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What Does Science Explain? Part 3 – The Mythos of Objectivity

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In parts one and two of my series “What Does Science Explain?,” I contrasted the metaphysics of the medieval world with the metaphysics of modern science. The metaphysics of modern science, developed by Kepler, Galileo, Descartes, and Newton, asserted that the only true reality was mathematics and the shape, motion, and solidity of objects, all else being subjective sensations existing solely within the human mind. I pointed out that the new scientific view was valuable in developing excellent predictive models, but that scientists made a mistake in elevating a method into a metaphysics, and that the limitations of the metaphysics of modern science called for a rethinking of the modern scientific worldview. (See The Metaphysical Foundations of Modern Science by Edwin Arthur Burtt.)

Early scientists rejected the medieval worldview that saw human beings as the center and summit of creation, and this rejection was correct with regard to astronomical observations of the position and movement of the earth. But the complete rejection of medieval metaphysics with regard to the role of humanity in the universe led to a strange division between theory and practice in science that endures to this day. The value and prestige of science rests in good part on its technological achievements in improving human life. But technology has a two-sided nature, a destructive side as well as a creative side. Aspects of this destructive side include automatic weaponry, missiles, conventional explosives, nuclear weapons, biological weapons, dangerous methods of climate engineering, perhaps even a threat from artificial intelligence. Even granting the necessity of the tools of violence for deterrence and self-defense, there remains the question of whether this destructive technology is going too far and slipping out of our control. So far the benefits of good technology have outweighed the hazards of destructive technology, but what research guidance is offered to scientists when human beings are removed from their high place in the universe and human values are separated from the “real” world of impersonal objects?

Consider the following question: Why do medical scientists focus their research on the treatment and cure of illness in humans rather than the treatment and cure of illness in cockroaches or lizards? This may seem like a silly question, but there’s no purely objective, scientific reason to prefer one course of research over another; the metaphysics of modern science has already disregarded the medieval view that humans have a privileged status in the universe. One could respond by arguing that human beings have a common self-interest in advancing human health through medical research, and this self-interest is enough. But what is the scientific justification for the pursuit of self-interest, which is not objective anyway? Without a recognition of the superior value of human life, medical science has no research guidance.

Or consider this: right now, astronomers are developing and employing advanced technologies to detect other worlds in the galaxy that may have life. The question of life on other planets has long interested astronomers, but it was impossible with older technologies to adequately search for life. It would be safe to say that the discovery of life on another planet would be a landmark development in science, and the discovery of intelligent life on another planet would be an astonishing development. The first scientist who discovered a world with intelligent life would surely win awards and fame. And yet, we already have intelligent life on earth and the metaphysics of modern science devalues it. In practice, of course, most scientists do value human life; the point is, the metaphysics behind science doesn’t, leaving scientists at a loss for providing an intellectual justification for a research program that protects and advances human life.

A second limitation of modern science’s metaphysics, closely related to the first, is its disregard of certain human sensations in acquiring knowledge. Early scientists promoted the view that only the “primary qualities” of mathematics, shape, size, and motion were real, while the “secondary qualities” of color, taste, smell, and sound existed only in the mind. This distinction between primary and secondary qualities was criticized at the time by philosophers such as George Berkeley, a bishop of the Anglican Church. Berkeley argued that the distinction between primary and secondary qualities was false and that even size, shape, and motion were relative to the perceptions and judgment of observers. Berkeley also opposed Isaac Newton’s theory that space and time were absolute entities, arguing instead that these were ideas rooted in human sensations. But Berkeley was disregarded by scientists, largely because Newton offered predictive models of great value.

Three hundred years later, Isaac Newton’s models retain their great value and are still widely used — but it is worth noting that Berkeley’s metaphysics has actually proved superior in many respects to Newton’s metaphysics.

Consider the nature of mathematics. For many centuries mathematicians believed that mathematical objects were objectively real and certain and that Euclidean geometry was the one true geometry. However, the discovery of non-Euclidean geometries in the nineteenth century shook this assumption, and mathematicians had to reconcile themselves to the fact that it was possible to create multiple geometries of equal validity. There were differences between the geometries in terms of their simplicity and their ability to solve particular problems, but no one geometry was more “real” than the others.

