Philosophy of the Modern Era

Philosophy of Logical Analysis

In philosophy, since the time of Pythagoras, there has existed a dichotomy between those whose thoughts were predominantly stimulated by mathematics and those influenced more by empirical sciences. Plato, Thomas Aquinas, Spinoza, and Kant belong to the faction that might be termed mathematical; whereas Democritus, Aristotle, and the empiricists of the modern era, from Locke to the present day, align with the opposing faction. In contemporary times, a philosophical school has emerged with the aim of eradicating Pythagoreanism from the principles of mathematics and uniting empiricism with an interest in the deductive components of human knowledge. The goals of this school may be less flamboyant than those of most philosophers of the past, yet many of its achievements are as significant as those of scientists.

This philosophy owes its origins to the accomplishments of mathematicians who sought to cleanse their discipline of errors and careless conclusions. The great mathematicians of the seventeenth century were optimistically inclined and pursued rapid results, thus failing to provide a reliable foundation for the calculus of infinitesimals and analytical geometry. Leibniz believed in the reality of infinitesimals, but while this belief corresponded with his metaphysics, it lacked a solid basis in mathematics. In the mid-nineteenth century, Weierstrass demonstrated how calculus could be grounded without infinitesimals, finally rendering it logically sound. Following him, Georg Cantor entered the realm of mathematics, developing the theory of continuity and infinite numbers. The term “continuity,” before Cantor offered a definition, was ambiguous, conveniently misused by philosophers like Hegel, who sought to introduce metaphysical confusion into mathematics. Cantor ascribed precise meaning to this term and showed that continuity, as he defined it, is a concept essential for mathematicians and physicists alike. As a result, many teachings of mystics, such as Bergson, were rendered obsolete.

Cantor also resolved an age-old logical conundrum concerning infinite numbers. Consider the series of whole numbers beginning with 1; how many are there? It is clear that their number is infinite. Before a thousand lies a thousand numbers, and before a million lies a million. No matter what finite number we name, it is evident that the total count of whole numbers exceeds this, since there are exactly as many numbers from one to the named number as the named number itself, and there are still others greater than that. Hence, the number of finite whole numbers must be an infinite number. Yet, an intriguing fact follows. The number of even numbers must be equal to the total number of whole numbers. Consider two series:

1. 1, 2, 3, 4, 5, 6...
2. 2, 4, 6, 8, 10, 12...

Each number in the upper series corresponds to a number in the lower series, thus the number of members in both series must be identical, even though the lower series contains only half of the members of the upper series. Leibniz, upon noticing this, deemed it a contradiction and concluded that while infinite sets exist, there are no infinite numbers. Conversely, Georg Cantor boldly denied the existence of a contradiction here. He was right: it merely appears strange.

Georg Cantor defined an “infinite” set as one whose subsets contain as many members as the entire set itself. On this basis, he was able to construct the most fascinating mathematical theory of infinite numbers, incorporating into the domain of precise logic an entire field previously filled with mysticism and confusion.

The next significant figure was Frege, who published his first work in 1879 and, in 1884, provided his definition of “number.” However, despite the fact that his inquiries heralded a new era, he remained unrecognized until I drew attention to his work in 1903. It is noteworthy that all definitions of number proposed prior to Frege contained elementary logical errors. Traditionally, “number” was identified with “a multitude, a set.” Yet a specific example of “number” is a definite numeral, say, 3, while a specific example of 3 is a particular trio. The trio is indeed a multitude, while the class of all trios, which Frege identifies with the number 3, constitutes a multitude of multitudes, and the number in general, of which 3 is a particular case, is a multitude of multitudes of multitudes. The elementary grammatical error of conflating a number in general with a simple collection of a given trio rendered the entire philosophy of number prior to Frege an entanglement of absurdity in the strictest sense. From Frege's works, it follows that arithmetic and pure mathematics, in general, are nothing more than an extension of deductive logic. This refutes Kant’s theory that arithmetic statements are “synthetic” and contain a reference to time. The further derivation of pure mathematics from logic was meticulously carried out by Whitehead and myself in Principia Mathematica.

It gradually became evident that a considerable portion of philosophy could be reduced to what is termed “syntax,” although this term should be employed here in a broader sense than has been accustomed thus far. Some scholars, particularly Carnap, proposed a theory that all philosophical problems are in fact syntactical; if one avoids errors in syntax, then any philosophical problem will either be resolved through syntactical means or its insolubility will be demonstrated. I believe, and Carnap would now concur, that this is an exaggeration, but there is no doubt that the utility of philosophical syntax for resolving traditional problems is considerable.

I shall illustrate this utility with a brief explanation of what is known as the theory of descriptions. By “description,” I mean a phrase such as “the current president of the United States,” where a particular person or thing is designated not by name but by some attribute believed or known to belong exclusively to that person or thing. Such phrases have caused much difficulty in the past. Suppose I say, “The golden mountain does not exist,” and you inquire, “What precisely does not exist?” It seems that if I respond, “The golden mountain,” I attribute some form of existence to it. Clearly, if I say, “The round square does not exist,” this would not be the same but rather a different assertion. Here, it appears to imply that the golden mountain is one thing and the round square is another, even though both do not exist. The purpose of the theory of descriptions is to overcome these and other difficulties.

