Mathematics: The Loss of Certainty by Morris Kline Reviewed by Southern Israelite

klineAfter studying through Dr. Clark’s Philosophy texts numerous times, his book The Philosophy of Science and Belief in God, and debating dozens of atheist educators about its contents, I found that I had a fundamental hole in my refutation of science: Mathematics. Gordon Clark, in his letter to Mr. M., admitted,

“Most unfortunately I did not end Chapter One properly. I should have pointed out that modern Calculus can no more explain how motion begins than ancient Zeno could.”

No need to despair. Morris Kline’s Mathematics: The Loss of Certainty (New York: Oxford University Press, 1980), satisfies this deficiency in full. This is a very difficult book in some sections but in all Kline gets his points across quite clearly. I recently finished reading through it. My readers will forgive my recent absence but this book took me about 35 hours to get through. It is not for the light of heart but very thorough.

Morris Kline (1908– 1992) was a Professor of Mathematics at New York University (NYU). Kline studied mathematics at New York University, earning his bachelor’s degree in 1930, his master’s degree in 1932, and his doctorate in 1936. From 1936-1942 he continued at NYU as an instructor. After WW2, Kline resumed his mathematical teaching at NYU, and became a full professor in 1952. He taught at NYU until 1975. All in all, the man had about 30 years of professional work in the field of mathematics and its application to Science.

This work is a summary of the nature and history of mathematics with its various failures, revisions and successes. All in all, the work is a complete refutation of the idea that mathematics is a demonstrable body of truths. As Kline mentions many times, mathematics is the essence of science (Kline says, “As we shall see, the most well developed physical theories are entirely mathematical.” [pg. 7]; “Descartes was explicit in his Principles that the essence of science was mathematics.” [pg. 43] ;). Given this premise, science falls completely out of the category of demonstration and is therefore impotent in its criticisms of religion. As Hermann Weyl said of The Principia Mathematica by Alfred North Whitehead and Bertrand Russell which based mathematics,

“not on logic alone, but on a sort of logician’s paradise, a universe endowed with an ‘ultimate furniture’ of rather complex structure…Would any realistically minded man dare say he believes in this transcendental world? …This complex structure taxes the strength of our faith hardly less than the doctrines of the early Fathers of the Church or of the Scholastic Philosophers of the Middle Ages.” (pg. 226)

Bertrand Russell said in Portraits from Memory, 1956,

“I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers wanted me to accept, were full of fallacies … I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was no more secure than the elephant, and after some twenty years of arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.” (pg. 229-230)

Kline begins his introduction by pointing out that there is no one theory of mathematics or geometry. There have been and are many different theories that men use to function in our physical world. This has been the difficulty for many serious mathematicians to take science seriously. Kline says,

“The loss of truth, the constantly increasing complexity of mathematics and science, and the uncertainty about which approach to mathematics is secure have caused most mathematicians to abandon science…The crises and conflicts over what sound mathematics is have also discouraged the application of mathematical methodology to many areas of our culture such as philosophy, political science, ethics, and aesthetics. The hope of finding objective, infallible laws and standards has faded. The Age of Reason is gone.” (pg. 7)

Kline summarizes the genesis of mathematics giving emphasis to the Greeks. The Pythagoreans’ metaphysics explained all reality through numbers. However, the materialism (atomism) of Democritus and its re-establishment through Galileo and Descartes replaced this idea. Yet, Kline reminds us that an objective materialism was refuted by David Hume. Kline says,

“Hume was equally dubious about matter. Who guarantees that there is a permanently existing world of solid objects? All we know are our own sensations of such a world. Repeated sensations of a chair do not prove that a chair actually exists. Space and time are but a manner and order in which ideas occur to us. Similarly, causality is but a customary connection of ideas…Man himself is but an isolated collection of perceptions, that is, sensations and ideas…Hence there can be no scientific laws concerning a permanent objective physical world; such laws signify merely convenient summaries of sensations…we have no way of knowing that sequences perceived in the past will recur in the future. Thus Hume stripped away the inevitability of the laws of nature, their eternality and their inviolability…Hume then, answered the fundamental question of how man obtains truths by denying their existence; man cannot arrive at truths. Hume’s work not only deflated the efforts and results of science and mathematics, but challenged the value of reason itself.” (pg. 74-75)