If you think about it, this should not be surprising. The basic objects of geometry — points, lines, and planes — aren’t floating around in space waiting for you to take note of them. They are concepts, creations of the human brain. We may see particular objects that resemble points, lines, and planes, but space itself has no visible content; we have to add content to it.  And we have a choice in what content to use. It is possible to create a geometry in which all lines are straight or all lines are curved; in which some lines are parallel or no lines are parallel;  or in which lines are parallel over a finite distance but eventually meet at some infinitely great distance. It is also possible to create a geometry with axioms that assume no lines, only points; or a geometry that assumes “regions” rather than points. So the notion that mathematics is a “primary quality” that exists within objects independent of human minds is a myth. (For more on the imaginary qualities of mathematics, see my previous posts here and here.)

But aside from the discovery of multiple mathematical systems, what has really killed the artificial distinction between “primary qualities,” allegedly objective, and “secondary qualities,” allegedly subjective, is modern science itself, particularly in the findings of relativity theory and quantum mechanics.

According to relativity theory, there is no single, objectively real size, shape, or motion of objects — these qualities are all relative to an observer in a particular reference frame (say, at the same location on earth, in the same vehicle, or in the same rocket ship). Contrary to some excessive and simplistic views, relativity theory does NOT mean that any and all opinions are equally valid. In fact, all observers within the same reference frame should be seeing the same thing and their measurements should match. But observers in different reference frames may have radically different measurements of the size, shape, and motion of an object, and there is no one single reference frame that is privileged — they are all equally valid.

Consider the question of motion. How fast are you moving right now? Relative to your computer or chair, you are probably still. But the earth is rotating at 1040 miles per hour, so relative to an observer on the moon, you would be moving at that speed — adjusting for the fact that the moon is also orbiting around the earth at 2288 miles per hour. But also note that the earth is orbiting the sun at 66,000 miles per hour, our solar system is orbiting the galaxy at 52,000 miles per hour, and our galaxy is moving at 1,200,000 miles per hour; so from the standpoint of an observer in another galaxy you are moving at a fantastically fast speed in a series of crazy looping motions. Isaac Newton argued that there was an absolute position in space by which your true, objective speed could be measured. But Einstein dismissed that view, and the scientific consensus today is that Einstein was right — the answer to the question of how fast you are moving is relative to the location and speed of the observer.

The relativity of motion was anticipated by the aforementioned George Berkeley as early as the eighteenth century, in his Treatise Concerning the Principles of Human Knowledge (paragraphs 112-16). Berkeley’s work was subsequently read by the physicist Ernest Mach, who subsequently influenced Einstein.

Relativity theory also tells us that there is no absolute size and shape, that these also vary according to the frame of reference of an observer in relation to what is observed. An object moving at very fast speeds relative to an observer will be shortened in length, which also affects its shape. (See the examples here and here.) What is the “real” size and shape of the object? There is none — you have to specify the reference frame in order to get an answer. Professor Richard Wolfson, a physicist at Middlebury College who has a great lecture series on relativity theory, explains what happens at very fast speeds:

An example in which length contraction is important is the Stanford Linear Accelerator, which is 2 miles long as measured on Earth, but only about 3 feet long to the electrons moving down the accelerator at 0.9999995c [nearly the speed of light]. . . . [Is] the length of the Stanford Linear Accelerator ‘really’ 2 miles? No! To claim so is to give special status to one frame of reference, and that is precisely what relativity precludes. (Course Guidebook to Einstein’s Relativity and the Quantum Revolution, Lecture 10.)

In fact, from the perspective of a light particle (a photon), there is infinite length contraction — there is no distance and the entire universe looks like a point!

