This is an essay written in early 2018 and has been published in Kalaya. I thought it will interest those philosophers who believe in "Mathematics is Ontology" and those scientists who believe in "A Scientific Method".
Those who cannot follow reasoning or give their own reasoning or consider this as gossip may add references.
Those who cannot follow reasoning or give their own reasoning or consider this as gossip may add references.
What Is Science (Pattapal Boru)
Is there a specific method of acquiring knowledge, which is peculiar to Western Science? Starting with Francis Bacon up to Thomas Kuhn, perhaps the Philosophers of Science had thought so, though the methods that they had identified were not the same. They agreed that Science differs from other subjects and systems of knowledge mainly because there was a “scientific method” in acquiring knowledge. Paul Feyerabend on the other hand was of the opinion that there was no scientific method as such.
Western Science that is supposed to “understand” an already existing nature, is not very much different from other disciplines but that does not mean that there is no difference at all between Science and other systems of knowledge. Western thinking is based on dualities and the so-called scientific methods are categorized under either empirical or rationalistic headings. In empiricism, sense experience is primary while rationalists are associated with the deductive method. Bacon is supposed to be an empiricist while Galileo is called a rationalist who followed the deductive methods.
This is an unnecessary categorization, as there are neither pure rationalists nor pure empiricists from the time of ancient Greeks in the western tradition. Induction (inductive reasoning) does not come under logic, but arises from the extension of a property of some members of a population from a limited number of observations in a sample(s) of the population to the entire membership of the population, whether on a statistical basis or not. The population could be infinite. It is not different from having an idea in rationalism in western tradition.
Rationalism begins with basic ideas that may be in the form of axioms or hypotheses or whatever one calls them, all of which will be called “axioms” hereafter. The jump from observations of a property of a limited number of members of a population to all the members of the population, whether it is finite or not, will be called a generalization. There is no empirical basis for this jump, and is based on an “axiom”. Neither is there any logic in an Aristotelian sense. It does not follow any rule of inference. It is an idea or a hypothesis as in western rationalism. Generalizations come under rationalism in Western Philosophy. Generalizations in a finite population are said to be concrete, when the generalization could be imagined. Otherwise, a generalization is said to be abstract. These generalizations are results of a process known as inductive reasoning.
When it is said that induction does not come under logic, it means that induction is not deductive starting from a set of “axioms” using rules of inference and Aristotelian two valued two-fold logic. Inductive statements are not different from “axioms”, which need not be observed for the entire population. They are ideas imposed on the population. For example, the oft quoted statement “All men are mortal” cannot be observed, and it is an abstract generalization.
The syllogism in a way can also be considered as a test. For example, consider the syllogism:
The syllogism in a way can also be considered as a test. For example, consider the syllogism:
men are mortal
Socrates is a man
Socrates is mortal.
The statement "All men are mortal" is obtained from a limited number of observations by generalization. It is an induction. If one considers syllogism as a deduction, it is not significant, as Socrates is mortal is included in the premise “All men are mortal”. It is transparent at the very beginning and the so-called deduction has no significance.
Socrates is mortal.
The statement "All men are mortal" is obtained from a limited number of observations by generalization. It is an induction. If one considers syllogism as a deduction, it is not significant, as Socrates is mortal is included in the premise “All men are mortal”. It is transparent at the very beginning and the so-called deduction has no significance.
However, one can consider it as a test of the generalization or the “axiom” all men are mortal. Here is Socrates. If Socrates is mortal as implied by the “axiom” then the axiom holds. Otherwise the “axiom” is invalid, and inductive knowledge “all men are mortal” is not consistent with observations.
The jump from observed to unobserved is not empiricism but rationalism, which cannot be solved in empiricism alone in Western Philosophy. This is one of the problems that David Hume associated with induction, which has no solution within the sphere of pure empiricism. Most of the problems of Hume including causality arose from being a pure empiricist, if one may say so. Western thinking, or more specifically Chinthanaya (it is much more than thinking) is dualistic, and empiricism and rationalism are considered as two tight compartments without any interaction or association (ontology and epistemology constitute another duality). Induction, as mentioned, however could be tested, which brings back us to a form of empiricism.
On the other hand, deductions in rationalism in Western Knowledge make use of rules of inference, which are obtained through induction. The rule if a=c and b=c, then a=b is nothing but a generalization from limited observations such as if two people are equal in height to a third person, then the two people concerned are themselves equal in height to one another. The rule if a=b then b=a, or the rule a=a are rules of inference obtained through induction by observing a limited number of cases. These rules are not God given. Reasoning which depends on rules of inference, obtained by induction cannot be justified without induction that does not come under logic.
