ANTI-MATHEMATICS

© Henry A. Flynt, Jr.

 

 

New Forward, 2011

 

       I completed the original of this manuscript in March 1980.  At that time, I promised the following contents.

 

Preface; Naiveté and the Critique of Mathematics

I.              Co-optation of “Failure Theorems” as the Sustaining Strategy of Mathematics

II.            Overview of the Trans-mathematical Critique of Mathematics

III.          Argument that the Meta-theory of Arithmetic and of Set Theory Is Inconsistent

IV.        Failure Theorems at the Research Frontier

       Appendix:  List of Established Failure Theorems

V.          Problematic Junctures in the Quantification of Nature

General References

 

Only the Preface and Chapter I (and General References) were completed in 1980.  Chapter II was a note, bridging Chapters I and III.  I incorporated it in the 1980s drafts of “Refutation of Arithmetic,” a separate study in lieu of Chapter III.  Chapter V was completed as a separate study, also serving as the conclusion of “Studies in Personness and Pre-Science.”  Chapter IV now exists as a draft.  So the undertaking has, in some fashion, been carried to completion.

 

                                           °

       Ever since the Greeks, the essence of mathematics has been the mystique of incontestability as much as it has been quantity or space.  It suffices to cite Georg Kreisel.  His Axiom of the Universality of Mathematics declares:  every thinking person must [sic] arrive at the same mathematical conclusions.  (How could the great Kreisel have been so oblivious to the history of mathematics?  How could he have been oblivious to the Cantor-Kronecker dispute?)

       We can get even more of a sense of the mystique by going beyond what mathematicians say in praise of themselves.  One window on the mystique is the pronouncements of the forgotten twentieth-century literary intelletual Eliseo Vivas.  In a throwaway line, Vivas said that the “higher activities” of contemporary civilization are “instinct with self-hatred.”  (While it would be too facile to draw radical conclusions from this thought alone, I think Vivas spotted something overwhelmingly important.)  Vivas became so estranged from scientism that he ended by railing against “Sovereign Reason” (Francis Bacon).  (Just to protect myself, I have to say that I am aware that Vivas’ answer to scientism was to slink back to the past.  What I think of that will become obvious.)

       Even as Vivas said these things, he offered an epistemology of logic and mathematics in a few throwaway lines, e.g. in The Moral Life and the Ethical Life (1950).  First, logic (which according to some is just the more general base of mathematics) explains the right way to proceed through the maze of ideas if you have the good will to want to be instructed.  It is an ideal for thinking—but you cannot hew to it unless you first commit to it.  You cannot be instructed unless you opt to listen.

       Continuing, Vivas cared enough about Dostoevsky’s protagonist who rejected 2 X 2 = 4 that he gave an answer:

 

… anyone who rejects [logic and mathematics] is crazy.

                                                  MLEL, 255

 

Vivas goes on to say (paraphrasing):

 

A refusal by a sane person to accept “the stone wall” of logic [or arithmetical propositions] is inconceivable.  It is impossible to change facts … .

 

Vivas’ rhetoric may be misleading.  He is so horrified by illogic that he hardly holds it out as a real possibility.  Unlike sin, which is a very real alternative to virtue, illogic is an almost unthinkable transgression.  Vivas almost forgets his pronouncement that logic is merely a guide for those who want guidance.  If you want to sin, Vivas can tell you exactly what to do.  If you want to transgress logic, there is no assurance that Vivas can tell you how to do so. 

       The lesson:  even an intellectual who took his syllabus from Dostoevsky told us that it is impossible to change mathematical truth. My citation of Vivas is not a digression into the milieu of the literary middlebrow.  Vivas is expressing the educated laity’s heartfelt faith in mathematics as a pillar of sanity—a footprint of God in the mind.  (A more scientistic way of saying the same thing would be:  the prime number series is the DNA of the universe, so the human brain must be hard-wired for it, otherwise we would not be able to live in this universe.)

       The ultimate step in ensconcing logic is taken by Wittgenstein.  Wittgenstein’s “psychological” aphorisms say that the world is logical because our minds cannot conceive an illogical world.  (Tractatus, 3.031, 3.032, 5.4731, 6.361, 6.362.)

       I would shortchange the reader if I did not mention that Vivas had another dispensation on logic.  He insisted vehemently that logic cannot guarantee itself.  It has to have a ground, a base.  Throughout his career, Vivas spoke in veiled language as if he had a faith he dared not confess in a secular milieu.  What Vivas was saying, when one gets underneath that language, is that we can only guarantee logic through faith in God.  Well, then!—logic cannot establish itself.  The universe’s DNA is not enough.  Logic needs a direct divine guarantee.

       Again there is something that can be compared with Wittgenstein.  One of Wittgenstein’s claims was that logical laws are like traffic lights we install.  It cannot be a mystery why it is “right” to obey them.  The Tractatus proved that twice two equals four (6.241):  so much for Dostoevsky. 

       But there is too much of a muchness in Wittgenstein.  It is a mere convention—but we have no choice.  The human mind cannot conceive an illogical world.  Even if we want to go crazy, Wittgenstein cannot tell us how, because it is not possible to do so.  Vivas almost converges with that Wittgensteinian outcome.

 

                                           °

       Around 1960, when I was between Israel Scheffler’s Philosophy of Science class and the first draft of Philosophy Proper, I conceived the goal of taking down mathematics.  If that is unheard-of, the foregoing remarks may indicate why.  Before me, everyone who was smart enough to learn any mathematics became a loyalist of the discipline. 

       One has to object to the civilization—in particular, to object to the regime of “mechanical” relationships among totally sterile objectivities which do not even exist (i.e. which are phantoms)—in order to view mathematics as a discipline deserving defeat.  (As to ‘mechanical’:  the lexicon is mechanical; the totality is not.  I spell this out in Supplement Two.)

       Then, in order for the attack not to be mere “primitivism,” one has to imagine that a technology far beyond the present manipulation of matter is possible.  In fact, one has to imagine a technology whose field of action is the “determination of reality.” 

       So:  my starting-point has two requirements in particular. 

a) It is necessary to reject the idea that the antidote to “scientific dehumanization” can be found in myth and superstition.  (More formally, one has to reject the idea that subjectivity and personness will be honored by believing in little people in the sky and other chimeras.) 

b) There has to be an attack on scientific objectification which complies with science’s demand for tangible effectiveness.  I want tangibly effective procedures which derive from unreduced subjectivity—procedures which “break the framework of objectivity.”  Such procedures merge two results into one.  i) They return us to our personness without mystification or delusion.  ii) If fully potentiated, they would give us ascendancy over scientific technology.

       The conventional question would be, “why don’t you support mathematics—everyone else does?”  “Why do you want to defeat mathematics?”  The explanations I have just made are as much of an answer as I want to give here.

 

                                           °

       Surprisingly, the next step does not consist in telling what I was able to contribute to defeating mathematics.

       The next step consists in recounting something I discovered—in the course of trying to contrive a defeat of mathematics—that I had not expected.  It came to me while walking in the city in 1980, and struck me with such force that I almost fell on the sidewalk.  Mathematics cannot be refuted from the inside.

       By the beginning of the twentieth century, any notion that mathematics was a science of the world had been abandoned.  Mathematics was a science of conventions, of their internal relationships.  The only merit that could be asserted for it was consistency.  So it was that Hilbert’s Second Problem asked for a proof of the consistency of arithmetic.  Hilbert considered the consistency of finite systems evident to inspection.  The problem was to prove the consistency of infinity.  He pontificated that a tribunal will judge the worth of mathematics once-for-all:  on the basis of whether an inconsistency cannot or can be found in it.  We have Hilbert in van Heijenoort, page 384.

 

Mathematics in a certain sense develops into a tribunal of arbitration, a supreme court that will decide questions of principle—and on such a concrete basis that universal agreement must be attainable and all assertions can be verified.

 

       So.  Hilbert’s Second Probem provided

 

(1) Mathematics can be proved meritorious by proving its consistency.

 

By implication,

 

(2) Proving the inconsistency of mathematics would discredit it.

 

       Shortly afterward, Gödel’s Incompleteness Theorems were announced, causing a great stir.  Hilbert’s Second Problem was shown unsolvable.  André Weil, one of mathematics’ most colorful and instructive commentators, wailed:  God exists since mathematics is consistent, and the Devil exists since we cannot prove it. 

       God?  Yes, we first saw the appeal to God here in connection with the literary intellectual Vivas.  I don’t know whether the appeal to God will seem ripe in this day and age, but Wittgenstein’s merciless naturalism was a puerile reductionism.  God lurks everywhere in mathematics.  We are reminded of it every time mathematicians say “natural number.”  We are reminded of it every time they say that they discover mathematical structures.  Without God, mathematics is concocted, expedient, and revocable.  Mathematicians want to picture their science as the touchstone of certainty.  They do not want to picture it as an expedient and they do not want to picture its subject-matter as a concoction.

 

                                           ° 

       (1) became inoperative because of Gödel, generally speaking.  The Incompleteness Theorems are heralded in popularizations as a barrier to human reason—à la Weil.  “Lower academics” say that the impact of “Gödel’s theorem” (shorthand, I trust) on twentieth-century thought is on a par with relativity, the uncertainty principle, etc. 

       We must supply a caution.  In my estimate, the Incompleteness Theorems are not like the Theory of Evolution, i.e. an unshakable orthodoxy that grips an entire civilization.  By no means did mathematicians stop investigating consistency proofs as a result of the Incompleteness Theorems.  Not only is there an entire literature which announces consistency proofs.  If these proofs are offered as contingent results that don’t get past Gödel’s barrier, the authors don’t say so.

       We have to be clear that Foundations of Mathematics was created for the benefit of a fringe that wanted an in-house substantiation of mathematical incontestability.  That fringe set about proving 1 + 1 = 2 — or proving that the truth is true — by making hundreds of pages of abstruse calculations.  The indispensable André Weil said, Gentzen is the lunatic who used transfinite induction to prove the consistency of ordinary induction.  But what Gentzen did is what Foundations is.  You prove the hand in front of your face by proving all the angels and archangels in Heaven before you know if there is a hand in front of your face.  That’s the only way they know how to think.

       If Foundations announces a crisis, that crisis is for the benefit of the adepts, not for the benefit of the educated laity.  The adepts can announce a new gimmick and announce that the crisis is over and done with any time they want to. 

       Beginning prior to the consistency stir of the twentieth century, a series of mathematicians launched a skepticism toward “infinitary extravagance” that I take as having a positivist flavor (even though Brouwer was a self-proclaimed mystic).  From the vantage-point of classical mathematics, the intuitionists or finitists wanted to impose strictures on mathematical thought. 

       That set up a whole new game for the adepts:  overrunning the strictures.  (Constructive mathematics.)

       I don’t believe I can overemphasize that:  these games are created by professionals for professionals.  They can declare a crisis when they want attention and then come up with a gimmick that overruns the crisis or allows them to ignore it.  The long-term import of mathematics culturally is not affected in the least.  Let us refer again to the indispensable Weil:  “God exists since mathematics is consistent.”  That does not change—because they need to believe it.

 

                                           °

       As a matter of usage, the Incompleteness Theorems should not be called a negative solution to Hilbert’s Second Problem.  They are an agnostic solution.  A negative solution to the Problem would be a proof that mathematics is inconsistent.

       What about (2) above?  There was a deep belief that a take-down of mathematics would look like an inconsistency proof involving technical quantitative considerations (a computational proof).  We find an emblematic declaration to this effect at the end of Brouwer (1933).  (It is necessary to paraphrase Brouwer, because he was either a terrible writer or else has been badly translated.)  “The most fashionable way of exposing errors of thought is the proof of a contradiction.”  That Brouwer speaks of the “fashionable way” is highly instructive.  What is at the forefront here is a professional cliché.

       What if a negative solution of the Second Problem were to appear?  Because it is, after all, a game that the professionals manage for their own benefit, a computational inconsistency proof of arithmetic would not especially be a setback to mathematics.  Culturally, Hilbert’s Second Problem is a hoax.  When Hilbert told us that mathematics would subject itself to a tribunal that judges its worth once-for-all, he was playing us for fools.  In fact, in 2011, I can be sharper than that.  Hegel knew far more about the institutional game than Hilbert.  Hegel, basing himself on the victory of infinitesimal calculus, told us:  they don’t care whether it’s inconsistent.

       All the while, mathematicians do not admit that their “knowledge” is inconsistent—alternatively, that they are irrationalists.  It is very rare for an insider to call the majority of mathematicians irrationalists.

 

       How can it be that mathematicians, equipped with such unusually sharp brains, can have so widely separate views on the most basic questions of mathematics?  Why is mathematics, supposed to be the most rational of sciences, presented as something deeply irrational by the many followers of Cantor now in control of mathematics departments and mathematics education?

