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Proof vs Probability

Scientific Reasoning, Proof and Falsification

Francis Bacon said that induction is an essential part of scientific reasoning. So what is induction? Induction is a way of reaching valid conclusions from the evidence. What do we mean by valid conclusions? It means that the conclusions need to logically flow from the evidence. This may seem obvious but poor science can result from invalid conclusions rather than bad evidence. In order to make sure that our conclusions are logical we need to understand how the arguments are made that lead us to the conclusion. So let’s look at how arguments are made. An example of an argument is:

All birds can fly

Geese are birds

All geese can fly

An argument has two parts:

  • 1. Premise - A premise is based on the observations we have made. There can be many premises. In the example above the premises are ‘All birds can fly’ and ‘Geese are birds’.
  • 2. Conclusion - What we can truthfully and logically conclude given that the premises are true. In the example above the conclusion is that ‘All geese can fly’.

The conclusions can be false even if the premises are trues and conclusions will be false if any of the premises are false. This is an area of science that can be misunderstood and misleading. So how do we make true conclusions from our evidence? There are two types of conclusions:

  • 1. Deduction - The conclusion is true given that the premises are true. This can be considered as a ‘proof'- The truth of the premises guarantee the truth of the conclusion.
  • 2. Induction - The conclusion is a generalisation based on a sample of possible objects. In our example above, the evidence we have for the premise ‘all birds can fly’ must be from a sample of all birds. We have not observed all the birds in the World, only a subset of them. Therefore, the conclusion we make is inductive.

An inductive conclusion is not a proof since it contains an assumption that all the objects are the same as the observed sample, it is a conclusion from the known to the unknown. Induction is a generalisation from the specific and assumes that the laws of nature don’t vary between the two. Most of science uses inductive conclusions.

Induction does not give us certainty but gives us sufficient probability not far short of certainty. In science you may think that we would only accept a deductive proof to give us certainty. Sometimes we are able to have that certainty but most of the time we cannot definitively prove something because we have to make assumptions. Scientists are often misreported as ‘proving’ something when in fact it has been inducted using a generalisation from the examined to the unexamined. This does not mean that the science is wrong or that the work is not scientific. It means that ‘proof’ is very hard to come by and we live with an element of uncertainty while seeking evidence to give us the best knowledge we can have at that time. It is an approach that has been successfully used over the past 400 years.

The use of induction in science was questioned by the philosopher David Hume (1711-1776) in his book “An Enquiry Into Human Understanding". Hume said that, although we use induction all the time in life, it is a “brute animal” and cannot be rationally justified so we should not use it. This is ‘Hume’s problem of induction’. It says that the process of induction, which is a key element of the scientific process, does not give us justified knowledge therefore it cannot be part of the scientific process. Hume’s argument is a blow to the scientific process as laid out by Francis Bacon. This philosophical problem is still debated todajy

Hume’s problem of induction provoked Karl Popper to explore how science could only use deduction. He wrote his findings in his book "The Logic of Scientific Discovery” (1934). He believed that Hume had shown the ‘unjustifiability of induction’ and therefore science should not rely upon it. Popper developed the concept of ‘falsification’ and argued that although we may never prove a theory is true by sampling a small quantity, it is possible to prove a

Flowchart showing the deductive and inductive processes for accepting or rejecting a theory. Credit

Figure 2.1: Flowchart showing the deductive and inductive processes for accepting or rejecting a theory. Credit: C. Devereux and P. Farrell.

theory false by just one example of it not being true. For example, the premise 'All geese are white’ is proved false by observing one grey goose. The conclusion becomes ‘It is false that all geese are white’: a deductive proof. The logic of Popper’s argument means that scientists should be working to show that their theories are false rather than that they are true. This is not how people work though. The focus of scientist’s research is, generally, to prove that their ideas are right. Despite this shortcoming, the concept of falsification still has a role in science; a theory can be disproved using falsification and experiments should be devised to test theories based on falsification. Publishing a theory allows others to see if there are ways to falsify it.

Popper’s theory of falsification did not change the need for induction in science but it did lead him to set out criteria to distinguish between science and pseudoscience by identifying ‘good’ science as:

  • • A critical attitude to the received wisdom.
  • • An insistence on evidence-based content that is precise and wide in scope.
  • • The use of creative thinking to solve problems with bold conjectures that open up radical new possibilities for experiment and observations.
  • • The ideas of ad-hocness, novel prediction and scientific corroboration.

Probability as Science

By now, you may be sceptical about how robust scientific knowledge really is. Inductive reasoning is the basis of most scientific work but cannot give certainty. Deductive reasoning is the basis of proof but is rarely possible in science. The concept of probability can help. Even if a theory cannot be shown to be true it can be shown to be highly probable.

