and often laborious enterprise.

For illustration consider group theory. A group is defined as a structure consisting of a non-empty set and a binary operation obeying certain laws. The theory of groups consists of all sentences true of all groups; its restriction to the formal "first order" language L determined by the group structure is called the elementary theory TG of groups. Here we have a formal proof procedure, proven complete by Gödel in his PHD thesis the year before his incompleteness proof was published. The elementary theory of groups is axiomatizable: it consists of exactly those sentences that are derivable from the axioms by means of the rules of first order logic. Thus TG is an effectively (recursively) enumerable subset of L; a machine, unlimited in power and time, could eventually come up with a proof of every elementary theorem of group theory. However, a human group theorist would still be needed to select the interesting theorems out of the bulk of the merely true. The development of TG is no mean task, although its language is severely restricted.

The axiomatizability of a theory always raises the question how to recognize the non-theorems. The set FF of those L-sentences that fail in some finite group is recursively enumerable by an enumeration of all finite groups, a simple matter, in principle. But, as all the excitement over the construction of finite simple monsters has amply demonstrated, that again is in reality no simple task.

Neither the theory of finite groups nor the theory of all groups is decidable. The most satisfying proof of this fact shows how to construct to every pair (A, B) of disjoint recursively enumerable sets of L-sentences, where A contains all of TG and B contains FF, a sentence S that belongs neither to A nor to B. This is the deep and sophisticated theorem of effective non-separability proved in the early sixties independently by Mal'cev in the SSSR and Tarski's pupil Cobham.

It follows that constructing infinite counter-examples in group theory is a truly creative enterprise, while the theory of finite groups is not axiomatizable and so, to recognize a truth about finite groups requires deep insight and a creative jump. The concept of finiteness in group theory is not elementary and yet we have a clear idea of what is meant by talking about all finite groups, a marvelously intriguing situation.

To wind up with a specific answer to the 2005 Question:

I do believe that every sentence expressible in the formal language of elementary group theory is either true of all finite groups or else fails for at least one of them.

This statement may at first sight look like a logical triviality. But when you try to prove it honestly you find that you would need a decision procedure, which would, given any sentence of L, yield either a proof that S holds in all finite groups or else a finite group in which S fails. By the inseparability theorem mentioned above, there is no such procedure.

If asked whether I hold the equivalent belief for the theory of all groups I would hesitate because the concept of an infinite counterexample is not as concrete to my mind as that of the totality of all finite groups. These are the areas where personal intuition starts to come into play.

REBECCA GOLDSTEIN
Philosopher and Novelist, Trinity College; Author, Incompleteness

I believe that scientific theories are a means of going—somewhat mysteriously—beyond what we are able to observe of the physical world, penetrating into the structure of nature. The "theoretical" parts of scientific theories—the parts that speak in seemingly non-observational terms—aren't, I believe, ultimately translatable into observations or aren't just algorithmic black boxes into which we feed our observations and churn out our predictions. I believe the theoretical parts have descriptive content and are true (or false) in the same prosaic way that the observational parts of theories are true (or false). They're true if and only if they correspond to reality.

I also believe that my belief about scientific theories isn't itself scientific. Science itself doesn't decide how it is to be interpreted, whether realistically or not.

That the penetration into unobservable nature is accomplished by way of abstract mathematics is a large part of what makes it mystifying—mystifying enough to be coherently if unpersuasively (at least to me) denied by scientific anti-realists. It's difficult to explain exactly how science manages to do what it is that I believe it does—notoriously difficult when trying to explain how quantum mechanics, in particular, describes unobserved reality. The unobservable aspects of nature that yield themselves to our knowledge must be both mathematically expressible and connected to our observations in requisite ways. The seventeenth-century titans, men like Galileo and Newton, figured out how to do this, how to wed mathematics to empiricism. It wasn't a priori obvious that it was going to work. It wasn't a priori obvious that it was going to get us so much farther into nature's secrets than the Aristotelian teleological methodology it was supplanting. A lot of assumptions about the mathematical nature of the world and its fundamental correspondence to our cognitive modes (a correspondence they saw as reflective of God's friendly intentions toward us) were made by them in order to justify their methodology.

