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Tuesday 14 May 2013

Observation, Measurement and Theory

Scientific theories (pravāda) are stories created to explain observations. Theories are created by making some observations and measurements, and making a mathematical model which generalizes the measurements. The model can then be used to predict (approximately) future observations and measurements within a limited problem domain. But what we are measuring and how it relates to the model is not always clear. It is interesting to study the sequence of activities which goes on to convert an observation into a measurement and from there to a theory.

Classification and logic

Classification is a natural function performed by the human mind. A child learns to classify before learning to speak or count. Classification of what we observe into concepts is based on a conventional agreement between observers. There is nothing inherent in what is being observed that classifies itself. When two observers agree that there is a chair in front of them, they are agreeing based on what they see or touch. They agree that what is being observed has certain properties in terms of dimensions, shape and feel, which allows it to be classified as an instance of the concept ‘chair’. The chair concept is an abstraction where possibly infinite number of ‘irrelevant’ micro (eg. its chemical composition) and macro (eg. its smell or color) properties are discarded to support practical classification.

The properties mentioned above (dimensions, color, shape, etc) are perceptions (sannā) and formations (sankāra), which exist only in the mind. For example, color is created in the mind of the observer, and depends on the observer’s range of visible light as well as the range of light being reflected by the observed object. It is not an inherent property of the object. We know that a colorblind person perceives color differently - but we consider that a distortion only because there are more people who perceive color ‘normally’. The perception of color in other species of animals is different. Even within one species, there is no direct correlation between the properties in the individual observers’ minds other than the agreement that they are observing the same properties. No one with normal mental powers can communicate the color they are observing (without referring to other concepts), so that another person can verify they are ‘seeing’ the same color the same way.

While classification may happen naturally in individuals, if we try to discuss and formalize classifications, we do that using logic. Logic is a set of rules created to allow individuals to reason individually but arrive at the same conclusion. When observers agree on classifications or properties, they are defining conventional logical truths. When two observers agree that they are observing a chair, they are agreeing that the statement “this is a chair” is true. Truth in this context is logical truth, and is nothing more than a concept which denotes a conventional agreement (sammuti) that a statement is in accordance with fact. We will go in circles if we try to define what fact is, but we all have a feel for what truth is. Similarly, ‘false’ in a logical context would mean that a statement is not in accordance with fact.

When two observers agree “this is a chair” to be true, one may logically may imply that they are agreeing that the statement “this is not a chair” is false. However on deeper analysis, this is not clear-cut, and it depends on the system of logic being used. For someone growing up thinking according to Aristotle’s two-valued logic, “this is not a chair” will naturally be false if “this is a chair” is true. But for someone thinking according to Buddhist Chathuskotika (four-valued) logic, both “this is a chair” and “this is not a chair” can be true. Both can also be false - a possibility which was exemplified by king Devanampiyatissa 2000 years ago, when he was asked the question “Is there anyone who is neither your relative nor non-relative”. His answer was “Yes, there is myself”, and according to Chathuskotika logic “I am neither a relative nor a non-relative” is perfectly valid, although it is not valid according to Aristotle’s logic.

In yet another system of logic, Jaina Syāvāda, a statement can be of seven ‘truth’ values:
Syād-asti—"in some ways it is"
Syād-nāsti—"in some ways it is not"
Syād-asti-nāsti—"in some ways it is and it is not"
Syād-asti-avaktavyaḥ—"in some ways it is and it is indescribable"
Syād-nāsti-avaktavyaḥ—"in some ways it is not and it is indescribable"
Syād-asti-nāsti-avaktavyaḥ—"in some ways it is, it is not and it is indescribable"
Syād-avaktavyaḥ—"in some ways it is indescribable"

If we rely on this English translation (there are other possibilities), the first three values are similar to values in Chathuskotika. However it is important to note that these expressions seem to explicitly indicate the conventional nature of logical truths. A Jain observer might say “in some ways this is a chair” or “perhaps this is a chair” rather than “this is a chair”.
The last four expressions are also interesting. They include the concept of being “indescribable”. In Buddhism we talk about concepts such as Nibbāna which can not be logically explained or understood (athakkāvachara). May be the “undescribable” is an attempt to include such concepts into logic. Or it may be a further division of the Chathuskotika value “it neither true nor false”. In any case I will not try to reduce the Jaina logic to Chathuskotika.

