The Limits of Numbers

The importance of numbers becomes even more mysterious if you begin to think about their limits. For all the things that numbers can do for us, they are dwarfed by the things that numbers can't do for us. Often

There is also the limit disguised as a need for large amounts of numbers. For instance, consider music CDs and their relation to actual performances. Although with good equipment a recording can really involve the listener, the experience of listening to a recording is worlds apart from being physically present at a performance. It would be easy to claim that the difference is in not having enough data - enough channels of music, the full sound (echoes etc.) of the performance site, the visual data presented to the eyes, etc. But in fact no amount of recording, no number of microphones and cameras, can duplicate the experience of a live in-person performance.

Another related issue - chaos, sensitive dependence on initial conditions.
Another - Numbers being unable to plumb human personality, morality, dignity, self, religious questions, etc.
Another - signals from within math - when infinity comes up.

The Mysterious Importance of Numbers

In fact each of these examples is counterintuitive. Imagine yourself five hundred years ago, before science. What did people back then use numbers for? Counting things of commercial import: money, seed, goods, lengths, people. Numbers were also used in symbolism and religion: the twelve tribes of Israel, the four winds. They had nothing to do with images, communication, weather, understanding your body, or the like. In daily life there were no calorie counts, no 98.6 degrees for the normal human temperature, no speed limits, no wind speed measurements, no measuring your pulse with a stopwatch. Numbers had their niche in daily life and they stayed there. Certain mystics and accountants might devote their lives to numbers, but that was their specialty.

No sensible person would have suggested that numbers are at the center of everything from understanding your own health and sickness to facilitating dialogue between cultures to agriculture. To get an idea of how incongruous that statement would have seemed, pick some other important part of daily life (fire, postage stamps, clouds, etc.) and substitute it for "numbers" in the preceding sentence. Sure, you can find connections between fire and practically everything else, but saying that fire is at the center of communication would be more than a little far fetched. Someone who is not simply holistic or poetic about fire, but really believes that it is the answer to everything, would be considered at best a dreamer or enthusiast or mystic, and at worst just crazy. That's the same reception that any visionary of today's numbers would have received before the success of numbers.

Even our own language betrays the unusual importance we give to numbers. We now speak of "quantum leaps", and the word "quantum" refers to counting by ones; a quantum simply means a single indivisible unit, the difference between one and two. Imagine encountering a culture where people spoke of "fire leaps" as the apotheosis of progress. What would you think? And what were the chances that someone before the victory of numbers would have thought our present attitudes as anything but crazy?

The worst of it was that there were no hints that numbers might be so useful for so many things. Throughout the ages, our visual experiences never gave us any hint of the billions of numbers that could be used to describe them. No doctor putting his hand on a fevered patient's forehead ever had the temperature suddenly revealed as a number. Even at the height of concentration no musician began hearing numbers emanating from her instrument. Our universe appeared then as it ever does, without numbers.

I don't think that science does anything to take away the mystery of the link between numbers and nature; nor does it give warning of the limits beyond which numbers provide no insight. Instead natural philosophers simply began with the intuition that numbers are the most important way of describing their experimental observations, and then began inventing and refining machines which attach numbers to the world around us. Every number we use to describe our actual physical world was mined from the world using special machinery: thermometers, speedometers, cameras, etc. (Footnote: There are two important numbers, pi and e, whose values can be derived independently of machinery, but neither are observed directly in the natural world - this is related to the issue of physical units, which we may discuss later in this blog.)

Physics and Numbers

We began this introduction by talking about how physics began as philosophy, a sort of thinking that tries to understand things more deeply, and doesn't have a results-oriented focus. But soon the first natural philosophers crossed a divide which continues to separate physics and all the sciences from philosophy. This divide has three hallmarks: experimental observation of nature's actual behavior, complete control over experiments including the ability to repeat them at will, and always using numbers to describe nature. We started by talking about experimental observation, and more recently discussed the focus on experimental control, how it results in a corresponding narrowing of "scientific truth" to those things that can be controlled and repeated at will, and how it unfortunately can be part of a vicious cycle which focuses on power and abuses both man and nature. Now we will turn to the third hallmark of natural philosophy: its focus on numbers.