I also believe that since not all of the properties of nature are mathematically expressible—why should they be? it takes a very special sort of property to be so expressible—that there are aspects of nature that we will never get to by way of our science. I believe that our scientific theories—just like our formalized mathematical systems (as proved by Gödel)—must be forever incomplete. The very fact of consciousness itself (an aspect of the material world we happen to know about, but not because it was revealed to us by way of science) demonstrates, I believe, the necessary incompleteness of scientific theories.

JONATHAN HAIDT
Psychologist, University of Virginia

I believe, but cannot prove, that religious experience and practice is generated and structured largely by a few emotions that evolved for other reasons, particularly awe, moral elevation, disgust, and attachment-related emotions. That's not a prediction likely to raise any eyebrows in this forum.

But I further believe (and cannot prove) that hostility toward religion is an obstacle to progress in psychology. Most human beings live in a world full of magic, miracles, saints, and constant commerce with divinity. Psychology at present has little to say about these parts of life; we focus instead on a small set of topics that are fashionable, or that are particularly tractable with our favorite methods. If psychologists took religious experience seriously and tried to understand it from the inside, as anthropologists do with other cultures, I believe it would enrich our science. I have found religious texts and testimonials about purity and pollution essential for understanding the emotion of disgust.

DONALD I. WILLIAMSON
Biologist, University of Liverpool; Author, The Origins of Larvae

I believe I can explain the Cambrian explosion.

The Cambrian explosion refers to the first appearance in a relatively short space of geological time of a very wide assortment of animals more than 500 million years ago. I believe it came about through hybridization.

Many well preserved Cambrian fossils occur in the Burgess shale, in the Canadian Rockies. These fossils include small and soft-bodied animals, several of which were planktonic but none were larvae. Compared with modern animals, some of them seem to have the front end of one animal and rear end of another. Modern larvae present a comparable set-up: larvae seem to be derived from animals in different groups from their corresponding adults. I have amassed a bookful of evidence that the basic forms of larvae did indeed originate as animals in other groups and that such forms were transferred by hybridization. Animals with larvae are "sequential chimeras", in which one body-form—the larva—is followed by another, distantly related form—the adult. I believe there were no Cambrian larvae, and Cambrian hybridizations produced "concurrent chimeras", in which two distantly related body-forms appeared together.

About 600 million years ago, shortly before the Cambrian, animals with tissues (metazoans) made their first appearance. I agree with Darwin that there were several different forms (Darwin suggested four or five), and I believe they resulted from hybridizations between different colonial protists. Protists are mostly single-celled, but colonial forms consist of many similar cells. All Cambrian animals were marine, and, like most modern marine animals, they shed their eggs and sperm into the water, where fertilization took place. Eggs of one species frequently encountered sperm of another, and there were only poorly developed mechanisms to prevent hybridization. Early animals had small genomes, leaving plenty of spare gene capacity. These factors led to many fruitful hybridizations, which resulted in concurrent chimeras. Not only did the original metazoans hybridize but the new animals resulting from these hybidizations also hybridized, and this produced the explosion in animal form.

The acquisition of larvae by hybridization came much later, when there was little spare genome capacity in recipes for single animals, and it is still going on. In the echinoderms (the group that includes sea-urchins and starfish) there is evidence that there were no larvae in either the Cambrian or the Ordovician (the following period), and this might well apply to other major groups. Acquiring parts, rather than larvae, by hybridization continued, I believe, throughout the Cambrian and Ordovician and probably later, but, as genomes became larger and filled most of the available space, later hybridizations led to smaller changes in adult form or to acquisitions of larvae. The gradual evolution of better mechanisms to prevent eggs being fertilized by foreign sperm resulted in fewer fruitful hybridizations, but occasional hybridizations still take place.

Hybridogenesis, the generation of new organisms by hybridization, and symbiogenesis, the generation of new organisms by symbiosis, both involve fusion of lineages, whereas Darwinian "descent with modification" is entirely within separate lineages. These forms of evolution function in parallel, and "natural selection" works on the results.

I cannot prove that Cambrian animals had poorly developed specificity and spare gene capacity, but it makes sense.

SETH LLOYD
Quantum Mechanical Engineer, Massachusetts Institute of Technology

I believe in science. Unlike mathematical theorems, scientific results can't be proved.They can only be tested again and again, until only a fool would not believe them.

I cannot prove that electrons exist, but I believe fervently in their existence. And if you don't believe in them, I have a high voltage cattle prod I'm willing to apply as an argument on their behalf. Electrons speak for themselves.