Conjectures and Refutations

Chapter 7 provides an overview of Kantian philosophy as described in Kant’s published work, Critique of Pure Reason (1781). Popper describes Kant as the embodiment of several modern ideals, including world citizenship and world peace, equality before the law, and human emancipation through knowledge. This chapter is separated into several short sub-categories, which have been enumerated and summarized below. Beyond a simplified overview of Kant’s Critique, Popper includes very little additional commentary.

1. Kant and the Enlightenment

Popper argues that Kant is the last great defender of Enlightenment ideals. He argues that Kant is wrongly associated with the German Romantic School of Johann Gottlieb Fichte, Georg Wilhelm Hegel, and Friedrich Wilhelm Joseph von Schelling in popular memory.

2. Kant’s Newtonian Cosmology

Throughout his life, Kant always defended spiritual and intellectual freedom, which are at the core of Enlightenment ideals. This is because Kant himself grew up in poverty and his family practiced a severe version of Puritanism. His best-known work, Critique of Pure Reason, was inspired by his interest in Newtonian cosmology and his probing the expanse of the universe. Kant is credited as one of the first to identify the nebulae as distant stellar systems, and he is famous for discussing the cosmological problem of the finitude or infinitude of the universe. He is also a firm believer in Newton’s theory.

3. The Critique and the Cosmological Problem

The Critique is a famous work published by Kant, in which he attempts to trace whether the world has a beginning. He provides a first proof in favor of the theory: An infinite sequence of years must never come to an end—otherwise it is self-contradictory. If this is true, then unless an infinite amount of time has elapsed since the present, it is impossible for the world not to have a beginning. This is reasoning based on pure logic.

Kant then provides a second proof against the idea of a beginning for the world. Before the world existed, there must have been what Kant coins as “empty time,” where there was nothing happening and thus time-intervals did not matter. However, if the world has a beginning, then the time-interval immediately prior to the world’s existence is suddenly significant, even though, at the time, it must have contained nothing, just as every time-interval prior to it. This paradox therefore favors the conclusion of the world not having a beginning.

These two contradictory proofs are dubbed “antimony” by Kant and are central to the cosmological problem. They prove that pure reasoning is not enough to establish facts, as they can be used to prove the validity of both a theory and its contradiction.

4. Space and Time

Despite the contradictions mentioned in the previous section, Kant concludes that the human idea of space and time must not be applicable to the entire universe—it can only be used to understand ordinary physical things and events. Time and space are therefore frames of reference used to describe the human experience. He names this theory “transcendental idealism.”

5. Kant’s Copernican Revolution

Kant believed, without question, in the truth of Newtonian theory. However, when he attempted to understand how it came about, he realized that Newton could not have based everything on observation. Observations only confirmed his theory, which came from his intellect—it was Newton that actively made an attempt to understand the data his senses collected.

Similarly, Copernicus could not have physically been to space to see the solar system as it was and had to rely upon his intuition and reasoning to formulate his theory, which was later confirmed through observation. Kant therefore concludes that people must actively seek to find the truth rather than be passive observers waiting for nature to reveal its regularities. Kant’s philosophy of actively seeking knowledge made an impression on the fields of physics and cosmology, and Popper believes that without this Kantian climate, it is much more difficult to conceive of Einstein’s theories nowadays.

6. The Doctrine of Autonomy

This subsection shifts from Kant’s physics to his moral philosophy. He believes in every individual’s capacity to discern the truth—people’s conscience is therefore their moral guidance. It reprises the Socratic idea of human self-sufficiency—the idea that everyone is capable of ruling themselves and upholding the rule of law. These ideas are the basis for Kant’s encouragement of the creation of a league of nations to maintain peace on earth.

Chapter 8 is a summary of a talk Popper gave on the topic of empiricism and metaphysics. In the first half, it discusses Kant’s philosophy and its basis in Newtonian dynamics. Its central argument is that experience, in the sense of empirical observation, is not at the basis of every scientific discovery—in fact, some of the greatest scientific revolutions, such as the Copernican, Newtonian, and Einsteinian revolutions, put forth hypotheses that could not have been the result of empirical evidence—it was only their validity that was confirmed with empirical evidence and testing.

In the second half, Popper discusses why philosophy and metaphysics are problematic when they profess to explain the proper functioning of the whole universe and human experience. His central argument is that philosophies that explain everything are not falsifiable, and there can exist logically-sound statements that are nevertheless false.

