Throughout most of our evolution as a species, our knowledge has been empirically obtained. That is, we observed our world and drew conclusions based on what we saw, heard and felt. We believed the earth was flat because we didn’t observe any curvature. We believed that the earth was the center of the universe because our ancient myths and religions had told us that we were important and everything moved around us. However, many millennia ago inquiring minds in various cultures began to question prevailing beliefs using deductive reasoning. The ancient Egyptians had discovered fractions and used them to distribute inherited shares of livestock. Then, around 600 BCE, the ancient Greeks used deduction to prove that some lengths cannot be expressed as whole numbers or fractions. These numbers like √2 and pi were called “irrational numbers.” In the celebrated Socratic dialogue with Meno, Plato showed how we can use deduction to discover things we never knew before. This was the new awakening of a human mental capability called “deductive reasoning” that is perhaps the greatest gift to human intelligence. Let’s look at a few examples.
The Deduction that the Earth is not Flat
The ancient Greeks believed that the earth was flat and that the god Helios drove a fiery chariot across the sky each day, from east to west, bringing daylight to the world. At night, he would journey back to the east under the ocean to begin his daily cycle again. However, the Greek mathematician, Eratosthenes, around 225 BC, used deduction to prove that the earth is a spheroid. By measuring at noon the length of the shadow cast by the sun at locations with the same longitude and different latitudes, he calculated the circumference of the earth to within 1% of its actual value.
In his letter, The Sand Reckoner, Archimedes of Syracuse revealed that Aristarchus had recognized that our solar system is heliocentric (anticipating Copernicus by almost two millennia):
Aristarchus has written a book in which he says … that the stars and the sun don’t move, and that the earth revolves about the sun and that the path of the orbit is circular.
Newton later used mathematical deduction to prove that the orbits are elliptical.
The Deduction that Genetically Rare Characteristics can survive for Generations
In the early 20th century, the swamping argument had asserted that any newly emerging hereditary trait would become diluted in successive generations by a blending of the parents’ traits in the offspring and would eventually die out. It would require a mathematical model to determine whether discrete hereditary elements could survive through several generations.
After a cricket match with the famous number theorist G. H. Hardy, the British geneticist Reginald Punnett described the swamping argument that was confounding their attempts to explain heredity using the Mendelian model of discrete alleles that transmit traits. Hardy immersed himself in the problem, and in 1908 he submitted his paper to the journal Science, stating that under a particular set of reasonable assumptions the proportions of all alleles in a population will stabilize and remain constant from the second generation onward. In what some scientists interpreted as a hint of condescension, he submitted his paper to that journal as a letter to the editor:
I am reluctant to intrude in a discussion concerning matters of which I have no expert knowledge, and I should have expected the very simple point which I wish to make to have been familiar to biologists. However, some remarks of Mr. Udny Yule, to which Mr. R. C. Punnett has called my attention, suggest that it may still be worth making…
The deductive proof the Hardy submitted and was later named the Hardy-Weinberg principle validated Mendel’s “particulate” model of inheritance, showing mathematically that rare alleles retain their proportion over successive generations and are not wiped out through blending. Mendel’s model, underpinned by the theorems of mathematical probability based on deductive thinking, paved the way for the study of heredity and genetics.
The EPR Paradox
In 1935, the year that Schrödinger introduced the gedanken experiment involving the superposition of the cat’s states of existence, Einstein and two colleagues, Boris Podolsky and Nathan Rosen proposed a gedanken experiment, now known acronymically as the EPR paradox. To show the perceived flaw in the Copenhagan concept of superimposed states, they proposed a situation in which two elementary particles locked in a no-spin pairing suddenly fly apart. Since the particles were originally part of a no-spin pair, they must have opposite spin. After the particles are miles apart, physicist Samantha measures the spin of one particle and discovers that it is spinning clockwise in a certain direction. This tells her that the other particle is spinning counterclockwise. When Sam investigates the spin of the other particle he verifies that, indeed, it is spinning in the opposite direction. There are two possible interpretations of what happened:
a) Each particle was simultaneously in two superimposed states: spin up and spin down. Samantha’s measuring of one particle caused the collapse of the two states into a definite state. This caused the other particle to collapse also into the opposite state, by some mysterious communication at a distance.
b) The two particles had definite spins that were opposite at the outset, so measuring the spin of one particle was enough to deduce the spin of the other. For example, if a pair of gloves were in a box that exploded, sending both gloves in opposite directions, the discovery of a right-handed glove at one location would indicate that the glove at the other location was a left-handed glove.
In their EPR paper, Einstein and his coauthors dismissed explanation “a” as untenable, since this would imply that previously connected particles could interact with each other when separated by great distances. But alternative explanation “b” left unanswered the question, “If quantum physics is complete, why can’t it predict the spin state of the particles before measuring them?” Einstein and others who were reluctant to abandon determinism, suggested that other hidden variables that quantum physics had not yet identified would eventually restore determinacy to quantum physics.
The Copenhagen interpretation languished in limbo for three more decades until 1964 when Irish physicist John Stewart Bell published a remarkable result now called Bell’s Theorem. In his paper On the Einstein-Podolsky-Rosen Paradox, he proved, using simple deduction and the quantum properties of spin, that if it were possible to predict the spin of an elementary particle without measuring, then it would lead to a logical contradiction. This invalidated interpretation “b” and with it, the EPR hope that a method of predicting the spin of an elementary particle was possible. Physicists were compelled to accept interpretation “a”, that the particles, even when separated by large distances, are interconnected. This interconnectedness, now called entanglement, was later verified by experiment.
The Ultimate Achievement of Deduction was to prove its own Limits.
In 1931, an Austrian mathematician, Kurt Gödel, at age 25 published his doctoral thesis, On Formally Undecidable Propositions of Principia Mathematica and Related Systems. In this epoch-making paper, Gödel enunciated a theorem and corollary that registered a magnitude ten on the seismometer of mathematical quakes.
Corollary (Gödel’s Second Theorem): The consistency of any mathematical system complex enough to contain simple arithmetic, cannot be proved within the system.
The programs of both the formalists and the logicians collapsed like giant skyscrapers whose foundations had crumbled under a tectonic shift. The impact of Gödel’s discovery on the mathematics community would eventually exceed the trauma felt two millennia earlier with the discovery of irrational numbers. Though it took many years for the mathematics community at large to appreciate what Gödel had achieved, those who were working in the foundations of mathematics soon recognized the far-reaching implications of his powerful theorem and its corollary. The brilliant wunderkind, John von Neumann, who had published The Axiomatization of Set Theory in 1928, was one of the first to perceive the “truth and importance of Gödel’s work.” Others soon followed and the initial shock wave eventually dissipated into a universal acceptance that absolute certainty of anything may be an illusion.
In 1978, an article in The New York Times10 described Gödel’s Theorem as “the most significant mathematical truth of this century, incomprehensible to laymen, revolutionary for philosophers and logicians.”
Gödel’s theorem implies that whether we choose the axioms of Peano, Hilbert, Russell, or anyone else, there will always be theorems that are true, but can neither be proved nor disproved using only those axioms and the rules of logic. In this sense, the system is incomplete. We can prove or disprove any particular theorem in the system by adding one or more axioms, but then the corollary asserts that we will not know whether or not our new set of axioms is consistent.
Deductive thinking was so powerful that it was able to prove its own limitations!
