Mathematics: The Loss of Certainty (Oxford Paperbacks)

I decided to write a review in 2006, after reading Walt Peterson's one, in which he says "My favorite mistake, and perhaps his [Kline's] most blatant blooper, is Kline's statement that the Loewenheim-Skolem Theorem implies Goedel's Incompleteness Theorem; he thinks that models with different cardinalities cannot satisfy the same sentences".

Then I was distracted by other matters, and it was only yesterday, when the book fell while I was rearranging others, that I remembered the issue.

Kline speaks about the L-S theorem in pp. 271-272 of the PB edition, and, in my opinion, nowhere does he say what Peterson accuses him of. The nearest he comes to it is in p. 273, where he writes: "The L-S theorem is not totally surprising. Gödel's incompleteness theorem does say that every [consistent] axiomatic system is incomplete. ... [six lines of text] ... . But the S-L theorem denies categoricalness in a far stronger or more radical way [than Gödel's]...". This is all. And, contrary to what Peterson says, in p. 272 Kline doesn't deny but on the contrary ASSERTS (using other words) that the L-S theorem means that models with different cardinalities CAN satisfy the same sentences: otherwise he wouldn't have mentioned it to reinforce his case about the loss of certainty. Did Peterson and I read the same edition of the book?

As for the rest, I readily admit that it may contain errors (that I, not being a professional mathematician, may have easily overlooked or not spotted). But Goethe was said -arguably- to be the last man who was able to know all that there was to know in his time. A mathematician nowadays can't possibly be at home in more than a small fragment of his chosen field. I remember the criticism levelled at Penrose by a logician in the number of the magazine "Sophia" totally dedicated to a discussion of his then new book "The Emperor's New Clothes". Or Fefferman's rebuttal, in "Psyché 2", of some inconsequential (for the sake of his central argument) mistakes Penrose had made in relation to Gödel's theorem in "Shadows of the Mind". Honest small mistakes are unavoidable in any single-person attempt to survey a large field; differences in interpretation even more so.

This said, I admit that Kline is maybe too inclined to slant the book toward his own point of view. But, as somebody -was it Nietzsche?- has said, impartiality is impossible to attain; the most that one can expect are honest biases, i.e. to interpret facts according to one's worldview (dishonest biases being suppressing known facts).

I think that overall Kline has done a splendid job: his book is eminently readable, concise, avoids major mistakes, and is one of the rare ones that, while written for laymen, is not dumbed down to them. So, four stars (not five, as he IS a little too opinionated for my taste in the last two chapters).

Preface explains: "This book treats the fundamental changes that man has been forced to make in his understanding of the nature and role of mathematics. We know today that mathematics does not possess the qualities that in the past earned for it universal respect and admiration. Mathematics was regarded as the acme of exact reasoning, a body of truths in itself, and the truth about the design of nature." This view drives modernity.

"How man came to the realization that these values are false and what our present understanding is constitute the major themes. . . . Recognition of the limitations, as well as capabilities, of reason is far more beneficial than blind trust which can lead to false ideologies and even to destruction."

Startling admission by devoted professor of math!

Outstanding evidence that math is not provable truth. When explaining Pascal's effort to prove calculus, he writes on page 134 . . .

"Pascal, too, had ambivalent feelings about rigor. At times he argued that the proper "finesse" rather than geometrical logic is what is needed to do the correct thing just as the religious appreciation of Grace is above reason."

The introduction presents the profund failure of mathematics to prove itself as "truth". Quotes Weyl, one of the greatest mathematicians of the twentieth century:

" 'Mathematizing' may well be a creative activity of man, like language or music, of primary originality, whose historical decisions defy complete objective rationalization." (6)

1) Genesis of Mathematical Truth
2) The flowering of Mathematical Truth
3) The Mathematization of Science
4) The First Debacle: The Withering of Truth
5) The Illogical Development of a Logical Subject
6) The Illogical Development: The Morass of Analysis
7) The Illogical Development: The Predicament circa 1800
8) The illogical Development: At the Gates of Paradise
9) Paradise Barred: A New Crisis of Reason
10) Logicism vs Intuitionism
11) The Formalist
12) Disasters
13) The Isolation of Mathematics
14) Whither Mathematics?
15) The Authority of Nature

About 1500 the rebirth of Greek belief in nature made from mathematics entered Europe. The scholars were devout Christians. How could they devote a life to mathematics (pagan) and the Christian God? Simple. . .