The final nail in the coffin of the metaphysics of modern science is surely the weird world of quantum physics. According to quantum physics, particles at the subatomic level do not occupy only one position at a particular moment of time but can exist in multiple positions at the same time — only when the subatomic particles are observed do the various possibilities “collapse” into a single outcome. This oddity led to the paradoxical thought experiment known as “Schrodinger’s Cat” (video here). The importance of the “observer effect” to modern physics is so great that some physicists, such as the late physicist John Wheeler, believed that human observation actually plays a role in shaping the very reality of the universe! Stephen Hawking holds a similar view, arguing that our observation “collapses” multiple possibilities into a single history of the universe: “We create history by our observation, rather than history creating us.” (See The Grand Design, pp. 82-83, 139-41.) There are serious disputes among scientists about whether uncertainties at the subatomic level really justify the multiverse theories of Wheeler and Hawking, but that is another story.

Nevertheless, despite the obsolescence of the metaphysical premises of modern science, when scientists talk about the methods of science, they still distinguish between the reality of objects and the unreality of what exists in the mind, and emphasize the importance of being objective at all times. Why is that? Why do scientists still use a metaphysics developed centuries ago by Kepler, Galileo, and Newton? I think this practice persists largely because the growth of knowledge since these early thinkers has led to overspecialization — if one is interested in science, one pursues a degree in chemistry, biology, or physics; if one is interested in metaphysics, one pursues a degree in philosophy. Scientists generally aren’t interested in or can’t understand what philosophers have to say, and philosophers have the same view of scientists. So science carries on with a metaphysics that is hundreds of years old and obsolete.

It’s true that the idea of objectivity was developed in response to the very real problem of the uncertainty of human sense impressions and the fallibility of the conclusions our minds draw in response to those sense impressions. Sometimes we think we see something, but we don’t. People make mistakes, they may see mirages; in extreme cases, they may hallucinate. Or we see the same thing but have different interpretations. Early scientists tried to solve this problem by separating human senses and the human mind from the “real” world of objects. But this view was philosophically dubious to begin with and has been refuted by science itself. So how do we resolve the problem of mistaken and differing perceptions and interpretations?

Well, we supplement our limited senses and minds with the senses and minds of other human beings. We gather together, we learn what others have perceived and concluded, we engage in dialogue and debate, we conduct repeated observations and check our results with the results of others. If we come to an agreement, then we have a tentative conclusion; if we don’t agree, more observation, testing, and dialogue is required to develop a picture that resolves the competing claims. In some cases we may simply end up with an explanation that accounts for why we come up with different conclusions — perhaps we are in different locations, moving at different speeds, or there is something about our sensory apparatus that causes us to sense differently. (There is an extensive literature in science about why people see colors differently due to the nature of the eye and brain.)

Central to the whole process of science is a common effort — but there is also the necessity of subduing one’s ego, acknowledging that not only are there other people smarter than we are, but that the collective efforts of even less-smart people are greater than our own individual efforts. Subduing one’s ego is also required in order to prepare for the necessity of changing one’s mind in response to new evidence and arguments. Ultimately, the search for knowledge is a social and moral enterprise. But we are not going to succeed in that endeavor by positing a reality separate from human beings and composed only of objects. (Next: Part 4)

Uncertainty, Debate, and Imprecision in Mathematics

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If you remember anything about the mathematics courses you took in high school, it is that mathematics is the one subject in which there is absolute certainty and precision in all its answers. Unlike history, social science, and the humanities, which offer a variety of interpretations of subject matter, mathematics is unified and absolute.  Two plus two equals four and that is that. If you answer a math problem wrong, there is no sense in arguing a different interpretation with the teacher. Even the “hard sciences,” such as physics, may revise long-established conclusions, as new evidence comes in and new theories are developed. But mathematical truths are seemingly forever. Or are they?

You might not know it, but there has been a revolution in the human understanding of mathematics in the past 150 years that has undermined the belief that mathematics holds the key to absolute truth about the nature of the universe. Even as mathematical knowledge has increased, uncertainty has also increased, and different types of mathematics have been created that have different premises and are incompatible with each other. The value of mathematics remains clear. Mathematics increases our understanding, and science would not be possible without it. But the status of mathematics as a source of precise and infallible truth about reality is less clear.