Numbers and the things we can do with them will be the daily bread of the rest of this blog. (That's why I discussed this hallmark last.) But before we get to know numbers I want to point out that there is a great mystery here: why are numbers useful for describing nature? Certainly this doesn't seem like a mystery today, when the success of numbers is a fact of life. The movie reproductions we watch via DVD or high definition television are composed of billions of numbers; indeed any image can be represented by numbers, as are the voice signals that we listen to on our cell phones and the songs we listen to on CDs. Mapquest and GPS have quantified our moving around: 5.3 miles north, then turn right and 1.4 miles east. And time also: we watches to tell the exact date and time. Clearly numbers have been very very successful, but their success alone does not explain why you're successful. If we think that the utility of numbers is obvious, it is only because we are prejudiced by our daily experience.

Control Over Experiments - Getting the Whole Picture

In previous blogs I discussed how physicists choose a role or social contract in which they are cast as specialists and professors, not as power brokers or politicians. The last blog argued that this casting choice is a way of cementing and deepening power, both because it gives scientists a special status and realm of control like that of priests in the temples of progress, and because it frees them from the obligations that would come from being responsible for the social and moral ramifications of their own work. Now I'll discuss two other ingredients in the power equation.

We have seen that physics strives for the power to make nature do something over and over again on demand according to our whims, and even to take nature apart; nature becomes our slave rather than our partner. Similar levels of control and power can not be obtained over even the most docile people. Consider the power given by a system of pipes and taps providing running water. Such a contrast with the power that would be offered by keeping servants or dedicated workers - they would talk to you as they brought the water, change their pattern slightly or greatly each day, not respond with the same alacrity as a water tap, and sometimes be unavailable, maybe even leave permanently or die. Clearly nature offers far greater opportunities for control and power than people do. A physicist who chooses to focus her energy on nature rather than people is making, from this perspective, a power-maximizing choice. A simple calculation is enough to see that the greatest success in obtaining control via politics and power games will still leave you at sea struggling against new waves; if science-based control is the only kind available, why not cling to this rock? Scientific knowledge can seem like the only certainty, the only lighthouse in an uncertain world; what does a scientist lose by abdicating her role and responsibility for society as a whole? In other words, if the overriding goal is power and control; if this goal is used as the measure of truth as exemplified by physics' emphasis on experimental reproducibility and analysis, then a policy of specialization, technical roles, and abdication of responsibility for society is the only reasonable choice.

Going further, it is fair to say that for many individual physicists (and scientists) science is the way of maximizing their own personal power and security. Whatever Einstein's actual character was, the popular caricature of Einstein as a man who was socially incompetent contains a grain of truth about physicists as group - many of us don't do as well with people as with experiments and equations; it is striking how many of us have poor emotional awareness, empathy, communication, leadership, and group skills. And even more deeply, there can be a lack of trust for people, a sense of alienation, a need for refuge and security. People like this can be found in all walks of life; however if they discover at a young age that they do well with math or machinery, and that not only nature but society rewards their scientific efforts, then science can easily become a holy grail. This powerful incentive results in the physics community having more socially challenged people than is normal in most other professions. What I am saying is that the professor/expert role maximizes power not only for physicists as a group, but also for many individual physicists who are much more comfortable with subservient nature up close and threatening society at arm's reach.

In summary, not only is the scientific method focused on obtaining total control over nature via experiments, but also physicists themselves seek out control and power, both by choosing physics in the first place and by insisting on a particular role in society which is carefully chosen to promote their own status, security, and freedom from social responsibility. One of the hallmarks of natural philosophy is its focus on experimental control. From a certain perspective that has been widely popularized, this looks noble and clear-sighted; one talks "laws of nature," the search for a "theory of everything," etc. But the unspoken truth is that down to its very heart science and the scientific community are alloyed, joined, infected with the search for power and security. The popularized story about science is a true but small part of the picture; from a more human perspective one sees a much larger tapestry, with the scientific community taking its place in the larger human community, and with many threads including responsibility and its absence, respect and disrespect, social contracts, personal talents and weaknesses, etc. As with all human stories, a lot of wisdom (wisdom! not only knowledge!) and love is needed to understand, care for, guide, regulate, and prune the scientific community and its work.

Control Over Experiments - The Scientist's Role vs. Responsibility

Why do physicists prize a role which renounces power or control in the political, economic, or environmental, or social arenas? Certainly the power we physicists have wrested from nature has transformed human affairs in these areas and others, and certainly physics itself is constantly driven from within to obtain more power and control over nature; this is at the center of our mission.

