Mathematics: The Loss of Certainty

Quotes

The first of the really troublesome mathematical contradictions was noted by Bertrand Russell (1872-1970) and communicated to Gottlob Frege in 1902. Frege at that time was just publishing the second volume of his Fundamental Laws in which he was building up a new approach to the foundations of the number system... Frege used a theory of sets or classes which involved the very contradiction Russel noted in his letter to Frege and published in his Principles of Mathematics (1903). Russell had studied the paradox of Cantor's set of all sets and then generated his own version.

Russell's paradox deals with classes. A class of books is not a book and so does not belong to itself, but a class of ideas is an idea and does belong to itself. A catlogue of catalogues is a catalogue. Hence, some classes belong to (or are included in) themselves and some do not. Consider N, the class of all classes that do not belong to themselves. Where does N belong? If N belongs to N, it should not by definition of N. If iN does not belong to N, i9t should by definition of N. When Russel first discovered this contradiction, he thought the difficulty lay somewhere in the logic rather than in mathematics itself. But this contradiction strikes at the very notion of classes of objects, a notion used throughout mathematics. Hilbert noted that this paradox had a catastrophic effect on the mathematical world.

- p. 247

The use of all three axioms, reducibility, infinity, and choice, challenged the entire logistical thesis that all of mathematics can be derived from logic. Where does one draw the line between logic and mathematics? proponents of the logistic thesis maintained that the logic used in Principia Mathematica was "pure logic" or "purified logic." Others, having the three controversial axioms in mind, questioned the "purity" of the logic employed. Hence they denied that mathematics, or even any important branch of mathematics, had yet been reduced to logic. Some were willing to extend the meaning of the term logic so that it includes these axioms.

In view of the controversies concerning Cantor's work and the axioms of choice and infinity, which reached high intensity during the early 1900s, Russell and Whitehead did not specify the two axioms as axioms of their entire system but used them (in the second edition, 1926) only in specific theorems where they explicitly call attention to the fact that these theorems use the axioms. However, they must be used to derive a large part of classical mathematics.

In his Philosophy of Mathematics and Natural Science (1949), Hermann Weyl said the Principia 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.

- MLOC p. 272

This science [mathematics] does not have for its unique objective to eternally contemplate its own navel; it touches nature and some day it will make contact with it. On this day it will be necessary to discard the purely verbal definitions and not any more be the dupe of empty words.

- MLOC p. 273

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 expected me to acceept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new field of mathematics, with more solid foundations than those that had hitherto been thought secure. ...after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.

- Bertrand Russell, Portraits from Memory (1958)

- MLOC p. 275

God made the integers; all the rest is the work of man.

- Leopold Kronecker (1823-1891)

...in 1882 Ferdinand Lindemann proved that $\pi$ is a transcendental irrational number. Apropos of this work Kronecker said to Lindemann, "Of what use is your beautiful investigation regarding pi? Why study such problems since such irrationals do not exist?" Kronecker's objection was not to all irrationals but to proofs that did not in themselves permit the calculation of the number in question. Lindemann's proof was not constructive. Actually, pi can be calculated to as many decimal places as desired by means of an infinite series expression but Kronecker would not accept the derivation of such a series.

Little by little we subtract

Fiath and fallacy from fact,

The illusory from the true

And then starve upon the residue.

- Samuel Hoffenstein

- MLOC p 289

{{Quote| Jacques Hadamard in The Psychology of Invention in the Mathematical Field (1945) investigated the question of how mathematicians think, and his findings were that in the creative process practically all mathematicians avoid the use of precise language, they sue vague images, visual or tactile. This mode of thinking was expressed by Einstein in a letter reproduced in Hadamard's book:

The words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought... The physical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be voluntarily reproduced and combined. The above-mentioned elements are, in my case, visual and some of muscular types. Conventional words or other signs ahve to be sought for laboriously only in a secondary state.

...Godel's results were shattering. The inability to prove consistency dealt a death blow most directly to Hilbert's formalist philosophy because he had planned such a proof in his metamathematics and was confident it would succeed...

