Knowing how to lose
in science: when you don't like the result.
(Appendix 1) Reduction to absurdity and square root
of 2.
In the first appendix of original “Cosmos”
book, Carl Sagan writes about square root of 2 (√2), whose solution is an irrational
number, that is, it can not be expressed as a fraction of two integers.
Pythagoreans were the first to discover that √2 was an irrational number,
through a geometrical approximation. That argument was based in a reduction to
absurdity, which, as Carl Sagan explains, is a way of reasoning in which we
accept an asseveration as correct, we follow their consequences and, at the
end, we arrive to a contradiction, showing the falsehood of that asseveration.
In this case, Carl Sagan also employs the reduction to absurdity, although from
an arithmetic point of view, to reach the same conclusion Pythagoreans
achieved. However, we will use this addendum for a little different purpose.
In order to understand a scientific discovery, you
have sometimes to realize what the one who made it was thinking about. Let’s
start by Pythagoreans then: Pythagoras of Samos is a figure whose life (and the
history of the school he created) is surrounded by legend. It is believed that,
when Pythagoras was a student of already famous philosopher Anaximander, he met
ancient Thales of Miletus and, then, he started to travel. It is said he
reached Babylon as a war prisoner, he visited India, and (more likely) he
arrived to Egypt. There are few and not trustable texts about his life, but the
ones we have tell us that, in all those countries, Pythagoras contacted with
magicians and priests in order to learn their knowledge, and then he travelled
to Croton, Italy, where he created his school. The way he and his students
(Phythagoreans) lived was astounding: they were vegetarians, refused to wear
any clothes made with animals, frequently meditated and aspired to dwell in a
permanent state of purity. Tradition has attributed to Pythagorean school an
eminent mathematical orientation, and it is true they made great contributions
to this field (although it is difficult to distinguish which achievements were
obtained by Pythagoras, and which by other members of the school, as all
discoveries were automatically attributed to the founder), including Pythagoras
theorem about right triangle. However, Pythagoreans worked on many more issues:
it could seem a reductionism to name them only mathematicians, but no doubt
they would agree with that, as they believed universe was based on cyphers.
From this starting point, their other ideas, related to religion and
metaphysics, were derived: soul immortality; general conception of an unlimited
universe; deep relation between astronomy, music, and medicine, sustained by
mathematics and their most perfect representatives, that is, numbers.
Perfection of those abstractions made their spirits feel drunk; therefore, √2
discovery, and all what it implied, was a double sip of bitter reality.
Right triangle.
Credit Wikicommons (Public domain): https://es.wikipedia.org/wiki/Ra%C3%ADz_cuadrada_de_dos#/media/Archivo:Square_root_of_2_triangle.png
The man who caused all these troubles was Hippasus of
Metapontum, the discoverer of √2; professor of Heraclitus, some authors say he
found the relation between bronze disks thickness and the sound they produce
when they are beaten (an idea which is related with the knowledge, acquired by
Pythagoreans, that the length of the strings of an instrument determines its
sound, and is connected with the Pythagorean theory about harmony of celestial
spheres). Hippasus belonged to a branch of Pythagorean school called the
Acusmatici, a section which had not the same level as Mathematici, who were
directly supervised by Pythagoras and learnt the whole doctrine of the school
(Acusmatici could not show off that privilege). That was the first cause of
misunderstanding. Discovery of irrational numbers happened by chance: value of
√2 was a result which had been searched for a long time because it is the
measure of the diagonal of a square with a side of 1 unit length (believe it or
not, that measure has a few practical implications). Hippasus employed geometry
and reached a conclussion that shocked to everybody: √2
was an irrational number. Pythagoreans’ dream was so nice that, when they woke
up, the situation transformed into a nightmare.
What was the importance, for Pythagoreans, that √2 can
not be expressed as a fraction of two integers and has an approximate value of
1.4 (Pythagoras, if you are reading this, forgive us)? For people who is
obssesed by beauty of mathematics, with planets harmony, truths had to be
represented by perfect numbers such as integers or, at least, fractions of them.
