Part II
Famous seventeenth century philosopher Descartes set out to discover what we can know with some certainty. He was familiar with the study of geometry because he taught it and wrote about it. He understood how it started with small number of known propositons and worked its way in tiny steps to a vast number of less obvious ones. We assume that he wished to take the techniques of geometry and apply them to other realms of knowledge. When one considers his philosophical writing, among what remains most useful today is his advocacy of starting with a very solidly known propostion or group of propositions and reasoning from known to known. It may be compared to crossing a creek by hopping from one dry stone to the next. Science has taken Descartes' advice and it has made much progress working the same way. It can be a painstakingly slow method, but it is a reliable one.
In his most famous work, Descartes starts out by distrusting his senses. As we worked out earlier, senses are unreliable. They depend on unreliable perceptions and on even less reliable interpretations that are shaped by language, culture, and belief. But it is worse than that. One can fall asleep and dream. Or, as Descartes argues, one could be deceived by a demon. In any case, Descartes imagines that anything one senses might actually be a product of the imagination. If that is so, what is left? Only cogitation. Descarte's thinks he thinks. And he concludes from this that he exists. Cogito ergo sum.
It's not much, but it is a start - that this one thing is knowable with a pretty high level of certainty by using the exercise of reason. Having not read Kant's Critique of Pure Reason we cannot argue from a strong position that this is what he argues, but we would be surprised if Kant, after saying that in reading Hume it "was as if scales fell from my eyes" that he would set out with vigor to argue something substantially different.
Sadly, with but a single known proposition, not much deductive philosophy can ensue. So with this we know as much as we can from the mere act of thinking. By thinking we learn to think that we think. We imagine so because we cannot imagine the products of another's dreams having dreams themselves. Descartes' work echoes work of previous philosophers in many ways. One way is a profound skepticism about what we know.
There are reasons why such a level of skepticism may be unwarranted, especially if one is a practical person just trying to get along in the world. But there are reasons for skepticism that are profoundly useful. What if there are things we assume to be true that are sometimes or always false? It would be useful to know this. But our cultures and languages tend to blind us to these problems because the conceptual blind spots are embedded in them. There are relationships that are rendered poorly in some languages. If one completes an engineering or physical science education, for instance, one knows that many of the relationships in those fields are better expressed in a different language, a mathematical one.
Skepticism, by systematically denying sacred assumptions embedded in what we do, think, and say, will sometimes root out problems in ways other methods cannot. So the practical reason for Descartes' exercise is to get us to a point where we are capable of discarding ideas and assumptions that we accept implicitly, not because they are correct or useful but because we have taken them as givens from authorities who may have been in other ways trustworthy.
One of the philosophical traditions Descartes appears to be following is the tradition of denying the external world. Until the rise of the empiricists in England, the Continental approach to philosophy was all there was, and this was part of that tradition.
We will take Descartes' word for it that there is no convincing proof for an outside world based in logic. But let us try an empirical approach. Let us suppose one were a philosopher who wished to use an empirical approach to discover if there were an outside world. How would one proceed? One would first start by realizing that a contrapositive proof would work nicely. Imagine that there is not an outside world and if that position becomes impossible to maintain, one must accept that there is an outside world.
One might stop eating. Food, after all, might be imaginary. One might stop drinking, fluid might be imaginary. The fastest means, of course, would be to stop breathing. So hold your breath. Now for so long as you, as a philosopher can hold your breath you can succeed in denying an outside world. But the moment you cease to hold your breath you implicitly assent to the fact that the exchange of air in the lungs is necessary. And if this exchange is necessary, it puts the air on the same level of existence as the being that requires it.
But, one might argue "What if my senses decieve me? What if I only imagine that I breathe or do not breathe?" There is only one response. "Hold your breath until you are convinced otherwise." It is not a very elegant body of reasoning, but the persuasive effect can be quite strong. Either one is persuaded that there is an external world, or one loses consciousness, stops thinking, and resumes breathing.