The one distinguishing feature of mathematics that it mgiht have claimed in this century, the absolute certainty or validity of its results, could no longer be claimed. Worse, since consistency cannot be proved, mathematicians risked talking nonsense because any day a contradiction could be found. If this should happen and the contradiction were not resolvable, then all of mathematics would be pointless...

Godel's incompleteness theorem is to an extent a denial of the law of excluded middle. We believe a proposition is true or false, and in modern foundations this means provable or disprovable by the laws of logic and any axioms of the particular subject to which the proposition belongs. But Godel showed that some are neither provable or disprovable.

- MLOC p. 316-317

The developments in the foundations of mathematics since 1900 are bewildering, and the present state of mathematics is anomalous and deplorable. The light of truth no longer illuminates the road to follow. In place of the unique, universally admired and universally accepted body of mathematics whose proofs, though sometimes requiring emendation, were regarded as the acme of sound reasoning, we now have conflicting approaches to mathematics.

- MLOC p. 330

What had mathematics been? To past generations, mathematics was first and foremost man's finest creation for the investigation of nature. The major concepts, broad methods, and almost all the major theorems of mathematics were derived int he course of this wrok. SCience ahd beent he life blood and sustenance of mathematics. Mathematicians were willing partners with physicists, astronomers, chemists, and engineers in the scientific enterprise. In fact, durign the 17th and 18th centureis and most of the 19th, the distinction between mathematics and theoretical science was rarely noted. And many of the mleading mathematicians did far greater work in astronomy, mechanics, hydrodynamics, electricity, magnetism, and elasticity than they did in mathematics proper. Mathematics was simultaneously the queen and handmaiden of the sciences.

- MLOC p. 334

Its [mathematics'] chief attribute is clarity; it has no symbols to express confused ideas. It brings together the most diverse phenomena and discovers hidden analogies which unite them. If matter evades us, such as the air and light, because of its extreme thinness, if objects are located far from us in the immensity of space, if man wishes to understand the performance of he heavens for the successive periods which separate a large number of centuries, if the forces of gravity and of heat be at work in the interior of a solid globe at depths which will be forever inaccessible, mathematical analysis can still grasp the laws of these phenomena.

- Fourier, The Analytical Theory of Heat (1822)

- MLOC p. 343

We cannot help feeling that in the rapid development of modern thought our science is in danger of becoming more and more isolated. The intimate mutual relation between mathematics and theoretical natural science which, to the lasting benefit of both sides, existed ever since the rise of modern analysis, threatens to be disrupted.

- Felix Klein, 1895

- MLOC p. 344

The mathematics of our day seems to be like a great weapons factory in peace time. The show window is filled with parade pieces whose ingenious skill, eye-appealing execution attracts the connoisseur. The proper motivation for and purpose of these objects, to battle and conquer the enemy, has receded to the background of consciousness to the extent of having been forgotten.

The most vitally characteristic fact about mathematics is, in my opinion, its quite peculiar relationship to the natural sciences, or, more generally, to any science which interprets experience on a higher than purely descriptive level.

- John von Neumann, "The Mathematician" (1947)

- MLOC p. 359

To nettle the purists, the applied mathematicians have remarked that the pure mathematicians can find the difficulty in any solution, but the applied men can find the solution to any difficulty.

- MLOC p. 361

To further belittle their opponents, the applied mathematicians tell another tale. A man has a bundle of clothes to be laundered and looks around for a laundry. He finds a store with a sign in the window, "Laundry Done Here," enters, and puts his bundle on the counter. The storekeeper looks a little astonished and asks, "What is this?" The man answers, "I brought these clothes in to be laundered." "But we don't launder clothes here," the storekeeper replies. This time it is the would-be customer who is astonished. He points to the sign and asks, "What about that sign?" "Oh," says the storekeeper, "we just make the signs."

- MLOC p. 362

Talleyrand once remarked that an idealist cannot last long unless he is a realist, and a realist cannot last long unless he is an idealist.

- MLOC p. 366

"In mathematics as elsewhere success is the supreme court to whose decisions everyone submits."

