This essay discusses the best current understanding of the relationship between mathematical and empirical knowledge.  It focuses on two questions:

1. Does mathematics have some sort of deep metaphysical connection with reality, and
2. if not, why is it that mathematical abstractions seem so often to be so powerfully predictive in the real world?

For two centuries after Newton, phenomenal science aspired to the kind of rigor and purity that seemed to be embodied in mathematics.  The metaphysical situation seemed simple; mathematics embodied perfect a-priori knowledge, those sciences able to most mathematicize themselves were the most successful at phenomenal prediction; perfect knowledge would therefore consist of a mathematical formalism, arrived at by science and embracing all of reality, that would ground a-posteriori empirical understanding in a-priori rational logic.  It was in this spirit that Condorcet dared to imagine describing the entire universe as a mutually-solving set of partial differential equations.

The first cracks in this inspiring picture appeared in the latter half of the 19th century when Riemann and Lobachevsky independently proved that Euclid's Axiom of Parallels could be replaced by alternatives which yielded consistent geometries.  Riemann's geometry was modeled on a sphere, Lobachevsky's on a hyperboloid of rotation.

The impact of this discovery has been obscured by later and greater upheavals, but at the time it broke on the intellectual world like a thunderbolt.  For the existence of mutually inconsistent axiom systems for geometry, any of which could be modeled in the phenomenal universe, called the whole relationship between mathematics and physical theory into question.

When there was only Euclid, there was only one possible geometry.  One could believe that the Euclidean axioms constituted a kind of perfect a-priori knowledge about geometry in the phenomenal world.  But suddenly we had three geometries, an embarrassment of metaphysical riches.

For how were we to choose between the axioms of plane, spherical, and hyperbolic geometry as a description of "real" geometry?  Because all three are consistent, we couldn't choose on any a-priori basis — the choice had to become empirical, based on their predictive power for a given situation.

Of course, physical theorists had long been accustomed to choosing formalisms to fit a scientific problem.  But it had been widely, if unconsciously, assumed that the need to do so ad hoc was a function of human ignorance; that, given good enough mathematics and logic, we could deduce the correct choice from first principles, producing a-priori descriptions of reality to be confirmed, as an afterthought, by empirical check.

But now, the Euclidean geometry that had been considered the model for axiomatic perfection in mathematics for over two thousand years, had been dethroned.  If one could not know something as fundamental as the geometry of space a-priori, what hope was there for a purely "rational" theory encompassing all of nature?  Psychologically, Riemann/Lobachevsky struck at the very heart of the enterprise of mathematics as it was then conceived.

Furthermore, Riemann/Lobachevsky called the nature of mathematical intuition into question.  It had been easy to believe implicitly that mathematical intuition was a form of perception — a glimpse of the Platonic noumena behind reality.  But with two other geometries jostling Euclid, nobody knew for sure what the noumena looked like any more!

Mathematicians responded to this dual problem with an increase in rigor, by trying to apply the axiomatic method throughout mathematics.  It was gradually realized that the belief in mathematical intuition as a kind of perception of a noumenal world had encouraged sloppiness; proofs in the pre-axiomatic period often relied on shared intuitions about mathematical "reality" that could no longer be considered automatically valid.

The new thinking in mathematics led to a series of spectacular successes; among these were Cantorian set theory, Frege's axiomatization of number, and eventually Russell & Whitehead's monumental synthesis in Principia Mathematica.

However, it also had a price.  The axiomatic method made the connection between mathematics and phenomenal reality narrower and narrower.  At the same time, discoveries like the Banach-Tarski Paradox suggested that mathematical axioms that seemed to be consistent with phenomenal experience could lead to dizzying contradictions with that experience.

The majority of mathematicians quickly became "Formalists", holding that pure mathematics could not be philosophically considered more than a sort of elaborate game played with marks on paper (this is the theory behind Robert Heinlein's pithy characterization of mathematics as "a zero-content system"). The old-fashioned "Platonist" belief in the noumenal reality of mathematical objects seemed headed for the dustbin, despite the fact that mathematicians continued to feel like Platonists during the process of mathematical discovery.