                                                  Claes Johnson

 

Let it be clear that Johnson, a Professor of Applied Mathematics in Sweden, writes as a disgruntled “Kroneckerite”; his lament is a mere curiosity.  Mathematicians have to claim to be surpassingly rational.

 

                                           °

       It is at this point that I enter the picture.  What I realized in 1980 is that no in-house proof will ever take mathematics down.  (What is really at issue is expressed by Vivas better than by any mathematician:  the childlike faith that mathematics can be trusted as an absolute, that it is God’s footprint.)  If mathematics is taken down, it will be by considerations that do not look like mathematics to a mathematician. 

       Let us say, temporarily, that “hypocrisy” continually erupts in mathematics.  (I say “hypocrisy” rather than inconsistency because the junctures in question include much more than inconsistency.  The word is an expedient we will dispense with in Chapter I.) 

       The revelation of 1980 was that the continual eruption of hypocrisy does not hobble mathematics.  Mathematics is nothing more nor less than the co-optation of its hypocrisy.  In the phrase to be introduced shortly, mathematics develops by co-opting its Failure Theorems.  Then to prove a Failure Theorem will not defeat mathematics:  and mathematics cannot be refuted from the inside.

       To reduce a vast and convoluted situation to an epigram, every time an in-house inconsistency proof appears, mathematicians simply move the goal posts and keep going.  The only way this can be appreciated, I suppose, is by being led through cases that comprise mathematical history.

       The mystique is embodied in the profession’s customs.  The disgruntled Kroneckerite Professor Johnson hints at an imposed orthodoxy.  It is far broader than he supposes.  Because one of the purposes of mathematics is to appear incontestable, mathematical expositions have to have a loyalist configuration to be publishable.

       Mathematics is in part driven by in-house “radicals” who become indignant at this or that aspect of mathematics.  Brouwer may have been the greatest of them.  Professor Johnson expresses the indignation of early constructivism.  Let it be clear:  all this indignation comes from loyalists who want to put mathematics right so that its victory will be assured.

       It is possible that new internal critiques will appear.  In the second half of the twentieth century, a few madmen essayed to deliver a death blow to mathematical orthodoxy.  If they succeeded, I would welcome it, but let us be clear that such a success would not do more than supply a moment of drama to an otherwise gray profession. 

       A decisive defeat of mathematics would have to come from a vantage-point that would be considered extra-mathematical. (At least, the professionals would consider it extra-mathematical in the beginning. For that matter, Foundations was considered extra-mathematical in the beginning.) 

 

                                           °

       The 1980 manuscript called Anti-Mathematics has a tilt which requires an explanation.  Much of my discussion keys on the milieu of late twentieth-century finitism, which eventually made contact with marginal professionals who believed that an inconsistency proof of arithmetic was possible.  Because of circumstances, I had a “front-row seat” for the stir around Yessenin-Volpin.  Yessenin-Volpin’s ideas achieved professional publication.  Hao Wang published a mainstream verdict on them:  they were a joke. 

       At some point Yessenin-Volpin began to believe that the fringe figure Eduard Wette had shown the way to a shattering anti-orthodox result.  (Wette was the subject of a section in Bernays’ famous 1971 Dialectica paper.  For a conventional commentator’s verdict on Wette, see Dennis Rohatyn.)  Yessenin-Volpin and his protégé C.C. Hennix began writing abstracts on the “Gödel-Wette Paradox.”  In the 1990s, Graham Priest, the exponent of paraconsistent logic, had a working conversation in New York with C.C. Hennix about the project of a computational proof of the inconsistency of arithmetic.  It was inconclusive at best. 

       C.C. Hennix and I were colleagues in several fields of endeavor beginning in the 1970s.  Hennix commended Yessenin-Volpin to me, and I ended up conducting a debate with Yessenin-Volpin in my mind—perhaps treating him as unavoidable when he wasn’t.  In-person exchanges came to include not only Hennix, but Yessenin-Volpin himself, as well as Jan Ovgard.  A sidelight was supplied by Remko Scha, who may have connected to the milieu via an art-and-technology premise.  I exchanged several letters with Wette.

       While I, a total outsider, do not work in the same way as the exponents of finitism and paraconsistency, I attended to what they were doing for many years.  Thus, they are given space in my discussion that may seem out of proportion to the professional verdict on them.  Moreover, my contact with the milieu induced me to emphasize Foundations, which is a sacred cow for Yessenin-Volpin.  The lunchpail mathematicians may not even consider Foundations to be mathematics.  Nevertheless, its methods have had overwhelming importance in reconstituting the way lunchpail mathematics is conceived and sectioned.  (Set theory.)

       I find it illustrative to cite exchanges with various players over and above what they placed on the record via publication.  When I cite a conversation that was not recorded, it can of course only be my impression of what was said.  Sometimes I can supply documentation in the form of my jottings about conversations immediately after they happened, which are more valuable than memories retrieved decades afterwards.  Then there are recorded conversations, of which I have many.  Then there are unpublished documents, including letters.  Then there are duplicated documents without a publisher’s imprint.  In the References, I have chosen to separate this side of things and to go deeper in the available documentation than I do with mathematics in general.             

 

                                           °

       The insight that mathematics develops by co-opting its “hypocrisies”—that the goal posts will be moved whenever it is expedient to do so—was important enough to me that I devoted Chapter I of Anti-Mathematics to it.  It is not aimed at a lay audience.  Some interest and training in mathematics is needed to appreciate the examples.  I am attacking the people I need to talk to.

       Chapter I was to be followed by chapters which commenced several lines of attack.  What happened was that drafts of the subsequent chapters were finished at such long intervals that Anti-Mathematics never became a unified document.  For all that, I still choose to call the body of Anti-Mathematics “Chapter I.”  The subsequent chapters turned out like this.   

 

II. Refutation of arithmetic by non-computational considerations that might be considered extra-mathematical (at first).

III. A review of Failure Theorems at the frontier.

IV. Fractures in the application of mathematics in physics—in the quantification of nature.

 

What are here called II and IV are finished as separate studies.  III has been drafted as a separate manuscript.  In Chapter I, I refer to these projects by their new chapter numbers.

       Anti-Mathematics became a sprawling project.  It is surrounded by a body of material some of which directly supports Anti-Mathematics, some of which directly pertains to it, and some of which is correlative to it. 

       The References will be found at the end.  It is divided into two sections, one general, one devoted to finitism.

       There is the intellectual autobiography, “Naiveté and the Critique of Mathematics.”  I no longer want this to be the text the reader has to wade through first.  That said, my testimony about the in-house norms I was snared in will be vital to the attentive student.

       When I revisited this manuscript in 2001, I had thoughts about what I had written that start off in new directions.  Those thoughts are gathered at the ends of the texts as Glosses; they are indicated in the body by parenthesized boldface numbers.

       There is so much other correlative or supporting material that it falls into several phases.  Much of meta-technology can be considered correlative.  A separate document is needed just to distinguish the phases and list the manuscripts.  So it is that I provide “Supplement Three:  The Correlative Manuscripts.”

 

 

Chapter I:  Co-optation of “Failure Theorems” as the Sustaining Strategy of Mathematics

1980/2001/2011

 

      What sort of critique of mathematics can be taken seriously?  As late as January 1980, I and my associates made two assumptions regarding this question which are probably representative of the culture’s myths about mathematics. 

       Assumption 1.  The derivation of internal contradictions via extended calculations—and only that—would be a significant objection to mathematics.  Such an objection, and only such an objection, would be compelling to professionals. 

       We find an emblematic declaration to this effect at the end of Brouwer (1933).  (It is necessary to paraphrase Brouwer, because he was either a terrible writer or else has been badly translated.)  “The most fashionable way of exposing errors of thought is the derivation of a contradiction.”  That Brouwer speaks of the fashionable way is highly instructive.  Brouwer adverts to the view of a professional majority.  (1)

       Assumption 2.  To derive contradictions as in Assumption 1 is largely a new and unsolved problem.  We refer to the same passage in Brouwer (1933).  He says that Excluded Middle, the key fault of classical mathematics, is not liable to an inconsistency proof.  (2)  (Therefore there is no way to smash it decisively.)  And Hao Wang tells us that “a genuine inconsistency proof for even one widely used formal system” has not appeared.  It would be shattering if it did.  As of 2001, I noted that a few on the fringe of metamathematics were awaiting a proof of the inconsistency of arithmetic that the profession would find compelling.  (Which means that they believed that no such proof had been obtained.)

       I now face the task of explaining that these assumptions are immense errors of judgment.  But first, I need to extend somewhat the concept of a “derivation of an internal contradiction” in mathematics.  What is of interest is the more general issue of whether mathematics satisfies reasonable norms of cognitive creditability; and there are several types of results which can count as evidence of speciousness.  In the 2011 Forward, I speak of “hypocrisies” as temporary shorthand.  We are concerned with all of the following situations. 

 

1. Desired assumptions yield a contradiction.

 

2. A desired or derived claim of existence of a mathematical object is absurd.

 

3. Desired or permitted assumptions yield an unwanted result—a result which is unwanted because it is patently absurd or paradoxical. 

 

4. A formal concept which is invoked to make an informal notion cogent fails to possess the properties which are to be transferred from the intuitive notion.

 

5. A formal concept which is invoked to make an informal notion cogent is generally insufficient or inconsistent.

 

       At any given time, mathematics has inherited intentions about its subject-matter.  A number (not a vector) is a “scalar.”  A line (not a segment) is infinite.  The nonnegative integers are a unique structure.  Infinite means endless, and there are no distinguishable infinities.  The volume of a solid is invariant.  Then it is often suitable to conceive results of the sort (1) – (5) as rebukes of traditional intentions.  The new result belies traditional intentions, or cannot be embraced unless traditional intentions are abandoned.

       The shorthand of the Forward can be supplanted by an analytical phrase.  Results of the sort (1) – (5) will be called Failure Theorems.  I can restate Assumptions 1-2 about the critique of mathematics as follows.  Failure Theorems, and especially a derivation of 0 = 1, are the only sort of objections to mathematics which are genuinely mathematical and will impress professional mathematicians.  And the problem of discovering Failure Theorems is new and unsolved.

       It was a misconception of immense proportions.  Mathematical activity has produced a steady stream of Failure Theorems from the very beginning.  (The earliest one ascribed to a specific mathematician is the Incommensurability Theorem for √2.)  Often Failure Theorems have sophisticated proofs involving technical quantitative considerations—cf. Gödel’s Incompleteness Theorems and the Hausdorff-Banach-Tarski paradox.  (Anything named a paradox is likely to be a Failure Theorem.  The Burali-Forti paradox.)  A considerable number of Failure Theorems are among the best-known theorems of mathematics.  Some Failure Theorems arose as embarrassments, but were subsequently re-conceived as opening up extraordinary new subject-matters—e.g. irrational numbers and imaginary numbers (the very name should be a tip-off).  A few Failure Theorems stand as as “setbacks”:  e.g. the discoveries that Cantor’s set theory, Frege’s set theory, and Quine’s ML were inconsistent; and the Incompleteness Theorems.

       One group of Failure Theorems is well-known to teachers of elementary mathematics because they correspond to the points at which the schoolchild may balk at mathematical indoctrination.  Stewart-Tall, pages 9-11, asks:  how can there be quantities less than nothing?—why does .999 … = 1?—why does

–1 X –1 = +1?  To this we might add:  how can zero exist as a number for nothing (to count)?  How can √-1 be meaningful when the operation cannot be performed (when the operation has no range)?  [For that matter, √2 does not have an explicit evaluation either.]  Knopp, pages 102-3, says, “It is also, perhaps, not superfluous to remark that it is really quite paradoxical that an infinite series … should possess anything at all capable of being called its sum.”

       There is a subtlety here.  Having done some tutoring, I I wish to distinguish reasons why the student may balk at mathematical ideas.  On the one hand, the equality of .999 … and 1 involves issues that divide sophisticated thinkers.  One asks the student to go along with the gag. 

       On the other hand, if you tell a student that f(x) is a function of one variable, the student may reply that it is a function of two variables, f and x.  There are conventions one has to grasp intuitively.  A proof that one is obliged to think in the proper way would presuppose far more than what is being taught—or to put it another way, you may not be able to prove that one is obliged to think in the proper way to somebody who does not intuit it at the outset.  To learn lower intermediate mathematics, one needs intuitions for the usage of abstractions.  There are mental leaps that define effective communities—but it cannot be proved that one ought to make those leaps. 

       I do not know that the two types of difficulties I have just mentioned are especially distinct.  I don’t know where the use of the notorious dots … falls, for example.  In any case, in 1995 I found the question of the faculty of abstraction so important that I wrote a memo on it; it is one of the correlative manuscripts here.      

       Wittgenstein referred to the obdurate student in Philosophical Grammar, pages 381-2.

 

A mathematician is bound to be horrified by my mathematical comments, since he has always been trained to avoid indulging in thoughts and doubts of the kind I develop.  He has learned to regard them as something contemptible and … he has acquired a revulsion from them as infantile.  That is to say, I trot out all the problems that a child learning arithmetic, etc., finds difficult, the problems that education represses without solving.  I say to those repressed doubts:  you are quite correct, go on asking, demand clarification!