And, by such an outcome, one learns that the assumption that there is an external world, if philosophically less robust than Descartes' argument regarding thinking, is empirically more so. We will admit that this is more a scientific proof than it is a philosophical one. There are dangers in confusing one with the other.
Furthermore, there may be compelling reasons to deny the external world. There is a whole branch of mysticism based on doing just that. Siddhartha, for instance, argued that all pain, sadness, dissatisfaction arose from human attachment to physical things. And the practice that he established, which came to be known as Buddhism, had as a central tenet the discipline of meditation aimed at denying the existence of the material world. It achieves mental quiescence by methods that move thought from the material world to other worlds. Whether his assumption is correct, many people find the practice derived from the assumption to be helpful in creating a calm effect in their minds.
The ultimate goal of the practice is to escape the displeasures of the world by nirvana, or non-existence. Not many of Buddhism's practitioners achieve this, but those few who do succeed realize, perhaps, what empirical philosophers might fail to do in attempting to deny an external world. Still, so long as we are convinced that we are happier eating and reproducing than we are simply dematerializing, there is something to be said for acknowledging a material existence.
Those of us who exist short of nirvana are left with two certainties. By thinking, we can be convinced that we think, and almost as convinced that we exist. By denying the existence of the external world we can prove its existence empirically. We have arrived at the two banks of the Humean stream: the one bank is "matters of fact" and the other is "relationships of ideas." It doesn't help us much. But it is something.
A person who studies science and then reads Aristotle might be struck with two antithetical responses. The first is a jaw-dropping surprise at how far Aristotle got using casual observation and the Socratic method. His work really was not surpassed until the English had been studying Greek for a few centuries and inventing their own empirical philosophy.
Aristotle is a great example of how one can push knowledge to its limits by working inside some logical framework. But his work is also a perfect example how one can sometimes drive well outside its operational bounds in the absence of experiment. Today's scientist can be amazed at how completly wrong some of his concepts are. As we mentioned above, Aristotle did science in a way that western scientists view as being odd. He thought carefully about the world he observed. He imagined that his powers of observation were sufficient to deduce all of the relevant scientific relationships.
We make observations today, too, and we think about them carefully. But today our methods of observing the world and of deducing scientific knowledge is purposefully controlled so that any obeservations we do make have some hope of having good predictive value. It was the English who established both a new way of doing philosophy and a new way of doing science, one that acknowledged the outside world and sought to understand it through systematic observation.
One early philosopher of this flavor was William of Ockham, known for "Occam's Razor." His observation goes to the heart of making models by which we interpret data. Loosely interpreted, he advises descriptive models be as general and as parsimonious as possible. They must explain as much as possible with the smallest possible expression - one using the least number of terms. This remains a primary standard against which all predictive models are judged. It is the standard for the Agatha Christie murder mystery just as it is for the scientific model.
It was Roger Bacon in 13th century England who proposed learning by controlled observation. He was the first prominent English philosopher to perform controlled experiments, and his methods of inquiry became the standards that inform science today. Controlled observation seems like an obvious notion now, but it overturned a more than a thousand years of scholastic practice that focussed almost exclusively on more informal observation and intense cogitation.
At least until the turn of the twentieth century, the continental methods dealt more rigorously with the model-making side of things, while English methods tended to be a bit more experimental. A rich interplay between these two sorts of approaches led to much progress, especially in the eighteenth and ninteenth century.
Once again, we see the two banks of the stream. On one bank, the bank of fact, stands Bacon advocating experimental observation. On the opposite bank, the bank of "relationships of ideas" we see Occam advocating interpretive methods that are simultaneously compact and robust. They stand near each other in sense, in geography, and in time. These ideas taken together establish the birth of both empirical philosophy and empirical science.