Philosophically, then, the axiomatic method lead most mathematicians to abandon previous beliefs in the metaphysical specialness of mathematics.  It also created today's split between pure and applied mathematics.

Most of the great mathematicians of the early modern period — Newton, Liebniz, Fourier, Gauss, and others — were also phenomenal scientists (i.e. "natural philosophers").  The axiomatic method incubated the modern idea of the pure mathematician as super-esthete, unconcerned with the merely physical.  Ironically, Formalism gave pure mathematicians a bad case of Platonic attitude.  Applied mathematicians stopped being invited to tea and learned to hang out with physicists.

This brings us to the early 20th century. For the beleaguered minority of Platonists, worse was yet to come.

Cantor, Frege, Russell and Whitehead showed that all of pure mathematics could be built on the single axiomatic foundation of set theory.  This suited the Formalists just fine; mathematics coalesced, at least in principle, from a bunch of little disconnected games to one big game.  It also made the Platonist minority happy; if there turned out to be one big, over-arching consistent structure behind all of mathematics, the metaphysical specialness of mathematics might yet be rescued.

Unfortunately, it turns out that there is more than one way to axiomatize set theory.  In particular, there are at least four major different combinations of assumptions about infinite sets that lead to mutually exclusive set theories (the Axiom of Choice or its negation; the Continuum Hypothesis or its negation).

It was Riemmann/Lobachevsky all over again, but on a much more fundamental level.  Riemannian and Lobachevskian geometry could be modeled finitely, in the world; you could decide at least empirically which one fit.  Normally, you could regard all three as special cases of the geometry of geodesics on manifolds, thereby fitting them into the superstructure erected on set theory.

But the independent axioms in set theory don't seem to lead to any results that can be modeled in the observable finite world.  And there's no way to assert both the Continuum Hypothesis and its negation in one set theory.  How's a poor Platonist to choose which system describes "real" mathematics?  The victory of the Formalist position seemed complete.

In a negative way, though, a Platonist had the last laugh.  Kurt Godel threw a spanner in the Formalist program of axiomatization when he showed that any axiom system powerful enough to include the integers would have to be either inconsistent (yielding contradictions) or incomplete (too weak to decide the truth or falsehood of some assertions in the system).

And that is more or less where things stand today.  Mathematicians know that any attempt to put forward mathematics as a-priori knowledge about the universe must fall afoul of numerous paradoxes and impossible choices about what axiom system describes "real" mathematics.  They've been reduced to hoping that the standard axiomatizations are not inconsistent but incomplete, and wondering uneasily what contradictions or unprovable theorems are waiting discovery out there, lurking like landmines in the noosphere.

Meanwhile, on the empirical front, mathematics continued to be a spectacular success as a theory-building tool.  The great successes of 20th-century physics (general relativity and quantum mechanics) wandered so far from the realm of physical intuition that they could be understood only by meditating deeply on their mathematical formalisms, and following through to their logical conclusions, even when those conclusions seem wildly bizarre.

What irony.  Even as mathematical 'perception' came to seem less and less reliable in pure mathematics, it became more and more indispensable in phenomenal science!

Against this background, Einstein's famous quote wondering at the applicability of mathematics to phenomenal science poses an even thornier problem than at first appears.

The relationship between mathematical models and phenomenal prediction is complicated, not just in practice but in principle.  Much more complicated because, as we now know, there are mutually exclusive ways to axiomatize mathematics!  It can be diagrammed as follows (thanks to Jesse Perry for supplying the original of this chart):

The key transactions for our purposes are C and D — the translations between a predictive model and a mathematical formalism.  What mystified Einstein is how often D leads to new insights.

We begin to get some handle on the problem if we phrase it more precisely; that is, "Why does a good choice of C so often yield new knowledge via D?"

The simplest answer is to invert the question and treat it as a definition. A "good choice of C" is one which leads to new predictions.  The choice of C is not one that can be made a-priori; one has to choose, empirically, a mapping between real and mathematical objects, then evaluate that mapping by seeing if it predicts well.

