Outer Limits of Reason (46 page)

One of the leading philosophers of science in the twentieth century, Karl Popper, believed that there is no real solution to the problem of induction. He claims that science does not work by scientists trying to verify laws made via induction. We don't just look at what happened before and generalize to make scientific laws. Rather, scientists make conjectures that can be shown to be wrong (i.e., they are falsifiable) and try to show that these conjectures are wrong. We will glimpse more of Popper's ideas shortly.

You might try to ignore the problem of induction and just say “it works!” After all, every time humans used induction in the past it worked, so induction must work. This pragmatic solution will not do. We are looking for a reason to believe in induction and you are saying that
it worked in the past so by induction it will always work.
But this is using circular reasoning: you are using induction to justify induction. David Hume summed up why this reasoning is not permitted: “It is impossible, therefore, that any arguments from experience can prove this resemblance of the past to the future, since all these arguments are founded on the supposition of that resemblance.”
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In other words, we are assuming that the universe stays the same in order to prove that the universe stays the same. That is not legal.

Another example that shows the seeming disconnect between reason and induction is called the
ravens paradox
or
Hempel's paradox
. Consider the following statement:

All ravens are black.
⁵

Every time you see a raven that is black you are confirming this statement. Assume this statement is true and consider an object that is not black. Because of the truth of the statement, this nonblack object is definitely not a raven. We are led to a statement that is logically equivalent:

All nonblack objects are nonravens.
⁶

These two statements say the same thing in different terms. If an observation confirms one of them then it automatically confirms the other one. On the other hand, if we find a nonblack raven then we have falsified both statements. Now, consider a green sweater. This object is not black and it is also not a raven so the green sweater is a confirmation of the second statement. Every time we see a green sweater—since green is not black and a sweater is not a raven—we are essentially confirming the second statement, which is equivalent to the original statement about black ravens. This is somewhat bothersome. How can it be that when we see a green sweater or a blue ball we are confirming the statement that all ravens are black?

We can even go further. When we see a green sweater we are also confirming the following statement:

All nonblue objects are nonravens.

The green (nonblue) object is a sweater (a nonraven.) This statement is equivalent to

All ravens are blue.

So by just looking at this sweater we are confirming that ravens are black and also blue. These are just two of the infinite number of statements confirmed by the observation of a green sweater. What is worse is that, as far as we know, ravens are not blue and these two statements are, in fact, false. How can green sweaters be so helpful in our ornithology observations?

There are various possible resolutions given for the ravens paradox. One solution is to simply agree with the conclusion of the paradox and say that when you observe a green sweater you are essentially confirming the statement that all ravens are black. However, one must think of it as a probabilistic confirmation. Say for a minute that there are a million ravens in this world. Every time you see a raven and it is black, you are one-millionth closer to confirming the statement. In contrast, there are vastly more nonblack objects in the world. When you see any of those objects you are getting closer to showing that the set of ravens is a subset of the set of black objects, as in
figure 8.1
. However, since there are so many nonblack objects in the universe, the confirmation of the statement is minuscule.

Regardless of which resolution one accepts for the problem of induction, you have to agree that inductive reasoning, the heart of the scientific endeavor, is beyond the bounds of reason. This is not to say that induction is false. Induction clearly works. We nevertheless have to be cognizant of the fact that it is not strictly a reasonable process.

Simplicity, Beauty, and Mathematics

Induction is not the only method that scientists use to find the laws of nature and to explain the inner workings of the universe. They also use other methodologies to select scientific theories. It is important to study these methodologies and their relationship with reason.

One of the oldest and most powerful tools that scientists keep in their toolbox is called
Occam's razor
or the
principle of parsimony
. William of Ockham (1285–1349) was an English philosopher who tells us not to assume more than we need to. That is, if we can explain something with fewer assumptions, then we should not assume more.
⁷
We should, metaphorically, use a razor to cut away any unneeded assumptions. There might be many different ways of explaining a phenomenon and we should always use the simpler explanation.

When Copernicus promoted the notion of a heliocentric world, it was not because of empirical evidence that implied the Earth moved. It certainly did not feel like the Earth was moving. Copernicus also did not emphasize the heliocentric worldview because it made better predictions about the universe. It did not (since he thought the planets traveled in a circle around the sun rather than in ellipses). Rather, his argument, which turned out to be correct, was that a heliocentric universe was simpler than a geocentric universe. There were no complicated epicycles in the heliocentric world as there were in the geocentric world.

A major problem with using Occam's razor is that it might not be correct. Choosing the simplest theory does not always work. For example, Copernicus thought that the planets follow a circular path around the sun, while Kepler showed that the path is actually an ellipse. Occam would have preferred the simpler circular path. Nevertheless, the laws of nature disregarded Occam's choice and follow the more complicated path. Another example of the failure of Occam's razor is that there are fewer equations in Newton's formulation of gravity than in Einstein's formulation of gravity. Fewer equations means it is simpler and so Occam's razor predicts it to be true. Nevertheless, physicists tell us that Einstein's theory is the right one to choose. Occam might counter that the number of equations is not the right measure of simplicity. Maybe he is right. Maybe not.