According to this theory, if a statement containing a phrase in the form “X is Y” is analyzed correctly, then the phrase “X is Y” disappears. For example, let us consider the statement “Scott was the author of Waverley.” The theory interprets this assertion as follows: “One and only one person wrote Waverley, and that person was Scott.” Or more fully: “There exists one object C such that the statement ’X wrote Waverley’ is true if and only if X is C, and false otherwise. Moreover, X is Scott.”

The first part of this statement up to the words “moreover” is defined as signifying: “The author of Waverley exists (or existed, or will exist).” Thus, “The golden mountain does not exist” means: “There is no object C such that the statement ’X is golden and has the form of a mountain’ is true if and only if X is C, but not otherwise.”

With this definition, one need not ponder what is implied when we say, “The golden mountain does not exist.” Existence, according to this theory, can only be affirmed concerning descriptions. We can assert: “The author of Waverley exists”; but to say, “Scott exists,” is either grammatically poor or syntactically very flawed. All this elucidates two millennia of foolish discourse concerning “existence,” initiated in Plato’s Theaetetus.

One of the outcomes of the philosophical endeavor we are considering is the dethroning of mathematics from the majestic pedestal it has occupied since the time of Pythagoras and Plato, along with the dismantling of the bias against empiricism that arose from this elevation. Indeed, mathematical knowledge is not derived from experience through induction; the foundation upon which we assert that 2 + 2 = 4 does not rest on the fact that we frequently observe through experience that one pair combined with another results in four. In this respect, mathematical knowledge remains non-empirical. Yet, it is also not an a priori knowledge of the world. It is, in fact, merely verbal knowledge. "3" signifies "2 + 1," and "4" signifies "3 + 1." Hence, it follows (albeit the proof is lengthy) that "4" means the same as "2 + 2." Thus, mathematical knowledge has ceased to be mysterious; it possesses the same nature as the "great truth" that there are 3 feet in a yard.

Physics, like pure mathematics, has also provided material for the philosophy of logical analysis. This is especially true concerning the theory of relativity and quantum mechanics. For the philosopher, the replacement of space and time with spacetime in the theory of relativity is of paramount importance. Common sense posits that the physical world consists of "things" that endure for a certain duration and move through space. Philosophy and physics have evolved the notion of "thing" into that of "material substance," asserting that material substance is composed of very small particles that exist eternally. Einstein, however, replaced particles with events; according to him, each event relates to every other event through what is termed an "interval," which can be decomposed in various ways into a temporal element and a spatial element. The choice among these various modes is arbitrary, and none of them is theoretically more preferable than the others. Given two events A and B in different regions, it may turn out that according to one convention they are simultaneous, according to another, A occurs before B, and according to a third, B occurs before A.

From all this, it follows that the material of physics should consist of events rather than particles. What was previously considered a particle should be viewed as a series of events. This series of events that replaces a particle possesses important physical properties and therefore must be examined. However, this series of events has no more substantiality than any other series of events we might arbitrarily select. Thus, "matter" is not a part of the finite material of the world but merely a convenient means of binding events together.

Quantum theory reinforces this conclusion, but its primary philosophical significance lies in its treatment of physical phenomena as potentially discontinuous. It suggests that within an atom (interpreted in the previously described sense), a certain stable state exists for some time, which is then abruptly replaced by another stable state differing from the first by a finite amount. It was long assumed that motion is continuous, but it has become clear that this was merely a prejudice. However, philosophy based on quantum theory has yet to be adequately developed. It seems to me that it will require an even more radical departure from the traditional teaching on time and space than that necessitated by the theory of relativity.

As physics has rendered matter less material, psychology has made spirit less spiritual. In the previous chapter, we compared the association of ideas with the conditioned reflex. It is evident that the latter, having replaced the former, is far more physiological (this is the only example; I do not wish to exaggerate the applicability of the conditioned reflex). Thus, from two opposing ends, physicists and psychologists are converging, making the concept of "neutral monism," proposed by W. James, who criticized the notion of "consciousness," more plausible. The distinction between spirit and matter entered philosophy from religion, although it long seemed sufficiently justified. I believe that both spirit and matter are merely convenient means of grouping events. I must concede that some individual events belong solely to the material group, while others belong to both groups and are thus simultaneously spiritual and material. Such a conception significantly clarifies our understanding of the structure of the world.