In contrast, a cypher followed by a list of decimals which never finishes
(Greek philosophers never saw this; their numbers did not use those kind of
tools)? What kind of abomination was that? Because of that way of thinking, the
solution accomplished by Hippasus made Pythagoreans feel uncomfortable; legend
says Pythagoras refused to talk about irrational numbers. For years,
Pythagoreans avoided any discussion using √2 as if it was an integer. Anyhow,
it is supposed that, about this issue, they imposed an absolute secret:
irrational numbers existence should never be revealed to the world. According
to the myth, Hippasus did not follow this rule and, as a punishment, he was
murdered.
However, we must not trust on every legend related
with Pythagoreans. Of course, there are rumours about Hippasus, claiming he was
on a ship which sank in strange circumstances: around them, the dark shadow of
suicide has always floated, or maybe the hand of the members of the sect, who
would have pushed him into the sea. An insaner version draws Pythagoras himself
throwing Hippasus from the deck of the ship, ashamed not only by being inable
to refute the finding made by Hippasus, but also because the truth came from
the leader of a branch of the school which was considered a lower one; Hippasus
was his nemesis, the only figure who could confront him. Independently of the
speculations, the hypothetical secret was broken and, today, we know irrational
numbers exist: in fact, many of them (as π, or phi, also
known as golden ratio) are very important to understand proportions
either in geometrical bodies or living beings. The conclussion, however, would
not have made Pythagoras feel satisfied.
Now, we are going to advance a few centuries, to
arrive to a new but not too different kind of discussion. Along the 1920s
decade, Albert Einstein and Niels Bohr were involved in a debate which
redefined fundamental ideas in physics. Einstein had previously ellaborated his
Theory of Relativity, which, although sustained on extremely abstract concepts,
explained much of the real behaviour of universe, as if Pythagoreans had
proposed it after a lightning of inspiration. That rabbit from the hat continues
resisting most of the experimental assays which attempt to deny it.
Nevertheless, Niels Bohr (the man who created a version of the atom which
overcame the one from his professor, Ernsest Rutherford) said once a sentence
which Carl Sagan tries to reduce to absurd in the Appendix 1 in Cosmos:
‘The opposite to a any great idea is another great idea’. In this case,
the motto is completely true, because the opposite to theory of relativity is
quantum mechanics, sustained by findings of Max Planck, that proposed a
radically different vision of physics, based on probabilities and how much we
are (or not) able to measure. Einstein always rejected that theory -causing a
great disappointing to its creators, who were partially inspired by him-, and
quantum mechanics became the central point in the discusssions Einstein
maintained with Bohr. We have the famous sentence from Einstein (‘God does not
play dice’), but even most shocking was the chain of events when Einstein tried
to reduce quantum mechanics to absurd, claiming that, according to that theory,
two particles, once they had contacted, would never be disconnected again.
Surprisingly, quantum theory followers analyzed that nonsense and discovered it
was right, changing the foundations of our complete knowledge, once again.
Today, rivalry between theory of relativity and quantum mechanics goes on:
relativity perfectly explains what happens with great masses (as a renewed
version of spheres harmony), while quantum mechanics describes with
mathematical certainty the behaviour at subatomic level; the problem is both
visions do not agree in any other point. A few people try to create an Unified
Theory which summarizes basic rules of universe, from which fundamental forces
could be deduced. Despite that purpose, nowadays, the end of that journey
appears far away, as the search for a Golden unicorn, or the quest for a scientific
Holy Grail.