There are different types of simplicity. On the one hand, there is
simplicity of hypothesis
: given two theories, choose the theory that uses fewer presuppositions. Another type of simplicity is
simplicity of ontology
: given two theories, choose the one that assumes fewer physical objects exist. For example, given two theories, one that assumes the existence of ether and one that does not, choose the one that does not need the ether. Fewer physical objects are better.

These different types of simplicity sometimes work against each other. It could happen that a theory that has fewer hypotheses demands more ontology and vice versa. A case in point is the theory of a multiverse.
⁸
This is the belief that the universe that we see around us is one of many universes. The mathematics describing a multiverse is simpler than the mathematics of a universe. However, there are obviously more physical objects in a multiverse than in a universe. Everett's multiverse increases the number of objects, but the amount or type of math needed to describe this universe is much simpler.

Why does Occam's razor work, in general? Why should we always choose the simpler theory? Many people believe that the reason Occam's razor is so effective is that the universe is simple rather than complex. Appropriately, we should choose the simpler explanation. However, there really is no logical or rational reason to believe that the universe is simple. Maybe it is, in fact, complex. It certainly looks complex.

 

Another methodology that scientists use to find and select different theories is beauty. Scientists insist that a theory must, in some sense, be beautiful. The world-famous physicist Hermann Weyl is quoted as saying, “My work always tried to unite the true with the beautiful, but when I had to choose one or the other, I usually chose the beautiful.” Paul Dirac had similar sentiments: “It is more important to have beauty in one's equations than to have them fit experiment . . . It seems that if one is working from the point of view of getting beauty in one's equations, and if one has really a sound insight, one is on a sure line of progress.”
⁹

What exactly is beauty? The term is just as hard to define in science as in regular life. Some physicists have equated beauty with elegance,
¹⁰
which is an equally indefinable concept. Some have said that beauty is related to simplicity, which is basically what Occam's razor is all about. And still others have said that a theory is beautiful if it exhibits a lot of symmetry or harmony. There is much disagreement because no one has a sure-fire explanation for what exactly to look for or why this property works at picking good theories.

One of the problems with beauty is that beauty does not always work.
¹¹
The universe is not as pretty as scientists imagine.
¹²
Bertrand Russell, in his inimitable humorous way, put it like this: “Academic philosophers, ever since the time of Parmenides, have believed that the world is a unity. . . . The most fundamental of my intellectual beliefs is that this is rubbish. I think the universe is all spots and jumps, without unity, without continuity, without coherence or orderliness or any of the other properties that governesses love.”
¹³
As with simplicity, there is really no reason to believe that the universe is always beautiful and symmetric.

 

Yet another sieve or tool that scientists use to select physical theories is mathematics. They want their theories to be as mathematical as possible. A theory is not really acceptable to physicists until they see nice equations. Whereas in earlier times math was considered a language or a tool to help with physics, nowadays mathematics is the final arbiter of a theory.
¹⁴
Physicists have placed their faith in the symbols and equations of mathematics. If the math works, then the physics must be correct. The culmination of this faith in the role of mathematics in choosing physical theories is the current hot physical theory termed
string theory
. This theory posits the existence of very small strings that wiggle, shake, combine, and separate. It is shown by looking at the mathematics of these strings that all the known forces in the physical universe can be explained with strings. This is one of the only theories that not only describe the forces of quantum mechanics, but also gravity that plays a major role in general relativity. Yet another advantage of string theory is that it deals well with problematic infinities. In other physical theories that try to unite quantum mechanics and relativity, the equations somehow develop uncomfortable infinities. String theory does not have those crazy infinities. For all these reasons, many physicists are excited about the developments in string theory. This seems to be the long-sought Theory of Everything. There is, however, only one problem with string theory: there is not a shred of empirical evidence that it is true. While it makes great mathematics, there is no observation that we can make (at present) that shows that the world is, in fact, made out of little strings. That does not mean the theory is false. Absence of proof does not mean proof of absence. It could very well be that the world is made out of very small strings and string theory is correct. On the other hand, string theory could just be an elaborate fiction. For the time being, we simply do not know. Can we just follow the mathematics without having empirical evidence?
¹⁵

In what way are we justified in using simplicity, beauty, and mathematics as heuristics? One possible pragmatic justification is that these heuristics have worked in the past and we should just continue to use them in the future. After all, for the most part, science has progressed fairly well using these methods, and we should expect that it will progress just as well if we continue using them. Alas, this justification does not hold water. This argument uses induction and, as we have seen, induction is not reasonable. Another possible justification is to say that the reason why these heuristics work is that the universe we live in is, in fact, simple, beautiful, and mathematical. While it definitely seems that this is true, it is far from certain. For all we know, the universe might be complex, ugly, and nonmathematical.

As with induction, we must recognize that these methodologies for advancing science are essentially beyond reason. They work and we will continue to use them, but there is no logical reason to do so.