These rules of inference are static in the sense “change” has been ignored. The rule a=a is valid only if a does not change. If a changes then what is identified as a does not exist as it is not the same a that is encountered. To say that a= a, one needs to consider a as an object that does not change. This is generalized from the observation of “non-changing” properties of objects.
Change itself is a property that cannot be deduced using two valued two-fold logic as exemplified by Zeno’s arrow. However, we shall not discuss it here as it is beyond the scope of this essay.
If one considers a finite population, makes a limited number of observations of a certain property of members of the population and then extend the observations for the unobserved as well, it may be called a concrete induction. However, if the population is infinite then the induction is abstract as the conclusion cannot be grasped through the senses or even imagined.
What we have tried to explain so far is that empiricism and rationalism do not belong to water tight compartments but depend on each other. Some people without being aware of it call themselves both empiricists and rationalists, which seems to be correct even though according to strict dualities in Western Chinthanaya, an empiricist cannot be considered as a rationalist as well. What is discussed here is not related to the so-called Copernican Revolution of Kant.
In western science, very often with or without experiments observations are made (In Astronomy no experiments as such are designed). Most of these observations depend on some basic knowledge that has been acquired by the observers (experimenters as the case may be) through concepts and “axioms”.
The observations are very often limited to samples of population, and induction is called into play in arriving at an “axiom” whether concrete or abstract. These inductive “axioms” could be tested by considering other members of the population, but until a member is found to the contrary the “axiom” is held to be valid. As Hume has observed people have a tendency to come to conclusions by induction.
Learning Process
People and animals learn by trial and error if left to themselves. It is revealed by experiments done with mice when they are left to find their way through a mesh with blocks and openings. They try one path, if they do not succeed they try another path and so on, until they find their way out. Children left on their own follow the same method of trial and error.
In western science scientists who want to “explain” nature adopt the same method in arriving at “axioms” that “explain” observations with respect to a property of the members of the population. This method is known as abduction after the American Philosopher and Logician Charles Sanders Peirce (1839 -1914). Abduction, though in the earlier formulations of Peirce had similarities with induction, is entirely different from induction. In abduction, in order to explain a set of observations a hypothesis, an “axiom” is guessed. If the “axiom” does not lead up to the set of observations, then the particular “axiom” is dropped and a different “axiom” is introduced. The process is repeated until a satisfactory “axiom” is obtained. The “axiom” if satisfactory, is sometimes called a theory. This is what happens in abduction but it has to be elaborated in comparison with induction and deduction. It has to be emphasized that satisfactory does not mean true or correct.
We shall first discuss briefly deduction, induction and abduction.
Deduction
In deduction there are definitions, “axioms” and rules of inference in a given population. We restrict ourselves to the use of the word inference only with respect to deduction. Depending on the population, we identify either concrete deductions or abstract deductions. The population has to be identified first, followed with definitions and “axioms”.
In deduction there are definitions, “axioms” and rules of inference in a given population. We restrict ourselves to the use of the word inference only with respect to deduction. Depending on the population, we identify either concrete deductions or abstract deductions. The population has to be identified first, followed with definitions and “axioms”.
Consider first, a concrete example. Let the population be a box of toys. It has to be understood that box, colour, white and toy have been defined in a larger context, in a larger population. Consider the property of colour of the toys. Let us begin with the “axiom” ‘All toys are white’.
Deductions are made using rules of inference. The rules of inference as shown above are arrived at by induction. It has to be emphasized that the “axioms” are mere statements unless they had been deduced in a different context. In any event the very first “axioms” are not deduced, and are mere statements. Thus, reasoning alone leads us nowhere, though many people have a kind of faith in the process. These people do not believe in a God, after life, but believe in reason. Can they show that reason is not a belief?
We shall consider two deductions one in a finite population and the other in an infinite population.
Syllogism is a rule of inference obtained by induction. If the objects in a population have the same property, then we know that any object of the population must have that property. It can be generalized as, if all M are P, all S are M, then all S are P. Applying this syllogism to the toys in the box we deduce that since all toys are white, and if some toys are selected from the bag, then the selected toys are white.
Let us now consider the case of the population of straight lines. This is a population of infinite number of members. Then starting with some of the definitions and axioms and theorems in Euclidean Geometry, and the rule of inference if a+b=a+c, then b=c, it can be shown that when any two straight lines intersect each other the alternate angles are equal. (It is easy to see that the rule of inference if a+b=a+c, then b=c, is also obtained by induction).
In deduction whether in a population of finite number of members or of infinite members one starts with definitions and “axioms” and conclusions are arrived at using rules of inference.
Induction
Induction as mentioned before is generalization of a property of a sample(s) of members of the population to the entire population. When the population is finite it is called concrete induction, whereas, when the population is infinite it is called abstract induction. The result by induction could be tested by considering members of the population, other than in the sample(s).