For example, the positive integers are a good formalism for counting marbles. We can confidently predict that if we put two marbles in a jar, and then put three marbles in a jar, and then empirically associate the set of two marbles with the mathematical entity 2, and likewise associate the set of three marbles with the mathematical entity 3, and then assume that physical aggregation is modeled by +, then the number of marbles in the jar will correspond to the mathematical entity 5.

The above may seem to be a remarkable amount of pedantry to load on an obvious association, one we normally make without having to think about it.  But remember that small children have to learn to count…and consider how the above would fail if we were putting into the jar, not marbles, but lumps of mud or volumes of gas!

One can argue that it only makes sense to marvel at the utility of mathematics if one assumes that C for any phenomenal system is an a-priori given.  But we've seen that it is not.  A physicist who marvels at the applicability of mathematics has forgotten or ignored the complexity of C; he is really being puzzled at the human ability to choose appropriate mathematical models empirically.

By reformulating the question this way, we've slain half the dragon.  Human beings are clever, persistent apes who like to play with ideas. If a mathematical formalism can be found to fit a phenomenal system, some human will eventually find it.  And the discovery will come to look "inevitable" because those who tried and failed will generally be forgotten.

But there is a deeper question behind this: why do good choices of mathematical model exist at all?  That is, why is there any mathematical formalism for, say, quantum mechanics which is so productive that it actually predicts the discovery of observable new particles?

The way to "answer" this question is by observing that it, too, properly serves as a kind of definition.  There are many phenomenal systems for which no such exact predictive formalism has been found, nor for which one seems likely.  Poets like to mumble about the human heart, but more mundane examples are available.  The weather, or the behavior of any economy larger than village size, for example — systems so chaotically interdependent that exact prediction is effectively impossible (not just in fact but in principle).

There are many things for which mathematical modeling leads at best to fuzzy, contingent, statistical results and never successfully predicts 'new entities' at all.  In fact, such systems are the rule, not the exception.  So the proper answer to the question "Why is mathematics is so marvelously applicable to my science?" is simply "Because that's the kind of science you've chosen to study!"

Eric S. Raymond

Bill Gates, 26 September 1997: “When putting this speech together, I’m thinking, boy, in the last 12 months I’ve given about 100 speeches. And I’m very careful to try never to say the same thing twice.”

Bill Gates, 1 October 1997: “In this 10-year time frame, I believe that we’ll not only be using the keyboard and the mouse to interact, but during that time we will have perfected speech recognition and speech output well enough that those will become a standard part of the interface.”

Bill Gates, 6 October 1997: “The PC five years from now — you won’t recognize it, because speech will have come into the interface, the screen will be a flat screen, the performance will be 20 times what it is today.”

Bill Gates, 7 October 1997: “What I am saying is that I’m optimistic enough to believe that within the next decade, we will see progress to a level that for things like dealing with data in a spreadsheet or text in a word processor, or navigating the Internet, you will find the speech interface has enough accuracy that it becomes a primary way of interacting with the machine.”

Bill Gates, 16 November 1997: “I talked about some recent advances that really have me excited. Here are some that are literally within the next few years. Speech recognition. A big, big breakthrough.”

Bill Gates, 10 January 1998: “When we start working on speech recognition, or artificial intelligence software, we know it’s going to be five to ten years before that’s having a major impact.”

Bill Gates, 27 January 1998: “In the next five years, in some ways there aren’t going to be that many surprises. Windows will have two very, very major releases-Windows in that time frame. You will start to see speech become a standard part of the interface.”

Bill Gates, 9 February 1998: “And the power that we’re talking about there, you know, there’s no doubt that machine is good enough for speech recognition, visual recognition, learning. I mean, even the machines six to eight years out, I feel very comfortable will be doing all of those things.”

Bill Gates, 4 March 1998: “We’re looking at speech is an add-on today, but we’re looking in this two to three year timeframe, whether we could put it in. Speech synthesis has gotten quite good, and our work has really put us out in the forefront of that. But we have not gone and put the speech recognition in.”

