The Many Varieties of Discontinuous Experience
As we awake each morning, we have the unconscious
expectation that we will awake where we fell asleep and that things will be
about the same as they were when we fell asleep. One might call this an
implicit assumption that the future will resemble the past in significant ways.
Buildings will be located where they were the day before. The people we meet in
the coming day will be many of the same people with the same names that we have
met previously. Gravity will continue to cause things that are tossed up to
come back down. The square root of two will still be an irrational number. The
expansion of the irrational number π (the ratio of the
circumference of a circle to its diameter) will continue without a repeating
sequence of digits. Some people will pass away. Some babies will be born. Such
is life. Things seem to proceed pretty much as they have in the past. The
future will resemble the past. So it would seem.
Several questions come to an idle mind, or perhaps to one
that has encountered an anomaly. First, and perhaps most disturbing, is this:
What evidence is there on which to base such a fundamental belief as the future
resembling the past? Second, and perhaps more profound, is this: Could we come
to know anything about the natural world without that fundamental belief? To
make this adventure more interesting, here is a third: Where will these
questions lead?
When departing on an adventure into the unknown, it is
generally a good idea to stock up on things known and well understood. Pack a
lunch and take along a good supply of truth. Truth? What is truth? What is
truth, Naomi asked Ruth. The truth in that case showed itself in the following
– where Naomi went, Ruth followed. In that case, the truth showed itself in a
way that others could discern. Others could see that Ruth followed Naomi.
Perhaps others could not see Ruth’s inner states of mind but they could see
that Ruth followed Naomi. That truth, then, can be stated in this way: Ruth
followed Naomi. Others can confirm this. We cannot say for certain what was in
her mind or heart. Adding motives, attributing reasons, and including
explanations adds unnecessary weight to the pack and is likely to overburden
limited working memory as well. It is best to stay with the simple truth that
Ruth followed Naomi. After all, this may be a long journey, and simple truths
are light, easily packed, and will not overburden one.
On the lighter side, and as a point of clarification, let’s
consider the mathematical examples mentioned – the square root of two and the
expansion of π. It seems simpler to prove that the square root of 2
is irrational (not representable as a ratio of integers) than it is to prove
that the expansion of π does not yield a repeating sequence. I like
the proof of the irrationality of the square root of 2 that proceeds indirectly – that is by first assuming that the
square root of 2 is rational and deriving a contradiction based on that
assumption. Arriving at a contradiction, one can then reasonably conclude that
the initial assumption was mistaken since each step followed rules that are
known to preserve truth. Preserving truth is a good thing to do. It is a
precious commodity – not in short supply, but precious nonetheless. If one
assumes that the square root of 2 is rational, then it is equal to a ratio of two integers, say
a / b, that have no common factors, and also no known enemies.
One can then square both sides of the equation to get a new equation: 2 = a2
/ b2. Squaring both sides maintains the equality and preserves
truth – treat the numbers and factors on both sides of the equation the same
and you will preserve truth, just as treating the people on both sides of an
issue the same will preserve peace and stability (First question of conscience:
Has that ever worked?).
We can next multiply both sides of the
equation by b2 (a step that also maintains the equality and
preserves truth) to get 2b2= a2. Now we know a2
is an even number because multiplying a number by 2 always yields an
even number – at least in all of the cases I have tested (e.g., 2 x 1 = 2; 2 x
2 = 4; 2 x 3 = 6; 2 x 4 = 8; 2 x 5 = 10; 2 x 6 = oops, ran out of fingers). The
only way that a2 can be an even number, though, is for a
to also be even, since the square of every odd number I have tested turns out
to be another odd number (e.g., 3 x 3 = 9; 5 x 5 = 25; 7 x 7 = 49; and so on
until you wake up tomorrow). Since a is an even number, it can be
replaced by another number multiplied by 2, again preserving that
precious commodity of truth: a = 2c, for example. And of course
we can substitute equals for equals (my grandmother often did this in her
recipes when cooking), to get 2b2= 4c2, which can
then be simplified to b2= 2c2. Seems like we have
seen something like this before. Since b2 is the result of
multiplying a number by 2, it must be an even number, which then means that
b is also an even number. What? Our original assumption was that a and
b had no common factors, but if both of them are even (even if they are
enemies), then they do have a common factor (2, for those of you keeping
score). Therefore, we got off on the wrong foot and our initial assumption was
mistaken. The square root of 2 cannot be the ratio of two integers. This
indirect proof by contradiction is a form of what is called reductio ad
absurdum for those who still speak Latin. It is also a useful technique in
exploring non-mathematical ideas – accept for the time being what is postulated
and then show, step by step, that the result leads one to a contradiction or
something extremely implausible or unlikely.
