Scientific Reasoning

Anirban Kundu
&
Partha Ghosh

Photo by Miguel Á. Padriñán and EKATERINA BOLOVTSOVA

 Science is based on experiments. Any theory has to be validated by experiments, which is the final arbiter. However beautiful a theory might look, if it is not supported by data, it has to be discarded, or modified. There are two ways to proceed:

• People perform experiments and get some data, and based on that, the theory is built. A very good example is the data taken by the Danish astronomer Tycho Brahe, based on which Kepler first gave his model of planetary motion, and then Newton formulated the theory of universal gravitation. Similarly, the study of the spectra of different gases led to the atomic model. Careful observations and logical deductions from them also fall in this category. Wegener’s theory of continental drift was the result of a study of the maps of the continents.

• People write down a model. As long as the theory is not verified by experiments, it remains a model. Then the model is tested in the laboratory. The data might support it, or refute it. An example is the currently accepted theory of elementary particles, called the Standard Model. The Model came first, and then it was verified by numerous experiments. It is now a theory, but the name “Model” still remains. Another example is the recently observed gravitational waves. It was a prediction made by Einstein more than 100 years ago, and its existence was verified only recently, by unimaginably precise experiments.

 This is a very simplistic description. Actual experiments never give a precise number; they always contain some error, or uncertainty, however small. Suppose a theory tells that the electron and the proton have equal and opposite electric charge. No experiment can however tell whether they are exactly equal in magnitude; all they can say is that the magnitude of the electron’s charge minus the magnitude of the proton’s charge (we take the magnitude, so the sign issue is irrelevant) is a very tiny number, consistent with zero. The finer an experiment is, the better the theory is tested. One important point to remember is that if a theory is vindicated by an experiment, it is never thrown out. Maybe better experiments in the future will find some discrepancy with theory; even then, we will say that the old theory is true up to a certain level, after which we need a new theory. Again, let us take Newtonian gravity as an example. The rule is very simple, and all the celestial bodies were found to obey the rule so that we can predict the time of eclipses with great precision. However, scientists found a very tiny deviation from the Newtonian theory for the orbital motion of Mercury. It’s very tiny, but it is unquestionably there, so Newtonian gravity would need some modification. That came through Einstein who gave the modern theory of gravity, and the theory successfully explained the motion of Mercury. This does not mean that Newton’s theory is wrong. It just means that under some extreme circumstances, or where we need extreme precision (like the GPS tracking in your phone), Newton’s simple theory may need a modification.

 This also gives the scientists a roadmap of how to develop new theories. Even the greatest scientists can be wrong, they can be challenged and their theories can be modified or even overthrown. However, most of the time their theories show shortcomings because of a lack of better experimental data. So, a new theory, any new theory, must first ensure that all the known results can be explained by that. There are very few examples where that is not possible because of a complete paradigm shift, like the development of quantum mechanics in the first half of the last century. The laws of Newtonian mechanics are simply not applicable to subatomic particles. Even then, you can restore Newtonian laws as the size of the object becomes bigger; a cricket ball definitely follows Newton’s laws, and nobody needs quantum mechanics to describe its motion.

 One thing is common to all forms of science: An ultimate goal to know. Curiosity & inquiry are the driving forces for the development of science and scientists seek to understand the world and the way it operates. Science, including mathematics, is based on strict logical reasoning. The reasons may come from experiment and observation, or from theory. One starts with some premises and reaches a conclusion. To do this, they use two methods of logical thinking; namely, inductive and deductive logic.

Inductive reasoning

 It is a form of logical thinking that uses related observations to arrive at a general conclusion. This type of reasoning is common in descriptive science. A life scientist such as a biologist makes observations and records them. These data can be qualitative or quantitative, and the raw data can be supplemented with drawings, pictures, photos, or videos. From many observations, the scientist can infer conclusions (inductions) based on evidence. Inductive reasoning involves formulating generalizations inferred from careful observation and the analysis of a large amount of data. Brain studies provide an example. In this type of research, many live brains are observed while people are doing a specific activity, such as viewing images of food. The part of the brain that “lights up” during this activity is then predicted to be the part controlling the response to the selected stimulus, in this case, images of food. Then, researchers can stimulate that part of the brain to see if similar responses result.