Bill Gates, 26 March 1998: “Well, the speech recognition will start off like most features do, as an add-on product, and you’re seeing a couple of those out there today … I think by next year, we’ll have a very rich API and then in the next major round of Windows, will have it as an optional but not required capability, and then maybe the major round after that will simply rely on it as the primary interface.”

Bill Gates, 25 June 1998: “The breakthroughs in interaction aren’t going to come in the next three years. We’ll have some additional speech recognition, but it won’t be the center of the interface. But in the three-to-six-year timeframe, I feel very confident that that will be not only a standard thing, but built into the operating system, and something that applications will sit on top of and take advantage of.”

Bill Gates, 24 March 1999: “Speech recognition … I don’t think you’ll see dictation as something that most people will use in the next couple of years. The extra processing power, getting the extra memory I think has us on a track to provide that, but for most people, I think it will be more like a five-year time frame before that’s a standard way of interacting.”

Bill Gates, 24 March 1999: “I also think speech recognition, although we’ve been talking about it for decades, is finally going to get to the point where it really is quite usable. That could be anywhere from, say, two to six years away, and the extra power, the software that understands the grammar, will help get us over the threshold where this is commonplace.”

Bill Gates, 26 March 1999: “Speech recognition will be part of the interface. That has been a tough problem because people are very demanding of high quality. The keyboard is not that bad. So, even though we are going to have this extra speed, probably we are four or five years away from that being the typical interface.”

Bill Gates, 10 March 2000: “You know, when I was a student at Harvard, the defense DARPA group was giving out money to universities that said, yes, in three years we’ll have great speech recognition … And so the frontiers that are out there — great handwriting recognition, great speech recognition, even having the computer have a visual capability so it can see who’s coming in, what’s going on, all of those things undoubtedly will be solved in the next decade.”

Bill Gates, 26 March 2001: “Because of where we’re going with real-time communications, including the instant messaging that will be included in Windows itself, voice annotation, voice communication and speech recognition are becoming mainstream capabilities. And so we believe that virtually all PCs should have that right out of the box.”

Bill Gates, 17 April 2002: “Likewise, in areas like speech recognition we have seen companies come and go. It’s a very tough problem, but it will yield itself to breakthroughs that will make that just common sense for every computer over the next five years.”

Bill Gates, 28 July 2003: “It’s all the dreams of software, of vision and speech recognition and business intelligence; those are within our grasp. Some people might say it’s three years, some people might say it’s 10 years to solve those things, but by and large, those very interesting things, put aside machine learning, the very interesting tool-based things, I think it’s very clear that we’re on a track to make some incredible advances.”

Bill Gates, 25 February 2004: “We believe that speech over the next several years will be ready for primetime.”

Bill Gates, 25 February 2004: “Now, with speech it’s not as easy. Speech is another one that will be solved, and will be solved for a broad range of applications within this decade.”

Bill Gates, 24 March 2004: “And the Speech Server that we’re announcing, and shipping this week, is about eliminating that problem, making it easy to write server-based voice recognition for domain-specific grammars … very, very modest … Now, Microsoft is very committed to speech. We see this as something that over the rest of this decade will simply become more and more mainstream.”

Bill Gates, 10 May 2005: “Windows Mobile 5.0 brings in some speech recognition capabilities and we see that for small command sets as being really ready today, something that is very powerful.”

Bill Gates, 28 July 2005: “And yet the really interesting problems — vision, speech, ink, security, learning — all these things, we’re going to have big breakthroughs on in the next 10 years.”

Bill Gates, 13 September 2005: “With microphones, things like speech recognition will eventually be commonplace.”

Bill Gates, 14 September 2005: “We totally believe speech recognition will go mainstream somewhere over the next decade.”