So much for deciding if a number is rational
or irrational. Imagine what it would be like if were trying to decide if a
person was rational or not. And if that seems complicated, just imagine what it
would be like to try to prove that the expansion of π
does not involve a repeating sequence. Oh, by the way, that is your homework
assignment – prove that the expansion of π does not involve a
repeating sequence.
The point of that mathematical excursion was to suggest
something about truth and to suggest that one should pack lightly and take
along as few truths as possible so as to lighten one’s load when searching for
new truths or when trying to determine if an old one is really true. The old
one, in case you already forgot, is this: The future will resemble the past
in significant ways. It is not much like the one about the square root 2 being an irrational number. With a little
effort, we were able to wrap our sticky fingers (a result of eating buttered
popcorn while working) around that mathematical proof.
Before addressing that old truth about the future resembling
the past, perhaps we ought to look at a few more examples. What about the
example of expecting that things tossed up will again come down? How shall we
proceed with that expectation? Is it true? Well, being a fan of science
fiction, I decided to conduct a controlled experiment. I took 10 unshelled
peanuts outside on a sunny day – five for my left hand, and five for my right
hand. Being right-handed, I decided to make the left hand the control group and
proceeded to toss up a peanut with my right hand, one at a time. Each one fell
to the ground, as expected. Then I proceeded to toss up the remaining five
peanuts with my left hand. They did not go quite as high as those tossed up
with my right hand, and the first four fell to the ground, as expected. However,
a strange and quite unexpected thing happened when I tossed up the last peanut.
It did not come back down. I looked around for it everywhere – no peanut. Then
I saw a raven sitting on the branch of a nearby tree munching on that peanut. I
expected the raven to say “nevermore” but it never spoke, and I never spoke
about that failed experiment to anyone until now.
That was the year I failed physics, by the way, and decided
to pursue a career in education rather than one in science. Good decision, I
think. The point, though, is that unexpected things do happen. I have since
learned a bit more science from watching the Discovery Channel. For example, I
learned that Isaac Newton made many observations (not so unlike my peanut
experiment but more properly controlled) that resulted in three laws of motion,
roughly stated as follows: (1) a body in motion tends to stay in motion (Newton
never observed my body) while a body at rest tends to stay at rest (maybe he
secretly observed me); (2) the force acting on an object is equal to the mass
of the body multiplied by its acceleration (hmmm … this is now getting more
complicated); and (3) for every action there is an equal but opposite reaction
(noticeable in many political debates). Isaac was not satisfied with these
observations about motion and conducted the famous falling apple studies,
noticing that apples falling from trees of different heights all fell to the
earth, concluding that there must be a force acting on the apple, otherwise it
would stay resting in the tree. He called that force gravity and argued that it
applied everywhere, all the time, such that it was directly proportional to the
masses of the objects involved (e.g., the apple and the earth) and inversely
proportional to the square of the distances between them (this accounts for a
difference in the force involved with apples falling from short and tall apple
trees).
I mention this case of falling apples for two reasons: (a)
most of us have been hit on the head by a falling object at one time or another,
and (b) the phrase ‘all the time’ jumped out and fell on my head. All the time?
How could he know that? Easily enough. He watched 100 or more apples fall – a
lot more than my ten peanuts. Every time in the past, the apple fell toward the
earth and not upwards or sideways (except once during a tornado). In the past,
they all fell down. In the past, the force of gravity has accounted for all of
those instances of falling apples and peanuts. In the past. But how is that
evidence for the future? What must one assume to project past behaviors into
the future? That is a second question of conscience – seems less serious than
the first one, but it is much more fundamental.
Being a visual learner, I drew myself a picture to see how
this argument must work, since it seems to be working for a lot of other
people. Here is what I drew in the sand (the observations are the premises or
evidence offered in support of a conclusion):
Observation 1: Apple1
fell earthward when dropped from a tall building.
Observation 2: Apple2
fell earthward when dropped from a tall building.
Observation 3: Apple3
fell earthward when dropped from a tall building.
Observation n: Applen
(as many apples as you like can be inserted here).
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Conclusion:
The next apple I drop will fall earthward.
Now, having graduated from high school, I knew that there
were two kinds of logical arguments: deductive (those which are intended to
establish their conclusions with certainty) and non-deductive (those which are
intended to establish their conclusions with less than certainty – i.e.,
probability). Suppose that drawing in the sand is intended to establish the
certainty of the conclusion. In that case, it is clearly deficient. It is
missing an important premise to be added to the observations that were recorded
– namely, the claim that all future observations of falling apples will behave
in accordance with those already observed – i.e., the claim that the future
will resemble the past in significant ways. Now we are getting somewhere.