Popper begins the first half by discussing the main purpose of the philosopher, which is to find a riddle, problem, or paradox previously unseen and undiscovered, and attempt to solve it. Kant is the first to attempt to tackle the riddle of natural science, about its empirical status and its ability to impart knowledge and truth.

Kant is a firm believer in Newton’s celestial mechanics, and his definition of natural sciences is synonymous with Newtonian law. He believes Newton’s theory to be a true reflection of how the universe functions. However, Kant points out that, despite their validity, Newtonian forces are not observable. He concedes that they may be measurable, but only if we presuppose the validity of Newtonian dynamics—after all, people cannot measure forces without a theory on their existence and application.

Kant concludes that certain theories must be the result of pure reason, and they cannot be logically derived only from observations. Popper concurs with Kant on this point and proclaims that the same could be said of Nicolaus Copernicus’s discovery of the earth revolving around the sun. It is clear Copernicus did not have the tools to observe the solar system from space; in fact, his imagination most likely originated from the Platonic and mythological idea that the sun fulfills an important role in the realm of visible things. After theorizing the centrality of the sun, Copernicus could use instruments to measure and attempt to prove his theory.

However, this line of reasoning gives rise to a paradox of experience: If Newton’s discoveries cannot be the result of pure experience and must have a theory to guide empirical observation, then the same could be applied to everyday life. People cannot make sense of their everyday experiences without a framework to guide their thoughts and actions. This leads to an unbridgeable gap between the physical, natural world and people’s capacity to understand it, which arises purely from abstract thoughts. Kant summarizes this paradox thus: “Newton’s dynamics goes essentially beyond all observations. It is universal, exact and abstract; it arose historically out of myths; and we can show by purely logical means that it is not derivable from observation-statements” (190). He concludes that humans must be imposing laws upon the natural world, and the world that humans know is but the result of their interpretation, based on observable facts.

Popper believes this paradox is actually a false dichotomy, a consequence of Kant’s unconditional belief in the truth of Newtonian dynamics. However, even parts of Newton’s law are now outdated, in light of Einstein’s theory of gravitation. Kant’s philosophy was therefore inadvertently based on a dogma, and if Newton’s observations (which Kant equates to the natural world) can be problematic, then Kant’s conclusion must be revised. Popper believes that, while it is true people impose their observations on nature, it does not follow that they are always successful at doing this—thus, human intellect is not drawn from nature but imposed upon it, with varying degrees of success.

Popper’s revision implies that reason is capable of multiple interpretations, all of which must be tested. Pure reason, then, is a free creation of the mind, a result of intuition and imagination, and an attempt to understand the laws of nature intuitively. However, rather than force these interpretations on nature, Popper encourages testing these creations and attempting to elicit from them negative answers. From this, he concludes that while pure observation may not reflect the true nature of the world, empirical tests based on experience and observation are reliable measures to falsify a theory. All empirical tests are therefore, in nature, attempted falsifications, and it follows logically that the objective state of nature can be best approximated through a process of trial and error.

The second half of this chapter tackles the problem of irrefutability of philosophical theories. Its central argument is that theories that proclaim to understand everything with no room for falsification must be wrong.

Popper defines philosophical theories as doctrines that attempt to predict the state of things. For example, Kant believed that a complete understanding of the psychological and physiological conditions of the human body, coupled with a full understanding of the environment, would make it possible to predict human behavior as assuredly as astronomy can predict solar eclipses. Popper summarizes this idea thus: “The future of the empirical world (or of the phenomenal world) is completely predetermined by its present state, down to its smallest detail” (193).

Post-Kantian philosophers have been inspired by his theories and attempted to reconcile, in their own ways, the bridge between subjective human experience and objective reality (which Popper calls “things-in-themselves”). Other examples of philosophical theories include idealism (the world is how people dream it to be); irrationalism (humans have irrational or supra-rational experiences in which they experience the real world, and so they have some kind of knowledge of reality); voluntarism (people have the ability to decide their visions and actions, therefore reality is the same as their will); and nihilism (human existence is characterized by meaninglessness and boredom, in which people are nothing in the grand scheme of things, therefore reality is nothingness).

Popper believes all of these theories are false. He is an indeterminist, a realist, and a rationalist. He does doubt that humans will ever gain complete knowledge of the real world, but this does not mean that reality is nothing but a reflection of human will. He pities nihilists who find no pleasure in anything.