It was not, however, until Newton's work on motion and on gravitation that this new method would prove its full worth. Newton's work is a kind of crown jewel of science because it answers a question that all men before him had asked when they observed the heavens: How does it all work? Newton, with a very tiny, very general set of equations - one for gravitational force and one for what he calls centrifugal force ( an artifact of the acceleration that arisis from circular motion ) - describes the motions of all observed heavenly bodies. And apples. The work is practically defines what Occcam's Razor is and does. And it describes some of the most painstakingly taken data taken to that time, measurements of planetary motion.
In order to describe these motions accurately Newton invents a new mathematical language that today we know as calculus. Calculus is a language that lives in the "relationship of ideas" and so, too, does Newton's description of gravitation, expressed in mathematical terms. Had Newton only invented calculus, he would be considered a towering genius; for calculus has proven to be of immense use in solving a host of physical problems. But his description of gravitation also led directly to much more profound and quantitative understanding of electrical and magnetic phenomena and to the understanding of heat transfer in solids. In short, Newton's work transformed our understanding of the universe.
Before Newton, the heavens were chaotic and so was the earth. Everything was chaos. But as people's minds grew accustomed to Newton's point of view, the universe itself became well ordered. It was understood that all of matter behaved precisely according to universal rules. And it only made sense that if one could understand those rules, one could know everything. It was the precise measure of time that allowed much of Newton's work on motion. And it was therefore the precise measure of time that allowed him to create his model of gravitation. The clock was rather new, and it still had the mystique of being somehow special, almost magical. And so it was predictable that the clock should eventually be used as a point of comparison, or a paradigm for a mechanical universe.
Newton's laws allowed one to predict the motions of heavenly bodies just like clockwork. In the case of heavenly bodies, if one knew at one point in time precisely where they were located and precisely their velocity (or all the things that affected this ) one could calculate where each heavenly body had been in the past and where it would be at any moment in the future. Like clockwork.
This tempted the notion that everything in the universe behaved the same way. If we could derive all of the governing equations that determine everything about the universe and one could know its state with high accuracy at one point in time, then one ought to be able to treat all physical phenomenon throughout the universe in the same manner. One writes the equations, solves them, knows everything. This idea has been central to or at least not far from the motivation of all scientific pursuits since Newton's. It is a kind of a "dream of knowing."
Newton's work changed the way Europeans saw the universe. And it changed the way Europeans saw man. An early example is Hobbes' view of society in Leviathan. He imagines society to run like clockwork. He imagines a man to be a kind of cog in a societal wheel. This, at least, is the underpinning of the work - it goes on to develop another metaphor that serves the purpose of the work better and the ends of man perhaps less well. But Hobbes is the first to apply a kind of Newtonian determinism to societal forces, and it is the start of a trend. By the time of the French Revolution, empiricists imagine the body to be nothing but a complex biological machine. It is a view that pervades science and popular culture to this day.
There are a number of compelling reasons to imagine that the universe is deterministic, and that by studying it more cleverly we shall come to know it better. Not the least of these reasons is that studying to know it better allows us to create a richer living environment; more stuff, less work. And this allows us to practice the arts, where man has traditionally found most meaning.
Science, by providing for our physical needs allows us, sometimes, to be more ourselves. But there are some compelling reasons to imagine otherwise, as well. The romantics imagined that reducing man to the status of a biological machine served man's psychological needs badly. Mary Shelly conjures up a vision of science, represented by Dr. Frankenstein, being overcome by its less than human creation, a monster without empathy. It is a scathing rejection of science's relentless reductionism. And appropriately, it hits home.
So we approach a fundamental problem about science : what is it for? Science and scientific knowledge are by definition about the factual world. They are about what is materially real, what materially exists. This allows us some amount of mastery of that material world, and this mastery allows, at least for some time, a comfortable standard of living. Mastery of the material world can allow us more, too.