Perhaps what looks on the surface like an abdication of power is actually at the heart a way of obtaining still more power, prestige, and security. Having an esteemed and needed role, and using this role to distinguish and divide themselves from others and their affairs, allows physicists to formalize a social contract with the world at large: society will provide generous funds, liberty, security, and respect, and in return physicists will tend the fountains of progress promising quality of life (power) for all but not asking any questions about how that power will be used. If faith in progress is akin to religion, then physicists might be compared to priests or shamans: people who promise and maybe deliver success and power, but wash their hands of profane things like politics and business, while expecting in return esteem, respect, regular support, and a realm of control.

This social contract between physicists and the larger society allows the two parties to absolve each other of responsibility. As I already sketched, physicists do not take responsibility for the ways that their focus on power has been imitated and continued in the world at large, or even for the moral issues of their own research activities; in other words physicists lay the responsibility with society. But in exchange society often lays the responsibility for its horrors and sins at the doors of science. This can transfer of responsibility can take one of two tracks; the "pessimistic" track attributes our ills to science and technology: some of the keywords are "nuclear", "genetically modified," "organic," "free radicals," "virus," "radiation," "chemical," and "pesticide," but there are many more suspicions ranging from the widely accepted to the fringe. Many of my Italian friends are still convinced that microwave ovens can cause cancer. While the attitude can be one of thoughtful discernment, at times the feeling is more one of terror about particular keywords and of blaming science for the world's problems and for life's insecurities, for instance the various ways we could die. While I believe that the scientific community should take responsibility for the moral weight of its actions, my expectation is that culpability will not center around mysterious technological keywords but instead around issues understandable to everyone like abuse of power and lack of respect.

There is another more "optimistic" way that society transfers the responsibility for its sin and failures onto science: faith in progress. If nuclear materials threaten to poison our world, we rely on our scientists to sooner or later provide a safe disposal technology. If we have poisoned our earth and water and stripped mined our land into lunar landscapes, then some day reprocessing and cleansing technologies will restore them. If the disenfranchised turn to terrorism or illegal migration, our scientists will develop new security technologies and smart bombs. If millions die yearly from wars, malnutrition, malaria, and political difficulties, then we will perfect the science of globalization. If we have an epidemic of obesity or AIDS, then we put people to work on new drugs, surgeries, exercise machines, prophylactics, immunizations, selective abortions, psychiatric and psychological techniques, mass media campaigns, and gene therapies. We can do no wrong as long as we continue giving money to scientists, because the scientists are sure to find a way to right all our wrongs. The "temporary" problems along the way are just the price of progress.

As a particular example, with the modern concern about global warming some people are popularizing the concept of the "carbon footprint," which means that your personal contribution to emission of carbon into the atmosphere. Along with this concept there is an organization which will allow you to pay them instead of reducing your footprint. If you go on an extra flight, that's OK: just give them a little bit more of your money. They promise to use your money to make a compensating reduction in the world's carbon emissions, perhaps by reducing someone else's footprint, or else by investing in technology that sooner or later will reduce emissions. This practice is identical to religious practices of sacrifice and atonement. People take what is precious to them: grain, cattle, money, clothing, in fact anything of value, and give the goods over to priests to either destroy or else devote them for sacred causes. In return for their sacrifices the people lose responsibility for their evils and are promised healing, success, and prosperity. Only two things have changed: (a) we have some new "sins" like pollution and the carbon footprint, and (b) now our priests are called scientists, experts, environmental advocates, etc. The substance is still the same: we give them our money, they give us freedom from any moral qualms about our actions and also a promise of prosperity.

The specific role that physicists cast themselves in allows them to confer moral responsibility to society, and in turn society transfers back its responsibility through faith in progress or else demonization of specific scientific keywords. In this exchange everyone is freed from responsibility, from qualms of conscience - which is another plus for the power seeker. Perhaps one of the greatest benefits which physicists derive from the role they cling to is precisely the way that it frees them to research what they like without moral or human considerations.

Control Over Experiments - The Scientist's Role

During the last years of World War II the U.S. government recruited many physicists to work intensely on developing a nuclear bomb. The problem's complexity and novelty, plus the camaraderie of a huge team of some of the world's best physicists and also the urgency given by military challenges and government backing, absorbed everyone's interest and supplied excitement and dedication. It was an adventure. Many of those physicists later explained that they were so excited about the breakthrough and loved their community problem solving so much that they didn't stop to think about the morality of the weapon which they were developing for the United States. It was only as the bomb was nearing completion that these natural philosophers became philosophical about things and began to have second thoughts. They later emphasized that the leadership role, moral weight, and international import of the Manhattan project had been a new thing for them. The move to ascendancy was exceedingly brief, they were not prepared for it, did not realize what had happened until years had passed, and were surprised at themselves afterwards. Who had ever told them that their work could have such moral weight and inspire such awe and terror?