Most importantly, Popper upholds that all these theories are false because they are irrefutable. They are irrefutable both in the purely logical sense and in the empirical sense. They are logically irrefutable because a theory’s validity cannot be inferred from its consistency. They are empirically irrefutable because they profess to be compatible with every possible experience. It is this irrefutability that makes them false, and this can be easily proven: If both determinism and indeterminism are logically and empirically irrefutable, then they both must be true, but this cannot be, as they defend opposite realities.

Popper concludes, therefore, that there must be irrefutable statements that are false. For example, “Today is Monday” and “Today is not Monday” are both logically irrefutable linguistically and based on pure logic, but both cannot be true on the same day. If today is Tuesday, then the first statement must be false, even if its logic is sound.

Existential statements about the whole universe, unlike mathematical or scientific theories, are irrefutable because they cover themes so broad that there exists no method by which they can be falsified. Since philosophies or metaphysical theories are typically existential statements, they cannot be falsified, and thus it is impossible to evaluate their validity.

The solution to this problem lies in rationalism. Popper points out that every observation meant to describe some aspect of reality must be an attempt to answer a question or tackle a problem. For them to hold any weight, they must be testable. By contrast, frivolous declarations based on fancy are not up for debate and are not attempts to understand reality. Thus, philosophical theories, insofar as they attempt to be rational, must try to solve some problems about the world. As long as they are thus tethered, they can be rationally discussed and falsified.

Theories that are non-empirical and irrefutable may still be discussed in relation to the very real problem they are trying to solve. People can ask whether one philosophical theory is better at answering the question than another. Thus, Kant’s problem explained in the first half of the chapter dissolves. Popper summarizes it thus: Philosophy’s purpose is to find problems to solve. Once the problem is found, it is final, but the solution to the problem is never final. Theories, empirical or not, philosophical or scientific, must be attempts to find the answer through conscientious critical examination and rigorous testing.

This chapter references British philosopher Gilbert Ryles’s Presidential Address to the Aristotelian Society, Knowing How and Knowing That (1945), to discuss how the rules of logic function and how language can at times affect its use. Popper defends the idea that rules of inference are meta-linguistic.

Popper first establishes that people can reason intuitively, without needing to be taught pure logic, just as a pianist may know how to perform without understanding the intricacies of music theory. Most ordinary people can apply the basic rules of inference in their reasoning. However, there is a need to fully understand these rules in order to ascertain their validity, and it is the job of the logician to break them down to their simplest forms. Although it is not necessary to understand them fully to appreciate Popper’s essay, he summarizes a few common rules to help the average reader.

Consider the following statements: “Xyler is the father of Yolanda” and “Yolanda is the mother of Zion.” Intuitively, people can infer that Zion is the grandson of Xyler. These statements are expressed using artificial symbols in pure logic:

‘x R y’

‘y S z’

‘R ‘S = T’

In this scheme, x, y, and z stand for the individuals Xyler, Yolanda, and Zion respectively. The symbols ‘R, ‘S, and T’ stand for the relation between the individuals and the symbol = expresses an equality of extension between the symbols ‘R, ‘S, and T’, in which the inferred T’ stands for grandson.

This formulation is unconditional, in that it asserts that the fundamental logic behind the very act of inferring information is sound. It is not hindered by language either—these rules are sound no matter the language they operate in. There are a multitude of common rules of inference, but for the purpose of this book, it suffices to understand that they prove the soundness of basic rules of logic and they are meta-linguistic. In sum, rules of inference are “statements about statements” and they remain valid no matter what language is used.

The rules of inference used by logicians should not be confused with the rules of arithmetic used in calculus, which are very similar in style. For example, consider the following scheme:

“For all R, S, and T; and for all x, y, and z: if x R y and y S z, and R’S=T, then x T z.”

This formula makes use of hypotheticals and is only applicable conditionally to the relations it describes. It does not assert anything about the validity of the inference itself and in fact assumes that it is logical, the same way the pianist performs well, without using music theory to explain why the piece sounds good. Symbols used by mathematicians therefore accomplish a different goal from those of the logician. The two should not be confused, because while the logician’s symbols make universal and unconditional claims about the soundness of logical reasoning itself by breaking it down into its simplest forms, arithmetic calculi can easily fall prey to logical truisms, which are statements that are true but do not impart much informational or logical weight (e.g., “all rocks are either heavy or they are not” or “if x is a table, then it is a table”) (202-203).

