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hear “proof” used in casual conversation, where it’s closer to “sufficient evi-
24
dence that we believe something is true.”
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In a court of law, where precision is a goal but metaphysical certitude can
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never be attained, the flexible nature of proof is explicitly recognized by
27
invoking different standards depending on the case. In US civil courts,
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proving your case requires that a “preponderance of evidence” be on your
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side. In some administrative courts, “clear and convincing evidence” is re-
30
quired. And a criminal defendant is not considered to be proven guilty
31
unless the case has been demonstrated “beyond a reasonable doubt.”
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None of these would impress a mathematician in the slightest; their first
33
instinct would be to start thinking about the unreasonable doubts. Scien-
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tists, who have often taken a few math courses in their day, tend to have a
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similar idea about what constitutes proving something— and they know
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that it’s not what they do for a living. So if a scientist says “Human activity
02
is heating up the planet,” or “The universe is billions of years old,” or “The
03
Large Hadron Collider is not going to make a black hole that will gobble
04
up the Earth,” all you have to do is innocently ask whether they can really
05
prove it. Once they hesitate, you will have won a rhetorical victory. (You
06
will not have made the world a better place, but that’s your decision.)
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•
08
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Let’s see the distinction more explicitly. Here is a mathematical theorem:
10
There is no largest prime number. (Primes are whole numbers greater than
11
zero that can be evenly divided by only one and themselves.) And here is
12
a proof:
13
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Consider the set of all the prime numbers: {2, 3, 5, 7, 11,
15
13 . . . }. Suppose that there is a largest prime, p. Then there are
16
only a finite number of primes. Now consider the number X
17
that we obtain by multiplying together all of the primes from
18
our list, exactly once each, and adding 1 to the result. Then X is
19
clearly larger than any of the primes in our list. But it is not di-
20
visible by any of them, since dividing by any of them yields a
21
remainder 1. Therefore either X itself must be prime, or it must
22
be divisible by a prime number larger than any in our list. In
23
either case there must be a prime larger than p, which is a con-
24
tradiction. Therefore there is no largest prime.
25
26
Here is a scientific belief: Einstein’s theory of general relativity (GR)
27
accurately describes how gravity works, at least within the solar system, and
28
at least to an extremely high accuracy. And here is the argument for it:
29
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GR incorporates both the principle of relativity (posi-
31
tions and velocities can be measured only relative to other ob-
32
jects) and the principle of equivalence (in small regions of space,
33
gravity is indistinguishable from acceleration), both of which
34
have been tested to very high precision. Einstein’s equation of
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GR is the simplest possible non- trivial dynamic equation for the
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curvature of spacetime. GR explained a preexisting anomaly—
01
the precession of Mercury— and made several new predictions,
02
such as deflection of light by the sun and the gravitational red-
03
shift, which have successfully been measured. Higher- precision
04
tests from satellites continue to constrain any possible devia-
05
tions from GR. Without taking GR effects into account, the
06
Global Positioning System would rapidly go out of whack, and
07
by including GR it works like a charm. All of the known alter-
08
natives are more complicated than GR, or introduce new free
09
parameters that must be finely tuned with experiment to avoid
10
contradiction. Furthermore, we can start from the idea of mass-
11
less graviton particles that interact with all sources of energy,
12
and show that the only complete version of such a theory leads
13
to GR and Einstein’s equation. Although the theory is not suc-
14
cessfully incorporated into a quantum- mechanical framework,
15
quantum effects are expected to be negligible in present- day ex-
16
periments. In particular, quantum corrections to Einstein’s
17
equation are expected to be unobservably small.
18
19
None of the details here is important; what matters is the difference in
20
underlying method. The mathematical proof is airtight; it’s just a matter of
21
following the rules of logic. Given the assumptions, the conclusion neces-
22
sarily follows.
23
The argument in favor of believing general relativity— a scientific one,
24
not a mathematical one— is of an utterly different character. It’s abduction:
25
hypothesis testing, and accumulating better and better pieces of evidence,
26
seeking the best explanation of the phenomena. We throw a hypothesis out
27
there— gravity is the curvature of spacetime, governed by Einstein’s
28
equation— and then we try to test it or shoot it down, while simultaneously
29
searching for alternative hypotheses. If the tests get better and better, and
30
the search for alternatives doesn’t turn up any reasonable competitors, we
31
gradually start saying that the hypothesis is “right.” There is no sharp, bright
32
line that we cross, at which the idea goes from being “just a theory” to being
33
“proven correct.” When scientists observed the deflection of starlight
34
during a total eclipse of the sun, just as Einstein had predicted, that
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didn’t prove that he was right; it simply added to a growing pile of evidence
02
in his favor.
03
It is an intrinsic part of this process that the conclusion didn’t have to
04
turn out that way. We could certainly imagine a world in which some more
05
complicated theory than Einstein’s was the empirically correct theory of
06
gravity, or perhaps even one in which Newtonian gravity was correct. De-
07
ciding between the alternatives is not a matter of proving or disproving; its
08
a matter of accumulating evidence past the point where doubt is reasonable,
09
updating our credences along the way like good Bayesians. This is a funda-
10
mental difference between the kind of knowledge given to us by mathemat-
11
ics/ logic/ pure reason and the kind we get from science. The truths of math
12
and logic would be true in any possible world; the things science teaches us
13
are true about our world, but could have been false in some other one. Most
14
of the interesting things it is possible to know are not things we could ever
15
hope to “prove,” in the strong sense.