Somewhere dark, that is. How was that missing premise established? Was it the
conclusion of a prior unstated deductive argument? If so, what were the
premises of (evidence offered in) that prior argument? Rather than waste more of
your time on tedious homework, let us simply admit that as a deductive
argument, the above argument sketched in the sand is deficient, even with the
added unsupported premise.
Our other choice is to treat it as a non-deductive argument,
one that only intends to establish its conclusion with probability. This seems
like a more reasonable path to follow. Most of the time, the apples and peanuts
have fallen to earth in the past, except for the relatively rare occurrences of
a hungry raven or an angry tornado. Is it not reasonable, then, to expect the
next apple or peanut to fall to earth as those before have fallen? Indeed, that
seems to be what people do expect. But is it reasonable? There have been
exceptions, after all, however rare. The rate of exceptions could change at any
time. More ravens might develop cravings for peanuts. Tornados might occur with
increasing frequency and intensity. Is the future as certain as we imagine?
The point here is simple. We live as if there will be no
more exceptions or no increase in exceptions. However, we have no evidence on
which to base those beliefs. It is convenient to believe that the future will
resemble the past in significant ways and ignore the possibility that the
future may suddenly become quite different. Creatures of convenience, we are,
as much as we are intermittently creatures of reason and evidence.
The unexpected will most certainly happen. Okay, maybe that
is an exaggeration of the kind I have been indirectly attacking. Unexpected
things may well happen that will upset our lives, our thinking, our perception
of ourselves and our worlds. As Oets Kolk Bouwsma remarked in an unpublished
journal entry: “The world may gladden your heart; it will surely make you cry.”
Perhaps the most remarkable thing of all is what Wittgenstein commented on in
the Tractatus Logico-Philosphicus, remarks 6.43 and 6.44: “The
world of the happy man is a different one from that of the unhappy man” and “It
is not how things are in the world that is mystical [mysterious] but that
it exists.” He also said in remark 7 that what we cannot speak about sensibly
we must pass over in silence. Few politicians have heeded that advice – most of
us, as well.
Indeed, there is much that we cannot know. According to
ancient Jewish sources, Job discovered the inherent limitation of being human
and having limited knowledge in a conversation with G-D speaking from a
whirlwind – a tornado. While the lesson of limited knowledge, especially with
regard to the future, dates back to ancient times, few act as if they really
believe it. It is convenient and comforting to believe that we know more than
we could possibly know. We simply do not know that the future will resemble the
past in significant ways. You may trip on a butterfly and fall into a rabbit
hole. You may swallow a grasshopper whole and lose your voice. Maybe you will
discover a repeating sequence in the expansion of π. While those
may seem unlikely, they are at least as likely as hearing a voice emanate from
a whirlwind.
Bertrand Russell, a British mathematician and philosopher,
called things unseen and unverified bare possibilities. Examples for
Russell include there being a direct causal connection between a physical state
and a mental state, or the voice heard from the whirlwind and the existence of
G-D. Bare possibilities can nonetheless be motivating. Abraham reported hearing
a voice telling him to get up and move to a far place, and that was followed by
Abraham getting up and moving to a far place. Is that another form of truth
showing itself in the following? What truth shall we take? Remember to pack
light.
In any case, while there are many, many bare possibilities,
there are also both apparent possibilities and practical probabilities. Would
not reason incline one to follow those practical probabilities? There is an
apparent possibility of winning a lot of money in the lottery. Then there is a
practical probability of saving one’s money to purchase a wedding ring or a
bottle of wine (always good to have choices). Many people opt for both – mixing
in a few bare possibilities with some apparent possibilities along with a dose
of practical probability. It could be an interesting exercise to see which
things people place in each of those categories. But then I am not a scientist
– just someone tripping on a butterfly and swallowing the occasional
grasshopper.
Strange and unexpected things do happen. They have happened
to me. Some have gladdened my heart. Some have made me cry. Altogether, they
have made me realize that I know far less than I am typically inclined to
believe. The future may well not resemble the past in significant ways. How so?
What has taken you by surprise and changed your life, your perceptions, your
beliefs? My guess is that this has happened to you more than once. As
Heraclitus reportedly said, all things flow. The nature of all things is
change. One cannot even step into the same river once, if that is so.
Try it. Get your feet wet for a change.
J. Michael Spector
March 2014