To the extent that we can understand our own actions, atitudes, behaviors, propensities, aptitudes, desires, sources of satisfaction, relational needs, social connections, societal problems, and so on by employing empirical methods, we need to do this; because while the process is painfully slow, and the results are frequently different from what we imagine, the process tells us important things about our lives. And they are things that we need to know. Humans are fundamentally social, relational.
While it may be that our chief dissatisfactions are material problems, our chief sources of satisfaction arise from interactions with others. Science has recently proven this. And the purely reductionistic point of view that characterized enlightenment science manages to deftly dodge this fact by understanding man as a collection of parts. Not only does it separate the whole man from his society, but it separates his nose from his face, his tongue from his mouth, his liver from his spleen. This is a necessary part of understanding man as a material object, but the relationships between the parts will often explain much more than the individual parts themselves. One cannot know everything about a liver when it exists isolation from all the other organs, floating in a preservative bath. The same is true of a human considered outside society.
The process cannot end with reductionism. It is the mystical arts that have the longest history of studying the sources of human affection and disaffection. It is the mystical arts that hew most closely to describing sources of happiness. One hypothesis for this we work out in a brief essay on mysticism and language. In it we suggest that mysticsm is well described as a kind of open puzzlement. And that its satisfaction sometimes lies in this state itself, and sometimes in the solution of puzzles. This state of being is a requirement for the early acquisition of language; without it, language would be too difficult to learn.
This is a hypothesis that science might be able to validate. But that validation would not be sufficient to create the same sense of satisfaction as a puzzle well solved. Solving puzzles brings a sense of joy. I remember my own sense of joy when, in the middle of a first year physics midterm I realized why tides were high twice a day rather than once a day. I will not ruin the fun for those who wish to work it out, but the hint I needed was that the earth and moon, in their dance, rotate around a common point that is not at the center of the earth, but is about 1000 miles beneath the earth's surface. Feyman's lectures on physics provides further hints. It helps to keep in mind the fundamental equilibrium problem that Newton was trying to solve when he invented the idea of gravitation. I remember getting more joy from solving this puzzle than from all the Christmas presents I have gotten, put together. This suggests that sometimes we do science not so much for the rewards of the discovery as for the rewards of discovering. The act of doing science - of working out an explanatory model for some phenomenon - is actually quite a satisfying thing. It is, in my own view of mysticism, a kind of mystical experience. And, in fact, Buddhist practice prescribes koans which are little puzzles whose solution provides the same sort of satisfaction.
My own advice is this: The subject of science is the material world. If we want answers about the material world we must turn to science. The world of spirituality is about our sense of well-being. To the extent that this is affected by the material world - our external conditions, the chemistries of our brains, and soon - again we must turn to science for help. But our satisfactions are relational and mystical. If we are to live lives of joy and satisfaction, we need to live lives that soar far above the reductionistic views of enlightenment scientists. In a material sense, it is what we know that matters. In a spiritual sense, it is what we do not know that keeps us interested. One does not use a sledgehammer to sew sutures; nor catgut to drive railroad spikes. It is essential that, when we consider issues of science and spirituality we make good distinctions.
The twentieth century dawned in Europe in a hopeful way. In physics, all the important problems had been solved; only a few loose ends remained to be cleaned up. Chemistry had been transformed from a black art to a science. And between Darwin and Mendell, all of the forces in biology seemed to be well understood. The head of the US patent office had, not long before, proposed closing it because "everything that would be invented had been." Railways and steamers moved cargo and people in unprecedented comfort and ease. And Bertrand Russell, a prominent mathematician worked on a set theory of everything. It was halcyon days for the world of knowledge.
Then all hell broke loose. At roughly the same point in history:
Each of these items limits what can be known. And each operates in a different domain of knowledge, attacking our sense of knowability at different levels. Quantum physics presents several 'knowledge' problems. One of the more troubling is a view of the world that raises questions about our understanding of the notions of cause and effect. It is unclear whether these questions arise more from the state of the universe - one in which time is not meaningful for certain kinds of processes - or whether they arise from some misapprehension we have about the very nature of cause and effect. These questions plague physicists today. Or would if they spent any time with them.