To illustrate this difference further, Popper establishes the following three rules:

1.  Arithmetic calculi are semantical systems and rely on language

2.  They are designed to facilitate the understanding of certain things, but do not serve all purposes

3.  Once calculus is applied to reality, it loses its logical status and becomes empirically refutable; if it is not applied to reality, then it remains logically true on a universal basis

The first rule states that arithmetic calculi, insofar as they try to help people make sense of the world, must be tethered to a language. The statement “200-120=80” would not be possible in a language lacking the ability to count beyond 5.

The second rule argues that not all of these mathematical rules are designed to be used in every scenario. For example, natural numbers are used to count entities, such as the number of crocodiles in a zoo whereas real numbers might be used to measure velocities or distances. It would be counterintuitive to use real numbers, such as pi, to count crocodiles. Thus, not all calculi of arithmetic are applicable to all instances of reality.

The third rule separates logic from applied mathematics. The rule “2+2=4” remains universally true insofar as it is not applied to physical objects or specific scenarios. This is because once it is applied, it can become empirically falsifiable. For example, if there are two apples in a basket and someone adds two more, then there will be four apples in the basket for the time being. However, if there are two male rabbits in the basket and two female rabbits are added, very soon, the basket will hold more than four rabbits. Popper concludes that, if these arithmetic rules are abstract, then they will always be true but reveal nothing about reality. If these rules are applied to reality, then they become conditional to the specific scenario at hand and lose their universality.

Rules of inference and calculi in general are thus not proven to be applicable to reality, since logic has no physical shape and things that have physical shapes might not follow universal logical rules, which raises the question of how rules of inferences are connected to reality.

Ryles theorizes that rules of inference fundamentally describe procedures, rather than objects, things, or facts. They are a guidebook created by people to help them navigate and make sense of the world. Their purpose is not to force reality to fit, but to help guide people’s actions. In fact, Ryles suggests that there is no way to figure out whether reality in its physical form really does “fit” these rules.

This conclusion bypasses the original question by making it irrelevant: It fundamentally asserts that whether reality fits human logic or not does not matter, since rules of inference are only there to help humans navigate the world. In other words, it modifies the original question from “why are the rules of logic applicable to reality?” into “why are the rules of logic useful to humans?” (205).

Popper accepts this conclusion for the time being. Despite its instrumentalist nature—it reduces rules of inference to their use instead of trying to probe the real question of whether our logic reflects the functioning of the physical world—it is a tentative answer that can give rise to more research on the topic of logic and arithmetic.

This chapter focuses on how to encourage scientific progress. Science’s capacity to grow is the fundamental factor in its empirical and rational character. Popper makes an important claim about scientific growth: It is not characterized by the accumulation of knowledge, but by the constant replacing of old, less accurate theories with new, more satisfactory ones. This is done primarily through intellectual reasoning and trial-and-error.

Popper begins by commenting on how to define progress. His first thesis is that people can discern whether a scientific theory might be an improvement or not, before tests are even performed, based on its potential satisfactoriness. The greater the explanatory and predictive power of a theory, the more severely it can be tested by comparing its predictions with observations. In other words, the higher the empirical content of a theory, the more testable it is, and the more satisfactory it will be if it withstands the tests.

Popper’s argument can be easily understood with a quick example: 

Let p(T) stand for the probability of it being Tuesday.

Let p(r) stand for the probability of it raining.

The probability of p(T) and p(r) taken individually is fairly high, with p(T) having a fixed 1/7 chance of occurring in any given week. However, the probability of both happening at the same time—that it will be a rainy Tuesday—is much lower. Thus, the higher the content of a theory, the less probable it is to happen, and the more valuable it is. A theory that is too poor in content (e.g., “it will rain sometime in the future”) has little predictive power and, thus, little scientific use. No tests need to be conducted for people to understand that such a theory has little scientific use.

On top of high informative content, the best scientific theories must have a high degree of falsifiability. Statements like “it will rain someday” are not very falsifiable and thus possess little worth. A theory that attempts to accurately predict the weather per hour for the coming week, on the other hand, has a high chance of being falsified, but it is also much more valuable if it ends up withstanding the tests. Popper concludes that the goal of scientific progress is to aim for high informative content and highly falsifiable theories. 

