Introduction:the nature of science 7 What is induction? To set the stage then for these puzzles,I shall say something about what induction is.As the nature of induction is so puzzling,it would be foolish to attempt to define it.But some rough descriptions will give you the idea.We use "induction"to name the form of reasoning that distinguishes the natural sciences of chemistry,meteorology,and geology from mathematical subjects such as algebra,geometry,and set theory.Let us look then at how the way in which scientific knowledge is gained differs from the reasoning that leads to mathematical understanding.One distinction might be in the data used.For instance, while knowledge in the natural sciences depends upon data gained by observation,the practitioners of mathematical subjects do not need to look around them to see the way things actually are.The chemist conducts experiments,the geologist clambers up a rock face to observe an unusual rock stratum,and the meteorologist waits for data from a weather station,whereas the mathematician is content to sit at his or her desk,chew a pencil,and ruminate.The mathematician seems to have no data,relying instead on pure thought to conjure up ideas from thin air.Perhaps this is not quite right,as the mathematician may be working with a set of axioms or problems and proofs laid down by another mathematician,or indeed with problems originating in real life or in the natural sciences,for instance Euler's Konigsberg bridge problem or some new theory required by subatomic physics.These things are not quite like the data of geology or the experimental results of chemistry,but let us agree that they at least constitute a basis or starting point for reasoning that will lead,we hope,to knowledge. What really is different is the next stage.Somehow or other the scientist or mathematician reaches a conclusion about the subject matter,and will seek to justify this conclusion.In the case of the scientist the conclusion will be a theory,and the data used to justify it are the evidence.The mathematician's conclusion is a theorem justified by reference to axioms or premises.It is the nature of the justification or reasoning that differentiates the two cases;the mathematician and the scientist use different kinds of argument to support their claims.The mathematician's justification will be a proof.A proof is a chain of reasoning each link of which proceeds by deductive logic.This fact lends certainty to the justification.If what we have really is a mathematical proof,then it is certain that the theorem is true so long as the premises are true.Consequently,a proof is final in that a theorem once established by proof cannot be undermined by additional data.No one is going to bring forward evidence contrary to the conclusion of Euclid's proof that there is no largest prime number.'This contrasts with the scientific case.For here the support lent by the data to the theory is not deductive.First,the strength of justification given by evidence may vary.It may be very strong,it may be quite good evidence,or it may be very weak.The strength of a deductive proof does not come by degree.If it is a proof,then the conclusion is established;if it is flawed,then the conclusion is not established at all.And,however strongly the evidence is shown to support a hypothesis,the logical possibility of the hypothesis being false cannot be ruled out.The great success of Newtonian mechanics amounted to a vast array of evidence in its favour,but this was not sufficient to rule out the possibility of its being superseded by a rival theory.What is induction? To set the stage then for these puzzles, I shall say something about what induction is. As the nature of induction is so puzzling, it would be foolish to attempt to define it. But some rough descriptions will give you the idea. We use “induction” to name the form of reasoning that distinguishes the natural sciences of chemistry, meteorology, and geology from mathematical subjects such as algebra, geometry, and set theory. Let us look then at how the way in which scientific knowledge is gained differs from the reasoning that leads to mathematical understanding. One distinction might be in the data used. For instance, while knowledge in the natural sciences depends upon data gained by observation, the practitioners of mathematical subjects do not need to look around them to see the way things actually are. The chemist conducts experiments, the geologist clambers up a rock face to observe an unusual rock stratum, and the meteorologist waits for data from a weather station, whereas the mathematician is content to sit at his or her desk, chew a pencil, and ruminate. The mathematician seems to have no data, relying instead on pure thought to conjure up ideas from thin air. Perhaps this is not quite right, as the mathematician may be working with a set of axioms or problems and proofs laid down by another mathematician, or indeed with problems originating in real life or in the natural sciences, for instance Euler’s Königsberg bridge problem6 or some new theory required by subatomic physics. These things are not quite like the data of geology or the experimental results of chemistry, but let us agree that they at least constitute a basis or starting point for reasoning that will lead, we hope, to knowledge. What really is different is the next stage. Somehow or other the scientist or mathematician reaches a conclusion about the subject matter, and will seek to justify this conclusion. In the case of the scientist the conclusion will be a theory, and the data used to justify it are the evidence. The mathematician’s conclusion is a theorem justified by reference to axioms or premises. It is the nature of the justification or reasoning that differentiates the two cases; the mathematician and the scientist use different kinds of argument to support their claims. The mathematician’s justification will be a proof. A proof is a chain of reasoning each link of which proceeds by deductive logic. This fact lends certainty to the justification. If what we have really is a mathematical proof, then it is certain that the theorem is true so long as the premises are true. Consequently, a proof is final in that a theorem once established by proof cannot be undermined by additional data. No one is going to bring forward evidence contrary to the conclusion of Euclid’s proof that there is no largest prime number.7 This contrasts with the scientific case. For here the support lent by the data to the theory is not deductive. First, the strength of justification given by evidence may vary. It may be very strong, it may be quite good evidence, or it may be very weak. The strength of a deductive proof does not come by degree. If it is a proof, then the conclusion is established; if it is flawed, then the conclusion is not established at all. And, however strongly the evidence is shown to support a hypothesis, the logical possibility of the hypothesis being false cannot be ruled out. The great success of Newtonian mechanics amounted to a vast array of evidence in its favour, but this was not sufficient to rule out the possibility of its being superseded by a rival theory. Introduction: the nature of science 7