The most easily understood of the problems raised by quantum phyics is the Heisenburg Uncertainty Principle. It is a principle that deals with empirical measurements involving electrons and other tiny particles. It says, in mathematical language, that the more precisely you would measure the position of an electron, the more you must alter its velocity. And vice versa. The practical upshot is that one cannot measure, with arbitrary precision, both an electron's position and its velocity.
If the electron were the next bus I could know either how close it was to the stop before mine, or how fast it was travelling. But I could not measure both to an arbitrary accuracy. There is a theoretical limit to how much time I can hope to shave from my wait at the bus stop.
Heisenberg's uncertainty principle then, tells us something about the limits we encounter in mensuration. It tells us that not only can we never simulteously know the state of every particle in the universe, but we cannot do so for one particle. It is like saying "we have a clock with lots of hands and some hands are pointing up and some hands are pointing down, but most of the hands we cannot see."
If one generalizes this principle a bit, it means that one cannot measure a system without affecting its state. In systems that are very large and very robust, this is frequently not the sort of problem that it is in quantum physics. A vague awareness of the problem and a wise choice of measurement techniques will frequently be more than enough to allow a useful measurement of macroscopic quantities. But not all macroscopic systems are that robust.
There is a West Wing episode in which a pollster is tasked with finding out what voters would think if they learned their President had MS. Now, if several thousand people got a call asking that question, and if several of them communicated it to someone else, it would not be long before the poll would start a rumor that the president had MS. In other words, the act of measuring would cause a change in the system. Pollsters are aware of this and some, I am led to understand, exploit it to mislead voters. In other words, a there are a number of polls that are more closely designed to be persuasive Socratic dialogues than they are to be polls. In such a case a persuasive device is disguised as an instrument of measurement.
It is notable that in quantum physics one is dealing with the states of a small collection of particles of matter and in elections one is doing essentially the same thing. It is not intuitively obvious that quantum physics would give us insights into societal behavior, but in this case it does. And it means that measurements of societal beliefs can change those same beliefs. In both cases, we are presented with a fundamental problem of mensuration.
The uncertainty principle dealt with problems of measuring. It is an idea that deals with the fundamental limits on controlled experiments. It limits what we can know of "fact." But there exist problems of knowability within the world of "relationaships of ideas." Goedel, a contemporary of Einstein, and a friend, worked out a novel proof. It said that any formal system sufficiently powerful to prove all true propositions within that system, must be capable also of proving false ones as well. Or, put another way, if one excluded from a formal system the capacity of proving false propositions, that system would be incapable of proving all true ones. Either way one looked at it, if one created a formal system to describe something of interest, one could not perfectly enumerate all true and all false propositions.
The form of Goedel's proof suggested that the problem lay in self-referential propositions such as "this proposition is true" and "this propositon is false" and "Cretins are only capable of speaking lies. I should know; I am one." It is possible that Goedel's theorem demonstrates a kind of singularity in mathematics and that its repercussions do not reach very far. It may be that what is unknowable in mathematics is so small as to be insignificant. Goedel's proof may be a trivial proof in the sense that it may apply to a tiny and insignificant nugget of propositional knowledge. It may be that it pertains to precisely one sort of self-referential propositon. Yet how can we know?
The simple existence of Goedel's proof succeeds in demonstrating that even if we confine ourselves to the world of "relationships of ideas" there are limits to where our models might take us. That is a profound insight.
Men of towering intellect who saw these first two knowledge problems were less troubled by them than they were by the third. Von Karman studied quantum physics before turning to fluid mechanics and he was more troubled by the lack of "knowability" in the latter. Fluid systems can be described with an extremely high level of accuracy using the Navier-Stokes equations. These equations account for all of the forces that are exerted on fluids, and describe fluid motions completely.