Immanuel Kant tried to answer this difficulty with his apriori forms. Kline summarizes Kant’s philosophy,

“He granted that we receive sensations from a presumed external world. However, these sensations or perceptions do not provide significant knowledge. All perception involves an interaction between the perceiver and the perceived. The mind organizes the perceptions and these organizations are intuitions of space and time. Space and time do not exist objectively but are the contributions of the mind. The mind applies its understanding of space and time to experiences which merely awaken the mind. Knowledge may begin with experience but does not really come from experience. It comes from the mind.” (pg. 231)

Kline then summarizes Kant’s attempt to answer Hume,

“Indifference to and even dismissal of God as the law-maker of the universe, as well as the Kantian view that the laws were inherent in the structure of the human mind, brought forth a reaction from the Divine Architect.” (pg. 78)

Kline then waxes prophetic stating, “God decided that He would punish the Kantians and especially those egotistic, proud, and overconfident mathematicians. And he proceeded to encourage non-Euclidean geometry, a creation that devastated the achievements of man’s presumably self-sufficient, all-powerful reason…It was precisely Euclidean geometry that ‘God’ attacked.” (pg. 78)

The problem that the Euclidean system faced was the parallel axiom or, as it is often referred to, Euclid’s fifth postulate. This postulate assumed upon the reality of infinite lines. Kline remarks, “Certainly experience did not vouch for the behavior of infinite straight lines, whereas axioms were supposed to be self-evident truths about the physical world.” (pg. 78) Thus this problem gave rise to non-Euclidean geometry: Hyperbolic and Elliptic. Kline admits, “the most significant fact about non-Euclidean geometry is that it can be used to describe the properties of physical space as accurately as Euclidean geometry does. Euclidean geometry is not the necessary geometry of physical space; its physical truth cannot be guaranteed on any a priori grounds.” (pg. 84)

Kline mentions that Einstein used a non-Euclidean geometry to create his theory of relativity. (pg. 344) Next Kline exposes the problems concerning the nature of numbers,

“Helmhotz made many pertinent observations. The very concept of number is derived from experiences. Some kinds of experience suggest the usual types of number, whole numbers, fractions, and irrational numbers, and their properties. To these experiences, the familiar numbers are applicable. We recognize that virtually equivalent objects exist and so we recognize that we may speak, for example, of two cows. However, these objects must not disappear or merge or divide. One raindrop added to another does not make two raindrops…Many examples may be adduced to show that the naive application of arithmetic would lead to nonsense. Thus if one mixes two equal volumes of water, one at 40 degrees Fahrenheit and the other at 50 degrees, one does not get two volumes at 90 degrees…We learn in chemistry that when one mixes hydrogen and oxygen he obtains water. But if someone takes two volumes of hydrogen and one volume of oxygen, he obtains, not three, but two volumes of water vapor. Likewise, one volume of nitrogen and three volumes of hydrogen yield two volumes of ammonia. We happen to know the physical explanation of these surprising arithmetic facts…Ordinary arithmetic also fails to describe the combination of some liquids by volume. If a quart of gin is mixed with a quart of vermouth one does not get two quarts of the mixture but a quantity slightly less. One quart of alcohol and one quart of water yield about 1.8 quarts of vodka.” (pg. 92-93)

Because of these difficulties and others Kline must admit, “Thus one cannot speak of arithmetic as a body of truths that necessarily apply to physical phenomena. Of course, since algebra and analysis are extensions of arithmetic, these branches, too, are not bodies of truth….It seemed as though God had sought to confound them with several geometries and several algebras just as he had confounded the people of Babel with different languages…Nature’s laws are man’s creation. We, not God, are the lawgivers of the universe. A law of nature is man’s description and not God’s prescription.” (pg. 95-98)

In passing, I find it curious that Kline mentioned that Euclid defined a point as “that which as no parts”. (pg. 101) If that is true, then things cannot be infinitely divisible. There must be a replacement of atomism for science to speak of points or coordinates. If monism is the response we are right back into the Pre-Socratic Era.