Even today, physicists in the United States generally do not seem comfortable seeing themselves as power brokers or political or moral leaders. There are few American physicists who are syndicated for nationwide publication or who hold political office. (Internationally there is an exception: Dr. Abdul Kamal, "Missile Man of India," held office as India's President from 2002 until 2007. This is primarily a figurehead role; his tenure was known more for idealism and technological enthusiasm than for government. Perhaps the exception proves the rule.)

In other words, there is a distinct tendency for physicists to cast themselves in a role as absent minded professors, geeks, specialists, experts who understand the universe but not society and power structures. This casting choice allows physicists to absolve themselves of responsibility for society, government, and human issues. The accounts of the Manhattan project physicists are just a convenient example of this attitude: during the war most of them simply did not even consider taking this sort of responsibility, and after the war they excused themselves for this lapse by saying they were not prepared, new to the game, etc. This excuse making was exceptional, since in most cases physicists never reach the point of reconsidering whether they might have taken greater responsibility for their influence on society at large. They are so very sure of their self-casted roles. They see no connection between themselves and the fact that modern society is incredibly power hungry and often uses its power over nature and over people in destructive ways.

Yet we have seen that some of the first natural philosophers were obsessed with power, and that a physicist's goal is a certain kind of knowledge that involves a huge amount of power over the thing known. Even in the years since World War II physicists have been very glad to receive privileges and pride of place from society in return for the great contributions they had made to military power (radar, the bomb, jets, rockets, submarines) and economic power (semiconductors, LCDs, the Internet). No one could accuse the physics community of being aloof to power; in fact in the current years we seem to be running ever faster to convince power brokers that we still belong on their team. It's just that we want to play the game our way, in the role of specialists and professors who are not responsible for the use and abuse of power in the world at large. Far be it from us to be power brokers or politicians!

Control Over Experiments - Analysis

Another hint about the defects of physics can be found in its reliance on the reasoning technique called "analysis." Analysis means that you solve problems by breaking them into pieces, and then handle the pieces one by one, or perhaps even break the pieces into smaller pieces, and so on. This approach is a key reasoning skill: any time you make a list you are probably performing analysis, and any time you count a group of things you have probably already done some analysis. What is strange is that physics relies so heavily, almost exclusively, on analysis. When trying to understand an object, material, phenomenon, etc., we start by trying to figure out what are its pieces and the rules governing their movements, and then based on the pieces we begin to reason about the behavior of the whole. For example, the solar system is conveniently broken into planets, asteroids, and the sun. Ceramics and metals may be understood as crystals packed against each other, or as large collections of atoms. Animals can be broken into collections of muscles, bones, and organs, or alternatively into cells, or alternatively into the building blocks of cells including the nucleus, DNA, cell wall, cell skeleton, etc. And any material as well as any force can be understood as "particles" - electrons, photons, gluons, quarks, etc. This sort of breaking things down - analysis - is always the starting point for physics. We will see later in this blog that matter and forces are not the only things physicists like to analyze - we also want to take apart time, space, and anything else we can think of. This attitude carries over into all of modern society; for instance analysis is at the root of how our workplace is organized, with many different specialties and degrees and professions and experts, so that one person knows how to solve one sort of problem and another person knows another sort of problem, and the general hope is that if we have enough specialists and if we train and pay them well enough then everything will come together into a great society.

Of course analysis has its limits. It's only one tool in the human toolbox. Almost every human activity, intellectual or not, involves breathtaking leaps far beyond the reaches of analysis. Following a path through the woods, writing a paragraph, figuring out whether a stranger is a friend or an enemy, tying your shoe, judging the beauty of a landscape, finding a concord with others, learning new ways of thinking - these may involve some element of analysis, some thinking about pieces and lists - but they go far beyond what analysis can do by itself. For instance, we are able to perceive a whole object, and make judgements about it, without consciously thinking about its pieces, and often enough we don't even know its parts. And organizations and societies based purely on specialization wouldn't get very far either - we all know where that would go: a lot of passing the buck when a problem doesn't fit into any particular specialist's competency, or when a new problem comes up, a lot of stereotyping, lack of humanity and compassion, bureaucrats not leaders, people falling through the cracks, and both big and small crimes against human dignity. The only way that our society functions is because a lot of people step outside of narrowly defined roles and let the buck stop with them: most importantly parents all of whom are jacks-of-all-trades for their children and and spouses, but also people in the workplace who do what it takes to make things work, secretaries who fill in wherever the specialists don't, those who truly innovate, those in politics who are truly not satisfied with the status quo.