 

Referring again to Stewart-Tall, the authors acknowledge that the study of Foundations will not produce correct answers for the questions they ask.

       The majority of Failure Theorems, on the other hand, are neither well-known nor elementary.  They are assimilated to mainstream subject-matter as positive results (e.g. Riemann’s Rearrangement Theorem, wherein addition fails to be commutative).  [The Cantor set of points in a finite interval is uncountable but has measure zero.  Cantor was the first to conceive uncountability as a defining property of the continuum.  But uncountability does not guarantee a continuum anywhere. Here a Failure Theorem and the intention it violates arrived at the same time.]  Otherwise Failure Theorems are buried in discussions of Foundations and other peripheral discussions which are ignored by the mainstream.  (3)

       Mathematics is sustained by processes of misrepresentation whose purpose is to systematically co-opt Failure Theorems.  The reason why it is so difficult to get a grasp of this state of affairs, and to write an exposé of it, is that it involves a configuration of mind-sets and human purposes which prevails without challenge.  Everybody who has the ability to understand some mathematics becomes an apologist for the doctrine.  Everybody who is involved in the discussion of mathematics is a loyalist, an apologist, a defensivist—and these attitudes are so unquestioned that mathematicians acknowledge them with no awareness that they are confessing a lack of intellectual integrity.  Mathematics unfolds under a stipulation that mathematics cannot be wrong (because it is God’s thought!); failures arise only in the human understanding of mathematics.

       There is no single course of action in which Failure Theorems are co-opted.  The sequences of Failure Theorems and the various processes of co-optation run through mathematical activity as a whole in a number of different directions.  In order to document the regime of co-optation thoroughly, it is necessary to look at the same mathematical history from a number of different perspectives.  (This is as good a place as any to acknowledge my debt to Hao Wang.  He did more than any other professional to expose the “hypocrisies” without claiming to have the answers that resolved all difficulties.)

       One perspective of co-optation is as follows.  When significant Failure Theorems arise, mathematicians react in certain complementary ways.  One group of mathematicians assign themselves the task of overmastering the Failure Theorems by placing mathematics on a firmer foundation—by getting mathematics on the right track.

       A second group of working mathematicians ignore the Failure Theorems on the grounds that they are specialized quibbles in Foundations.  For example, Eilenberg, page 102, says, “The algebraic topologists are not bothered by [Foundational troubles] and apply the new concepts to their problems, well confident that the foundational difficulties will be cleared up.”

       Textbook writers and teachers do their part by assimilating the Failure Theorems of previous eras as positive subject-matter, or by suppressing mention of them.

 

                                           •

 

Real numbers

 

       The subject-matter which involves irrational numbers, infinitesimals, and classical analysis has a history in which all these types of co-optation play a part.  When Pythagoras proved the incommensurability of 1 and √2, he correctly recognized the result as a Failure Theorem.  The Greeks also derived results about the conception of lines as aggregates of unextended points, and about infinite sums of decreasing positive terms, which they correctly recognized as Failure Theorems.  Then Descartes, Newton, and Leibniz willfully disregarded these Failure Theorems and invented coordinate geometry and the infinitesimal calculus.  The calculus was recognized to be absurd from the beginning, but mathematicians willfully extended it in spite of its unresolved difficulties.  Meanwhile, a few mathematicians sought to place real analysis on a firmer foundation and thereby to vindicate it. But the Cauchy-Weierstrass definition of a limit proved, in turn, to be unsatisfactory.  [In 1980, I claimed to derive contradictions from this definition.  Wang(C), pages 75-81, may offer some perspective on my brainstorm.  I opt to hold that claim in suspension now.]  As for irrational numbers, how many students have been told that Cantor and Dedekind vindicated real numbers?  They didn’t.  How many students have been told that the least upper bound concept suffices to define real numbers?  It doesn’t.  (A useful review of attempts to vindicate real numbers from the time of Dedekind is provided by Wang(C), pages 75-81.)

 

Impredicativity

 

       Then we come to the attempts to avoid impredicativity in the definition of real numbers.  (Roughly, defining a set by invoking a set that contains it.  Note that the strong Cantor diagonal argument is impredicative.  Wang(C), page 77.)  It seems to me that impredicativity is usually presented from a loyalist standpoint.  The threat posed by impredicativity is, for example, the threat of inconsistency:  that is not always underlined.  Wang(C) is helpful here.  There is no informative proof of the consistency of impredicative definitions (page 79).  The problem of the consistency of classical analysis is the problem of the consistency of impredicative definitions, and the latter is not assured (page 247).  All the same, after the initial stir over impredicativity died down, there haven’t been aggressive attempts to give inconsistency proofs for impredicativity.

       Have students been told that to avoid impredicativity in defining real numbers, the l.u.b. of a bounded set of reals has to be assigned to a higher order than the members of the set?  This qualification violates the intention that the reals be symmetrical and makes it impossible to quantify over all the reals.  [Wang(C), page 78, has a “simple” solution for this difficulty which involves the truism that the set of all [positive integers] includes n+1 for each n.  In “Failure Theorems at the Research Frontier,” this “truism” is re-examined in connection with finiteness.  Finiteness appears to be brought to the forefront by Impredicativity.  Wang(C), page 125.]

       Let me skip the rest of the increasingly arcane difficulties for real numbers and just quote this summation from Wang(C), pages 244-5:

 

… we still do not have any definitive theory of what a real number … is.  Perhaps we can never have a definitive theory.  It seems quite unknown how this fundamental unclarity affects the rest of mathematics and the novel applications of mathematics in physics.

 

That begins to be a syllabus for this study.

 

Intuitionism

 

       Intuitionism had its beginnings in one of the most pronounced divisions in the mathematics profession.  Cantor proved Failure Theorems which Kronecker, Poincaré, etc., found abhorrent; he advanced them as a new and even primary subject-matter.  Subsequently, the resistance to Cantor was crystallized into a school with its own syllabus by Brouwer.

       But intuitionism, like every other faction in the profession, offers nothing but a different flavor of loyalism.  Roughly, intuitionists strip away what their arbitrary-dogmatic restrictions tell them is extravagant, and then reconstruct as much of mathematics as they can with methods which Brouwer himself called “more forced.”  To repeat what we said at the outset, the intuitionists cannot hope to impose an intellectual penalty on classical mathematicians for ignoring their restrictions.     

       After Hilbert, the pose was that mathematicians were free to fantasize at will just as long as they did not contradict themselves.  Again, intuitionists do not claim that ignoring their restrictions will land you in contradiction.  “Even the alternative mathematicians” are witnesses for loyalism and apologism.  After a certain point, say 1950, intuitionism became nothing more nor less than a weaker formalism used indifferently by investigators aligned ideologically with the mainstream.  (Did the turning point come with the announcement of an intuitionist formal logic?)

       Historically, there is more to intuitionism than constructivism.  Indeed, while we are usually told that mathematics is simply the truth, its past literature is so far from simply being the truth that a philologist is needed to sort it out.  On the one hand, Brouwer—and his “allies” such as Kolmogorov and Weyl—assure us that use of the hated Excluded Middle will not create contradictions.  <FN In addition to what was already cited, see papers in van Heijenoort.>  On the other hand, late in his life, Brouwer offered proofs purporting to smash classical analysis.  The others who used Brouwer’s name all sided with the less confrontational Brouwer—and nobody (including Brouwer) insisted on the distinction.

       One time when the two Brouwers are counterposed, and the comparative consequences are explored, albeit briefly, is in D. van Dantzig, especially pages 348-49.  (“My death creates the possibility of a mathematical process which is both infinite and finite.  Death is a contribution to mathematics.”)  Anyone who hasn’t seen that paper doesn’t know how dangerous it got before Brouwer’s followers clamped down.  Kreisel explicitly brought the heretical threat to an end by announcing his Axiom of the Universality of Mathematics:  every thinking person must arrive at the same mathematical conclusions.

       To the school of Brouwer, Brouwer’s inconsistency proofs were just embarrassments—and anyway, Brouwer’s followers did not want to offend their classical colleagues.  The intuitionists who mattered most concentrated on reconciling intuitionist and classical mathematics; their greatest success was to replace intuitionism with constructivist mathematics.  As Lorenzen notes, page 133, “Constructivism means nothing else than criticizing this so-called “classical” tradition—and trying to save its achievements as far as this can be done by reconstructing systematically the historically given.”  As Goodstein notes, page 44, the requirement of constructibility only requires the logical possibility of constructability [not its actuality].  (The reality-type of abstractions still governs.)

       Both the late Brouwer and his conciliatory followers are relevant in this study.  In the case of the late Brouwer, one has to excavate his “lost” Failure Theorems and the very limited discussion which addressed them directly.  I devote a manuscript separate from this study to that task.  The conclusion, roughly speaking, is that Brouwer gave mathematics a new set of headaches, but nobody finds his proofs decisive.  (4)

       We may refer back to Hao Wang on saving real numbers and tie it to intuitionism.  Wang is perfectly aware of the intuitionist proposal to define real numbers in terms of free choice sequences.  As I said, I find the old and not-so-old intuitionistic attempt to salvage something of the real numbers to be tediously contorted.  Brouwer said:  “more forced.”  In this case, I like “thievery better than honest toil.”  Cf. Brouwer in van Heijenoort; Kleene-Vesley; Troelstra.  These attempts declare that many of the intentions surrounding classical analysis cannot be upheld. 

All the same, their mission is to “save” the doctrine.

 

                                            °

       Let us again take the long view of the attempts to reclaim the real numbers.  All the attempts are said to be unsatisfactory.  What that can mean is that inconsistency proofs are not precluded.  The attempts do not, however, describe the situation in that way.  The vulnerability in the foundations of real analysis is not treated as an opportunity to prove Failure Theorems.  The attempt to reclaim real numbers has finally led us into a labyrinth of disputes in Foundations—specialized disputes which working mathematicians willfully ignore—without ever producing a creditable result.  Mathematical activity could not have traveled this route and arrived at this point if it were not under an unchallenged regime of apologism.  It requires a blindly apologetic mind-set not to see that this course of events is self-discrediting.  And it is self-discrediting at the level of an entire social practice, not the level of one formal contradiction.

       To repeat, three thousand years of attempts to deal with √2 have eventuated in a sidelined dead-end.  But is there any textbook in real analysis which says in the very first sentence that the subject-matter of the book is specious in every known formulation?  And what does it mean when mainstream mathematicians pointedly disregard the negative outcomes of the attempted vindications?  And why does every failure to vindicate real numbers evoke a more tediously contorted attempt at vindication?  What we have here is a fanatical commitment never to acknowledge that real analysis is simply specious (by reasonable standards of cognitive creditability).

       Once more, mathematics began with a Failure Theorem.  When the Pythagoreans discovered that √2 was irrational, they recognized that this result was a Failure Theorem and a defeat for mathematics.  The Pythagorean cult sought to conceal the result, and murdered Hippasus for divulging the secret to outsiders.  Three thousand years later, the difficulty exposed by that result has not received a definitive resolution:  the Failure Theorem is as effectual as ever.  Yet G. H. Hardy can say in A Mathematician’s Apology that the proof of the irrationality of √2 is one of two examples of beautiful and significant mathematics.  And this propaganda is possible in turn because for several centuries irrational numbers have been willfully re-conceived as opening up a new subject-matter.  Further, once irrationals were conceived as a new subject-matter, further evidence that they were inconsistent led not to their abandonment but to increasingly contrived, contorted, counter-intuitive attempts to salvage them. 

       It is time to characterize this state of affairs more bluntly.  Why is it permissible to ignore the intractability of the problem of validating the irrationals?  And why is it necessary to save them, on penalty of having to appeal to ever more contrived constructions? 

       I could make a suggestion:  admit that irrationals cannot satisfy conventional norms of cognitive creditability, and that they are retained because of their practical efficacy.  But we already know better:

 

First, mathematical practice outruns any rule-of-thumb basis.  The expectations have outrun accepting π = 22/7, for example.

 

Secondly, while mathematicians have been willing to renounce the traditional intentions which made irrationals and imaginaries problematic, they are totally unwilling to relinquish the legitimating ideology of mathematics as incontestable knowledge.  (Think God.)  They insist on the claim of cognitive creditability and they insist on retaining real analysis, even though the two demands are incompatible.  They are left with only one course:  to protect the doctrine through a policy of obscurantist rationalization.