For over 2000 years, the geometrical conclusions of the Greek mathematician Euclid were regarded as the most certain type of knowledge that could be obtained. Beginning with a small number of axioms, Euclid developed a system of geometry that was astonishing in breadth. The conclusions of Euclid’s geometry were regarded as absolutely certain, being derived from axioms that were “self-evident.”  Indeed, if one begins with “self-evident” truths and derives conclusions from those truths in a logical and verifiable manner, then one’s conclusions must also be undoubtedly true.

However, in the nineteenth century, these truths were undermined by the discovery of new geometries based on different axioms — the so-called “non-Euclidean geometries.” The conclusions of geometry were no longer absolute, but relative to the axioms that one chose. This became something of a problem for the concept of mathematical “proof.” If one can build different systems of mathematics based on different axioms, then “proof” only means that one’s conclusions are derivable from one’s axioms, not that one’s conclusions are absolutely true.

If you peruse the literature of mathematics on the definition of “axiom,” you will see what I mean. Many authors include the traditional definition of an axiom as a “self-evident truth.” But others define an axiom as a “definition” or “assumption,” seemingly as an acceptable alternative to “self-evident truth.” Surely there is a big difference between an “assumption,” a “self-evident truth,” and a “definition,” no? This confusing medley of definitions of “axiom” is the result of the nineteenth century discovery of non-Euclidean geometries. The issue has not been fully cleared up by mathematicians, but the Wikipedia entry on “axiom” probably represents the consensus of most mathematicians, when it states: “No explicit view regarding the absolute truth of axioms is ever taken in the context of modern mathematics, as such a thing is considered to be irrelevant.”  (!)

In reaction to the new uncertainty, mathematicians responded by searching for new foundations for mathematics, in the hopes of finding a set of axioms that would establish once and for all the certainty of mathematics. The “Foundations of Mathematics” movement, as it came to be called, ultimately failed. One of the leaders of the foundations movement, the great mathematician Bertrand Russell, declared late in life:

I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new kind of mathematics, with more solid foundations than those that had hitherto been thought secure. But as the work proceeded, I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was no more secure than the elephant, and after some twenty years of arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable. (The Autobiography of Bertrand Russell)

Today, there are a variety of mathematical systems based on a variety of assumptions, and no one yet has succeeded in reconciling all the systems into one, fundamental, true system of mathematics. In fact, you wouldn’t know it from high school math, but some topics in mathematics have led to sharp divisions and debates among mathematicians. And most of these debates have never really been resolved — mathematicians have simply grown to tolerate the existence of different mathematical systems in the same way that ancient pagans accepted the existence of multiple gods.

Some of the most contentious issues in mathematics have revolved around the concept of infinity. In the nineteenth century, the mathematician Georg Cantor developed a theory about different sizes of infinite sets, but his arguments immediately attracted criticism from fellow mathematicians and remain controversial to this day. The central problem is that measuring infinity, assigning a quantity to infinity, is inherently an endless process. Once you think you have measured infinity, you simply add a one to it, and you have something greater than infinity — which means your original infinity was not truly infinite. Henri Poincare, one of the greatest mathematicians in history, rejected Cantor’s theory, noting: “Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already.”  Stephen Simpson, a mathematician at Pennsylvania University likewise argues “What truly infinite objects exist in the real world?” Objections to Cantor’s theory of infinity led to the emergence of new mathematical schools of thought such as finitism and intuitionism, which rejected the legitimacy of infinite mathematical objects.

Cantor focused his mental energies on concepts of the infinitely large, but another idea in mathematics was also controversial — that of the infinitely small, the “infinitesimal.” To give you an idea of how controversial the infinitesimal has been, I note that Cantor himself rejected the existence of infinitesimals! In Cantor’s view, the concept of something being infinitely small was inherently contradictory — if something is small, then it is inherently finite! And yet, infinitesimals have been used by mathematicians for hundreds of years. The infinitesimal was used by Leibniz in his version of calculus, and it is used today in the field of mathematics known as “non-standard analysis.” There is still no consensus among mathematicians today about the existence or legitimacy of infinitesimals, but infinitesimals, like imaginary numbers, seem to be useful in calculations, and as long as it works, mathematicians are willing to tolerate them, albeit not without some criticism.