What I'm saying is that analysis, by itself, gives a very incomplete vision of life and a very limited ability to understand or adapt to it. While physics doesn't rely exclusively on analysis, it is so focused on a component-wise view of the world that its vision is skewed substantially, and many things are just outside of physics' possibilities - for instance questions like communication between persons, beauty, ethics, etc. Consider the knowledge you might get by smashing a vase on the ground and looking at the fragments. That's a whole different ball game than what you will learn from looking at the vase's beauty, or its symbolism, or the functional experience of using the vase, or considering its place in artistic history, or in the culture, or as an expression of the particular artist. Not to mention just looking at the vase, touching it, smelling it. The knowlege physics gives us is often a lot more like the knowledge obtained by smashing the vase than like any of the others.


Analyzing something implies that you have a lot of power over it. You are taking it apart, either mentally, or in real life. Something that you can take apart is very much at your mercy. Moreover it is not your equal; it is subject to to you, and does not receive your full respect. You probably feel free to manipulate, modify, use, and discard it as you wish. For instance, if have an egg, you may wish to separate the yolk from the egg white and perhaps throw one away, or scramble them together, or leave them as is. Similarly, if you take an analytic approach to the earth then you may wish to dig up certain "valuable" parts of it, often destroying much more in the process. Even with human beings an analytic approach - whether in terms of organs and arteries and hormones, or in terms of psychological drives and hangups and instincts - is always accompanied by temptations to disrespect, toward forgetting that the person in front of you is far more than you could ever begin to discover by these analytical approaches. You may begin to think that you know more about the person you're analyzing than they do themselves, or that you should "fix" them, or find yourself reluctant to take to heart their opinions about yourself or others, their recommendations to you, and their wishes even about their own lives.

In other words, analysis is an inherently aggressive kind of reasoning, that runs hand in hand with power and control. Even the words that define analysis are telling: dividing something up, pieces, parts, taking something apart, breaking it apart, etc. All of these phrases have connotations of control, power, division, even destruction. If analysis is actually carried out physically then the object being analyzed ceases to exist for the duration of the analytic procedure. If we do not or can not put it back together then the analysis was the same as destruction. The sort of knowledge that can only be bought at the price destroying the thing known is very strange indeed. Have we really learned anything about the object that was destroyed?

As a matter of fact physics experiments often do destroy the things being measured. We need only think of the accelerators and colliders used to study high energy particle physics - the whole point of these experimental devices is to destroy atomic nuclei by hitting them with various things, and then see what fragments come out of the collision. This is physics' main way of finding out about atomic nuclei, and it involves the total destruction of the nuclei concerned.

While the collision experiments of particle physics are the natural end point of the analytic method, there are many other experimental techniques that cause permanent alterations to the thing being studied. Many observation techniques are based on throwing things at the object under study (this is called bombardment, another aggressive word) and seeing what happens. In some experiments the typical outcome is only that what is thrown just bounces off and you learn from how it bounces - this is what happens when you look through a microscope: you are seeing light that has bounced off of the thing you're studying. However in quite a few other experiments the bombardment process breaks things off of the thing under study, or makes cracks and other defects in it, etc, and then you learn from the debris and defects. Still other experiments take apart methodically the object under study rather than colliding it. Or perhaps a piece is removed (added, substituted) from the studied object, and then you observe changes in its behavior. In all these cases knowledge is obtained only at the cost of substantially changing or even destroying the object of study.

The physicist's theoretical understanding of nature always involves an analysis of things into their components, and physics experiments often are also analytical, even destroying the thing under study. While analysis is an important tool for solving problems and understanding things, it is particularly aggressive, based on our power to manipulate, and often ending in the destruction of the thing being analyzed. Analysis dovetails very well with physics' passion for experimental control, for being able to get nature to do the same thing, over and over again, on demand. Like magicians we are interested in power over nature, but unlike them we want only the power that works on demand for anyone who does the right manipulations. This focus on total control and power means that we who drink from the well of natural philosophy drink tainted water. We really need the water from that well, but unless we carefully watch and control its side effects, we should not be surprised by when we fall sick. The seeds of the abuses and horrors of this age, especially of destruction of the environment and manipulation of human beings, are part and parcel of natural philosophy.