Before Nicolaus Copernicus, geocentrism was the dominating theory in modeling planetary motions. Geocentrism (such as Ptolemaic system) can predict fairly well the motion of the stars. But for the motion of the planets, Ptolemaic system fails to predict their motion. That’s why planets means “wandering stars” (Ancient Greek: ἀστέρες πλανῆται asteres planetai).

To overcome this limitation, the concept epicycle was introduced. Shown by Norwood Russell Hanson in “The Mathematical Power of Epicyclical Astronomy”, Epicycle can be mathematically modeled as complex Fourier Series. Therefore, compounded epicycle model can give arbitrary accuracy in predicting planetary motion.

In contrast, Johannes Kepler and later Isaac Newton‘s description of the heliocentric system consisting of elliptical orbits can accurately predict the motion of the planets with a single elegant ellipsoid equation rather than many complexly nested epicycles. To many, this is another confirming case for the Occam’s razor. More on that later. But to me, the story of epicycle shows an important lesson in science:

Validity is not truth.

Even if a scientific theory (such as the epicycles) can accurately predict the future, does not mean that the scientific theory in anyway reflect the reality of the world. Based on Karl Popper’s definition of scientific method, i.e. empirical falsification, scientific method does not prove any truth regarding the natural world. It can only be served as a verification that the proposed model or theory’s ability to predict the events in future given the empirical evidence. In other words, by assuming the theory is true, scientific method only shows that the empirical data does not contradict the theory. But it can not prove a theory to be true by assuming it is true in the first place (this fallacy is referred to as petitio principii, or the mistranslation “begging the question”). Scientific method can not prove the theory to be true.

Further more, by the definition empirical falsification, only theories that are falsifiable by empirical evidence may be considered scientific. At the time of proposal, Karl Popper witness the birth of quantum mechanics, psychology, and social sciences, which in comparison to traditional “sciences” (as there wasn’t a formal definition of science before Popper proposed it), deal with probabilities rather than certainty. Though the principle of falsification is the same, it would require knowledge of probability to understand which propositions about a probability is falsifiable. Therefore, here I will use traditional deterministic scientific theories as examples.

Deterministic scientific theories (and scientific theories in general) use formal logic, also known as deduction for reasoning. In contrast to abduction and induction, which are informal logic and widely used in our daily arguments (, which I think should be used very carefully with good understanding of their limitation because they do not guarantee true premise would lead to true conclusion), formal logic guarantees that if the premise is true, the conclusion would also be true. We may wonder why this is the case. I will talk about this fantastic question later. But for now, we will just accept this property.

Here is an example of a theory:

All cats have one tail.

The word “All” used in a theory is called a universal quantification. It is very easy to see that we can falsifies this theory if we found a cat with 0 or 2 or more tails. Therefore, all cats have one tail is a scientific theory.

Another example of a theory:

Some cats have one tail.

The word “Some” is now the existential quantification. We can easily prove this statement to be true if we found even just one cat that have one tail. But we can not prove this statement to be false, as it require us to examine all cats from the past, present, and the future to conclude its truth value. Therefore, some cats have one tail is not a scientific theory.

This example demonstrated an important quality of scientific theories defined by falsification:

A scientific theory has to apply to the potentially infinite cases in the future.

It may not be surprising, as science aims to study general and repeatable universal laws. But it implies that forming theories for a past event is not science, until we propose how it can be reproduced in the future. Scientific theories with universal quantification have worked very well with fundamental sciences: physics, chemistry, and to some degree, biology in the past years. With many repeatable and reproducible experiments, we have gained much confidence in those theories. But for many modern scientific fields, the sheer complexity of the subjects they study, has limited our ability to create models that capture the universal quantification. (In other words, we generally can not make claims about certainties in those fields.) This has raised many concerns in the scientific field in recent years. But I will delay the discussion to the future.

So, now the question:

Why does deductive reasoning work the way it is?

The reader may find this answer unsatisfying, but formal logic works the way it is, because we defined it that way. Formal logic, as a mathematical model, its behavior is given by its definition we constructed.

But if formal logic is a pure abstract construct, how does it reflect the causality of the physical world?

This is a question that has vexed me for a long time, and many people may find my answer unsatisfying. But logic, just as any other mathematical models we use for scientific theories, is valid in predicting the future, but we have no evidence that it is in any way reflects the physical world. We have accepted that it as true, just as any other scientific theories we have accepted, because, so far, from the time that we started to reason, we have no evidence contradicting its validity. (I want to point out here that when I said logic, I meant formal logic or deduction. There are many critiques upon induction and abduction.)

But this leaves me with an uncomfortable cyclical reasoning, that the tools we use for scientific method, is verified using scientific method. But maybe how we get out of this labyrinth, is to accept that we can never be certain what the absolute truth might be. The best proof we have for our belief, is just the lack of evidence against it.

---

P.S. If reader is interested to learn more on scientific method, SCIENCE: CONJECTURES AND REFUTATIONS by Karl Popper is a great reference.