Bill Gates, 14 October 2005: “Another big change you’ll see is that we’ll have microphones on PCs and the speech recognition will be built-in as a standard feature. And that’s probably two to three years from now that that really becomes mainstream…”

Written by Matthew Paul Thomas in his weblog

The Machinery of Thought

By Tim Beardsley

In a darkened basement laboratory on the campus of the National Institutes of Health in  Bethesda, Md., volunteers earn $100 by lying for two hours with their head inside a huge magnetic resonance imaging (MRI) machine while they gaze at a screen reflected in a mirror. The screen periodically displays black-and-white pictures: some are faces, others scrambled blocks of light and shade. When a face appears on the screen, the subject signals by pressing buttons whether the face is a new one or the same as one that was shown a few seconds earlier as a "target" to be remembered.

As the test proceeds, the MRI machine bombards the volunteer's brain with radio-frequency waves that excite hydrogen atoms in the bloodstream, causing the atoms to emit signals of their own. Later, the machine transforms the resulting electromagnetic cacophony into color-coded maps of oxygen consumption levels throughout the subject's brain. Because increased oxygen consumption results from heightened neural activity, researchers can analyze these brain maps to learn what parts of the brain work hardest when a person recognizes a face.

With experiments such as these, researchers are beginning to fathom the neural processes underlying "working memory"—the limited, short-term store of currently relevant information that we draw on when we comprehend a sentence, follow a previously decided plan of action or remember a telephone number. When we bring to mind the name of Russia's president, for instance, that information is temporarily copied from long-term memory into working memory.

Psychological studies have demonstrated that working memory is fundamental to the human ability to reason and make judgments that rely on remembered contextual information. There are compelling humanitarian reasons for understanding working memory. Schizophrenia, one of the most devastating mental illnesses, is believed to be caused in part by a defect of this system. Studies of the molecular basis of working memory "have implications for drug treatment in mental illness," says Patricia Goldman-Rakic of Yale University, one of the most prominent investigators of working memory.

An intensive research effort has started to produce detailed information about the areas of the brain involved when we engage this vital intellectual faculty and is illuminating the patterns of neural activity that allow it to operate. The important role of specific brain chemicals in working memory is also becoming clear. Yet for all the progress, researchers have still to agree on how working memory is controlled and organized.

From Electrodes to Fast MRI

The prototypical test for working memory involves what is called delayed choice. An animal or a person signals where some specific cue was previously seen, before an imposed period of waiting. Thus, a monkey might be given a choice of two jars in separate positions and be rewarded for pointing to the one in which it previously saw food placed.

The task provides no clue to the correct response at the time of testing, so the monkey must rely on its recollection of the correct location. A related challenge rewards an animal for remembering which of several images it saw presented initially as a target. The NIH volunteers who were recalling faces were engaged in a variant of this test.

Technological advances have greatly enhanced researchers' ability to probe the neural underpinnings of such capacities. Investigators began studying cerebral activity in working memory some 40 years ago by inserting electrodes into individual neurons within the brains of monkeys. This method has its limits, however. Although monkey brains have clear anatomical similarities to human brains, the animals' behavior is vastly simpler, making detailed comparisons with human thinking problematic. Lacking language, the animals must be patiently trained over a period of weeks to master tasks that a person would pick up in a minute.

Electrode-recording techniques are also ethically unacceptable for use on people. Researchers try to learn which parts of our species' brain do what by studying the effects of damage caused by injury, disease or therapeutic surgery. Yet patients have different medical histories—and their brains vary in exact shape—so interpreting this clinical data is tricky at best.

Earlier this decade, positron emission tomography, or PET scanning, made enormous strides by showing which parts of the human brain are busiest when performing different tasks, such as hearing words or speaking. But PET requires exposing the human subjects to radioactive tracers, and to keep radiation doses within acceptable levels, researchers have to use techniques that can resolve brain areas only about a centimeter apart. Also, during a delayed-choice task, PET scans are too slow to distinguish between the neural activity pattern of a target being held in mind and the pattern that follows a few seconds later when the target is recognized.

The new technique used at NIH and elsewhere, called functional MRI, can resolve the position of active neurons to about two millimeters and is fast enough to study activity before and after the brain recognizes a cue on a screen. The rapidly improving technique has over the past two years become the state of the art for functional brain imaging.