The existence of different types of mathematical systems leads to some strange and contradictory answers to some of the simplest questions in mathematics. In school, you were probably taught that parallel lines never meet. That is true in Euclidean geometry, but not in hyperbolic geometry. In projective geometry, parallel lines meet at infinity!

Or consider the infinite decimal 0.9999 . . .  Is this infinite decimal equal to 1? The common sense answer that students usually give is “of course not.” But most mathematicians argue that both numbers are equivalent! Their logic is as follows: in the system of “real numbers,” there is no number between 0.999. . . and 1. Therefore, if you subtract 0.999. . .  from 1, the result is zero. And that means both numbers are the same!

However, in the system of numbers known as “hyperreals,” a system which includes infinitesimals, there exists an infinitesimal number between 0.999. . .  and 1. So under this system, 0.999. . .  and 1 are NOT the same! (A great explanation of this paradox is here.) So which system of numbers is the correct one? There is no consensus among mathematicians. But there is a great joke:

How many mathematicians does it take to screw in a light bulb?

0.999 . . .

The invention of computers has led to the creation of a new system of mathematics known as “floating point arithmetic.” This was necessary because, for all of their amazing capabilities, computers do not have enough memory or processing capability to precisely deal with all of the real numbers. To truly depict an infinite decimal, a computer would need an infinite amount of memory. So floating point arithmetic deals with this problem by using a degree of approximation.

One of the odd characteristics of the standard version of floating point arithmetic is that there is not one zero, but two zeros: a positive zero and a negative zero. What’s that you say? There’s no such thing as positive zero and negative zero? Well, not in the number system you were taught, but these numbers do exist in floating point arithmetic. And you can use them to divide by zero, which is something else I bet you thought you couldn’t do.  One divided by positive zero equals positive infinity, while one divided by negative zero equals negative infinity!

What the history of mathematics indicates is that the world is not converging toward one, true system of mathematics, but creating multiple, incompatible systems of mathematics, each of which has its own logic. If you think of mathematics as a set of tools for understanding reality, rather than reality itself, this makes sense. You want a variety of tools to do different things. Sometimes you need a hammer, sometimes you need a socket wrench, sometimes you need a Phillips screwdriver, etc. The only true test of a tool is how useful it is — a single tool that tried to do everything would be unhelpful.

You probably didn’t know about most of the issues in mathematics I have just mentioned, because they are usually not taught, either at the elementary school level, the high school level, or even college. Mathematics education consists largely of being taught the right way to perform a calculation, and then doing a variety of these calculations over and over and over. . . .

But why is that? Why is mathematics education just about learning to calculate, and not discussing controversies? I can think of several reasons.

One reason may be that most people who go into mathematics tend to have a desire for greater certainty. They don’t like uncertainty and imprecise answers, so they learn math, avoid mathematical controversies or ignore them, and then teach students a mathematics without uncertainty. I recall my college mathematics instructor declaring to class one day that she went into mathematics precisely because it offered sure answers. My teacher certainly had that much in common with Bertrand Russell (quoted above).

Another reason surely is that there is a large element of indoctrination in education generally, and airing mathematical controversies among students might have the effect of undermining authority. It is true that students can discuss controversies in the social sciences and humanities, but that’s because we live in a democratic society in which there are a variety of views on social issues, and no one group has the power to impose a single view on the classroom. But even a democratic society is not interested in teaching controversies in mathematics — it’s interested in creating good workers for the economy. We need people who can make change, draw up a budget, and measure things, not people who challenge widely-accepted beliefs.

This utilitarian view of mathematics education seems to be universal, shared by democratic and totalitarian governments alike. Forcing students to perform endless calculations without allowing them to ask “why” is a great way to bore children and make them hate math, but at least they’ll be obedient citizens.