Modern physics and physiology shed new light on a very old problem of perception. If there is something that can be termed "perception," it must, to some extent, involve the influence of the perceived object, and it should resemble the object in order to serve as a source of knowledge about it. The first condition can only be fulfilled if there exist causal chains that are more or less independent of the rest of the world. According to physics, this is indeed the case. Light waves travel from the Sun to the Earth, following their own laws. However, this is only approximately true. Einstein demonstrated that gravitational force acts on light waves. Upon reaching our atmosphere, they undergo refraction, with some scattering more than others. When they encounter the human eye, certain phenomena occur that do not exist elsewhere, leading to what we call "seeing the sun." Yet, although the sun as perceived by us differs greatly from the sun as understood by astronomers, it still serves as a source of knowledge about the latter because "seeing the sun" differs from "seeing the moon" in such a way that this distinction is causally related to the difference between the sun and the moon as perceived by astronomers. However, what we can know about a physical object in this manner consists only of certain abstract properties of its structure. We may discern that the sun is, in some sense, round, though not strictly in the sense that what we see is round. Yet we have no grounds to assume that it is bright or warm, as physicists can explain why it appears bright or warm without positing that it is inherently so. Thus, our knowledge of the physical world comprises only abstract and mathematical knowledge.

Contemporary analytical empiricism, which I wish to present in this chapter, diverges from the analytical empiricism of Locke, Berkeley, and Hume in that it incorporates mathematics and develops a powerful logical technique. Consequently, it is capable of arriving at certain answers to questions of a scientific rather than philosophical nature. In comparison to philosophers who construct systems, logical empiricism holds the advantage of addressing each of its problems individually rather than inventing a general theory of the entire universe in one sweeping motion. Its methods in this regard resemble those of science. I have no doubt that, to the extent that philosophical understanding is possible, it will need to be sought through precisely such methods. I am equally confident that many very old problems may be fully resolved using these methods.

However, a vast domain traditionally included within philosophy remains where scientific methods are inapplicable. This realm encompasses finite problems of value; for instance, one cannot prove through science alone that enjoying the suffering of others is wrong. Everything that can be known can be known through science, but matters that are legitimately the domain of feeling lie beyond its scope.

Throughout its history, philosophy has consisted of two parts that have failed to harmonize with one another. On one side lies a theory of the nature of the world, and on the other, ethical and political teachings on how best to live. The inability to clearly distinguish these two facets has been a source of great confusion in thought. Philosophers, from Plato to William James, have allowed their beliefs about the structure of the universe to be influenced by the desire to instruct; believing (as they did) that certain convictions would lead to virtuous behavior, they devised arguments—often quite sophistic—to prove the validity of those beliefs. In my view, such bias is condemnable on both moral and intellectual grounds. Morally speaking, a philosopher who employs his professional abilities for anything other than impartial searches for truth commits an act of betrayal; and if he preemptively assumes, prior to inquiry, that certain beliefs—regardless of their truth—promote good conduct, he restricts the domain of philosophical reasoning to such an extent that philosophy becomes trivial; a true philosopher is prepared to examine all assumptions. When any restrictions—conscious or unconscious—are placed on the pursuit of truth, philosophy is paralyzed by fear, and the ground is prepared for governmental censorship, punishing those who express "dangerous thoughts"; indeed, the philosopher has already imposed such censorship on his own investigations.

Intellectually, the influence of erroneous moral considerations on philosophy has largely impeded progress. Personally, I do not believe that philosophy can prove or disprove the truth of religious dogmas; yet, beginning with Plato, most philosophers have felt it their duty to devise "proofs" of immortality and the existence of God. They found errors in the proofs of their predecessors: St. Thomas refuted St. Anselm’s arguments, and Kant critiqued Descartes. However, in doing so, they committed new errors of their own; to make their arguments appear valid, they had to distort logic, flood mathematics with mysticism, and insist that deeply rooted prejudices were divine revelations.

Philosophers who have made logical analysis the core of philosophy reject all of this. They openly acknowledge that the human intellect is incapable of providing definitive answers to many of the most significant questions for humanity, yet they refuse to believe in the existence of some "higher" mode of knowledge by which we can unveil truths hidden from science and reason. For this refusal, they have been rewarded by discovering that many questions, previously shrouded in the fog of metaphysics, can be answered with precision, and that there exist objective methods devoid of the philosopher's temperament, save for the desire to understand. Consider such questions as: What is number? What are time and space? What is spirit? What is matter? I do not claim that we can now provide final answers to all these ancient queries, but I assert that methods have been uncovered by which we can, as in science, progressively approach the truth, with each new stage emerging as a refinement rather than a rejection of the previous one.

In the tumult of conflicting fanaticisms, one of the few unifying forces is scientific truthfulness, by which I mean the habit of grounding our beliefs in observations and conclusions that are as "impersonal" and devoid of local biases and temperamental inclinations as possible for a human being. It is precisely in insisting on the incorporation of these admirable traits into philosophy and in devising a powerful method to render philosophy fruitful that the fundamental merit of the philosophical school to which I belong lies. The habit of rigorous truthfulness acquired through the practice of this philosophical method can extend to all areas of human activity. Wherever it exists, it will lead to a reduction of fanaticism and an increase in the capacity for empathy and mutual understanding. By relinquishing part of its dogmatic claims, philosophy does not cease to offer and inspire ways of living.