Monkey Puzzle

Experiments involving electrodes implanted in monkeys still provide crucial information, however, because they reveal in fine detail and on a millisecond-by-millisecond timetable what happens as these primates respond to cues and rewards. When animals perform such feats of working memory, several brain regions can play a role, but as Joach’n M. Fuster of the University of California at Los Angeles showed in the 1970s, one area that is always involved is the prefrontal cortex.

The prefrontal cortex is a layer of tissue that lies just behind the forehead. With neural connections to almost all the areas of the brain that process sensory information, it is well situated to maintain a flexible store of information relevant to any task at hand. It is also the part of the brain that has grown the most in humans, as compared with monkeys. Monkeys missing some parts of their prefrontal cortex preserve their long-term memory but perform miserably on delayed-choice tests. Humans similarly afflicted suffer a reduced attention span and ability to plan.

Fuster and, separately, Kisou Kubota and Hiroaki Niki of the Kyoto Primate Center made electrical recordings from a variety of neurons in the monkey prefrontal cortex, including some that apparently were active only while the animals were holding information in working memory. Subsequently, Goldman-Rakic and her colleagues have explored working memory in monkeys with more sophisticated tests. They established that prefrontal neural activity during a delayed-choice task indeed corresponds well to the functioning of working memory.

Goldman-Rakic and her associate Graham Williams have taken the analysis all the way to the subcellular level, showing that receptors for the neurotransmitter dopamine pivotally influence the responsiveness of cells in the prefrontal cortex and their actions in working memory. "There is no other example I know" of research that spans the gulf between behavior and subcellular function, Goldman-Rakic notes. She and her colleagues have recently shown that administering antischizophrenic drugs to monkeys for six months leads to specific changes in the numbers of two different types of dopamine receptors in that region, further evidence that schizophrenia—or its treatment—alters normal function there.

Research by other scientists supports the view that the prefrontal cortex could sustain working memory. Robert Desimone of the National Institute of Mental Health, along with Earl K. Miller, Cynthia Erickson and others, has discovered in the monkey's prefrontal cortex neurons that fire at different rates during the delayed-choice task, depending on the target the animal saw previously. Neurons in other parts of the brain generally "forget" the target when a distracting stimulus appears—their rate of firing changes. Prefrontal neurons detected by Desimone and his colleagues, in contrast, maintain their rate of activity during a delayed-choice task even after the animal is presented with irrelevant, distracting stimuli.

Activity in some prefrontal neurons, then, appears to embody directly the temporary working memory of the appearance of a target the animal is seeking. Other researchers have found prefrontal neurons that seem to maintain locations in working memory: Giuseppe Di Pellegrino of the University of Bologna and Steven Wise of the National Institute of Mental Health have found prefrontal neurons that are busiest when an animal has to remember where it saw a cue. Stimuli fail to excite the same frenzy unless they are in the location that is the current target for the task.

Neurons in the prefrontal cortex could thus apparently control how animals respond in a delayed-choice task. Fuster, one of the pioneers in the field, says the prefrontal cortex "serves the overarching function of the temporal organization of behavior" by driving networks that maintain currently important information in an active state. And neurons in the prefrontal cortex might exert their influence in more subtle ways, too.

Besides controlling directly the responses in delayed-choice tests, Desimone believes, the prefrontal cortex might tune the visual and possibly other perceptual systems to the task at hand. "What's loaded into working memory goes back to sensory processing," he suggests. Hundreds of experiments with both animals and people have shown that organisms are far more likely to perceive and react to cues relevant to their current needs than to irrelevant stimuli. This effect explains why we are more likely to notice the aroma wafting from a neighbor's grill when we are hungry than just after eating. If Desimone is right, the prefrontal cortex could be responsible for focusing an animal's attention and thus possibly steering awareness.

Imaging studies with PET and functional MRI corroborate the evidence from brain injuries that the human prefrontal cortex, like that of monkeys, is central to working memory. Several research groups have now imaged activity in the prefrontal cortex when people remember things from moment to moment. Different tasks may also require various other brain regions closer to the back of the head, but for primates in general, the prefrontal cortex always seems to be busy when target information is kept "in mind."