The Copernican, Newtonian, and Einsteinian revolutions were all instances of scientists formulating highly unlikely theories that overthrew long-held assumptions and have, for the most part, withstood the most rigorous tests. Popper then asks how these leaps in progress come about. Popper suggests they were not purely discovered by chance, but were the result of rational thinking, imaginative power, and most importantly, the consequences of disappointed expectations for existing theories.

Thus, scientific progress can be hindered by a lack of imagination or interest; a misplaced faith in formalization; or by authoritarianism. Structural rigidity and the inability to think freely—two consequences of formalization and authoritarianism— prevent scientific growth because they hinder imagination and do not tolerate attempts at falsification. Thus, there is no need to pursue a formalized deductive system, as establishing axiomatic rules of deduction can easily lead into the trap of dogmatism. What is now accepted as common-sense facts (such as the earth revolving around the sun) might have been considered so imaginative as to be heretical when they were discovered (such as during Copernicus’s time). 

The author also stresses the importance of considering scientific growth not as a process of accumulating more and more theories, but one that progresses from problems of ever-increasing depth. In other words, the basis of science is the process of solving problems. It is not the result of observing the world. Scientific progress is, in sum, the process of solving ever more intricate problems using theories of high-informative content that are easily falsifiable.

In the second half of the chapter, Popper defines what he means by “truth.” While it is possible to discuss scientific progress without mentioning the elusive idea of “truth,” and Popper used to avoid the topic for lack of a definite answer, he has since amended his views.

There are two major competing views for what consists of “truth.” The first is objectivism and the second is subjectivism. To understand their differences, Popper proposes replacing the word “truth” with “corresponding to the facts.” Consider the following two statements:

1. The statement “snow is white” corresponds to the facts (i.e., reflects the “truth”) if, and only if, snow is indeed white.

2. The statement “snow is red” corresponds to the facts (i.e., reflects the “truth”) if, and only if, snow is indeed red.

According to subjectivism, both 1 and 2 can correspond to the facts (reflect the “truth”) if it fits somebody’s criteria. The theory proposes that knowledge is a “special kind of mental state” or “disposition” characterized by history or physiology or some other belief system. In other words, subjectivism defines truth in terms of “the sources or origins of human beliefs” (225).

Objectivism, on the other hand, argues that a theory may be true (i.e., correspond to the facts) even if no one believes it. Alternatively, a theory can be false (i.e., not correspond to the facts) even if it is widely believed to be true.

Popper has firmly rejected subjectivism because it cannot be falsified. If snow can be at the same time white and red if it fits specific individual experiences, then no theory can ever be considered false under these circumstances. In other words, the very idea of falsifiability implies there exists an objective truth, which can be approximated if not reached, with testing.

There is a third, less popular view on the topic of “truth” that Popper disagrees with. Irrationalists and skeptics believe truth does not exist, or if it does, it is unknowable to humans. To disprove this idea, Popper proposes the concept of “verisimilitude,” which calculates the “degree of truth” predicted by a theory. Verisimilitude can be tangibly measured: The more number of assertions a theory makes, the more precise they are, the more details they describe, the more rigorous tests it passes, and the more problems it can answer, the greater its explanatory power. To reprise an earlier example, both “it will rain some day” and “it will rain at 6 PM next Friday” can be “true,” but their degrees of truth (their verisimilitude) differ. Through verisimilitude, it can be proven that there is indeed such a thing as being “closer” or “farther” to an objective truth. Popper underlines that this is not merely a semantic trick, but a logical deduction.

The concept of verisimilitude can be easily applied to the social sciences. For example, when discussing why a specific historical event happened, it is often not possible to come up with a simple explanation. However, some theories will always have more explanatory power than others. They are therefore better approximations and closer to the “truth.”

Popper concludes by listing three requirements for the progress of science and knowledge. The first is that new theories should be new, powerful, unifying, and simple ideas that connect two hitherto unconnected things. In other words, they must have high informational content and high degrees of falsifiability. The second is that it must be independently testable. Any theory that cannot be tested cannot be falsified, and thus their truth content cannot be verified. The third is that, for the theory to be considered good, it must pass severe empirical tests. It must not be too safe, for science would stagnate if scientists were content with making easily- confirmed predictions. Moreover, considering science is a constant progress of refinement, researchers should not be deterred by the idea that their theory is short of flawless. However, the test should not be falsified too quickly—it must pass some severe tests for it to have widespread applicability.