Thoughts on Logical Leap by David Harriman, Part II
Can Full Context of Knowledge be Defined?
In my last essay, which you can read here, I gave a short comparison of Deductive reasoning and Inductive reasoning, pointing out that Inductive reasoning had largely been dismissed in the field of Philosophy of Science. Leonard Peikoff resurrected the topic in 2002 with his lecture series titled "Induction in Physics”. David Harriman, with Peikoff’s assistance, provides a theoretical framework for Inductive reasoning in his book “Logical Leap: Induction in Physics”. I ended that essay wondering if the effort fell short in one aspect. I address that in this essay.
David Harriman lays out several criteria for Inductive reasoning in his 2010 book, Logical Leap: Induction in Physics. The criteria I focus on in this essay is “Full Context of Knowledge”. Harriman says,
“When a scientist confronts some aspect of nature, . . . ; he enters his investigation armed with a vast context of knowledge that precisely delimits the possibilities. A factor qualifies as relevant to his investigation only if there is some reason to suspect that it plays a causal role.” pg 73
A valid scientific conclusion must include the scientist’s full context of knowledge applied to the subject being studied. The concept seems perfectly clear as the author relates historical examples of scientific discovery. This analysis benefits from hindsight – it is easier to see what knowledge applies and what doesn’t after the fact. But how could one define this in general terms so that scientists can approach new subjects? Let’s start with Harriman’s retelling of some historical examples to see if there are some guidelines to follow.
Galileo noticed that the chandeliers in the church appeared to have the same period regardless of how wide they swung. This simple observation set Galileo on a path to study pendulums. To investigate this phenomenon, he thought of all the factors that might contribute to this result. It was obvious that the length of the pendulum and the object swinging played a role. Did the material make a difference? He decided to include experiments that used different material for the bob and the string. Did the weight of the bob make a difference? He ran experiments with different weights. Did the angle at which the bob started its swing make a difference? He conducted experiments where the bob was positioned at a small and large angle with respect to the vertical.
Those were the obvious factors that were included in the “full context”. Galileo surveyed his understanding of motion in general and found two other concepts that he decided to include but weren’t so obvious.
Friction was the first issue Galileo identified. His experience riding horses or walking into a stiff breeze taught him that air resistance was a force that could retard movement. He reasoned that the pendulum bob would likely experience a similar force as it moved back and forth. He did his best to diminish its influence on his results but kept this issue in the back of his mind.
Impetus was the second issue Galileo identified. He made the bold decision to reject impetus, declaring it an invalid concept. The scientific consensus of the day said that objects move because they have impetus, an ill-defined attribute of an object in motion. As the impetus diminishes, the object slows to a stop and then drops straight down. Galileo rejected this notion based on other experiments he had performed using balls rolling down inclined planes.
This example provides a good sketch of what it means to include “full context” when crafting an experiment. It seems sensible to say that Galileo had all his bases covered. What else might affect the experimental results? The time of day? The color of the bob? Silly. He conducted experiments over a range of test configurations that covered his ‘full context’ and used the method of agreement and differences to arrive at the startling conclusion that the period of a pendulum was proportional to its length. Furthermore, the proportion was constant across different experiments. Based on Harriman’s criteria, Galileo was justified in making the generalization “The period of oscillation for a pendulum is proportional to its length (across all times and place)”.
Galileo did notice one anomaly, however. When he swung the pendulum in a wide arc, the period was slightly longer than predicted by his formula. He attributed this to air resistance. After all, the greater the angle, the faster the bob was moving, and he knew that air resistance increased with speed.
Many years after his death, other scientists conducted the same experiments at sea level and in the mountains of Italy, finding that the proportionality was not constant. Where Galileo had reported the same value in all trials, experiments at sea level resulted in a smaller value while it was slightly larger when conducted in the mountains. It appeared that the measured results were affected by altitude, something no one could have expected.
These two anomalies appear to be “black swans” that call into question the validity of Inductive reasoning. That would be a misunderstanding of the concept “full context of knowledge”. First, “full context” does not require omnipotence. The discrepancy found at sea level and at altitude indicated that there was an unknown factor influencing the pendulum’s motion. It wasn't until Newton derived his Theory of Gravity several decades after Galileo’s death that scientists understood the reason – the force of gravity diminishes as 1/r2. The slight change is altitude reduced the force of gravity just enough that the proportionality constant of the pendulum changed measurably.
When Galileo surveyed all that was known in his time, he had no reason to suspect altitude above sea level would have any effect on his results. So, it cannot be considered an error that he didn’t conduct experiments at different altitudes. I would offer the following as a point of clarification:
Induction: Fine Point #1
Conclusions reached by Inductive reasoning are not invalidated by new knowledge. More specifically, the “full context of knowledge” includes only the knowledge known at the time and does not rule out changes as new discoveries are made.
The scientific transition from Newton’s Theory of Gravity to Einstein’s Theory of General Relativity provides another refinement to the concept of “full context”.
Newton’s Theory of Gravity involves a wide array of observations that cover falling objects, pendulums, ocean tides and planetary orbits. I won’t recount the “full context of knowledge” Newton used to arrive at his theory. Suffice it to say that he included a vast amount of knowledge when he came to his conclusions.
Despite this, some have pointed out that there was at least one piece of evidence, known to Newton, that casts doubt on the universal nature of his theory. Remember that Induction arrives at generalizations that apply to all things covered by the theory. Well, the planet Mercury didn’t get Newton’s memo. Mercury’s actual orbit did not match the orbit predicted by the Theory of Gravitation. Predictions using the theory were off by just enough to be measured by astronomers of the day.
This raises the question of whether Newton had discovered a universal law or not. Was it appropriate for him to say his law applied to all objects? The answer, I think, is a qualified yes. It was appropriate. Newton had managed to integrate many different kinds of motion, demonstrating that they all conformed to his laws of motion. His theory explained the motion of pendulums, apples falling from trees and projectiles on Earth. He identified the causal mechanism, the force of gravity, to explain observations of the ocean tides and (most) orbits of planets around the Sun (and the moons of Jupiter. He had also created a new type of mathematics, Calculus, that aided his investigations. His theory was grounded in observation, mathematical and identified a universal cause. It was not subjective and arbitrary.
I would add a caveat, however. Given that there was a known case where his theory did not work, Newton should have limited the scope of generalization. He could not find a reason why Mercury behaved as it did. Newton was known for avoiding any form of speculation. So, he refrained from speculating at all (a subject I intend to expand upon in the future). I do believe it would have been better if this problem with Mercury’s orbit had been carved out as an exception or anomaly.
Induction: Fine Point #2
Induced generalizations need to explain observed phenomena from the full context of knowledge. Any discrepancies should be examined and specifically called out for further investigation.
Einstein’s Theory of General Relativity finally explained Mercury’s orbit. His theory demonstrates that Mercury’s orbital procession is caused by the Sun’s strong gravitational field.
It is a common error to say that General Relativity replaced Newton’s Gravitational Theory. But this isn’t entirely true. In weak gravitational fields, General Relativity collapses down to Newton’s equations, demonstrating that scientific knowledge is hierarchical.
With the knowledge provided by Einstein’s General Relativity, Newton’s Theory of Gravity can be seen in context. Newton’s theory described motion outside a strong gravitational field. Mercury is too close to the Sun and therefore is outside the scope of Newton’s theory.
I offer these two minor clarifications to Harriman’s concept of “Full Context of Knowledge” hoping they provide a better guide. There is a great deal more to discuss regarding the practice of Inductive reasoning. Future essays will dive into more detail.