Einstein was disgusted with quantum theory because, as
a faithful determinst, he was uneasy with premises which gave so much relevance
to probability and the magnitudes measured by observer. However, he didn’t try,
as Pythagoras, to forbid the spreading of such a terrible idea. In any case, it
would be neither the first nor the last time that opposition from veteran
scientists prevents a new theory from being accepted. A recent study says that
certain scientific fields feel a blooming when famous experts in the area die,
indicating that the presence of such figures complicates the acknowledgement of
younger scientists who challenge stablished dogmas. Max Planck, the father of
quantum theory (the first one who became puzzled by it) said: New ideas in
science advance not because they are right, but because their enemies die. Maybe,
the best example can be found in the physics with another debate, the one
between Rutherford and Lord Kelvin. The last had made great contributions to
science, but he was very old and, from his powerful position, he refused to
admit those results which indicated the age of the Earth was higher than the
one he had suggested (under 20 million years old). So, when a young Rutherford
standed in one of the Royal British institutions, in front of a public composed
by 800 people, to expose how radiactivity supported the concept of an Earth of
hundreds of millions years, his only concern was the opinion from Lord Kelvin.
The key meeting had different stages: first, while Rutherford was speaking,
Lord Kelvin fell asleep. Later, he woke up and showed a happy smile -what could
you expect from a restorative nap? Then, Rutherford found the way to convince
the prominent figure: he mentioned a sentence from Lord Klevin himself which
stated that age of the Earth should just be a few million years, as far as
there was not a new source of heat which was able to explain data. Rutherford
claimed, therefore, that Lord Kelvin was the first to anticípate the existence
of that new source of heat (radioactivity) and, so, he was a co-participant in
the discovery. He was trying to flatter the vanity of an old man but, as it
usually happens in these cases, it worked, and Lord Kelvin nodded as a sign of
agreement. The obstacle had been erased, and not using reason and the behaviour
of facts, as science does, but employing psychology and the behaviour of
humans, as personal interactions do. Science, after all, has its deffects, and
those (as fails which are shared by almost every human activity) are mainly
caused because the ones who make science are humans too.
In this case, we have been talking about new findings
but, sometimes, the most appropiate contribution next generations can give to
the future is a novel perspective. For that, the best example we can think
about is an episode, attributed to many couples of master/student, the most
known of which is the one formed by Niels Bohr and Ernest Rutherford. According
to the story, Rutherford asked, in an exam, how to find the height of a
building with the help a barometer. The myth says Bohr ellaborated dozens of
answers (we can not reproduce all them by lack of space; one included offering
the barometer as a gift in exchange for the information), none of them,
although correct, corresponded to the canonical solution. You know this is
not the answer I’m looking for, was presumably affirmed by Rutherford, to
whom Bohr would have replied: Then, you should maybe reformulate the
question. Sometimes, the biggest favor we can do to science is to change
the matters to be discussed, in order to avoid that answers become rotten
before starting. It is the only way to get a different fruit from Newton’s
tree, hopefully more tasteful or, at least, different from an apple.
Science, in some aspects, means that: people who
arrive with new concepts which, for the previous generations, seemed
irreverent, disrespectful, even irrational. But which fit better with how
nature works or, alternatively, with how we work. Scientists come and go:
today, defenders of quantum mechanics and relativists secretly spy each other,
while Pythagoreans were expelled from Croton because they were not wise enough
to keep out from the irrational world of politics. In spite of those events,
science and everybody’s contributions get accumulated, and biographical
episodes and temporal disputes are forgotten to create what, with caution, we
can name as “the truth”; even if it is imperfect, uncomplete and defying. After
all, the main quality of a scientist is curiosity, and a scientist must be
always be prepared to get surprised. Carl Sagan knew that; he transmitted us a
portion of his joy when he got impressed by universe miracles. He gave us this
gift, and the best tribute or addendum our generation can make to Cosmos,
that wonderful legacy, is (from a different point of view) to continue
wondering. To be delighted, if required, with the perfection of the
imperfection.
Emilio Tejera Puente.
Doctor
(Ph.D.) en Bioquímica, Biología Molecular y Biomedicina.
Instituto Cajal
(CSIC), Madrid.
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