They are taken to be correct. And in all known formal senses thay are so. But the first problem with fluid systems is that there are stable systems that admit multiple solutions. The classic is a toroidal convective cell. An annular space - it is shaped like a doughnut - is fitted with an inner wall that is heated and that rotates. If one makes the space tall enough, several doughnut shaped convection cells will form. The fluids in the apparatus self organize into doughnut shaped flows, flowing upward at the center, outward at the top, downward on the outside and inward on the bottom. But under identical conditions, the number of such doughnut shaped cells that form naturally and stably is indeterminate. In one experiment it may be 3. In another, using identical conditions it may be 4. There is no way of knowing in advance of doing the experiment. Nobody can predict exactly how many will form.
Here, we have a problem that we can describe with almost perfect mathematical accuracy using reliable mathematical tools, but we know that we cannot know the answer. We know what can happen. We know what cannot happen. We know what is likely to happen. And what is unlikely. But we cannot know what will happen.
More vexing still is the phenomenon of "turbulence." Fluids flowing along a solid boundary will flow smoothly for some distance. Then, wierd things happen. Patches of fluid lift up off the surface and form whirling dervishes above it. This is the transition from laminar to turbulent flow. One can predict where along the surface it is likely to start; but once it has started, there is no analytical way to predict the motion of the fluid. Nor is there a way to predict it with arbitrary accuracy using numerical models. One can predict average behavior, but not actual instantaneous behavior at a point.
This all sounds very abstract, but the equations that describe this turbulent flow are the ones that describe the motion of air in weather. Weather prediction relies on being able to predict the fluid motions of air. And it fails to do so in a robust way over long time intervals because of some of these problems.
The good news is that the effect is not huge. This means that one can predict weather with suitable accuracy for some number of days. And past that point one can still use methods that are more stable to produce guesses that are, on average, helpful. Still, as a practical matter, weather is not knowable far into the future. And there are very good reasons to believe that it cannot be. Small perturbations can lead to significant differences in a distant location at a distant point in time. The effect is sometimes known as the "butterfly effect" because in some sense, a butterfly's motions in Guatemala may produce changes of state that ripple through the system changing whether there is a thunderstorm in St Louis half a year or a year later.
The turbulent flow model is an example of a system that can be described mathematically with very high precision but whose mathematical formulation cannot be solved exactly, or analytically within mathematics. It can, however be solved approximately. But the form of the equation is such that small inaccuracies accumulate and cause the calculated solution to diverge from the real world phenomena it is predicting. This happens with a huge number of problems.
So we have seen how quantum physics predicted a theoretical limit to what we can measure about the physical world. We have seen how mathematics itself tells us that it cannot tell us everything. And we have seen how real world systems demonstrate behaviors that defy prediction using mathematical tools that are appropriately suited to describing them.
With these examples we have seriously bounded the grand vision of a completely predictable mechanical universe. We know that our knowledge is limited. But we cannot be sure from these examples whether this is a serious practical limitation or just an interesting thing to know. But we do know that the vision of the mechanical universe is a doomed artifact of history. We need to be able to adapt its strengths without falling prey to its weaknesses.
In this brief essay, we started out with the notion of certainty. We developed two propositions about which we can be pretty certain; one in the field "relationships of ideas" and one in the field of "fact." We gave reasons why propositions beyond these are more speculative. Then we briefly developed the conditions that gave rise to the Newtonian revolution, and touched on some of its dissatisfactions. Finally, we discussed some fundamental limits of knowledge. The hope is to develop a sense of what we mean when we say "know." We intend to develop the sense that when we say "know" we say it in a guarded and functional sense. We say it only to allow ourselves to act. And when we act, we must do so subject the awareness that we risk error, even in cases where we know quite a bit. The proper relationship between knowledge and action will be the subject of a subsequent piece.
Copyright: Stephen R. Brubaker, 2007. All Rights Reserved