- David Hilbert (1925)

- MLOC p. 394

But does mathematics need absolute certainty for its justification? In particular, why do we need to be sure a theory is consistent or that it can be derived by an absolutely certain intuition of pure time, before we sue it? In no other science do we make such demands. In physics all theorems are hypothetical; we adopt a theory so long as it makes useful predictions and modify or discard it as soon as it does not. This is what happened to mathematical theories in the past, where the discovery of contradictions has led to modification in the mathematical doctrines accepted up to the time of that discovery. Why should we not do the same in the future?

- Haskell B. Curry, Foundations of Mathematical Logic (1963)

- MLOC p. 396

the role of the alleged "foundations" is rather comparable to the function discharged, in physical theory, by explanatory hypotheses... The so-called logical or set-theoretical foundation for number theory or of any other well established mathematical theory is explanatory, rather than foundational, exactly as in physics where the actual function of axioms is to explain the phenomena described by the theorems of this system rather than to provide a genuine foundation for such theorems.

- Godel (1950)

- MLOC p. 397

Many other prominent workers in the foundations have accepted as a practical solution the same test of what sound mathematics is. Mathematics can be firmly, if not absolutely, secured by its applicability even if occasional corrections are required. AS Wordsworth put it, "To the solid ground of nature truss the Mind that builds for aye."

- MLOC p. 397

The "correctness" of mathematics must be judged by its applicability to the physical world. Mathematics is an empirical science much as Newtonian mechanics. It is correct only to the extent that it works and when it does not, it must be modified. it is not a priori knowledge even though it was so regarded for two thousand years. It is not absolute or unchangeable.

- p. 398

If mathematics is to be treated as one of the sciences, it is important to be fully aware of how science operates. It makes observations and experiments and constructs a theory, a theory of motion, or of light, sound, heat electricity, chemical combinations, and so forth. These theories are man-made and are tested by checking their predictions with further observation and experiment. If the predictions are verified at least within experimental error, the theory is maintained. But it may be overthrown later and must always be regarded as a theory and not as truth imbedded in the design of the physical world. We are accustomed to this view of scientific theories because we have had many examples of scientific theories being overthrown and rejected for new ones. The only reason men did not accept this view of mathematics is, as Mill pointed out, the basic arithmetic and Euclidean geometry were effective for so many centuries that people mistook it for truth. Btu we must now see that any branch of mathematics offers only a theory that works. As long as it works we shall hold to it, but a better one may be needed later. Mathematics does mediate between man and nature, between his inner and outer worlds. It is a bold and formidable bridge between ourselves and the external world. It is tragic to have to recognize that the bridge is not firmly anchored in reality or in human minds.

- p. 398

Other contemporary mathematicians are aware of the uncertainties in the foundations but prefer to take an aloof attitude toward what they characterize as philosophical (as opposed to purely mathematical) questions. THey find it hard to believe that there can be any serious concern about the foundations, or at least about their own mathematical activity.... What matters for them is new publications, the more the better. If they respect sound foundations at all, it is only on Sundays and on that day they either pray for forgiveness or they desist from writing new papers in order to read what their competitors are doing. Personal progress is a must - right or wrong.

Are there then no authorities who might urge restraint on the ground that foundational issues remain to be resolved? The editors of journals could refuse papers. But the editors and referees are peers who take the same position as mathematicians at large. Hence papers that maintain some semblance of rigor, the rigor of 1900, are accepted and published. If the emperor has no clothes and the court also has none, nudity is no longer astonishing, nor does it cause any embarrassment. As Laplace once wrote, human reason has less difficulty in making progress than in investigating itself.

- p. 401

The philosopher Goerge Santayana in his book Skepticism and Animal Faith pointed out that while skepticism and doubt are important for thinking, animal faith is important for behavior. The values of much mathematical research are superb and if these values are not be nourished, research must go on. Animal faith supplies the confidence to act.

- p. 402

Mathematicians and theoretical physicists speak of fields - the gravitational field, the electromagnetic field, the field of electrons, and others - as though they were material waves which spread out into space and exert their effects somewhat as water waves pound against ships and shores. But these fields are fictions. We know nothing of their physical nature. They are only distantly related to observables such as sensations of light, sound, motions of objects, and the now perhaps too familiar radio and television. Berkeley once described the derivative as the ghost of departed quantities. Modern physical theory is the ghost of departed matter.