The next topic of interest that Kline presents is the calculus. Kline says, “calculus utilizes the concept of function, which, loosely put, is a relationship between variables.” (pg. 127) Yet Kline admits, “all of the logic that was missing in the treatment of number was also missing in the work with functions.” (pg. 128) Kline mentions that the difficulties with calculus concerned the nature of the derivative (Which deals with velocity. The problem is deriving a conclusion when the time is at 0) and the definite integral (Which involves deriving the area of figures bounded by curves. No matter how many intervals the area is divided up into, the sum is never reached). Many atheists appeal to Newton as the solution to Zeno’s paradox of the infinite. This is curious because Kline admits that Newton,

“abandoned the infinitely small quantity (ultimate indivisibles [Which I am assuming refers to atomism-DS])…then proceeded to give a new explanation of what he meant by a fluxion.’Fluxions are, as near as we please, as the increments of fluents [variables] generated in times, equal and as small as possible, and to speak accurately, they are in the prime ratio of nascent increments…’ Of Course such vague phraseology didn’t help much. As to his method of calculating a fluxion, Newton’s third paper is as crude as the first one…Undoubtedly he realized that his explanation of a fluxion was not satisfactory and so, perhaps in desperation, he resorted to physical meaning…Since the results of his mathematical work were physically true, Newton spent very little time on the logical foundation of the calculus…Newton had faith in Euclidean geometry but no factual evidence that it could support the calculus.” (pg. 135-136)

Next, Kline deals with Leibniz’s attempts to answer his critics regarding his calculus. Kline says, “In an article in the Acta of 1689, he said that infinitesimals are not real but fictitious numbers.” (pg. 137) Kline says, “Until the end of his life in 1716 Leibniz continued to make explanations of what his infinitely small quantities (infinitesimals, differentials) and his infinitely large quantities were. These explanations were no better than what we have seen above. He had no clear concepts or logical justifications of his calculus.” (pg. 139) Yet, as science does, it points the critics to the results and science pushes on with the subsequent generations believing that they have inherited knowledge when in fact they have not. Bishop Berkley criticized these men and especially Newton’s derivatives as the ratio of the evanescent quantities, “They are neither finite quantities, nor quantities infinitely small, not yet nothing. May we not call them the ghosts of departed quantities?” (pg. 146)

Next Berkley strikes the heart of the scientific excuse: “Berkley asked rhetorically ‘whether the mathematicians of the present age act like men of science in taking so much more pains to apply their principles then to understand them.’ ‘In every other science.’ he said, ‘men prove their conclusions by their principles, and not their principles by their conclusions.” (pg. 147) Kline adds,

“Even the one solace mathematicians derived from their work, namely, its remarkable effective applicability to science, can no longer be a comfort because most mathematicians have abandoned applications.” (pg. 307)

Kline provides a fascinating historical event on pages 149-151. The Mathematics section of the Berlin Academy of Sciences held a contest in 1784 for the contestants to provide a solution to the problem of the infinite in mathematics. In all, 23 papers were submitted and all were found completely incompetent. To further complicate calculus “Emile Picard said in 1905, ‘If Newton and Leibniz had known that continuous functions need not necessarily have a derivative, the differential calculus would never have been created.” (pg. 177)