 

                                           •

 

Repairs and loyalism

 

       Taking a different perspective, consider what has happened whenever entire systems have been found to be inconsistent.  Infinitesimal calculus was recognized at its inception to be inconsistent; but mathematicians continued to extend it and then launched a massive effort to vindicate it.  Euclidean geometry was found to be inconsistent, or the inconsistency became an issue, with the publication of Moritz Pasch, Vorlesungen über neuere Geometrie (1882).  Dedekind’s theory of the continuum would have been contradictory because of its vagueness concerning the formation of sets.  (See Wang(C), page 76.)  Cantor’s set theory was recognized to be contradictory.  Frege’s theory was recognized to be contradictory in the original version, and (according to the standard account) in the hastily repaired version.  Quine’s ML was found to be contradictory by Rosser.  The response, by Pasch, Russell, Wang, and others, was to modify the discredited systems in order to eliminate the located contradictions.  (It is an irony that the Wang whom I cite so often for his exposés was the savior of Quine’s ML.)  Gödel, meanwhile, proved the Failure Theorem that derailed Hilbert’s program:  the theorem that the sort of consistency proof for mathematics that Hilbert had hoped for was not possible.  [Gödel proved it to the profession’s satisfaction.  Correlatively to what I say in Supplement Two, it is the history of the professional mainstream that is being reviewed here.  Doubts about Gödel’s results are not entertained in the mainstream.]

       It is with the advent of Gödel’s Theorems that you really have to separate slogans from behavior.  In 1980, I wrote as if the Gödel Theorems didn’t even matter except as incidents.  That’s right.  Mathematicians still march under the banner “Consistency!”  Gödel’s proof(s) can be deemed valid; that does not rule out an inconsistency proof.  Meanwhile, mathematicians simply have faith that the theories they like are consistent.  Meanwhile, as I observed in the Forward, there an entire literature which announces consistency proofs.  If these proofs are offered as contingent results that don’t get past Gödel’s barrier, the authors don’t say so.  Wang(C), pages 199-200, discusses a proposed informal consistency proof for set theory.

       As a supplementary observation, you can get famous for consistency proofs which are preposterous or which seem preposterous to the mainstream.

 

i) Andre Weil:  Gentzen was the lunatic who used transfinite induction to prove the consistency of ordinary induction. 

ii) Wang(C), page 290:  “some people regard [Yessenin-Volpin’s consistency proof of ZF set theory] as an elaborate joke.”

 

       Referring back to the identification of contradictions in this or that system—by Russell, Rosser, or whoever—a further observation should be made.  The contradictions in question were discovered in theories formulated by logicians whose purpose was to place mathematics on a firmer foundation.  Yet the profession is not at all dismayed by these stumbling blocks.  No pipsqueak with his little paper in a journal is going to bring mathematics to a halt.  Hegel already said it:  [sanctimony notwithstanding,] they don’t care if its inconsistent.

       Taking yet another perspective, consider the succession of radical critics of mathematics and their motivations.  Weierstrass, Dedekind, Cantor acknowledged the unsatisfactory state of mathematics in preparation for attempts to validate mathematics.  (A significant number of mathematicians believed that Cantor’s validation made matters much worse.)  Frege criticized mathematics in preparation for an attempt to validate it.  Russell criticized Frege in order to get mathematics on the right track.  Brouwer criticized mathematics in preparation for getting it back on the right track.  Yessenin-Volpin criticizes mathematics in preparation for the most loyalist goal of all:  giving an absolute consistency proof for set theory in spite of Gödel.  Only fanatics who seek to uphold the vision of mathematics as incontestable knowledge are accepted as radical critics.

 

                                           •

 

Foundations of arithmetic

 

       We have reviewed the vicissitudes of the mathematics of real numbers, or what is the same thing, the successive attempts to manage Failure Theorems regarding irrational numbers.  We should take a similar look at the arithmetic of the natural numbers.  And the condition of arithmetic is a test case of one issue regarding Failure Theorems.  There are many Failure Theorems which, because of their simplicity, are dismissed by mathematicians on the grounds that increased sophistication has disposed of them or rendered them ineffectual. 

       One example of such a Failure Theorem is the proof of 0 = 1 by cancellation of zero in the true equation 0 X 0 = 0 X 1.  Is the latter operation prohibited for no other reason than that it yields the contradiction, or can it be proved to be wrong on the basis of the fundamental structure of multiplication, zero, etc.?  (In Supplement One, I set up the question via Birkhoff-MacLane and Wang.)  We need to bring in the actual course of research in foundations of arithmetic.  Attempts by Leibniz, Grassman, Dedekind, Peano, Frege, and Russell to place nonnegative-integer arithmetic on a firm foundation have led into a labyrinth of specialized research.  (Wang(C), pages 62-68, provides a useful summary of this research.)  Because recursive definitions are inexplicit, there is a dispute over whether they are legitimate.  Research by classical mathematical logicians (mainly Gödel and Skolem) on relations between formalization, categoricity, and completeness for Peano arithmetic has yielded mainly negative results.  Beth (A), pages 514-6, provides further discussion of the negative classical results on the categoricity of arithmetic.  In turn, the positive results mentioned in Wang(C)—such as the classical proof of the categoricity of the nonnegative integers on page 65—are repudiated by Yessenin-Volpin, whose approach Wang finds “rather obscure.”  Yessenin-Volpin even repudiates the axiomatic approach.  [Yessenin-Volpin’s approach has in turn elicited a series of objections on my part, to be found in various correlative manuscripts.]  We are led into a labyrinth of arcane disputes at the research frontier in Foundations, rather than to a definitive theory.

       In that sense, no law of arithmetic has been given an indisputable cognitive foundation.  That entitles me to say:  the reason for prohibiting zero-cancellation is the same now as it was originally, to forestall an easy inconsistency proof.  [Compare the way the Burali-Forti paradox was blocked by simply banning unrestricted use of e.g. “all sets with the property P.”]

 

                                           •

 

New subject-matters

 

       Let us adopt another perspective, and consider some cases in which mathematicians had to debate explicitly whether a particular Failure Theorem represented a defeat, or a new subject-matter.  Galileo’s observation that the even integers can be paired with all the integers was a major Failure Theorem.  In the late nineteenth century, there was a disagreement between Bolzano and Cantor over what conclusion should be drawn from this result.  Bolzano argued that the pairing of two (infinite) sets should not be accepted as proof that the sets are equinumerous.  Cantor argued that the whole can be equinumerous with a (proper) part of itself.  Mathematicians subsequently sided with Cantor even though Bolzano’s position is closer to common usage.  (Wang(C), page 70.)  Mathematical history is, for example, a series of debates in which the unreasonable side usually wins.

       The observation that √-1 could not be an “actual” number was a much-disputed Failure Theorem until around 1800, when complex numbers were rationalized as “two-dimensional real numbers.”  (Cf. Wang(C), page 61.)  For this rationalization to be accepted, two traditional intentions had to be suppressed:  that operations have to have evaluations; that the numbers which solve polynomial equations have to be scalars.

       An even more instructive case study is the difference between the reaction to Leibniz’s series and the reaction to Euler’s algebraic paradoxes.  The views I find most curious are those of Eves, the author of a popular undergraduate text on the history of mathematics.

       The three different sums for Leibniz’s series which can be obtained by grouping were retrospectively proclaimed not to be contradictions, but rather new consistent subject-matter which invalidated the associative law relative to infinite series.  (5) 

       Euler’s paradoxes,

 

-1  =  1 + 2 + 4 + …

 

0  =  … + x² + x + 1 + 1/x + 1/x² + …

 

 

are mathematically connected with Leibniz’s series (as divergent series) and represent quite similar embarrassments. 

       But Eves, who accepts Leibniz’s series as legitimate (page 447), dismisses Euler’s results, sharply censuring eighteenth-century algebraists for using algebraic operations without the necessary restrictions (page 349).  On the other hand, we learn from Kline’s vastly more learned treatise on the history of mathematics that Euler’s results are by no means mere mistakes.  Euler himself interpreted –1 = + ∞ as a reasonable result indicating that + ∞ is, like 0, a limit between positive and negative numbers (Kline, page 447).  What is more, Leibniz’s and Euler’s results led into a long controversy over divergent series.  In the first half of the nineteenth century, French mathematicians rejected divergent series, while German and English mathematicians accepted them—with De Morgan defending –1 = + ∞ as no more absurd than √-1.  (Kline, pages 975-6.)  As for contemporary opinion, Kline says (if I understand him correctly) that the experts (of whom Eves is obviously not one) accept –1 = + ∞ as a legitimate result when it is conceived as an expansion of the appropriate f(x) in the neighborhood of x = + ∞, with the sum of a series being defined independently of convergence (Kline, pages 1096-1120).  Both the disparity in Eves’ reactions to infinite series and to Euler’s paradoxes, and the discrepancy between Eves and Kline, are of interest.  Eves evidently finds –1 = + ∞ to be patent nonsense, so he treats it as the contradictions resulting from cancellation of zero are treated.  He blocks off the result by retrospectively stipulating that the operations which produced it are prohibited.  The attitude which Kline reports, on the other hand, is that divergent series are pragmatically valuable and thus must be rationalized by whatever means are necessary.  (Cf. especially Kline’s summary, page 1120, which I will quote shortly.)

       What is the overall situation here?  There is a mindset which stipulates that mathematics can never be wrong.  Thus, NO “EXPOSÉ OF A HYPOCRISY” CAN CONSTITUTE A REFUTATION OF MATHEMATICS.  Indeed, it is not meaningful to ask whether a given result in mathematics is a contradiction, as if there were an objective test for contradiction—because the status of a “hypocrisy” is settled by a more or less political altercation among mathematicians.  If the result is sufficiently unpopular, then a retrospective stipulation is made which prohibits the operation that produced the result.  But if the result is sufficiently popular, then it is retrospectively stipulated to be a consistent new result.  How can it fail to be a contradiction that the sums computed for Leibniz’s series are unequal?  Because one can, if one wishes, distinguish among the groupings which produce the different sums—and suppress the traditional intention that addition of integers is associative.

 

                                           •

 

       What attitude is expressed in the mathematical literature itself concerning these matters?  A compendium of Failure Theorems, showcased as such, would be unpublishable.  Mathematicians would define it as unprofessional.  Histories of mathematics sometimes discuss many Failure Theorems, but because of the prevailing regime of loyalism and apologism, the authors imagine that they are reporting a processing approach to the truth—not a continuous relabelling of defeats as victories.  Kline’s descriptions of Failure Theorem controversies would make a large book.  But Kline, and also Eves, are confident that the profession’s current opinion on the neutralization of any given Failure Theorem is the once-for-all right answer, that our immense sophistication has disposed of all embarrassments (although Kline’s and Eves’ accounts of current opinion sometimes diverge).  Then, Wang (C) serves as a compendium of Failure Theorems as they were known to a mid-twentieth century classical logician.  But Wang also was unaware that he was reporting a continuous relabelling of defeats as victories.

       Let me conclude these citations with some instances in which mathematicians explicitly acknowledge their loyalist-apologetic commitment—without realizing that this commitment makes it impossible to controvert mathematics from within.  The most helpful admission is made by Kline, page 1120, in summing up the outcome of the divergent series controversy. 

 

… when a concept or technique proves to be useful even though the logic of it is confused or even nonexistent, persistent research will uncover a logical justification, which is truly an afterthought.  It also demonstrates how far mathematicians have come to recognize that mathematics is man-made.  The definitions of summability are not the natural notion of continually adding more and more terms, the notion which Cauchy merely rigorized; they are artificial.  But they serve mathematical purposes, including even the mathematical solution of physical problems; and these are now sufficient grounds for admitting them into the domain of legitimate mathematics.

 

Wang(C), discussing the reactions to the paradoxes in set theory, pages 190-193, says:

 

… the proponents of the “misunderstanding theory” propose to uncover flaws in seemingly correct arguments, while the “bankruptcy theorists” find our basic intuition proven to be contradictory and seek to reconstruct or salvage what they can, by ad hoc devices if necessary.

 

Yes, these are indeed the strategies by which mathematics neutralizes Failure Theorems and sustains the illusion that it never contradicts itself.  Then, Abraham Robinson, page 188, says:

 

There are but few mathematicians who feel impelled to reject any of the major results of Algebra, or of Analysis, or of Geometry and it seems likely that this will remain true also in the future.  Yet, paradoxically, this iron-clad edifice is built on shifting sands.  … it is hard, and perhaps even impossible, to present a satisfactory viewpoint on the foundations of Mathematics today … .

 

                                           •

 

       Let me return to my assessment of the overall situation in mathematics.  In 1977, Hennix seemed to promise to renounce mathematics if I could produce one internal inconsistency proof.  But there are already hundreds, if not thousands, of internal inconsistency proofs, many known to Hennix, which are still operative.  Consider again the successive discoveries of inconsistencies in set theory and the attempts to assure the elimination of those inconsistencies.  Hennix implicitly fantasized that mathematics is like a court which makes once-for-all judgments of cognitive creditability and which regards internal inconsistency as decisive evidence of speciousness.  Well, there is nothing idiosyncratic about that picture; it is the official picture. Hilbert in van Heijenoort, page 384, is worth repeating.