The Devil in the Details

Having shown that the prefrontal cortex is crucial to working memory, investigators naturally want to understand its internal structure. Goldman-Rakic and her associates at Yale have found evidence that when an animal retains information about a spatial location, the prefrontal activity is confined to a specific subregion. A separate area below it is most active when an animal is remembering the appearance of an object. These findings, together with observations of the anatomy of neural pathways, led Goldman-Rakic to propose that the prefrontal cortex is organized into regions that temporarily store information about different sensory domains: one for the domain of spatial cues, one for cues relating to an object's appearance and perhaps others for various types of cues.

There are, moreover, some indications that the human prefrontal cortex may be organized along similar domain-specific lines. A PET study reported last year by Susan M. Courtney, Leslie G. Ungerleider and their colleagues at the National Institute of Mental Health found that in humans, as in the monkeys studied earlier by Goldman-Rakic, certain brain areas are especially active during exercises that challenge working memory for visual details and for locations. Moreover, the most active brain regions lie in similar relative positions in both species.

Goldman-Rakic's proposal about the organization of the prefrontal cortex argues against the standard view of the various components of working memory. The British psychologist Alan Baddely proposed in 1974 that working memory has a hierarchical structure, in which an "executive system" in the prefrontal cortex allocates processing resources to separate "slave" buffers for verbal and spatial information. The memory buffers were supposed to be well behind the prefrontal cortex. But Goldman-Rakic is unconvinced that the brain's executive processes are confined to any particular location. Moreover, in the traditional model, memories organized by domain would lie somewhere behind the prefrontal cortex, not within it.

The high-speed imaging capability of functional MRI is now able to help resolve the question. A study that Courtney and Ungerleider and their colleagues published in April in Nature pinpoints the part of the brain that is liveliest while working memory holds an image of a face. That region—the middle part of the prefrontal cortex—has been fingered as the crux of working memory in a variety of studies.

Yet the face-recognition task Courtney and company used does not involve any obviously executive functions, Ungerleider notes. Their findings thus contradict the view that only executive functions reside within the prefrontal cortex, but they do fit with Goldman-Rakic's scheme. Similarly, Jonathan D. Cohen of Carnegie Mellon University and his co-workers found a region of the prefrontal cortex partly overlapping the one identified by Courtney that is active while subjects remember letters seen in a sequence. The more the subjects had to remember in the Cohen experiment, the more active their prefrontal regions. So Cohen's result also suggests that working memories are actually stored, in part, in the prefrontal cortex. Domain-specific organization "is the dominant view" of the prefrontal cortex, Wise says.

Wise himself does not subscribe to that dominant view, however. He points, for example, to a study reported in Science in May by Miller and his associates at the Massachusetts Institute of Technology. The researchers recorded from neurons in the prefrontal cortex of monkeys while they solved delayed-choice tasks that required them to remember information about both the appearance and spatial locations of objects. Over half the neurons from which Miller recorded were sensitive to both attributes, a result not expected if domain-specific organization prevails. "It argues against Goldman-Rakic's view that identity and location are processed in different parts of the prefrontal cortex," Miller says.

Goldman-Rakic responds that she and her colleagues have recently found hundreds of cells in part of the prefrontal cortex that respond selectively even in untrained animals to objects or faces—further evidence, she asserts, that the information in that area is organized in part by sensory domain. "We do feel the evidence is overwhelming that the functions of neurons in the prefrontal cortex are dictated in large part by the neurons' sensory inputs," she says. Moreover, Goldman-Rakic believes technical problems cast doubt on Miller's experiment. She maintains the targets he used were too close to the center of the visual field, which could produce spurious firings.

Keeping Self-Control

Michael Petrides of McGill University, another leading figure in the field, has mounted a different challenge to the standard view. Petrides's studies point to two distinct levels of processing, both within the prefrontal cortex. In his view the levels are distinguished primarily not by whether they maintain information about place or objects, as Goldman-Rakic holds, but rather by the abstractness of the processing they perform. The lower level in the hierarchy—physically lower in the brain as well as conceptually lower—retrieves data from long-term memory storage elsewhere. The higher "dorsolateral" level, in contrast, monitors the brain's processes and enables it to keep track of multiple events. This higher monitoring level is called on when subjects are asked, for example, to articulate a random list of each number from 1 to 10, with no repetition: a subject has to remember each digit already chosen.