 From several specific cases, we formulate a general conclusion in inductive logic. Thus, this is the logic of evidential support. In a good inductive argument, the truth of the premises provides some degree of support for the truth of the conclusion, where this degree of support might be measured via some numerical scale. By analogy with the notion of deductive entailment, the notion of inductive degree-of-support means: among the logically possible states of affairs that make the premises true, the conclusion must be true in (at least) some fraction \(r\) of them —where \(r\) is some numerical measure of the support strength. Let us look at some examples. We look at a sample of \(1000\) crows and find all of them to be black, and we reach the general conclusion that all crows are black. This conclusion will remain true till we find a white crow, at which point the probabilistic measure \(r\) falls from one. The closer it is to one, the stronger the general conclusion is. For the case of the crow color, \(r\) maybe something like \(0.999999\), so “all crows are black” is a pretty robust statement, if not completely true. Inductive reasoning is therefore weaker than deductive reasoning, but in science, we almost always use inductive reasoning. We try to reach a general conclusion, a universal law, from specific observations and experiments. Examples are galore, from Newton’s law of gravity to Darwin’s theory of evolution. Inductive conclusions change when we find exceptions, and that is how science progresses. We have observed the scattering of one billiard ball from another billions of times and we can formulate an inductive law of scattering. When we apply this law to an electron scattering from another, the result is at variance with the data, so here is an exception, and we need to change the law. As we have said before, this does not invalidate the law of billiard ball scattering --- it just says that we need a new law for the scattering of tiny objects like electrons.

 Inductive reasoning is almost always based on experimental data, and there lies the importance of experiments in science.

Deductive reasoning

 Deductive reasoning or deduction is the type of logic used in hypothesis-based science. In deductive reasoning, the pattern of thinking moves in the opposite direction as compared to inductive reasoning. Deductive reasoning is a form of logical thinking that uses a general principle or law to forecast specific results. From those general principles, a scientist can extrapolate and predict the specific results that would be valid as long as the general principles are valid. Studies in climate change can illustrate this type of reasoning. For example, scientists may predict that if the climate becomes warmer in a particular region, then the distribution of plants and animals should change. These predictions have been made and tested, and many such changes have been found, such as the modification of arable areas for agriculture, with changes based on temperature averages.

 In deductive logic, the premises of a valid deductive argument logically entail the conclusion, whereas “logical entailment” means that every logically possible state of affairs that makes the premises true must make the conclusion true as well. Thus, the premises of a valid deductive argument provide total support for the conclusion. In other words, we go from a general proposition to a specific proposition. Suppose we start with two premises: All human beings are mortal, and Mr. X is a human being. This leads to the irrefutable conclusion, provided both the premises are true, that Mr. X is mortal. If we drop any one of the premises, say the second one, then the conclusion does not follow; Mr. X can very well be an immortal robot. The first principles in any discipline are in general some sort of premises. They cannot be derived, but their validity can be checked through what we obtain assuming they are true. Needless to say, they are never formulated arbitrarily, there are always some logical processes, some experimental facts, that lead to the first principles.

 Newton’s laws of motion are the first principles of classical mechanics, the Schrödinger equation is the first principle of quantum mechanics, and so on. There can be equivalent first principles, like Heisenberg’s uncertainty relation for quantum mechanics, but at least one of them has to be assumed. Now consider the general premise: For every action, there is an equal and opposite reaction, and the specific case: The rifle recoils when it is fired. This is also a case of deductive reasoning. The best-known example of deductive logic is Euclid’s Geometry. It consists of a collection of undefined terms like point and line, and five premises, or axioms, from which all other properties can be obtained through deductive logic.

 Four of the axioms are so self-evident that it is unthinkable to call any system “geometry” unless it satisfies them:

1. A straight line may be drawn between any two points.
2. Any terminated straight line may be extended indefinitely.
3. A circle may be drawn with any given point as center and any given radius.
4. All right angles are equal.
The fifth axiom was a different sort of statement:
5. If two straight lines in a plane are met by another line, and if the sum of the internal angles on one side is less than two right angles, then the straight lines will meet if extended sufficiently on the side on which the sum of the angles is less than two right angles.