 

Mathematics in a certain sense develops into a tribunal of arbitration, a supreme court that will decide questions of principle—and on such a concrete basis that universal agreement must be attainable and all assertions can be verified.

 

But if anybody were serious about this, set theory would be banished on the basis of its vicissitudes.

       Another illustration of a subject-matter for which every explanation is “unsatisfactory” (i.e. yields contradictions) is furnished by irrational numbers.  All the same, mathematicians continue to use set theory and real analysis.  But that is not all.  Faced with Gödel’s “magisterial” proof that set theory cannot be proved consistent, Yessenin-Volpin set about to change the rules so that he could prove consistency.  Hennix has been one of a minute minority who would claim that Yessenin-Volpin has succeeded, and has thereby rendered the agnostic results on set theory unimportant.  For myself, I must contend that Yessenin-Volpin’s methods expose mathematics to a new kind of “internal” objections.  But nobody would change the rules to save the much-debunked set theory if mathematics were the disinterested court of cognitive creditability proclaimed by Hilbert.  Nobody would make such an attempt if mathematics were not perpetuated by a regime of loyalism and apologism.

       Mathematics began with a Failure Theorem about √2, an embarrassment which after three thousand years has not been disposed of satisfactorily.  As the case of set theory shows, many important mathematical doctrines have been multiply discredited by Failure Theorems.  If mathematics were a disinterested court making once-for-all judgments, each of those doctrines would have been rejected in turn.  But the opposite happened.  Mathematicians plunged ahead, disregarding Failure Theorems, misrepresenting some of them as successes (again cf. Hardy on √2), burying others, and making ever more contrived, counter-intuitive attempts to salvage mathematical concepts.  For matters to develop in this way, there has to be a consensus that mathematics cannot be wrong, that failures can arise only in the human understanding of mathematics.  (Then in Whose Mind does mathematics make perfect sense?)

       Again, the processes of co-optation of Failure Theorems are diverse.  Overall, there is a sort of divide and conquer regime.  First off, Failure Theorems are not a field of specialization in the profession.  There are no compendiums of Failure Theorems presented as such.  Journals do not publish Failure Theorems except as digressions in papers which attempt to justify mathematics in some new way.  Those who deal with Failure Theorems at the research frontier—the specialists in Foundations—are an eccentric, segregated minority in the profession; and they examine Failure Theorems in the context of the avowed goal of vindicating mathematics.  Working mathematicians willfully disregard the embarrassments uncovered in Foundations, and sometimes manage to be ignorant of established Failure Theorems.  Some Failure Theorems are willfully misconceived as successes and become the basis of mainstream subject-matter.  As for mathematics education, a beginning student who balks on a point that may be questionable, such as .999 … = 1, is not interviewed to discover how he or she sees it; he or she is washed out of the program.  Mathematics textbooks present the subject-matter in a triumphalist manner, as Kline’s preface hints.  The last thing one will find in a mathematics textbook is an up-to-date case against its subject-matter.  Failure Theorems are included only if they have been long assimilated to the subject-matter.  Students are told that a rigorous basis for the subject-matter they are learning is provided on an advanced level by specialists.  But the specialists are just the segregated eccentrics who know that virtually all of the issues in Foundations are still in dispute.  Finally, these specialists dedicate themselves to making ever more contrived, contorted, counter-intuitive attempts to salvage mathematical concepts.  And the thinkers who are publicized as radical critics of mathematics are precisely the most committed loyalists, the people who want to get mathematics back on the right track.

       Thus, Hennix’s challenge to give an inconsistency proof using genuine mathematics is a trap in two respects.  First, mathematics is nothing but “failure results” which have been co-opted.  Secondly, a new internal inconsistency proof would be co-opted.  Hennix’s challenge leads one who seeks the defeat of mathematics into futility.  [I don’t want to rule out Failure Theorems as a goal.  However, when the Cantorians ran roughshod over the Kroneckerites, it showed that Failure Theorems cannot subjugate the science.] 

       Consider the issue of attitude in my 1979-1980 “Problematic Junctures.”  The manuscript was clumsy—but the profession would reject it even if its technical points could be sharpened.  It is the attitude that would never be acceptable.  Mathematics is sustained by a consensus of loyalism and apologism—by processes of misrepresentation whose purpose is to co-opt Failure Theorems.  To collect Failure Theorems is unprofessional by definition. 

 

MATHEMATICS IS AN ACTIVITY IN WHICH ONLY EFFORTS WHICH JUSTIFY MATHEMATICS COUNT AS ACHIEVEMENTS.  

 

We have spoken of God.  Can theology be far behind?

       One cannot directly acquire comprehensive familiarity with the historical totality of known Failure Theorems because mathematics education and publishing and mathematical discourse in general are organized to prevent one from doing so.  Every known Failure Theorem not at the research frontier has been co-opted and willfully misconceived as a success of mathematics; or else it has led into a labyrinth of tediously contorted attempts to neutralize its import—ending at the research frontier in Foundations.  And to repeat, research in Foundations does not count professionally unless it aspires to vindicate mathematics.  Attempts to prove new Failure Theorems will not be welcome professionally unless they appear in a context of apologism.

 

Do we want to race a team of apologists?

 

       My 1980 effort to derive contradictions from Cauchy-Weierstrass calls certain generalities to mind.  The rationalization of mathematical doctrines changes—as Wang(C), Kline, and others so often tell us.  Thus, the would-be critic who has only seen the undergraduate treatment of a given subject, and attempts to derive Failure Theorems regarding the subject, will be told that the subject was given a more sophisticated basis.  It may or may not take the critic longer to rederive the results relative to the newer basis.  Then one will be told that a yet newer basis has been provided in Foundations of mathematics.  It may or may not take the critic longer to rederive the results relative to that basis. 

       Eilenberg, page 98, says of one branch of mathematics that “the whole field changes radically over every ten-year period, and someone who has been away from it for any length of time might not understand a single word if he tries to read a paper.”  Likewise, the rationalizations of mathematics shift ground all the time.  The would-be critic simply races a team of apologists.  If your goal is to defeat mathematics, this is not a promising way to go about it.  (6)

      An element of the larger lesson is this.  The undergraduate exposition tempts you to waste your time, because the authors don’t disclose that they no longer believe it.  Wang(C) will tell you that the cliché justifications are already known to fail and have already been abandoned in Foundations; the textbooks say no such thing.  In particular, the student does not need to suspect or dispute Dedekind’s theory of continuity:  Foundations has already abandoned it!  But the instruction I encountered didn’t say that.

       All the while, something stays the same.  Foundations constantly shifts its ground; that does not mean that it rids us of past Failure Theorems.  Wang(B), pages 334-340 and 341, and Wang(C), page 74, observe that the sophisticated concepts which supposedly legitimate arithmetic and real numbers actually assume notions at least as problematic as the ones they supposedly legitimate.  (In “Problematic Junctures,” I made parallel observations regarding nonstandard analysis.  I also observed that to explain non-Euclidean geometry as trigonometry on a sphere leaves the most important intentions unsatisfied.  I have repeatedly made a corresponding observation regarding the explanation of complex numbers as “two-dimensional real numbers.”)  As a matter of fact, issues in Foundations end in a labyrinth of unresolved, eccentric disputes. 

       To repeat, elementary students may have two types of difficulties with mathematical subject-matter.  What matters here is that students may balk at expedient falsehoods.  (7)  Whatever the advantages are of acquiescing to these notions, cognitive creditability is not one of them.  As Stewart-Tall (and Wang, if you will), tell us, there is no decisive rebuttal of these beginners’ objections to mathematical notions.

 

                                           •

 

       We conclude that mathematics a an activity which arises and subsists under, and is sustained by, a social imperative that it must be devoted to its own intellectual legitimation—that it must be defended from reasonable, cogent objections.  But this means in turn that mathematics, as an institutionalized intellectual activity, is profoundly different from what it pretends to be, and functions in a way which has never been analyzed.  What does it mean when an intellectual activity is directed by mandatory self-legitimation?  What motivates the loyalist-apologetic-defensivist consensus?  And what features are specific to mathematics as a self-legitimating doctrine?

       I can only provide provisional answers to these questions.  At some point in the rise of human culture, mathematics is established as a regime of sterile phantom objects which can be

 

i) the occasion of speculative play

 

and which can also be applied to:

 

ii) the anthropomorphic chimeras of occultism;

 

iii) instrumental activity which would be recognized as efficacious by modern standards.

 

       Mathematics is unique among established cultural activities in uniting these possibilities.  Of all societies, ancient Greece was the one which placed the most emphasis on the speculative aspect of mathematics.  Nevertheless, Greek mathematical activity was stringently guided by the occult and technological aspects of mathematics as well.  Much of Greek mathematical activity consisted of the rationalization of results established by earlier cultures for non-speculative reasons.  The content of Greek mathematics did not run nearly as far beyond applications as it has in the modern era.

       If mathematics were nothing but a mercenary social practice—if its subject-matter had no intellectual rationale of its own—then mathematicians would not stumble across unwanted results (of which the irrationality of √2 continues to be the all-time classic).  [That was said too hastily.  We should learn from Hegel that expedient beliefs stumble across unwanted results upon elaboration.]  The suitability of mathematics to the manipulation of matter is also a challenge we must acknowledge.

       In passing I must mention an egoic motivation for mathematical loyalism.  Mathematics is unique in being a mental game of great technological efficacy which, because of its perversely abstract and “mechanical” character, is easy only for a minority of people.  [Again, for ‘mechanical’, see Supplement Two.]  Mathematics texts and treatises are available to everyone (in our culture—not before), but few people can comprehend the message, and far fewer can add to the doctrine.  The ability to understand even a little mathematics is a source of immense self-esteem.  I can think of no other doctrine whose acquisition by an individual almost never leads the individual to rebel against the doctrine.  [There are many, many renegade theologians, but no renegade mathematicians.]

       The motivation for mathematical loyalism which matters is its instrumental uses.  But this instrumental utility is not straightforward.  For one thing, some scholars argue that mathematics was originally developed to manipulate the chimeras of occultism—in other words, that it was originally valued for “practical” applications which are a laughing-stock today.  [Abraham Seidenberg.  Cf. Kepler’s occult applications of mathematics.  As a larger point, when positivists write history of science, they downplay occultism, and that is quite misleading.]  A contemporary survival is the practice of skipping 13 in numbering the floors of office buildings.  But there is a question which remains to be answered.  Why is there an essential overlap between the “factual content” of mathematics motivated by occultism and mathematics motivated by materialist instrumentalism?  One has to know how to count in the standard way to omit just 13, in the right place.

       Taking another angle, the alternative to a regime of apologism in mathematics need not be anti-apologism. There is another possibility which I call stagnationism.  It is so important that that it is worth spelling out the case of late ancient Christianity.  The Christians were against mathematics as a secular research activity.  But their ideological objections to mathematics were totally extra-mathematical and dogmatic—and their principal means for suppressing mathematics was physical force. 

       To elaborate.  The Christians co-existed uneasily with the elementary mathematics which they inherited, while giving greatest priority to a realm which was indifferent to that mathematics.  Elementary mathematics was also applied to ends determined by the non-mathematical realm.  (Calendrical science was important to the early medieval Church.)  New mathematics was discouraged. 

       It was physical force and not the Christian idea which the Christians used to suppress mathematical discovery.  The Christians’ choice of means to suppress secular mathematics research was an admission that their doctrine was more vulnerable than mathematics was.  In short, the Christians fought obscurantism with greater obscurantism.  What they did not have was manipulative power over mathematics, the ability to suspend any or all mathematical laws.  (It seems that I once read that medieval theologians argued over whether God has the power to make 2 plus 2 equal 5.  But that is not a very interesting question, because my program can deliver this result in various ways.)  And to be specific, what the Christians counterposed to mathematics was credulity toward anthropomorphic chimeras—not tangibly effective procedures deriving from unreduced subjectivity.  To me, this state of affairs is completely inconclusive.  It allowed pagan mathematical orthodoxy to survive like a dormant virus. 

       Thus, the motivation for mathematical loyalism which we have to consider is associated with the instrumental uses of mathematics which are recognized as real by contemporary standards.  Somehow mathematics can have technological efficacy whether it satisfies the norms of cognitive creditability or not.  Mathematics can contribute to the accomplishment of materialistic results.  And this state of affairs raises two questions.  The first is whether we want such results.  But the second and more productive question is how mathematics achieves those results, wanted or unwanted.  How can a doctrine which is so dependent on a fraudulent claim to cognitive creditability be so efficacious in the manipulation of matter?  What is especially outrageous is that calculus was invented as a practical science after the Greeks had thoroughly established the absurdity of the notions on which the calculus depends.  Cf. Resnikoff-Wells, pages 209-210; Kline, pages 383-389.  A minor point is that when calculus is used, it is implemented by discrete procedures.  Thus, the problem of the utility of the calculus reduces in one respect to the problem of the utility of discrete mathematics.  On the other hand, the discrete procedures by which calculus is implemented are guided by intuitions which derive from the investigation of the infinite and of continuity in medieval Christian scholarship.  Reductivism is unhelpful if it makes it impossible to understand the historical genesis of a technique.

