The Logic of Scientific Discovery

Popper describes the theories he outlines in previous chapters as tools he intends to apply to modern quantum theory. His study seeks four conclusions: (1) ranges of uncertainty in quantum physics are singular probability statements, (2) quantum theory can still maintain a high degree of precision in its statistical analysis, (3) limitations on that precision is not a logical assumption, and (4) this assumption contradicts the exact measurements of quantum theory.

Heisenberg’s Programme and the Uncertainty Relations

One of the dilemmas facing physicists is the unobservable components of an experiment. Einstein faced this problem while trying to analyze experiments using unobservable magnitudes. In many cases, scientists have developed assumptions about the physical world based upon unobservable data. Popper describes an experiment attempting to measure dispersed light. However, the incalculable interference develops a principle of uncertainty. Popper explains that by combining multiple forms of measurement one is able to calculate what previously seemed incalculable. The philosopher argues that quantum theory still relies on metaphysical principles that render the field non-scientific.

A Brief Outline of the Statistical Interpretation of Quantum Theory

Prior to Popper’s work, quantum theory was split down two separate paths. The first follows the “classical particle theory of the electron,” and the other follows wave-theory (222). The two ideas were later resolved as wave-theory revealed itself as an element of particle theory. Probability played an important role in this reconciliation. The validity of quantum theory is dependent upon its statistical analysis. Some argue that the uncertainty of quantum theory can be overcome by the performance of multiple experiments under identical conditions. Popper disagrees with this assertion and argues for a new way of thinking about uncertainty.

A Statistical Re-Interpretation of the Uncertainty Formulae

The scientific community has long held the belief that if multiple experiments utilizing the same conditions produce the same results, then the uncertainty principle can be overcome. Although this approach may appear to provide precision, Popper argues that the interpretation does not follow the formulae. Any assumptions must be statistically interpreted. By eliminating unmeasurable aspects that contribute to uncertainty, Popper proposes that the scientist can then focus on what can be determined through testing.

An Attempt to Eliminate Metaphysical Elements by Inverting Heisenberg’s Programme; with Applications

Popper engages in a thought experiment to examine the challenges facing quantum theory. He emphasizes the use of logical analysis and the distinction between statistical and probability concepts to interpret quantum theory. He rejects the assumption that statistical statements found in quantum theory are probability statements.

Decisive Experiments

The Heisenberg formulae can be interpreted statistically, but this interpretation comes with certain limitations that limit the ability to be precise. Heisenberg asserted that it was impossible to make “exact singular predictions” in the field of quantum theory; he argued that there were too many uncertainties that would make it impossible to attack the subject with any form of precision (237). Popper’s complaint with the Heisenberg formulae is that, while singular predictions cannot be made, there is no attempt to eliminate singular predictions using falsifiability. When experiments are designed to test falsifiability, the scientist gains better and more reliable information. Popper describes several simple experiments to show how this might work. His experiment shows that falsifiability creates the opportunity for the development of singular predictions.

Indeterminist Metaphysics

Scientists who study the natural world have two goals: (1) to discover laws that lead to single predictions, and (2) to advance probabilities. Popper argues that these tasks function together. He warns against metaphysical statements that deflect from a focus on logic and deductive reasoning, such as asking whether the world is governed by strict laws. Popper asserts that laws are nothing more than hypotheses; there is no way to verify them or ordain them with a seal of absolute truth. He also suggests that causality is a metaphysical concept; causality, like a law, relies on an assumption of truth. He admonishes the Heisenberg formulae, however, for attempting to limit the scope of scientific inquiry. Popper believes that science can encompass both prohibition through the study of hypotheses through observable data and the limitless possibility of learning.

Theories should not be discussed in terms of “true” or “false.” Instead, they should be approached as falsifiable or corroborated via probability. However, Popper diverges in this chapter from his attention to probability; he argues that probability should not be the direct focus of the scientist. Instead, attention should be centered on the rigor of testing.

Concerning the So-Called Verification of Hypotheses

The verification of a prediction does not correspond to the verification of a hypotheses. New information in the form of new testing may falsify the hypotheses of old experiments. Therefore, no test can ever prove total verification. Popper emphasizes the importance of non-verifiability in the methodology of scientific experimentation. He challenges the idea that nature has a uniformity that can be translated into laws of truth.