Suppose we take a box of toys and select a few members and find them to be white. Then by induction, we may say that all the toys in the box are white. If we come across even one toy in the box, which is not white, then our statement obtained from induction becomes invalid. Thus, induction can also be taken as a test for the statement arrived at. Though this is a population of finite members by considering the more famous examples such as all swans are white or all crows are black we can say abstract induction can be considered as a test as well.
Abduction
Abduction was introduced by Pierce, though at first he himself could not see the difference between abduction and induction. Abduction is guessing of an “explanation” (reason) of an observed property of the members of a population.
Let us consider the case of falling apples. One has to give a reason for the falling of the apples. One could say that there is an affinity for the apples to fall to the earth, with heavy apples having a strong affinity. However, apples come down with the same acceleration, and one gives up one’s guess. Newton comes and guesses something else, which happens to work, may be under certain conditions. Newton’s guess is taken as the “reason” for the falling of the apples.
Induction and Abduction in Western Science
Abduction is relevant to western science as it is the method that is used in arriving at “axioms” that explain generalized observations through induction. We may call it the scientific method if such description is needed. However, in western science, there are two stages. In the first stage observations and induction take place in that order, and in the second stage abduction and observations take place in that order. Having said that it has to be emphasized that in the first stage the observations are dependent on theory or one’s beliefs, and that the observations could have been proceeded by thoughts. There are neither observations independent of conceptions, nor perceptions without conceptions, and the idea of perceptions without conceptions stem again from the dualities in western Chinthanaya.
In western science it is said that the observations are important, but in Cosmology and modern Theoretical Physics the ideas are considered to be supreme. The symmetry of theories, Mathematical consistency and beauty whatever it means, have become very important. It is as if the Universes or Multiverses have been created Mathematically, and one may be inclined to think that God is a Mathematician, assuming that God exists.
Western science is very much concerned with observations but observations of multiverse is beyond the capacity of the scientists. It is only an idea, and Mathematics is all about ideas arrived at by induction or by construction. Axioms in Euclidean Geometry have been arrived at by induction, while those of Non- Euclidean Geometry are constructions. What I mean by Mathematics, is what is sometimes known as Pure Mathematics, excluding anything that goes as Applied Mathematics. However, this does not mean that ideas exist independent of observations, when sensory perceptible conceptions are considered.
In the second stage, we said that abductions and observations take place in that order. The guesses are made to explain the generalized observations from the first stage, and having made guesses or “axioms” observations are usually carried out in order to arrive at an “axiom” that works. The word ‘usually’ has been used as it is not the case with multiverses and string theory.
In western science induction is called into play in generalizing from a limited number of observations of a property of a sample(s) to the entire population very often infinite in numbers. These are usually abstract inductions. A good example is the falling of objects to the earth.
Newton would have observed some apples falling when released from the trees. What he did was to generalize this experience to all objects (not merely apples) near the earth, and to make the generalized statement that all objects near the earth fall to the earth when released. That was abstract induction, and the population was the objects near the earth. Then of course he had the problem of the moon that did not fall.
This is the first stage of western science. The population is identified, samples are considered, some property of the sample(s) is observed, and the property is generalized to the entire population, very often of infinite number of members. With respect to populations with infinite number of members we are dealing with abstractions.
As can be seen in induction it is assumed that the relevant property is common to the entire population. As has been said induction belongs to rationalism in western Philosophy. In western science it is assumed that with respect to a certain property the entire population behaves the same way. This is nothing but another “axiom”.
In the second state scientists attempt to give an “explanation” of the generalized abstract observation. This is where abduction comes in. Abduction is guessing and nothing else. It belongs to rationalism in western Philosophy and by abduction scientists arrive at “axioms”. In guessing it is assumed again that the “axiom” applies to the entire population.
Why do the objects near the earth fall to the earth? Newton had an answer to the question. Others might have given different answers to the question, which did not work. Finally, it was Newton’s guess on gravitation that worked.
However, it did not work for the entire population. The moon does not fall. Newton could have excluded the moon from the population by taking the moon not to be near enough to the earth. However, he did not do that, as he wanted his guess to be universal, and gave an explanation as to why the moon does not fall to the earth. It was with his laws of motion and by considering circular motion.
Newton’s guess could be considered as an abstract generalization valid for any two objects in the universe. However, it has to be mentioned that Newton did not explain how this force operates, and it was nothing but spooky action at a distance, if we use a later expression by Einstein with respect to Quantum Mechanics.
Abductions are guesses, and the scientists go on guessing until they come across a guess that works. However, these guesses are culture dependent, and abstract. Very often, it is those scientists in cultures that help abstract thinking who come out with successful abstract guesses. The guesses are made in a certain paradigm in Thomas Kuhn’s sense, and when a guess cannot be made in the existing paradigm, the scientist has to make a guess with respect to the paradigm as well. Paradigms are also guessed and guessing of a new paradigm is considered as revolutionary science by Kuhn.