       A more profound perspective on the utility of mathematics is as follows.  Mathematics “works”—but in some respects it is not possible to verify, or to discern, that mathematics works unless one accepts in advance the very doctrines which are at issue in the Failure Theorems and other objections to mathematics.

 

Rows of windows

 

       One day, while I was walking along the sidewalk, I found myself looking at the windows in the multistory buildings and counting them.  It seemed that this procedure was so ingrained in my ordinary apprehension of the world that I could not imagine a thought-style which would preclude it.  But further reflection established that on the contrary, the most elementary judgments of quantity are extremely problematic.  A row of windows on a building is a manifestation of simultaneously present, persistent “things.”  But I count them by pairing them with a succession of thought-events which appear and disappear in time. By the time I think the enumerative token “two,” “one” is gone.  Why should the result of this procedure be considered meaningful?  One has to assent to the meaningfulness of enumeration in order to verify that enumeration is consistent.

 

Pieces of cloud

 

       Wang(C), page 26, recognizes that elementary arithmetic would be falsified if it were “misapplied” to pieces of cloud.  (What about soap bubbles?  Soap bubbles are a metaphor for non-conservation of entities.  Pieces of cloud conserve “mass” but not plurality.)  But everything might have the character of pieces of cloud or soap bubbles if one willfully apprehended “the world” in that way.  E.g. I can discard distant windows by turning my head.  But to implement such a “determination of reality” seriously would require a break with established claims of the intersubjective consequentiality of “physical” concepts.  (Compare the dream reality.  “Dreams and Reality,” in Blueprint for a Higher Civilization.  Proposal for a Geniuses’ Liberation Project.)  And in this connection we can better understand why the Christian misgivings about mathematics were so inconclusive.  The Christians were committed to the pedestrian reality insofar as it related to societal regimentation.  They were in the position of wanting to join a compensatory fantasy to a pedestrian reality.  Thus, they were too vulnerable to go beyond a state of surly co-existence with mathematics.

       Wang’s example of the evanescence of plurality depends on counting empirical subjects which have some but not all of the attributes of the reality-type “object.”  The counter-example is inadequate because plurality is defined by clouds, a population in the “material world.”  It is really a counter-example in quantification of nature (applied mathematics, natural philosophy).

       I had already gone beyond Wang’s example in 1972, by writing and circulating “Subjective Propositional Vibration.”  (8)  I define plurality by a population which exists between notation and the perceiving consciousness.  [After Hilbert focused on the notation-token, that comes within the range of the mathematical.]  But never mind whether I was early; Wang is important because he said it from inside the Establishment.  With the passage of years, I have developed evanescent plurality so far that Wang no longer comes into it.  “The Apprehension of Plurality” and “The Counting Stands,” for example.  Hennix had a try in 1982 without having a suitable motivation (“Necker-numbers”) and the exercise failed to be a springboard to something else.

       We can understand better why mathematics is instrumentally efficacious.  The Self transforms its encounter of its world in accord with the tenets of mathematics; a shared objectivity is willfully erected which accords with the tenets of mathematics.  The Babylonians loaned money at compound interest and made discretized exponential function tables.  Especially in the 1970s, university students of musical composition were encouraged by Milton Babbitt and others to pursue music as a mathematically regulated system; and university students in literature were encouraged to peruse texts by making computerized statistical studies of their word patterns.  The shared world is willfully remolded in the image of quasi-consistent abstract formal objectivity.  Career rewards then become a steadily stronger motivation for mathematical loyalism.  [So, the self-fulfilling prophecy explanation.]

 

                                           •

 

       Having said something about the motivation of mathematical apologism, I must now begin to analyze the functioning of apologism in relation to the content of mathematics.  Mathematics is replete with contradictions, fallacies, and absurdities—but it is not out of control.  It successfully pretends to be perfect knowledge, secure against all contradiction.  (Wang(C) says on page 48 that “No formal system which is widely used today is under very serious suspicion of inconsistency”; and on page 239 that “The most impressive features of mathematics are its certainty, its abstractness and precision … .”)  There clearly are constraints which prevent the acceptance of every proposition as a theorem.  With respect to desired propositions in established subject-matter, to negate any one of them is an error by definition.  As for long-standing critiques of desired results, mathematical discourse is organized so to as to bury them.  A beginning student who balks on a point that admits dispute, such as .999 … = 1, is washed out of the program.  When a desired subject-matter is supported by pragmatic motivations but is visibly fallacious, there is a mind-set to the effect that vindication of the subject-matter is a mandatory task.  When a desired subject-matter is assailed by a flurry of inconsistency results, the option of repudiating the subject-matter is ruled out; instead, vindication again becomes a mandatory task.  The refutation of mathematics is not a permissible field of specialization; it is defined to be unprofessional.  When a specific inconsistency result is newly discovered, sometimes the procedure which produced it is retrospectively stipulated to be wrong.  (This is a second constraining principle which prevents some propositions from becoming theorems.)  At other times a specific new inconsistency result is reinterpreted by use of sophisms so that it becomes consistent and can be conceived as a new subject-matter.  The choice of whether to treat a specific new inconsistency result as the outcome of a retrospectively prohibited operation or as a consistent new subject-matter is made on the basis of:  personal taste, sophistry, pragmatics, tyranny of the majority.

       We may well consider what Hilbert had to say about the intuitionists.

 

Because of his authority, Poincaré often exerted a one-sided influence on the younger generation.

                                           van Hiejenoort, page 473

 

And as to Brouwer,

 

I am astonished that a mathematician should doubt that the principle of excluded middle is strictly valid as a mode of inference.  I am even more astonished that, as it seems, a whole community of mathematicians who do the same has now constituted itself.  I am most astonished by the fact that even in mathematical circles the power of suggestion of a single man, however full of temperament and inventiveness, is capable of having the most improbable and eccentric effects.

                                           van Hiejenoort, page 476

 

       In my terminology, mathematics is a biased inconsistent theory.  All the while, it presents itself as “mechanical,” abstract, sterile, objectified.  The irony is that the processes which sustain, guide, and constrain the content of the doctrine are interested social processes, of which the enforcement of orthodoxy is the most important.  The important properties of mathematics are not intellectual but social.

       Yet, again, mathematics is typified by the claim of cognitive credibility for a “mechanical,” sterile, phantom subject-matter—and by its appropriateness for the manipulation of matter.  Insofar as mathematics is separable from, and unlike, astrology (which uses it), and Christian dogma (which co-existed with it during much of European history), we must try to understand the rationale which is specific to it. 

       [2001.  We must do better than the model of mathematical thought presented by mathematical logic, whose lexicon is mechanical and whose approach is reductivist.]

       In this connection, let us again recall some cases in which fallacies surfaced in mathematics and became the occasions for episodes of overt co-optation.  And let us view these cases in as narrowly intellectual a perspective as possible without denying the involvement of social sanctions in the co-optation process.  Calculus was invented for pragmatic reasons after its presuppositions had been decisively refuted, and it was supported even though it was fallacious.  The eruption of paradoxes in set theory evoked the dispute between the “bankruptcists” and the “misunderstandingists,” as Wang calls them.

       Let me focus on the attitudinal threshold between

 

1.  Retrospectively prohibiting an operation to prevent a contradiction.

 

2.  Legitimating the contradiction as new consistent subject-matter.

 

Again let me contrast the reaction to Leibniz’s series with the reaction to Euler’s algebraic paradoxes.  After a period of controversy, Leibniz’s series was retrospectively proclaimed to be new consistent subject-matter.  Euler’s paradoxes are mathematically connected with Leibniz’s series and represent quite similar embarrassments.  We recall that Eves accepts Leibniz’s series as legitimate.  On the other hand, he finds Euler’s results to be patent nonsense.  Apparently  –1 = + ∞ demands a little more absurd faith than Eves can muster.  (Yet Euler himself, De Morgan, and others were prepared to accept  –1 = + ∞  as a meaningful equality.)  Thus, Eves blocks off the results by retrospectively stipulating that the operations which produced them are prohibited.  Indeed, he scolds eighteenth-century algebraists for using algebraic operations without necessary restrictions.  On the other hand, the vastly more learned historical treatise by Kline, who evidently is a specialist in pre-twentieth century calculus and differential equations, recounts a history of active re-examination of divergent series in which  –1 = + ∞  eventually acquires a stable meaning (if I understand Kline correctly).  Kline concludes in effect that mathematicians are capable of rationalizing anything they want to.  This material (including the discrepancy between two contemporary mathematician-historians) illustrates clearly and thoroughly the attitudinal threshold between prohibition and legitimation.

       Can we discern a few recurrent sophisms by which mathematicians transmute contradictions into new consistent subject-matter?  This question is simplistic.  In the historical cases, a fallacious subject-matter is legitimated by an integrated readjustment of the doctrine which requires a knowledge of how a wide diversity of considerations impact on each other, an ability to judge priorities, and an ability to judge plausibilities.  (Also, we must judge that Hennix’s explanation of contradictions as improper collations is reductivist.  Again, the metamorphosis of a contradiction into a consistency in a historical case involves an integrated process which is a kind of coercive indoctrination.  It is not only a matter of a formal stipulation.)

       All the same, as a heuristic exercise, it is worthwhile to describe two ploys for legitimating absurdities.

 

1. Forgetting intentions

 

Suppose that it is desired to claim existence for an entity with incomprehensible or contradictory properties.  This claim can be legitimated if a construct can be devised, from legitimate entities, which replicates some formal aspects of the behavior of the questionable entity.  More is involved here than the formal notion of a model.  The model will fail to satisfy many of the intentions or demands which surrounded the problematic entity when it first surfaced.  To decide which of those intentions should go unsatisfied, and then to cause the mathematical community to forget the unsatisfied intentions, is a matter not of model theory but of salesmanship and intimidation. <FN As another example for the untutored layperson of what is possible, “spherical” non-Euclidean geometry provides a system in which straight lines—I don’t mean rectilinear segments—are of finite length.  This result is just a consequence of redefinitions.>

 

2. Segregating operations

 

Suppose that a system of elements and operations arises historically.  Suppose this system is surrounded with intentions or demands which establish (autonomous) relations of equality among the elements; and which require that certain operations, while formally distinguishable, produce equal outcomes when applied under certain conditions.  These distinguishable operations may then be called ‘symmetrical.”  Suppose, then, that the system is extended to new permutations of elements and operations or larger numbers of elements or operations.  Suppose that cases are found in which symmetrical operations produce unequal outcomes.  I am going to give an entirely fictitious example in the hope that it will suggest to the reader how disconcerting such a case would be to one who was not prepared for it.  If the reader finds the example unhelpful, he or she may ignore it—since I have already given the actual examples repeatedly.  Suppose it were found that substitution of 1+3 for 4 in 1+4 = 5 set 5 equal to 1+1+3 (which was known independently to equal 5); but that substitution of 2+2 for 4 in 1+4 = 5 set 5 equal to 1+2+2 (which was known independently to equal 6).  The extended system has produced a contradiction.  Suppose that it is desired to convert this contradiction to a new consistent subject-matter.  There are two ploys by which this feat could be accomplished—and an Alternative.

 

a.  Retrospectively stipulate that the outcomes are correct and unequal, but that the operations, while correct, are no longer symmetrical.  That would be like deciding that substitution of 1+3 for 4 is not symmetrical with substituting 2+2 for 4.

b.  The second ploy is more complicated.  First, retrospectively stipulate that the outcomes are correct and equal.  In our fictitious example,  5   =  1+2+2 [= 6]  is correct.  However, the outcomes constitute a separate subsystem because the operations which yield them, while correct and symmetrical with each other, are not symmetrical with other operations on other elements.  Instead of being “forgotten” once they are used, the operations become “markers” which distinguish the results obtained with them as a separate subsystem.  5 = 6 is true, but only in the 4-substitution universe, say.  In contrast, substitution of 1+4 for 5, and substitution of 2+3 for 5, would be required to yield outcomes equal by the original autonomous definition of equality of elements.

Alternatively. One could save mathematics by banning the bothersome result:  retrospectively stipulating that e.g. 4-substitution is an erroneous operation, a violation of the restrictions which must “obviously” be placed on substitution. <FN Compare the ban on cancellation of 0 in e.g. 0 X 0 = 0 X 1.>

 

       The example of 4-substitution serves to illustrate the most important aspect of these ploys for neutralizing a contradiction.  THE CONTRADICTION CAN BE NEUTRALIZED BECAUSE THE MATHEMATICIAN CAN CONVINCE THE COMMUNITY TO FOREGO IMPLICIT, ESTABLISHED INTENTIONS OR DEMANDS ABOUT THE BEHAVIOR OF THE SYSTEM.  Would it be ridiculous to ask people to agree that substitution in simple arithmetic is erroneous for 4 only?  Would it be ridiculous to ask people to agree that 5 = 6 is not a contradiction but a many-valued number which exists only in a special 4-substitution universe and therefore does not contradict the established results outside that universe?  Mathematicians have repeatedly gained acceptance for comparable notions.