The Probability of a Hypothesis and the Probability of Events: Criticism of Probability Logic

In this section, Popper asserts that a focus on the probability of a hypotheses—rather than the probability of a predictive statement—has a foothold in inductive psychologism. For the philosopher, the line between positivism and probability of hypotheses is too narrow. The probability of a statement is an important tool of measurement in experimentation, but the probability of a hypothesis asserts that a theory has a determinate level of truth. The inductivist approach creates symmetry between verifiability and falsifiability, and Popper has already established that the relationship between the two is asymmetrical.

Inductive Logic and Probability Logic

It is a mistake to assume that the probability of a statement proves the probability of an event. Popper considers what it means to assign probability to the appraisal of a hypotheses. All appraisals, he argues, rely on concepts like “truth” or “falsity,” or adequacy and inadequacy. The problem with this approach is that it is untestable. There is no way to test the truth of a hypotheses.

The Positive Theory of Corroboration: How a Hypothesis may “Prove its Mettle”

Popper then flips his own criticisms of inductive logic to his theory of corroboration. He argues that his approach differs because rather than focusing on truth, his approach utilizes “provisional conjectures” (265). A hypothesis is corroborated when it stands up to testing; however, its status is provisional. It can be falsified at any time by any new experiment. Corroboration is also dependent upon a high degree of falsifiability and simplicity.

Corroborability, Testability, and Logical Probability

In this section, Popper opens by asserting that “a theory can be the better corroborated the better testable it is” (269). The testability of a theory is directly related to its degree of falsifiability. Corroboration increases each time another event or experiment upholds the hypotheses. Corroboration also increases when the hypotheses contain greater degrees of universality and precision. Auxiliary hypotheses are difficult to test and, therefore, should be used sparingly.

Remarks Concerning the Use of the Concepts “True” and “Corroborated”

Popper clarifies that the terms “true” and “false” may sometimes be necessary in the line of scientific inquiry. His admonishment of these terms applies only to their application toward the hypotheses itself. Corroboration differs from truth because it does not assert total verification. Its definition is limiting; corroboration affirms but does not confirm.

The Path of Science

Science must shift focus from inductive reasoning to a focus on testability, falsifiability, and corroborability. Popper addresses the question of why scientists do not begin with statements of greater universality. Hypotheses of all levels of universality arise in the scientific community; the philosopher suggests that this should be encouraged because these lines of inquiry may lead to strict universal statements with high degrees of testability.

Popper concludes by outlining a scientific future based on his theories. This science does not have an end. It carries on forever and recognizes the limitations of human understanding. He rejects inductive reasoning that utilizes guessing and belief over logic and experience. Science is never certain.

Appendix i. Definition of the Dimension of a Theory

Popper proposes a provisional definition to the dimension of a theory. His previous model did not account for occurrences with spatio-temporal co-ordinates. His new definition includes these occurrences and limits the possibility of a theory with two fields of application.

Appendix ii. The General Calculus of Frequency in Finite Classes

In this appendix, Popper provides a multiplication theorem that indicates the finite classes of frequency.

Appendix iii. Derivation of the First Form of the Binomial Formula

Popper then presents both the first binomial formula and its derivation that can be used for “finite sequences of overlapping segments” (290).

Appendix iv. A Method of Constructing Models of Random Sequences

Popper constructs an approach that eliminates aftereffects in random sequences.

Appendix v. Examination of an Objection. The Two-Slit Experiment

An imaginary experiment is performed to address Popper’s change in assertion that precise simultaneous measurements correspond with quantum theory. He statistically re-interprets a theory set forth by Heisenberg and obtains exact patterns. However, in the third instance, he obtains a prediction that contradicts the theory.

Appendix vi. Concerning a Non-Predictive Procedure of Measuring

Popper looks at two ways of predicting measurement in a non-monochromatic beam of particles. Popper suggests a way of designing a theory that utilizes a theory of indeterminacy—or a non-predictive procedure.

Appendix vii. Remarks Concerning an Imaginary Experiment

The philosopher carries forward with an imaginary experiment outlined in Chapter 9. This addition to the experiment shows that a non-predictive measurement can be precise and symmetrical.

By 1956, Popper still agrees with most of the ideas presented in his work. However, he never stops analyzing, critiquing, and testing his ideas. One idea not included in the original work is the propensity interpretation of probability. He explores these ideas in a later book entitled: Postscript: After Twenty Years.