"The answer was to add a new doctrine, that the Christian God had designed the universe mathematically. Thus the Catholic doctrine postulating the supreme importantance of seeking to understand God's will and His creations took the form of a search for God's mathematician design of nature. . . . Mathematical knowledge, the truth about God's design of the universe, was as sacred as any line of scripture." (35)

Kline notes that the unquestioned faith in divine mathematics was essential for the flowering of mathematics at that time. Copernicus and Kepler, both deeply devout Christians, demonstrate this clearly. Their faith drove their life-long labors.

Newton's effort to justify calculus also failed. It was used and loved because was physically true, page 135, not because it could be logically proved. Why was Newton convinced mathematics is true? Newton believed God used mathematics to design the world. Newton wrote in "Opticks" -

"What is there in places almost empty of matter, and whence is it the sun and planets gravitate towards one another, without dense matter between them? Whence is it that nature doth nothing in vain; and whence arises all order and beauty we see in the world? . . . How came the bodies of animals to be contrived with so much art, and for what ends were their several parts? Was the eye contrived without skill in optics, or the ear without knowledge of sounds? How do the motions of the body follow from the will, and whence is the instinct in animals? . . . And these things being rightly dispatched, does it not appear from phenomena that there is a being incorporeal, living, intelligent, omnipresent, who, in infinite space, as it were in his sensory, sees the things themselves intimately, and thoroughly perceives them." (59)

This unshakeable conviction of a mathematical designer, created unshakeable faith in mathematics.

Newton and Leibniz both pursued science for the glory of God. Leibniz wrote:

"It seems to me that the principal goal of the whole of mankind must be the knowledge and development of the wonders of God, and that this is the reason that God gave him the empire of the globe." (60)

Page 162, "Students of the calculus remains perplexed, and the best they could do was to follow the advice of d'Alembert : "persist and faith will come to you."

"Indifference to and even dissmissal 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. 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." (78)

This new mathematics (geometry) is key to understanding modernity.

Einstein's famous comment in 1921 makes the point: "Insofar as the propositions of mathematics give an account of reality they are not certain; and insofar as they are certain they do not describe reality." (97)

Deep insight that is not well known.

The dream of safe, certain, provable mathematical truth, was - well - a dream. "Mathematicians were mislead by that 'evil genius' Euclid." (313)

Quotes Schopenhauer: "it is necessary to demand above all that one abandon the preconception that consists in believing that demonstrated knowledge is superior to intuitive knowledge."

"The concept of proof then, large as it has loomed in the public mind and in the publications of mathematicians, has not played the role commonly assumed. . . . As far back as 1928 , G. H. Hardy poke out with his usual bluntness: There is strictly no such thing as mathematical proof; we can, in the final analysis, do nothing but point. . . . Rhetorical flourishes designed to affect psychology, pictures on the board n lectures, devices to stimulate the imagination of pupils." (314)

Whitehead: "The conclusion is that Logic, conceived as an adequate analysis of the advance of thought, is a fake. . . . My point is that in the final outlook of philosophic thought cannot be based upon the exact statements which form the basis of special sciences. The exactness is false." (315)

Stunning. The modern world is lead by trust in such "false exactness". This is "faith" without reason, not "faith" in reason.

Ends by quoting Alford North Whitehead, " "let us grant that the pursuit of mathematics is a divine madness of the human spirit." Madness, perhaps, but surely divine."

Explains that the loss of faith in math began with the loss of faith that God is, as Galileo taught, a mathematician who used math to create the world.

Perhaps trust in math will return with trust in the Creator.

Kline is a daring man. In the face of all of his own mathematical genius, he comes to terms with the specific advances, not just in mathematics, but in human thinking and discovery, and the result is, I would say, superficially a loss, but actually--due to his assessments and honesty--an exceptional gain.

Kline is a really clear and entertaining writer. He doesn't patronize the less math inclined reader.
One aspect of the math debate is whether real math only deals with theory. This book would enable someone who hasn't studied this and similar issues to understand the grounds for the debate.

I stumbled across this book in a footnote in another book that I read. I certainly enjoyed this book. It provides an interesting and somewhat disturbing look at the history of mathematics. I would recommend this book for its honesty, thoroughness and expressiveness. Kline is a very good writer and he definitely made me think hard about a difficult subject.

If you are interested, I have posted a longer review on my blog. See my profile for the blog address.

Morris Kline knows how to take the gas out of noise makers who idolize math.

He does it with grace and a sincere appreciation of the subject's true historical and practical worth.
Any math educator worth their salt would make this book a must read.

Amazing unsettling attack on our oblivious acceptance of certain seemingly solid fields of thinking. College professors beware. Students -here's your chance ot one-up your profs.

Math, strengthened by schisms with God yet weakened by schisms with Truth, forfeited her rule of science without religious and secular rhetoric, Kline laments, by further breaking with natural 'subjects' like physics for cloistered realms like topology.