What is probability? It is how likely something is to occur. Flipping a coin is a random process and has an equally likely chance of landing on a head or a tail. If we flip the coin once we do not know which way it will land. If we flip the coin many times then half the time it will land on heads and half the time it will land on tails. The more we flip the coin the closer the result will be to exactly 50% heads and 50% tails. We can now put a number on the probability and we have an objective measure of the outcome of a flip of a coin. Probability can also be a subjective measure. If we say that finding alien life on Mars has a very low probability then it will mean different amounts to different people. There is no objective probability, we cannot put a number on it, we just know that it is very unlikely to occur.

Thomas Bayes was a 17th century English clergyman who pioneered probability theory and discovered the principle of conditionality; the probability of something occurring is based on whether it has occurred before. As more evidence supports a theory, there is more confidence in that theory and a scientist’s rational belief in the theory should change. A difficulty with this approach is knowing what probability to put on the theory before the evidence is collected, often this is a subjective best guess. This subjective element can be unwelcome by some scientists but it does not stop Bayes’ approach being widely used particularly in analysing the statistics of experiments and is an important part of the scientific method for many disciplines.

Using probability within scientific reasoning provides an adaptation to the scientific method:

  • 1. have an initial probability that the theory is true,
  • 2. get evidence to support or refute the theory,
  • 3. update the probability in light of new evidence by following the principle of conditionality.


Changing scientific ideas

How Do We Make Theories?

A scientific theory is an explanation that has been shown to fit the evidence, it answers the why question. Why does the World behave as we are observing it? In science, a theory and a model mean different things. The following are definitions of types of scientific explanations.

  • • A Scientific Fact is a statement supported by evidence. An example is ‘everyday the sun rises in the East and sets in the West’. Facts can change as observations and measurement techniques improve.
  • • A Scientific Law is a description of the relationship between things that is supported by lots of evidence from different sources. Laws are often mathematical statements. Newton’s Law of Gravity is a law because it provides a description of how the Earth orbits the Sun but not why. Laws do not explain the causes and can have exceptions.
  • • A Scientific Theory is an in-depth explanation of why something behaves as it does that has been extensively tested. It gives the causes for the laws based on evidence. Einstein’s theory of relativity is a theory because it determines the causes of gravity as the distortion of spacetime. A theory can be tested to determine how accurate it is. If there is more than one theory to explain a piece of evidence it is said to be ‘under- determined’. In this case we cannot determine the true causes of the evidence we are observing and we need more evidence.
  • • A Scientific Principle is a rule by which a phenomena works. A principle is more specific than a law and may be more restricted when it can be used. An example is the cosmological principle which describes the uniformity of the universe. There is no mathematical equation for this, it is a description of a concept that is supported by evidence.
  • • A Scientific Hypothesis is a statement, such as, if A happens then В will happen, that has not yet been shown to be true. Scientists work on hypotheses derived from theories, laws and evidence in order to extend knowledge. An example of an hypothesis is that small variations in the orbits of the outer planets of the Solar System could mean that an unseen planet exists that has yet to be found called Planet 9. An hypothesis needs to be testable.
  • • A Scientific Model is a tool to help aid scientific thinking about how the World works. There are different purposes to developing a model:
  • — To simplify the World or phenomenon so we can understand it.
  • — A starter model that can be built on to develop the full theory.
  • — A specific case of a phenomenon.
  • — To represent reality until a theory is established.
  • — A computer simulation using the laws to provide tests of theories.

Models are used in the daily work of a scientist. In cosmology, ACDM is a model that is being built on to develop a full theory.

Now we know what a scientific theory is, the next step is to know how to develop a theory and, of course, there are theories on how to develop theories. Here are four important ideas about how to make scientific explanations.

  • 1. Hempel’s Covering Law. The German philosopher Carl Hempel (1905-1997) said a scientific explanation must include reference to a scientific law and some initial conditions or particular facts. This is called the Covering Law because the thing to be explained is ‘covered’ by at least one general law of nature. Hempel asserted that an explanation is potentially a prediction and a prediction based on evidence is potentially an explanation.
  • 2. Harman’s Inference to Best Explanation (IBE). American philosopher Gilbert Harman (1938-) proposed that theories are chosen by picking the best explanation to fit as much of the evidence as possible. Although this may seem obvious it has been an influential method use widely by scientists.
  • 3. Occam’s Razor (also called the Law of Parsimony). William of Occam (1285-1349) was an English philosopher and monk who stated the principle “pluralitas non est ponenda sine necessitate” (plurality should not be posited without necessity). It is interpreted as ‘do not make things more complicated than necessary’. If there are two theories, then the simplest explanation is the one to pick. This rule is widely quoted and used. We have to be careful with this approach though; the universe is a complicated place and the simplest answer may not be the right one.
  • 4. Kuhn’s Ideas on Scientific Theory. Thomas Kuhn (1922-1996) was an American philosopher who developed radical ideas about scientific revolutions and how scientists develop theories. Kuhn said that a scientific theory should:
    • (a) Be based on accurate observations.
    • (b) Fit with other accepted theories in the subject.
    • (c) Be able to explain more than just what it was designed for.
    • (d) Be as simple as possible (Occam’s Razor).
    • (e) Be able to lead to other research.
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