 Once you assume these premises to be true (it is hard to see how any of the first four can be false) everything about Euclidean geometry follows. Only in the 19th century, it was shown that one can still formulate a consistent geometry without the fifth axiom; such classes of geometries are called non-Euclidean. The fifth axiom tells us that the internal angles of a triangle sum up to 180 degrees, but if you try to draw a triangle on the surface of a sphere, the sum of the internal angles will definitely be greater than that; this is an example of non-Euclidean geometry. Thus, deductive logic essentially follows the logical chain for any three statements \(A\), \(B\), and \(C\): If \(A\) implies \(B\) and \(B\) implies \(C\), then \(A\) implies \(C\). This chain can be continued ad infinitum. Let us have an example. Consider the three statements: \(A\): The sky is cloudy. \(B\): The night is hot. \(C\): We do not sleep well. The chain means: If the sky is cloudy, the night is hot. If the night is hot, we do not sleep well. So, this means: If the sky is cloudy, we do not sleep well. Note that \(A\) implies \(B\) does not mean \(B\) implies \(A\), we cannot say “If the night is hot, the sky is cloudy”, because the night can be hot even on a clear summer day. However, if \(A\) implies \(B\), the negative of \(A\) implies the negative of \(A\): If the night is cold, the sky is clear. The negative of \(A\) is mathematically written with a bar above 'A' (\(\bar{A}\)) and called \(A\)-bar. Those who know Boolean algebra must be familiar with this notation.

Relationship Between Science and Logic:

 A valid hypothesis must be testable. It should also be falsifiable, meaning that it can be disproven by experimental results. Importantly, science does not claim to “prove” anything because scientific understandings are always subject to modification with further information. This step—openness to disproving ideas—is what distinguishes sciences from non-sciences. The presence of the supernatural, for instance, is neither testable nor falsifiable.

 To test a hypothesis, a researcher will conduct one or more experiments designed to eliminate one or more of the hypotheses. Each experiment will have one or more variables and one or more controls. A variable is any part of the experiment that can vary or change during the experiment. The control group contains the entire same conditions EXCEPT for the variable being tested. Therefore, if the results of the experimental group differ from the control group, the difference must be due to the manipulation, rather than some outside factor.

 Both types of logical thinking are related to the two main pathways of scientific study: descriptive science and hypothesis-based science. Descriptive (or discovery) science, which is usually inductive, aims to observe, explore, and discover. Hypothesis-based science, which is usually deductive, begins with a specific question or problem and a potential answer or solution that can be tested. In hypothesis-based science, specific results are predicted from a general premise.

 The boundary between descriptive science and hypothesis-based science is often blurred, and most scientific endeavors combine both approaches. Inductive and deductive reasoning are often used in tandem to advance scientific knowledge. The fuzzy boundary becomes apparent when thinking about how easily observation can lead to specific questions. For example, a gentleman in the 1940s observed that the burr seeds that stuck to his clothes and his dog’s fur had a tiny hook structure. On closer inspection, he discovered that the burrs’ gripping device was more reliable than a zipper. He eventually developed a company and produced the hook-and-loop fastener popularly known today as Velcro. Descriptive science and hypothesis-based science are in continuous dialogue.

 Any scientific statement has to follow the logical reasoning process, either deductive or inductive, and a statement that follows neither is simply not science. Logical reasoning means the conclusion is potentially falsifiable; if a statement is not falsifiable, it is vacuous and cannot be called science. Thus, “you will die if you consume poison” is a scientific statement, based on inductive reasoning, with a very high probability of the premise “if a person takes poison, that person dies” being true. But “everything is destiny and pre-determined” is not a scientific statement, because you can never falsify it, it is a vague philosophical statement and nothing more. This is why the intermediate steps are equally, if not more, important than the final result. They not only show how the logic worked but also make it easy for others to reproduce the result. A key ingredient of any scientific result is its reproducibility by others. A good example is Fermat’s Last Theorem. In 1637, Pierre de Fermat mentioned an important theorem about positive integers in the margin of a book and commented that the proof was too big to fit there. The proof was given by the mathematician Andrew Wiles in 1994, more than 350 years later, and using techniques that were unknown in Fermat’s time. That is why Wiles, and not Fermat, is credited with the proof of the celebrated theorem. (The theorem says that one cannot find three positive integers \(a\), \(b\), and \(c\), for which \(a^n + b^n = c^n\) is satisfied for any positive integer \(n\) greater than \(2\).)

 The scientific method may seem too rigid and structured. It is important to keep in mind that, although scientists often follow this sequence, there is flexibility. In fact, the scientific process is much more complex in practice. Sometimes an experiment leads to conclusions that favor a change in approach; often, an experiment brings entirely new scientific questions to the puzzle. Many times, science does not operate linearly; instead, scientists continually draw inferences and make generalizations, finding patterns as their research proceeds. Scientific reasoning is more complex than the scientific method alone suggests. Notice, too, that the scientific method can be applied to solving problems that aren’t necessarily scientific in nature.

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