A paradigm prescribes the Game Rules that have to be adhered to in making guesses of “axioms”. A change of paradigm or paradigm shift is a change of the Game Rules. In the Newtonian paradigm all velocities were relative to the so-called inertial frames, but in Einsteinian paradigm this rule was changed, and the velocity of light remained constant in all so-called inertial frames of reference.
“Axioms” and paradigms are guessed in a culture. Both Newtonian paradigm and the Einsteinian paradigms were created in western Judaic Christian culture, while Quantum Mechanics was created outside Judaic Christian culture with its Aristotelian Logic. Bohr who was a pioneer in creating Quantum Mechanics was influenced by Ying -Yang idea in Chinese culture.
Any guess is subject to correction, and would not hold for the entire population for all situations. Guesses, and hence “axioms” which are sometimes called theories, are valid only for limited cases. It can be said that the guesses work only for a limited number of cases, approximately, and one should expect them to fail in some cases. This is somewhat similar to Popper’s falsification, and guesses or “axioms” or theories are subject to falsification, after particularization as explained below. The guesses are never right but only “work” under certain circumstances. However, an “axiom” is not thrown away, simply because one of its particularization does not work. It is used wherever it works, leaving aside the case when it does not work. The theory of gravitation due to Newton was not thrown away just because its particularization with respect to the orbits of the planets around the sun did not work. It is still used wherever it works.
This is based on pragmatism, and abduction is based on pragmatism as a Philosophy. It does not come as a surprise to note that abduction was introduced in US that follows a pragmatic philosophy. Quantum Mechanics, though not “understood” by western scientists within their culture continues to be used for its pragmatic features in western science mainly because US is the dominant force in science today.
Verification of “axioms”
Guessing of “axioms” belong to rationalism and not empiricism in Western Philosophy. The “axioms” are abstract statements, and one can go on deriving results as in Mathematics using rules of inference, as Greeks used to do with Euclidean Geometry. However, Science is not Mathematics, and one is interested in finding out whether the “axioms” have any sense with respect to observations.
We have previously said that “axioms” work in certain situations, and one should have wondered as to what is meant by working. This is a tricky question and involves a jump from general to particular. It is the reverse of induction, and may be called particularization, for want of a better word.
As “axioms” are abstract statements, what are deduced from them using rules of inference are also abstract statements. For example, from Newton’s theory of gravitation one could derive that an object is attracted towards another object with a certain acceleration that increases as the distance between them decreases. From this abstract statement scientists jump to the particular case of an apple falling to the earth, and say that the apple falls to the earth with increasing acceleration.
The “axioms” are not verified by observations. The “axioms” are abstract statements in western science, while observations are concrete experiences. An “axiom” has to be first particularized before an observation is made. Thus, what is verified or falsified is not the “axiom” but a particularization of the “axiom”.
Karl Popper’s falsification of theories could be valid only for particularizations. However, even then the “axioms” are not thrown away completely, but are made use of under special circumstances. This is a consequence of the nature of “axioms”. As “axioms” are guesses, they can be guessed only as far as certain situations are concerned, and not to cover the entire ambit of the population.
In western science the “axioms” are not supposed to be true or false but to work under certain conditions. Firstly, the “axioms” do not reflect a reality as such and the old “inference” that if a theory P implies a certain result Q, and if Q is observed, then P is valid does not hold. It is not obtained from any rule of inference as such, as the rule of inference states that if P is valid and P implies Q, then Q is valid. Secondly “axioms” can be guessed only within a limited range of observations to work.
The so called Scientific Method
Feyerabend said anything goes in science. However, there is a method in western science, though that method is used by others as well. It is a guessing game called abduction, that is practiced by rats in finding out the way to escape from a maze, by the children in learning a new technique or acquiring new “knowledge”, by search engines that throw out thousands of guesses, by artificial intelligence people etc. The difference between the others and the western scientists is that the guesses of the others are concrete, while the guesses of the western scientists are abstract. Western doctors in diagnosing use the method of abduction, though concrete.
In western science from a limited number of observations of a property of the members of a very large population, very often infinite, by induction generalized abstract statements are made. In generalizing it is implicitly assumed that the property holds for the entire population. Having made generalized statements with respect to the relevant property, western science looks for explanations for the property. These explanations are not causes as such but some guesses that work. The guesses unlike in the case of rats and ordinary people are abstract. Having guessed working “axioms” a jump is made through particularization to test whether the “axiom” in a concrete form works in a limited range. No “axiom” will work in the entire range.