       As for Hennix’s dichotomy of correct and incorrect collations, it is irrelevant here, because what is happening is that the community is being pressured to abandon traditional intentions in order to gain an absurdity as a new consistent subject-matter.  Again, the decision as to which intentions should go unsatisfied, and the ability to cause the mathematical community to forget the unsatisfied intentions, is a matter of salesmanship and intimidation.

       These ploys must not be taken as evidence that mathematical sophistry can be mechanized.  Indeed, mathematics cannot afford to codify its interested social practices explicitly.  Mathematics must pretend that it is a disinterested tribunal which makes once-for-all judgments of cognitive creditability.  Again, Hilbert.  And in historical cases, the legitimation of a fallacious subject-matter is accomplished by an intricate, integrated readjustment under intense social sanctions.

 

       Rosser-Turquette, page 2, says that “a theory of many-valued numbers has not yet been constructed.”  I don’t know what Rosser-Turquette had in mind, but a non-trivial, “mad” theory in which 5 = 6 and 1+4 = 7 [evidently from using the absurd equality twice] most certainly could be devised if the demand for the novelty overcame the resistance to the sophistry required to justify it. 

       I may mention that when I first proposed a “logic of contradictions,” I wanted to devise a theory of many-valued numbers.  My “logic of contradictions” dates from 1970.  Perhaps I started thinking about many-valued numbers then, but I can’t find a record of what I had in mind. 

       A reckoning in which 0/0 is the only element might do it:  it amounts to an argument, n, that can have a different value every time it appears.  It might be a novelty of redefinition as per “Forgetting intentions” above.  However, inasmuch as this was an idea for a pencil-and-paper game with the received pencil-and-paper tokens, I chose not to invest any time in it.

       I quickly realized that what I wanted a logic of contradictions to do is to generate experiential anomalies as “picturable contradictions.”  Or to expose hidden contradictions in the shared conceptual system through which we apprehend “the world.”  To break the framework of objectivity.  Alternatively, to produce or expose an anomaly in our numerical intuition.  (9)  See below.

       On the basis of the understanding gained so far, it is possible to give a second heuristic exercise, a political algorithm for making mathematical revolutions.

 

Algorithm for Mathematical Revolutions

 

1. Choose a thesis which is considered to be fallacious (absurd or contradictory).

2. Claim that it is not a fallacy but a new subject-matter.

3. Persuade people that it is not absurd or contradictory.  Utilize ploys such as (1)-(2) above.  Plunge ahead, readjusting the content of the rest of mathematics to make it complement your chosen absurdity.

4.    a. Make the new theory professionally attractive.  Use it to solve an unsolved problem regarded as mandatory by the profession: accomplish this feat by changing the meaning of the problem and causing other mathematicians to forget every established intention which you cannot satisfy.

b. Promote the theory at the mercenary level, exploiting peer pressure, intimidation, etc.

 

       While this algorithm is an artificially simplified one, does it not resemble the co-optation of any number of Failure Theorems as new subject-matter?  Looking forward instead of backward, let us resume with Yessenin-Volpin, whom Hao Wang called a clown (in effect).  Imagine that Yessenin-Volpin’s doctrine were to prevail.  The doctrine’s superstructure rests on the tenet that 2 can be infinite and is the smallest positive integer that can be infinite.  [For the uninitiated:  literally.]  He willfully readjusts the rest of mathematics to accord with his chosen provocations, neutralizing the resulting classical contradictions with discernable ploys.  His inducement to mathematicians is that his theory can prove the consistency of ZF, positively solving Hilbert’s Second Problem or providing a proxy for such a solution.  In asserting the infinitude of 2, he establishes multiple versions of the nonnegative integers which have classically finite upper bounds.  [Again for the uninitiated:  literally.]  Thereby does he proclaim himself to be radical.

       What is of most interest here is not that Yessenin-Volpin has chosen to work on an idea which is outrageous.  (While being better connected to the going thing in Foundations than the uninitiated would expect.)  What makes my point is that he has sugar-coated the poison pill by promising greater certainty and by promising a consistency proof for mathematics.  Radicalism is loyalism and apologism.

       Immediately after drafting this manuscript in 1980, I typed a review of Yessenin-Volpin’s methods entitled “A Logical Arithmetic with Absorbed Contradictions:  Computing on Clashing Enumerations.”  It proposed to tell how you could get away with anything in arithmetic.  But even Yessenin-Volpin maintains the professional guise.  As noted above, Wang says that the laws of arithmetic fail if applied to pieces of cloud.  It is not an adequate counter-example.  I had already gone much more deeply into evanescent plurality with “Subjective Propositional Vibration,” and subsequently developed the line of thought so far that Wang no longer comes into it.  Wang could publish professionally (never mind that his example fails to be adequate); I couldn’t.  

       When writing “Problematic Junctures,” I was searching for new Failure Theorems.  To recur to Assumption 1—the refrain of this study—a single internal inconsistency proof would supposedly crush mathematics.  But now I am certain it wouldn’t play out like that.  A presented Failure Theorem, specifically, an internal inconsistency proof, would not be judged once-for-all according to disinterested standards of verity.  When a new Failure Theorem is unpopular, then obvious objections are enough to stop it.  When a new result comes from a loyalist, or tells the majority something that pleases them, then obvious objections have no force.

       I experienced one sort of disillusionment with the vocation of learning or proving mathematical theorems by 1960 or 1961.  In 1980, that vocation lost its glory with me in a more intimate way, so to speak.  (Notwithstanding that I still say that the critic should pay attention to Failure Theorems—cf. Chapter III.)  To play the game merely indoctrinates oneself with an official delusion.  Nothing of importance behaves like the “mechanical” formal sterile abstract phantom objectivities of mathematics—not even mathematics.  What is more, mathematics is so malleable that for any given theorem, the negation of that theorem can also be established.  I am not referring to the cliché that if {A, B} is one axiom system, then {A, ¬B} can be chosen as a different axiom system which will yield a different theory.  No, I am saying that once you understand the malleability of mathematics, you can prove ¬B from {A, B} as axiom system—if you are willing to engage in salesmanship and coercive indoctrination.  After all, hasn’t Yessenin-Volpin proved that 2 can be infinite?  And he has done so not with respect to some non-constructive nonstandard integers—but rather with respect to the naive integers, by changing the rules of the game.

       Thus, the fixed-point theorem could perfectly well be falsified in the “same” topology that now prevails.  And we get a further hint as to how this and other tricks might be accomplished from Eilenberg, who gives a proof by contradiction of the fixed-point theorem.  Proof by contradiction is often used in classical mathematics; and everywhere it is used, there is a vulnerability which can be manipulated.  [Is this just the intuitionist rejection of the Excluded Middle?  No, because the intuitionist rejection of the excluded middle is agnosticism, which proceeds by inventing cases in which neither A nor ¬A can be verified.  As I said earlier, the typical intuitionist position is that use of the Excluded Middle will not create contradictions.]  My claim is confrontational.  One proves A by contradiction by assuming ¬A and producing a contradiction.  But this only proves that ¬A is “wrong”; it does not prove that A is “right.”  If you had started by assuming A, you could have produced a contradiction also—if you really wanted to.  Mathematics is theoretical quicksand—and there is no reason why a thesis and its negation should not both lead to contradictions.

       As a matter of fact, the proof that √2 is irrational is a proof by contradiction; the assumption that x, for x² = 2, is rational gives a contradiction.  Then what is the contradiction that follows from the assumption that x, for x² = 2, is not rational?  It is, first of all, the body of absurdities that have come to be known to us as real-number mathematics.  <FN If I were ready to insist on them, I could bring in my Failure Theorems concerning Cauchy-Weierstrass limits here.>

       In the October 1979 conversation, Hennix proposed that Yessenin-Volpin’s approach would allow numbers which are strictly between finite and infinite [ordinals?].  [Not referring to nonstandard arithmetic, surely.]  Can this claim be rationalized?  Of course it can.  I wanted to have a logic of positional relationships in the visual field which is distinct from coordinate geometry.  Can such a logic be rationalized?  Of course it can.  The foregoing discussion assures us that anything can be made to work in mathematics if you are willing to expend the effort.

       [In “Rational Constructive Analysis,” James Geiser derived, as a consequence of Yessenin-Volpin’s theory, that if we have two natural number series of different lengths, the longer plays the role of rationals relative to the shorter, and real analysis or infinitesimals can be introduced by calculating on the longer series restricted to the shorter.  The radicals, the self-styled anti-traditionalists, find a way to salvage received results.  But I have a further observation.  On the basis of my understanding of the arbitrariness of mathematics, I speculated (in 1980, without having seen Geiser’s work) that his results are not necessary and unique conclusions from Yessenin-Volpin’s theory.  But if I wish to dispute Geiser, I had better do it while ultra-intuitionism is still considered to be a joke.  If ultra-intuitionism were to become the new orthodoxy, then Geiser’s results would ossify into laws of mathematics whether they are necessary and unique derivations from ultra-intuitionism or not.]

 

                                           •

 

The logic of contradictions

 

       In the Seventies, I proposed a logic of contradictions; I have been adding to the proposal ever since.  <FN If somebody supposes that I am competing with paraconsistency, I may reply that there is no need for me to grapple with paraconsistency.  I explain that in a comment in Supplement One.>

       To summarize what I spell out elsewhere.  My logic of contradictions starts from experiences of the logically impossible in perceptual illusions, in dreams, and as they are associated with the shared conceptualization through which we apprehend “the world.”  (I characterize these experiential contents as “logical impossibilities” because that is the appropriate characterization relative to prevailing cognitive discourse—as all the perceptual psychology textbooks agree.)  With respect to shared conceptualizations, I place special emphasis on “common sense.”  I define common sense as the conceptualization (or system of assertions) which is the medium of ordinary apprehension of the world and ordinary social interaction.  Our enculturation with common sense, or equivalently, with the natural language, is the source of our ability to use concepts at all; and common sense is inescapable so long as we do not escape the Lebenswelt which it codifies.  In addition, I contend that common sense is inescapably paradoxical.  It is crucial that the paradoxes of common sense are content-determined and qualitatively specific.  [They pose a problem not solved by the trifling with inconsistency which has gone on in mathematical logic.]

       One of the aims of my logic of contradictions is to provide the ability to manipulate contradictions deliberately, as a means of dismantling the prevailing “determination of reality.”  Indeed, even though the logic of contradictions starts from inherited delusions (the least escapable delusions!), it is so far from being a confirmation of those delusions that it unravels reality to the point where the deluded conceptualization “blurs” or “liquefies.”

       In some sense, a shared conceptualization has the logical structure traditionally attributed to systems of propositions.  Associated with the conceptualization is the set of all grammatical assertions; and a given assertion may be (thought to be) true, may be false, may or may not be deducible from other assertions (which serve as premises).

       I must now devise some definitions which allow for a proposition-system like common sense to be inconsistent without being “out of control.”  [I will not keep repeating that mathematical logic has not modeled this phenomenon.]  I consider logic to be the calculus of the consequence-relationship, but the consequence-relationship need not be governed exclusively by syntax.  Let some subset of grammatical propositions be specified as premises.  The (inferential) closure of the set of premises is the set of conclusions of the premises—that is, the set of propositions deducible from the premises under the system’s rules of inference.  (Premises are trivially conclusions of themselves.)  The system is graded if every conclusion is designated as “pertinent” or ‘extraneous” (not both).  (The rigorous meaning of grades is left open pending subsequent explanations.  Heuristically, the reader may imagine that a proposition is pertinent if it is authentically descriptive of a world-state.  Here pertinence is an extension of the concept of truth and is therefore a semantic notion.)  The system is biased if some proposition is not a conclusion; or if the system is graded and not every conclusion has the same grade.  The system is inconsistent if A and ¬A (or A&¬A) are conclusions, a state of affairs which means that every proposition will be a conclusion under the (unconditional, once-for-all) application of classical rules of inference.  [By the way.  The proof that “from a contradiction, everything follows” is very elementary.  But it draws a blank intuitively.  It calls for an unforgiving re-examination.]

       To continue to explain why I introduce these labels:  it has become urgently necessary to be able to understand a belief-system as being contradictory but able to sideline many grammatically correct propositions as “wrong.”  When the illusions are stripped away, mathematics discloses itself as such a belief-system.  Any doctrine of logic which cannot analyze mathematics accordingly, because it has to maintain that mathematics is a paragon of certainty (and consistency), is worthless.  Somehow we have to take hold of the possibility that an important and supposedly inescapable belief-system can be suffused with contradictions. 

       We are concerned with shared conceptualizations or proposition-systems.  Here I do no more than develop the notion of a biased inconsistent system a little further.  (In other words, the shared proposition-system is a case in my logic of contradictions.)