16
Even when we do believe a theory beyond reasonable doubt, we still
17
understand that it’s an approximation, likely (or certain) to break down
18
somewhere. There could very well be some new hidden field that we haven’t
19
yet detected that acts to slightly alter the true behavior of gravity from what
20
Einstein predicted. And there is certainly something going on when we get
21
down to quantum scales; nobody believes that general relativity is really the
22
final word on gravity. But none of that changes the essential truth that GR
23
is “right” in a certain well- defined regime. When we do hit upon an even
24
better understanding, the current one will be understood as a limiting case
25
of the more comprehensive picture.
26
•
27
28
These features of science— a form of knowledge gathering that we under-
29
stand relatively well— apply more broadly. The basic recognition is that
30
knowledge, like most things in life, is never perfect. Inspired by logically
31
rigorous proofs of geometry, Descartes wanted to establish an absolutely
32
secure, bedrock foundation for our understanding of the world. That’s just
33
not how knowledge of the world works.
34
Think about Bayes’s Theorem: the credence we place in an idea after re-
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ceiving some new information is the prior credence we started with for that
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idea, times the likelihood of obtaining that new information if our idea was
01
correct. At first glance, it seems easy to achieve perfect certainty: if the like-
02
lihood for a particular outcome is exactly zero according to some idea, and
03
we observe that outcome to occur, our credence in that idea gets set to zero.
04
But if we’re being scrupulous, we shouldn’t ever think that the likeli-
05
hood of observing a particular outcome is precisely zero. You might think
06
something like “In special relativity, particles never travel faster than light,
07
so I have zero credence that I would ever observe a faster- than- light particle
08
if special relativity were correct.” The problem is that your observations
09
could always be mistaken. Maybe you think you’ve seen a particle traveling
10
faster than light, but instead your apparatus was faulty. This is always pos-
11
sible, no matter how careful you are. We should always imagine that there
12
is some nonzero likelihood for absolutely any observation in absolutely any
13
theory.
14
As a result, our credences never go all the way to zero— nor precisely to
15
100 percent, since there are always competing possibilities. And it’s a good
16
thing that credences never reach these points of absolute certainty; if they
17
did, no amount of new evidence could ever change our minds. That’s no
18
way to go through life.
19
20
•
21
Not everyone agrees, of course. You may have heard that there is a long-
22
running dispute about the relationship between “faith” and “reason.” Some
23
argue that there is perfect harmony between them, and indeed there have
24
historically been many successful scientists and thinkers who have been
25
extremely devout. Others argue that the very notion of faith is inimical to
26
the practice of reason.
27
The discussion is complicated by the presence of multiple incompatible
28
notions of what is meant by “faith.” A dictionary might define it as “trust”
29
or “confidence” in a belief, but it will go on to offer meanings along the lines
30
of “belief without justification.” The New Testament (Hebrews 11:1) says
31
“Now faith is the substance of things hoped for, the evidence of things not
32
seen.” For many, faith is simply a firm conviction in their religious beliefs.
33
The word “faith” is highly charged, and this isn’t the place to argue over
34
how it should be defined. Let us merely note that sometimes faith is taken
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as something that is absolutely certain. Consider these statements from the
02
Catechism of the Catholic Church:
03
04
• The faithful receive with docility the teachings and directives
05
that their pastors give them in different forms.
06
• To obey (from the Latin ob-audire, to “hear or listen to”) in
07
faith is to submit freely to the word that has been heard, be-
08
cause its truth is guaranteed by God, who is Truth itself.
09
Abraham is the model of such obedience offered us by Sacred
10
Scripture. The Virgin Mary is its most perfect embodiment.
11
• Faith is certain. It is more certain than all human knowledge
12
because it is founded on the very word of God who can-
13
not lie.
14
15
It is this kind of stance— that there is a kind of knowledge that is certain,
16
which we should receive with docility, to which we should submit— that I’m
17
arguing against. There are no such kinds of knowledge. We can always be
18
mistaken, and one of the most important features of a successful strategy for
19
understanding the world is that it will constantly be testing its presupposi-
20
tions, admitting the possibility of error, and trying to do better. We all want
21
to live on a stable planet of belief, where the different parts of our worldview
22
fit together harmoniously; but we want to avoid being sucked into a black
23
hole of belief, where our convictions are so strong that we can never escape,
24
no matter what kind of new insight or information we obtain.
25
You will sometimes hear the claim that even science is based on a kind
26
of “faith,” for example, in the reliability of our experimental data or in the
27
existence of unbreakable physical laws. That is wrong. As part of the prac-
28
tice of science, we certainly make assumptions— our sense data is giving us
29
roughly reliable information about the world, simple explanations are pref-
30
erable to complex ones, we are not brains in vats, and so forth. But we don’t
31
have “faith” in those assumptions; they are components of our planets of
32
belief, but they are always subject to revision and improvement and even, if
33
necessary, outright rejection. By its nature, science needs to be completely
34
open to the actual operation of the world, and that means that we stand
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ready to discard any idea that is no longer useful, no matter how cherished
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and central it may once have seemed.
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•
01
02
Because we should have nonzero credences for ideas that might seem com-
03
pletely unlikely or even crazy, it becomes useful to distinguish between
04
“knowing” and “knowing with absolute logical certainty.” If our credence
05
for some proposition is 0.0000000001, we’re not absolutely certain it’s
06
wrong— but it’s okay to proceed as if we know it is.
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The Big Picture Page 22