Kline reviews Mathematics from the late 19th century to the 20th century,

“While logicism was in the making, a radically different and diametrically opposite approach to mathematics was undertaken by a group of mathematicians called intuitionists. It is a most interesting paradox of the history of mathematics that while the logicists were relying more and more on refined logic to secure a foundation for mathematic, other were turning away from and even abandoning logic. In one respect, both sought the same goal. Mathematics in the late 19th century had lost its claim to truth in the sense of expressing laws inherent in the design of the physical universe. The early logicists, notably Frege and Russell, believed that logic was a body of truths, and so mathematics proper if founded on logic would also be a body of truths, though ultimately they had to retreat from this position to logical principles that had only pragmatic sanction. The intuitionists also sought to establish the truth of mathematics proper by calling upon the sanction granted by human minds. Derivations from logical principles were less trustworthy than what can be intuited directly. The discovery of the paradoxes not only confirmed this distrust but accelerated the formulation of the definitive doctrines of intuitionism.” (pg. 230)

Intuitionism was an attempt to provide the basis for mathematics with a Kantian like philosophy of apriori forms. Kline says, “Some of intuitionism’s opponents agree that mathematics is a human creation, but believe that correctness or incorrectness can be objectively determined, whereas the intuitionists depend on self-evidency to fallible human minds…Another criticism of intuitionism is that it is not concerned with the application of mathematics to nature. Intuitionism does not relate mathematics to perception.” (pg. 242)

Kline criticizes the Kantian approach,

“this Kantian explanation that we see in nature what our minds predetermine for us to see does not fully answer the question of why mathematics works. Developments since Kant’s time such as electromagnetic theory can hardly be endowments of the human mind or the mind’s organization of sensations. Radio and Television do not exist because the mind organized some sensations in accordance with some internal structure which then enabled us to experience radio and television as consequences of the mind’s conception of how nature must behave.” (pg. 342)

This problem of how mathematical law relates to the physical world is a constant thorn in the flesh of science. Kline says, “The questions of how new ideas could enter mathematics and how mathematics can possibly apply to the physical world if its contents are derivable entirely from logic are not readily answered and were not answered by Russell and Whitehead.” (pg. 228)

There is no, one mathematics. Kline says, “In short, no school has the right to claim that it represents mathematics…since 1930 the spirit of friendly cooperation has been replaced by a spirit of implacable contention (pg. 276)…Thus mathematicians reached the stage where men held conflicting views of what may properly be designated mathematics-logicism, intuitionism, formalism, and set theory (pg. 309)…The formalists believe that logic alone does not suffice and axioms of mathematics must be added to axioms of logic in order to found mathematics. The set-theorists are rather casual about logical principles and some do not specify them. The intuitionists in principle dispense with logic…Mathematics grows through a series of great intuitive advances, which are later established not in one step but a series of corrections of oversights and errors until the proof reaches the level of accepted proof for that time. No proof is final (pg. 313)”

I will conclude this book review with a quotation by Albert Einstein,

“as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” (Geometry and Experience, Address to the Prussian Academy of Sciences in Berlin on January 27th, 1921)

I once asked “How did science recover from the second refutation of atomism (Zeno produced the first) in the 1930s, namely the splitting of the atom?” I asked this question to a friend of mine, Dr. James Wanliss who is the resident space physicist at Presbyterian College in South Carolina; from his site, “In 1995 he received an M.S degree in geophysics from the University of the Witwatersrand. In 2000 he received the Ph.D in physics from the University of Alberta, in Canada. He is the recipient of several awards and honors, notably an NSF CAREER award. His research work has been almost entirely supported by U.S. government agencies: NASA, and NSF.” Dr. Wanliss is the man who really turned me on strong to Dr. Gordon Clark. Wanliss is a Presbyterian and a member of the Church that I used to be a member of in South Carolina. I asked him once what he thought about Dr. Clark’s book on the Philosophy of Science. He said, “it is irrefutable”. Recently I asked him my question about science post-atomism and he told me that Monism replaced it as Science’s theory of metaphysics. Corporeal Monism is just sophistry for Pantheism.


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