Petrides finds that both humans and monkeys with lesions in the dorsolateral part of the prefrontal cortex are crippled in their ability to monitor their own mental processes: they perform badly on special tests he has devised that require subjects to remember their earlier responses during the test. He also cites PET studies of healthy humans that find heightened activity in the same region when subjects are performing the tasks he uses. The finding is the same whether the tasks involve spatial cues or not. "The material does not seem to matter—the process is crucial," Petrides says.

Other researchers have found evidence to support the notion that the higher parts of the prefrontal cortex are key for self-monitoring. In an experiment by Mark D'Esposito and his associates at the University of Pennsylvania, volunteers performed either one or both of two tasks that, separately, did not require working memory. One task required subjects to say which words in a list read aloud were the names of vegetables, whereas the other asked them to match a feature of a geometric figure seen in different orientations. Functional MRI showed that the dorsoventral prefrontal cortex became active only when subjects attempted both tasks simultaneously. And in April at a meeting of the Cognitive Neuroscience Society, D'Esposito presented a meta-analysis of 25 different neuroimaging studies. The analysis supported Petrides's general notion that tasks involving more computation involve higher regions of the prefrontal cortex. "It was amazing that this came out," D'Esposito says.

D'Esposito's analysis also confirmed earlier indications that humans, far more than monkeys, represent different types of information in different halves of the brain. The meta-analysis did not, however, detect the upper/lower distinction between spatial and object working memory that Goldman-Rakic espouses.

Asymmetry of the human hemispheres is becoming apparent to other researchers as well. John D. E. Gabrieli and his colleagues at Stanford University have used functional MRI to study the brains of volunteers who were solving pictorial puzzles such as those often found on intelligence tests. The puzzles were of three types. One group was trivial, requiring the subject simply to select a symbol identical to a sample. A second group was a little harder: people had to select a figure with a combination of features that was absent from an array of sample figures. The third group contained more taxing problems that required analytical reasoning.

Gabrieli's study sheds some light on the debate over the organization of the prefrontal cortex. When volunteers pondered the intermediate class of tasks, which most resembled the tasks other investigators have used when studying working memory, the right side of the higher part of the prefrontal cortex was prominently active. Moreover, the activity was in areas that other researchers have found to be used when cues about spatial location are stored. This result fits Goldman-Rakic's idea that working memory for spatial location is stored in the higher regions of the prefrontal cortex, because these intermediate tasks all demanded that subjects visualize features in different locations.

When the volunteers in Gabrieli's experiment worked on the hard problems, however, the prefrontal cortices of the subjects became even more active, on the left as well as the right side. The added complexity produced a pattern of activation like that Petrides has found during his tests of self-monitoring.

Gabrieli's data thus provide some support for Petrides's theory of a higher executive level in the prefrontal cortex, as well as for Goldman-Rakic's view that domain-specific regions exist there. "There are definitely domain-specific places," Gabrieli says. "And there are others that rise above that." In other words, both sides in the debate over domain-specific organization of the prefrontal cortex may have a point. Yet in June, Matthew F. S. Rushworth of the University of Oxford and his colleagues reported in the Journal of Neuroscience that monkeys with large lesions in their lower prefrontal cortex could still perform well on delayed-choice tests. The finding casts new doubt on the theory that object working memory resides there and seems to support Petrides.

It may take years before the outstanding questions about the prefrontal cortex are settled and the operation of the brain's executive functions are pinned down to everyone's satisfaction. "If you put a theory out, people will attack it," Goldman-Rakic muses. "Everyone is contributing." And the modus operandi of the brain's decision-making apparatus is slowly becoming visible. "We are getting," Goldman-Rakic observes, "to the point where we can understand the cellular basis of cognition."

Source: Reprinted with permission. Copyright © August 1997 by Scientific American, Inc. All rights reserved.