 

—With respect to the proposition-system in question, given that it is inconsistent, I want it to have the traditional inconsistency property that every proposition is deducible.  [Subject to re-examination of the proof of that principle.]

 

—On the other hand, I want the possibility that some conclusions in the system are affirmed while the rest are sidelined.  Hence the concept of grading. 

 

       I suggested that grading is a semantic concept—but might grading be defined syntactically, as a by-product of the action of non-traditional rules of inference?  Given a proposition-system, I characterize the semantics as follows for the present purposes.  [There is a treatment different in connotation, at least, in my paraconsistent logic series.]  The word “true” becomes shorthand for “authentically descriptive of a world-state.” (Acknowledging that a world-state can require a contradiction as its description.  This shorthand extends the traditional meaning of “true.”  To let “authentically descriptive” be a supplemental meaning of “true” seems reasonable to me, but it involves a “change of ideology,” and to anyone who objects to that, I offer a neologism, the contraction audes.)  In a consistent system it is just the conclusions which are true.  In a biased inconsistent system it is just the pertinent conclusions which are ‘true.”

       What I have said is merely a suggestion—but I do not concede for a moment that the loyalists resist it because it is skimpy.  No, the loyalists resist it because they cannot tolerate the first step, the presumption that an important or inescapable belief-system can be contradictory.  “Why would you want a logic of contradictions—which would be a logic of untrustworthy knowledge—when you can obtain absolutely trustworthy knowledge from mathematics?”

 

Answer:  Because mathematics is the worst sort of logic of contradictions.  It is a disguised biased inconsistent system.  Namely:

 

—The pertinent conclusions are misrepresented as the system’s only conclusions.

—The assignment of grades is misrepresented as the classical deductive structure of the system.

—The simultaneous presence of A and ¬A as premises (for some proposition A) is concealed. 

 

Mathematics generates its content by exploiting contradictions while concealing their presence and concealing the circumstance that every proposition is classically provable.

       I find it germane to say more about when I came to certain views.  My manuscript “Mathematical Logic, Common Sense, and the Logic of Contradictions” (1977) announced disguised biased inconsistent systems, and announced that mathematics was such a system.  Thereafter I spent a year and a half on the paper “Can the logic of contradictions be formalized?” which attempted to develop notions presented above abstractly and formally.  (Mainly I tried to link the assignment of grades to the action of non-traditional rules of inference.)  It is clear from the present discussion that that paper, which had the format of a paper in mathematical logic, was a major error of naiveté.  (And I must say that Hennix advised me all along that my effort was ill-conceived.)  But my error was not accidental, nor was it the result of unnaturally poor judgment.  The problem was that I was trying to match the precision and generality which contemporary mathematics and symbolic logic pretend to possess.  Mathematics says that symbolic logic is the whole of logic; that the reasonings and successive discoveries of mathematics are codifiable in symbolic logic; and that consistency of a theory is a matter subject to formal, disinterested, once-for-all judgment. 

       Mathematics as it is taught today is identified with symbolic logic plus certain nonlogical axioms.  The objective formal once-for-all judgment that <there is no grammatical statement A in a system such that both A and ¬A are “provable”> is the verdict that the system is consistent.  (Cf. Tarski, pages 10-12.)  “It may not be possible to prove consistency of a system within the system; but once A and ¬A are proved, this state of affairs can be checked mechanically and the system stands refuted once-for-all.”  <FN More examples of symbolic logic’s present-tense, formal-mechanical-abstract-sterile conception of the action of contradictions in theories can be found in Shoenfield, pages 42-52, 65 (problem 2), 79-80, and 97 (problem 11).>

       Inasmuch as contemporary mathematics claims that logistic expresses the practice of mathematics, contemporary mathematics is engaged in a monumental act of misdirection.  Mathematics is the most “mechanical,” the most abstract, the most sterile, the most objectified of all established intellectual activities—but the overall processes which sustain mathematical subject-matter and control its contradictions are not intellectual.  They are interested social processes.  Mathematical logic conjures with the notion of a consistent theory.  Consistency is a property to be found by a formal, disinterested, once-for-all judgment.  (In practice mathematicians have no compunction about operating in the absence of a consistency proof.)  This notion of a consistent theory is … not incomplete but worthless … as an overall conception of how important intellectual systems function.  And the claim that mathematics does function like that is a part of the social process of deception which actually sustains mathematics.

       My assessment of mathematics as a biased inconsistent system was more frank than the official claim that mathematics is a consistent formal first- or second-order system.  But in 1977 my surmise was not cynical enough.  I did not realize that whenever mathematics is discovered to be inconsistent, metamathematics grants a reprieve until the system is patched up.  (Kleene never said anything about this.)  I did not realize that mathematics is at its most typical when it is converting a contradiction to a consistency by suppressing the intentions according to which the result in question was an inconsistency.  I did not realize that mathematics involves a mind-set which stigmatizes the attempt to locate a contradiction in mathematical subject-matter as unprofessional.  (10) (Tarski never said that social sanctions were deployed to render contradictions inaccessible.)  I did not realize the influence of pragmatic considerations on the decision to defend a result even though it is contradictory. 

       At the same time, we must remember that mathematics cannot be just a mercenary social practice.  And its suitability to the manipulation of matter is a challenge which must be acknowledged.  Further investigation of mathematics as an actively evasive, socially regulated logic of contradictions should stress the interaction of its intellectual rationale, its technological efficacy, and its guidance by social influences.  But why do we want to investigate it?  If it is a theoretical quicksand—if it is untrustworthy—why dwell on it?  I can give three reasons for doing so.

 

1. An understanding of how mathematics functions as a logic of contradictions is one source of power over mathematics, of power to suspend any or all laws of mathematics at will.

 

2. Mathematics is an extraordinary instance of a historical, disguised logic of contradictions.  It teaches us to look beyond formal algorithms in searching for the mechanisms which control contradictions.  It teaches us that we should expect a logic of contradictions to be something more worldly than an abstract intellectual game whose worth is decided once-for-all.

 

3. When mathematics is viewed in the full context of its misrepresentations and its social ramifications, then (and only then) it does break the framework of objectivity—inasmuch as it represents the determination of a phantom objectivity by reciprocal subjectivity.  But mathematics is an obscurantist framework-breaking process because it acts by palpable deceit and delusion and thus cripples its subjects.  <FN It must conceal and deny the procedure wherein it breaks the framework.>  It is “a cruel experiment” [S. Lem, P. Berenyi].  Mathematicians engage in a sort of witless negotiation with one another and create a realm of mechanical relationships among sterile phantom objectivities which then usurps control and turns the mathematicians into its slaves.

 

       Let me refer again to what Hennix said in 1977 about the logic of contradictions.  “Why would you want a logic of contradictions which would lead you to give priority to the study of common-sense delusions, when mathematics enables us to rise above all social delusions and to achieve absolutely certain knowledge?”  This question expresses a monumental illusion, and indicates the extent to which the culture’s myths have rendered many unable to see the obvious.  (The circuit of disillusionment is finally competed, and “antitheses” turn out to be identical.)  Mathematics, supposedly the means for transcending the logic of contradictions, turns out to be the worst sort of logic of contradictions.  And the notion that mathematics is more rarified and noble than common sense turns out to be another monumental illusion—as I repeatedly tried to advise Hennix. 

       Mathematics is an extrapolation of natural language and common sense which is so one-sided that its refinements are often counter-intuitive.  Nevertheless, any number of primitive notions necessary to mathematics are contained in common sense; and the problem of the inconsistency of those notions is the same for common sense and for mathematics.  Then, the processes which sustain and control the inconsistencies in mathematics are interested social processes. 

      Common sense is the medium of apprehension of “the world” which must be learned like natural language.  Everybody has to be versed in it.  Mathematics, while technically indispensable to many civilizations, remains a province of adepts.  Those adepts are devotees of sterile “mechanical” phantom objectivities.  (Cf. every mathematician who says that mathematical results are discoveries.)  The obscurantist rationalization is on behalf of that devotion.

 

       The foregoing results make this study a contribution to my logic of contradictions, itself a branch of meta-technology.

 

                                           •

 

Afterward

 

       Hennix said in 1977 that the trans-mathematical direction of criticism is unimportant because it does not involve genuine mathematics.  (Cf. Supplement One.)  Wette and I crossed swords on the same issue in written exchanges early in 1984.

       But on the contrary.  The assessment of mathematics in this chapter constitutes an overwhelming case that in-house results cannot comprise a decisive critique of mathematics.  The profound direction of criticism is the trans-mathematical direction. 

 

                                           •    

 

Glosses

 

(1)  There is one gap in Brouwer’s formulation.  He does not demand that the contradiction issue from extended calculations.  It is a safe guess, however, that that is what he assumed.  Neither he nor any other professional was thinking about finding extra-mathematical contradictions in mathematics.  Of course, the boundary of mathematics became movable with the announcement of the figures of the syllogism, and later with the announcement of Boolean algebra, set theory, “the paradoxes,” and formal languages.  [A fringe example would be Yessenin-Volpin’s ascription of tenses to integers.]  Brouwer was insensible to these considerations because of his prejudicial elementarity beliefs about quantity.  (By the way, was the formalization of intuitionistic logic the crucial revision of Brouwerism?) 

 

(2)  “As long as only finite groups of properties are allowed,” says Brouwer.  What is a group of properties?—is this an obscure reference to set theory?  It needs a full explanation.  Hilbert, in his 1934 Introduction to Grundlagen der Mathematik, Vol. 1, said that one could get past the Gödel obstacle by using the finite standpoint more sharply.  Is this strange, coming from the man who said that we would not be driven from Cantor’s paradise?  There was a folklore that unwanted outcomes would disappear if one were finitist.  I suggest on the contrary that matters become worse that ever when one takes one’s stand on the generic quantification ‘finite’.

 

(3)  In this latter case, mathematics accomplishes what it wants to accomplish by running roughshod over reasonable objections.

 

(4) All of my philosophical distaste for intuitionism [and for ultra-intuitionism] remains.  Intuitionism’s claim to possess the true truth is repellant to me.  The prejudicial skepticism is repellant to me.  One may, if one wishes, directly challenge Brouwer’s prejudicial elementarity claims or prejudicial skepticism: 

a. The solipsistic conception of the Creative Subject.  The solipsistic self, by assumption, always has perfect reason and perfect memory.  This condition is a stronger standard of correctness than “what seems to be, to the self.”  This objective reliability thus transcends solipsism, contradicting the assumption that we are confined in solipsism.  Objective solipsism!

b. Brouwer says that mathematical knowledge can only be afforded by an achieved construction.  Nevertheless, he allows abstractly possible constructions and does not restrict constructions to those that are (empirically) feasible.

 

(5) Knopp’s intermediate textbook, page 102, does not present it that way, displaying instead an attitude like Remko Scha’s.  “You can’t apply a standard manipulation to a new problem until you prove that you are entitled to do so; the sums which can be obtained by grouping are not alternative results but mistakes.”  If it were simply forbidden to extend mathematics by analogy—which is what Knopp and Scha think they want—then Liebniz’s series should not have been permitted in the first place, and no infinite series should have been permitted.  The expedient

 

…

 

probably the most important symbol in mathematics-the-discipline, should not have been permitted.  Cantor’s diagonal argument, which is nothing but a two-dimensional use of ‘…’, should not have been permitted.

 

(6) I managed, after two long sessions (in the winter of 1997-98), to convince Graham Priest that Craig Smorynski’s proof of the Diagonalization Lemma in Self-Reference and Modal Logic is a bad job.  But Priest said, nobody would care that you discovered this.  No journal would print your objection.  Priest also said, if you manage to find something wrong with the Diagonalization Lemma, they will just change it.  (He couldn’t imagine that the objection would dog the proof in its new guise.)

 

(7) Civilization is an edifice of expedient falsehoods and the purpose of education is to discover those students who enjoy being deranged in approved ways.

 

(8) “Subjective Propositional Vibration” was published in 1975 in Blueprint for a Higher Civilization.

 

(9)  Plainly put, the “many-valued number” results that matter to me come in a range of meta-technological contributions.

 

(10)  I say this after observing that “destructive mathematics” has an “underground character.”  The goals of Wette, Yessenin-Volpin, Hennix placed them on the side of the Devil (about Priest I can’t say).  Hennix wondered in a typed document why there had been no appraisal of Wette’s “self-contained” paper (International Logic Review, 1974)—why the substantial appraisal of Wette was limited to the famous Bernays paper in Dialectica 1971.  Wette is very conscious of having violated a taboo.  He wrote to Hennix, 4 July 1996, “I am Beelzebub.” 

For completeness, I have to note that the theatrics do not always play out this straightforwardly.  It was Cantor who proved Failure Theorems, and Kronecker who railed against him on behalf of tradition.  History judges Cantor, the troublemaker, to be the hero—while tolerating Kronecker as a cramped alternative.