Appendix i. Two Notes on Induction and Demarcation, 1933-1934

Popper responds to the misunderstanding that his work advocates for replacing verifiability with a falsifiability that is concerned with meaning. Popper argues that his focus is on demarcation, not on meaning. Popper develops a set of criteria for demarcation. First is the preliminary question that inquiries into the “validity of natural laws” (312). Second is the main problem that is the attempt to find criteria for demarcation. Some philosophers have proposed that meaning represents demarcation, but Popper argues that this is a positivist concept. Statements must be falsifiable and decidable to have validity.

Appendix ii. A Note on Probability, 1938

Popper includes a publication called “A Set of Independent Axioms for Probability” he published in 1938 (318). He proposes a formal system for the theory of probability. This formal theory would put an end to scientists constructing their own theory of probability based on three different interpretations.

Appendix iii. On the Heuristic Use of the Classical Definition of Probability, Especially for Deriving the General Multiplication Theorem

A theorem with heuristic value can be applied to measurements that have no inconsistencies, such as the throws of a die. However, it cannot be applied to develop theories of probability for biased data points such as a die with weights on one side. Therefore, an older definition may be applicable in these instances.

Appendix iv. The Formal Theory of Probability

Popper recognizes that probability can be interpreted in many ways, so he sets forth a formal system for interpreting probable statements. This system is autonomous and symmetrical. Popper uses an axiom system he illustrates in detail.

Appendix v. Derivations in the Formal Theory of Probability

This appendix provides various derivations from the system outlined in the previous section. In the introduction to the New Appendices, Popper describes Appendices ii-v as “somewhat technical—too much so for my taste, at least” (309). Popper shows the various ways of interpreting probability statements.

Appendix vi. On Objective Disorder or Randomness.

Popper defines randomness as a type of order. An example of randomness in science is the distribution of molecular velocities. Popper argues that randomness can be tested and can be defined as an “absence of regularity” (359). An ideally random sequence provides insight into how randomness functions.

Appendix vii. Zero Probability and the Fine-Structure of Probability and of Content

In this appendix, Popper returns once more to the probability of a hypothesis and how this connects to the degree of corroboration. Corroboration is not, however, probability. Corroboration cannot be measured using calculus in the same way probability can.

Appendix viii. Content, Simplicity, and Dimension

New language systems obscure the philosophy of science because they place limitations on new ideas that may advance the field. This corresponds with Popper’s disagreement with absolute atomic statements over strict universal statements. Atomic statements also should not be used to measure the degree of simplicity or content of a theory.

Appendix ix. Corroboration, the Weight of Evidence, and Statistical Tests

This appendix is an abbreviated version of Popper’s 1954 publication in The British Journal for the Philosophy of Science. He explores how to calculate the degree of corroboration and how to illustrate that this does not correspond with probability. He closes by determining that a few theories may be used and that—as is true with all scientific inquiry—it is impossible to eliminate all but one theory for corroboration.

Appendix x. Universals, Dispositions, and Natural or Physical Necessity

Inductive reasoning relies on repetitiveness to draw its conclusions. The more repeated a generated condition is, the higher its level of inductive doctrine. Some scientists argue that, although one cannot prove the justification with impunity, inductive logic creates a belief in a universal law. Popper illustrates several arguments to refute an adherence to inductive doctrine. He argues that all repetitions are approximate but not guaranteed. He argues that “universal laws transcend experience” because they embrace a degree of falsifiability (425). Once more, Popper reminds the reader that science is never conclusive.

Appendix xi. On the Use and Misuse of Imaginary Experiments, Especially in Quantum Theory

Popper attempts to show which methods of argument are deductively logical and which are inductively logical. He draws a distinction between imaginary experiments that utilize deductive versus inductive logic. The classical argument between Aristotle and Galileo is used to illustrate this idea.

Appendix xii. The Experiment of Einstein, Podolski, and Rosen

Popper closes by including a letter from Albert Einstein that rejects Popper’s imaginary experiment outlined in Chapter 9. Einstein is complimentary of Popper’s work. Einstein explains his own thought experiment with Podolski and Rosen. While Popper admires Einstein’s work, he disagrees with Einstein’s ideas about probabilistic conclusions.