- p. 404

According to Newton's system, physical reality is characterized by the concepts of space, time, mateiral point, and force (reciprocal action of material points)...

After Maxwell they conceived physical reality as represented by continuous fields, not mechanically explicable, which are subject to partial differential equations. This change in the conception of reality is the most profound and ffruitful one that has come to physics since Newotn...

The view I have just outlined of the purely fictitious character of the fundamentals of scientific theory was nby no means the prevailingg one in the eighteenth and nineteenth centureis. But it is steadily gaining ground from the fact that the distance in thought between the fundamental concepts and laws on one side and, on the other, the conclusions which have to be brought into relation with our experience grows larger and larger, the simpler the logical structure becomes - that is to say, the samller the number of logically independent conceptual elements hwich are found necessary to support the structure.

- Einstein (1931)

- MLOC p. 404/405

One modern explanation stems from Kant. Kant maintained (Chapter IV) that we do not and cannot know nature. Rather we have sense perceptions. Our minds, endowed with established structures (intuitions in Kant's terminology) of space and time, organize these perceptions in accordance with what these built-in mental structures dictate. Thus we organize spatial perceptions in accordance with the laws of Euclidian geometry because our minds require this. Being so organized, the spatial perceptions continue to obey the law of Euclidian geometry. Of course, Kant was wrong in insisting upon Euclidiean geometry but his point that man's mind determines how nature behaves is a partial explanation. The mind shapes our concepts of space and time. We see in nature what our minds predetermine for us to see.

- p. 409

We have found that where science has progressed the farther,st, the mind has but regained from nature that which the mind has put into nature. We have found a strange footprint on the shores of the unknown. We have devised profound theories, one after another, to account for its origin. At last we have succeeded in reconstructing the creature that has made the footprint. And Lo! It is our own.

- Arthur Stanley Eddington

- p. 409

The key idea is that mathematics is not something independent of and applied to phenomena taking place in an external world but rather an element in our way of conceiving the phenomena. The natural world is not objectively given to us. It is man's interpretation or construction based on his sensations, and mathematics is a major instrument for organizing the sensations. Almost automatically then mathematics describes the external world insofar as it is known to man.

- p. 409

And yet science would perish without a supporting transcendental faith in truth and reality, and without the continuous interplay between its facts and constructions on the one hand and the imagery of ideas on the other.

- Hermann Weyl, Philosophy of Mathematics and Natural Science

- p. 417

Endowed with a few limited senses and a brain, man began to pierce the mystery about him. By utilizing what the senses reveal immediately or what can be inferred from experiments, man adopted axioms and applied his reasoning powers. His quest was the quest for odder; his goal, to build systems of knowledge as opposed to transient sensations, and to form patterns of explanation that might help him attain some mastery over his environment. His chief accomplishment , the product of man's own reason, is mathematics.

- p. 424

Just as a single human being, restricted wholly to the fruits of his own labor, could never amass a fortune, but on the contrary the accumulation of the labor of many men in the hands of one is the foundation of wealth and power, so, also, no knowledge worthy of the name can be gathered up in a single human mind limited to the span of a human life and gifted only with finite powers, except by the most exquisite economy of thought and by the careful amassment of the economically ordered experience of thousands of co-workers.

- Ernst Mach, The Economical Nature of Physical Inquiry (1892)

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Economics

Nothing can have value without being an object of utility.

- Karl Marx

Capital is money, capital is commodities. By virtue of it being value, it has acquired the occult ability to add value to itself. It brings forth living offspring, or, at the least, lays golden eggs.

- Karl Marx

A commodity appears at first sight an extremely obvious, trivial thing. But its analysis brings out that it is a very strange thing, abounding in metaphysical subtleties and theological niceties.

- Karl Marx

Art is always and everywhere the secret confession, and at the same time the immortal movement of its time.

- Karl Marx

Machines were, it may be said, the weapon employed by the capitalists to quell the revolt of specialized labor.

- Karl Marx

The relevance of Marxism to science is that it removes it from its imagined position of complete detachment and shows it as a part, but a critically important part, of economy and social development.

- John Desmond Bernal