In the development of mathematical and scientific knowledge, how problematic is the problem of induction?

by sofiyakoval (student)

International Baccalaureate Theory of Knowledge

11.04.2012

Sofiya Koval

In the development of mathematical and scientific knowledge, how problematic is the problem of induction?

People always have been curious about the world and universe around. New theorems, laws and improvements to previous pieces of knowledge are found almost every day. We acquire new knowledge by using reason: wonder why the Earth is moving around the Sun and not vice versa, made hundreds and hundreds of scientists to devote their lives for observations and analyzing how both the planet and the star behave in the universe.

Finding the “reason”, therefore a cause, always has been the interest of mathematicians and scientists. There are 3 known forms of reasoning: informal reasoning, deductive and inductive. The most popular, widespread and, unfortunately, most unreliable form of reasoning in the development of knowledge in science and mathematics is induction. Induction implies going from particular to the general, for example if we know that aluminum can be expanded, iron can be expanded and nickel can be expanded and these three matters are metals, we can conclude that all the metals can be expanded. Actually, we know from chemistry that all the metals can be expanded and we have no doubts that the conclusion is valid. But… What if one day chemists will find the metal, which cannot be not expanded? Therefore the conclusion that has been achieved above will no longer be valid. A lot of philosophers argue, whether the pieces of knowledge that are obtained by induction can or cannot be justified.

Most of laws and theorems in science were made using inductive reasoning, therefore generalizing. Here is the main problem of induction – generalization. Inductive reasoning maintains that if a situation holds in all observed cases, then the situation holds in all cases. Newton’s Third Law is believed to be true because there was no evidence that this law can be false. So, after completing a series of experiments that support the Third Law, one is justified in maintaining that the Law holds all cases. But there is always a chance, that tomorrow, this month or next year, someone will provide ...

This is a preview of the whole essay

Most of laws and theorems in science were made using inductive reasoning, therefore generalizing. Here is the main problem of induction – generalization. Inductive reasoning maintains that if a situation holds in all observed cases, then the situation holds in all cases. Newton’s Third Law is believed to be true because there was no evidence that this law can be false. So, after completing a series of experiments that support the Third Law, one is justified in maintaining that the Law holds all cases. But there is always a chance, that tomorrow, this month or next year, someone will provide the evidence of its invalidity. Falsifiability can be considered both as a positive and a negative side of one coin. On the one hand, till knowledge is valid and useful for everything where it has been used and if it was based on good observations under high level control we can continue using it, but on the other hand, if knowledge is falsifiable and there is a chance of it being invalid in different conditions, is it possible that all our previous results obtained by this knowledge were actually invalid but only seemed for us as valid? Does it mean that all knowledge that we have can become invalid one day and that all efforts of mankind were all in vain? Uncertainty about whether the truth in science and mathematics exists or not makes us concern about understanding of the world around. Induction gives a chance to generalize all observed pieces of knowledge in one field according to one or several common features and provide a general law or definition/description of interrelationship. Inductive method is very useful if for general law all the premises/assumptions are observed, but when unobserved is generalized as well as observed there is a possibility that the statement will wrong for other examples/in different periods of time/for different surrounding atmosphere.

In mathematics there is a way of proving that a given statement is true for all natural numbers – proof by mathematical induction. “The main idea of this is to prove that the first statement is true and then to prove that if any one statement in the infinite sequence of statements is true, then so is the next one.” (, 23.04.2012). Therefore, if a mathematician is able to prove something for a given series of problems, and one always gets results that correlate to each other, then one can deduce that there is a pattern found and therefore generalize a rule for all statements. In the documentary about Andrew Wiles, he is proving Fermat’s theorem, it was shown that many mathematicians before Andrew were testing this theorem for one number a time, and there were no number for which theory didn’t work, but no one was able to formulate a general law, that will prove that this theorem is working for every number from range of infinity. For Andrew it took ages to prove and to find this general law. According to Karl Popper’s theory “human knowledge in general, is irreducibly conjectural or hypothetical, and is generated by the creative imagination in order to solve problems that have arisen in specific historico-cultural settings” (, 23.04.2012). The mentioned-above statement gives us an example of critical way of thinking: from this point of view, all pieces of knowledge, which are mostly achieved by induction reasoning, in sciences and mathematics have no sense.

In conclusion I would like to point out that nevertheless without inductive reasoning most of the “ocean” of knowledge we have about the universe around us and our behavior will not exist, we shouldn’t rely with no doubts on these knowledge and always do best to prove, make improvements to it and try to justify our conjectures to understand the world around more precisely.