Counting and natural numbers

Let us next consider counting, which can be considered a precursor of measurement. A child learns to count items which are classified into concepts. Counting chairs first involves classifying observed items into chairs and non-chairs. In that sense counting can have a tenuous relationship to what is being observed. However we have to keep in mind that what we are counting are instances of concepts, classified in our mind according to some logic. We are not directly counting anything ‘out there’. We can also count purely mental creations which are unrelated to observations.

Many cultures have, probably independently, created natural numbers (represented by the symbols 1, 2, 3, 4..) to support counting. If you look at the numbers 1, 2 and 3 we use today (known as hindu numbers), one can see the possibility that they originated from one vertical line, two horizontal lines and three horizontal lines respectively (I , = , ), and evolved to ease the effort of writing. Natural numbers can be seen as a further abstraction that can represent instances of any countable concept.

It is easy to count chairs. But how do we count objects which are both chairs and not chairs? It would be interesting to see whether there was any historical attempt to include such observations in counting. Is one chair and one “chair and non chair” two chairs or one chair? One possibility which preserves the consistency of Chathuskotika is that it is both one chair and two chairs.

This is not without practical application - particularly in the area of computing. Digital computing uses boolean logic (which has its origins in Aristotle’s logic), and uses the concepts of bits for counting and by extension higher-level mathematics. A bit can only have the values 1 or 0 (which can be interpreted as true and false). So two bits have possible values of 00, 01, 10 and 11. They can represent numbers and hence used for counting, for example the above combinations can represent zero chairs, 1 chair, 2 chairs and 3 chairs. However with digital computing the two bits can only be in one of the above states.

Quantum computing (which is still mostly in the theoretical stage) uses the concept of a qubit. A qubit can be in a ‘superposition’ of both 1 and 0 state at the same time. Two qubits which are ‘entangled’ can be in any combination of the four states described before. One can see the parallel with “both one chair and two chairs”, but we have to note that it is still only using the first three possibilities of Chathuskotika. I am not sure if the fourth state of Chathuskotika can be applied in counting at all - maybe there is some parallel to complex numbers. Looking into the future, quantum computing can theoretically solve mathematical problems which are intractable with digital computing. It may be an area of study which comes naturally to cultures conversant with Chathuskotika.

Counting to Measuring

While counting arose from the need to quantify observed objects, measurement probably arose from the need to quantify concepts which can not be directly counted, such as distance, length and time. Such concepts can be quantified only relative to other quantities, which lead to the creation of accepted standard quantities against which others are compared. Measurement in its basic form is an attempt to count the number of standard quantities to which a particular quantity can be considered equivalent. When we measure the distance between two cities in meters, we are counting the number of the standard units (a meter is defined by a piece of metal stored in a museum Paris) to which it can be considered equivalent.

The need also arose to quantify ratios encountered in geometry. However this can also be seen as a comparison lengths, without referring to a standard.To deal with not being able to count perfect ‘wholes’ when making measurements and calculating ratios, mathematicians have created rational numbers and real numbers.

It can be argued that counting and measuring are the only mathematical operations which have a somewhat direct relationship with observation. All other mathematical operations and rules exist only in human minds, and were created to support models able to generalize past observations and measurements, and predict future observations and measurements. The operations and rules themselves have no direct relationship to the observation.

Measurement to Theory, Modelling and Induction

As mentioned before, when we measure the distance between two cities, we are counting the number of the standard units to which it can be considered equivalent.This assumption of equivalency as well as the concept of distance are creations of the mind, and not something inherent in the observation. However it will allow us to calculate approximately, for example, how long it takes us to travel between the two cities. So how do we take that leap?

This involves creating a theory. We create a theory that if a person takes s seconds to travel 1 meter, it will take d x s seconds to travel between the cities (d being the distance). Note that to arrive at this theory and it’s mathematical model, we already use the abstract concepts of time and distance, as well as higher-level mathematical rules and operations,which are purely creations of the mind.

We have also used another logical process, induction. Induction is the process by which we generalize and predict the time taken to travel any number of meters using the measured time taken to travel one meter. We can verify the theory by measuring the time taken to travel various distances, and checking their consistency with the induced value. However we have to note that a theory can never be proven correct by a limited number of verifications - it might prove to be wrong the next time we measure. In a strict sense, theories can only be disproven, and never proven.

Limits of theory

We have seen how theories depend on measurement and measurement on counting, and counting on classification and abstraction of observations. While less-abstract theories can be applied to specific problems, abstraction allows us to create theories which are useful in a wider problem domain, albeit a limited one. As you increase the level of abstraction, it may become applicable in wider domains. But as abstraction removes from consideration properties which are considered irrelevant for a particular set of problems, the theories developed on that abstraction have a high chance of being invalid in situations where those properties are relevant.

Thus, most theories are not universal, although they may claim to be. This limitation is relevant not only in the basic western sciences of physics and chemistry, but in medicine, economics or psychology. We now know of the limitations of Netwton’s theory of gravitation, which have been shown to be inapplicable in some astronomical and quantum domains, although it was originally called the Universal Theory Of Gravity. Considering the higher-level ‘sciences’, history is littered with cases of ‘solutions’ which worked in one country being applied in others with disastrous consequences. This is normally due to the theory behind the solution not taking into account factors which are irrelevant in the first country but very much relevant in the second country. The Kidney disease in Sri Lanka, which was found to be due to the use of fertilizers and insecticides containing Arsenic, is an example of such a scenario.

Correlation and Causation

Another important limit of theories is that while they can explain correlations between properties, they do not necessarily explain causation. While someone may say a force applied on a body causes an acceleration, the model of Newton’s second law, F=ma, does not imply any such causal relationship. One can equally say that acceleration causes the force. While this may look like a moot point in a simple scenario, the lack of causal models in theories become important in areas such as economics and medicine.

Finding correlations between observed properties lead scientists to develop many theories which are wrongly presented or interpreted as having causal relationships. Whether it is about smoking causing lung cancer, or red wine reducing the risk of heart disease, correlations can be made or lost depending on the population being observed. When correlations are found between two properties, they can be due to other common factors rather than one causing the other. Birds singing in the morning can be correlated to flowers blooming. However a theory about that correlation should not imply one causes the other, ignoring the common factor which is Sunlight. In more complicated and long-term scenarios with many factors, it becomes harder to prove or disprove causal relationships purely through observations. Historically many theories have presented correlations incorrectly as causal relationships. On the other hand, many studies have exonerated unhealthy products due to the inability to prove they cause diseases.

A Theory of Everything

We hear of attempts by physicists to come up with a Theory Of Everything which explains observations in all domains. This assumes that there is a level of abstraction which generalizes sufficiently to allow mathematical modelling of all domains, but takes into consideration all properties relevant to observations in all domains. So far such a level of abstraction remains elusive, and it is not known whether such a level of abstraction is possible.

There is also an assumption that properties discarded as irrelevant for such a theory of everything are emergent properties which are dependent on the ones considered relevant (and therefore can be derived). However, with complex systems it remains unproven that higher level properties can be completely modelled as the sum-of-parts. While it is easy to model a neuron, it is not possible yet to completely model a human brain which consists of billions of neurons.

It is important to consider what it really means even if such a theory of everything is created. It should not be considered as an explanation of everything, it merely shows apparent (ie. created in the human mind) yet useful relationships between observable properties.

There are many things which will be knowingly not included in a Theory Of Everything presented by physicists. Needless to say, it will not include anything which is athakkāvachara or beyond the system of logic underlying the model. We also have to consider whether there are properties which are not known to be observable. Physicists would say that such properties will be considered only if they affect observable properties. String theory, for example, assumes the existence of dimensions of space which are not (at least for now and not by a normal mind) observable, but are needed to model some observable properties. So the theory will only be ‘everything’ relative to the observations of a human (directly or via machines) - and not relative to other possible forms of life which may observe properties which we are not aware of.

Most importantly, does the Theory of Everything explain the mind? The answer would be a resounding no unless we consider mind is nothing more than a chemical activity or process in the brain. Regardless, the hypothesis that a mind which contains a theory can be explained by the theory itself assumes that the mind itself is an observation. Then who is the observer?

Manjuka Soysa