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**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]

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.)

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.

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.

•

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.

•

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.”]

•

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 = … + x^{2} + x + 1 + 1/x + 1/x^{2} + …

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.

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.

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.

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.

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 x^{2} = 2, is rational gives a contradiction. **Then what is the contradiction that follows
from the assumption that x, for x ^{2} = 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.]

•

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

**—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.

•

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.

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__

__GENERAL
__

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W.
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Michael
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E.
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Arithmetic,” Studies in Logic, Grammar and Rhetoric (2001)

Moritz
Pasch, *Vorlesungen** über neuere Geometrie* (1882), page
21

Carl
J. Posy, “Brouwer’s Constructivism,” Synthese 27, 1974,
pages 125-159 AP1.S9

Dag Prawitz, Natural Deduction (Stockholm, 1965) BC71.P73

W.V.O.
Quine, From a Logical Point of View BC71.Q48

W.V.O.
Quine, Word and Object B840.Q5

W.
V. O. Quine, Selected Logic Papers (1966), try page 119

W.V.O.
Quine, The Ways of Paradox (Revised Edition, 1976) B945.Q5 W3

W.
V. O. Quine, Ontological Relativity (1969) B840.Q49

H. Rasiowa and R. Sikorski, The Mathematics of Metamathematics
(1963)

H.
L. Resnikoff and R. O. Wells, Jr., Mathematics in
Civilization (1973) QA21.R43

Fred
Richman, ed., Constructive Mathematics

Abraham
Robinson, “Some thoughts on the history of mathematics,” in Logic and
Foundations of Mathematics (Groningen, 1968), pp. 188-193 QA9.L62

Dennis
Rohatyn, Philosophy/History/Sophistry (year given as 1997 and 1999) [seems a rare book]

J.B.
Rosser, Logic for Mathematicians (1953)

J.
B. Rosser and A. R. Turquette, Many-Valued Logics
(1952) BC135.R59

H.
L. Royden, Real Analysis (1963)

Abraham
Seidenberg, The Ritual Origin of Geometry, Archive for History of Exact
Sciences (1960)

Abraham
Seidenberg, The Ritual Origin of Counting, Archive for History of Exact
Sciences (1962)

Stewart
Shapiro, ed., Intensional Mathematics (1985)

Joseph
R. Shoenfield, Mathematical Logic (1967) QA9.S52

Jeffrey Sicha, A Metaphysics of Elementary Mathematics (1974) QA8.4.S54

Craig Smorynski, Self-Reference and Modal Logic (1985)

Ian
Stewart and David Tall, The Foundations of Mathematics (1977) QA9.S755

Gabriel Stolzenberg, “Can an Inquiry into the Foundations of
Mathematics Tell Us Anything Interesting about Mind?” Psychology and Biology of Language and
Thought (1978)

Alfred
Tarski, R. M. Robinson, and A. Mostowski, Undecidable
Theories (1953) QA9.T33

Robert Tragesser, “Gödel’s Paradox of Geometric Intuition,
Speculative Cosmology, and Zeno’s Paradoxes” (1978)

A.
S. Troelstra, Principles of Intuitionism (1969)
QA3.L28 no. 95

A.
S. Troelstra, Choice Sequences (1977) QA9.47.T77

A.
Turing, “Systems of Logic Based on Ordinals” in The Undecidable (1965) [page
166 has the Oracle]

Dirk
van Dalen, Logic and Structure (1983)

D.
van Dantzig, “Comments on Brouwer’s Theorem on Essentially-negative
predicates,” *Indagationes** Mathematicae*,
Vol 11 (1949) QA1.
I48

Jean
van Heijenoort, ed., From Frege to Gödel (1967) QA9.V3

W.P.
van Stigt, Brouwer’s Intuitionism (1990)

François
Vieta, Isagoge in artem analyticam (1591)

Eliseo Vivas, The Moral Life and
the Ethical Life (1950), for pages 254-55

Hao
Wang (A), A Survey of Mathematical Logic (1964) BC135.W3

Hao
Wang (B), ‘Process and existence in mathematics,” in Essays on the Foundations
of Mathematics, ed. Y. Bar-Hillel (Jerusalem, 1966), page 328 QA9.J45

Hao
Wang (C), From Mathematics to Philosophy (1974) BD161.W27

Ludwig
Wittgenstein, Remarks on the Foundations of Mathematics (revised, 1978)

Ludwig
Wittgenstein, Philosophical Grammar (1974)

Ludwig
Wittgenstein, Wittgenstein’s Lectures on the Foundations of Mathematics, ed.
Cora Diamond (1976) [page
237, intuitionism]

Ludwig
Wittgenstein, Philosophical Remarks (1975)

•

__FINITISM ETC., INCLUDING
UNPUBLISHED DOCUMENTS
__

__publications
__

Paul
Bernays, [section on Wette’s paper “Vom Unendlichen zum Endlichen”], Dialectica, 1971,
pages 191-193

A. Erenfeucht, “Logic Without Iterations,” Proceedings of the
Tarski Symposium (1974), p. 265

James
Geiser, “Rational Constructive Analysis,” in Constructive Mathematics (1981),
p. 321

David
Isles, “Remarks on the Notion of Standard Non-Isomorphic Natural Number
Series,” in Constructive Mathematics (1981), p. 111

Jan Mycielski, “Analysis Without Actual Infinity,” Journal
of Symbolic Logic, Sept. 1981, p. 625

Rohit
Parikh, “Existence and Feasibility in Arithmetic,” Journal of Symbolic Logic,
September 1971, pages 494-508.

Rohit
Parikh, Effectiveness, The Philosophical Forum, 1980

Graham
Priest et al., ed., Paraconsistent Logic (1989) [affirms that early calculus and naïve
set theory are non-trivially inconsistent]

Graham
Priest, “Is Arithmetic Consistent?” Mind, 1994, page 337

Graham
Priest, “What Could the Least Inconsistent Number Be?” Logique et Analyse, 1994, pages 3-12 BC1.L6

Graham
Priest, “Inconsistent Models of Arithmetic: Finite Models,” Journal of Philosophical Logic (1997), page 223 BC51.J68

Graham
Priest, “Perceiving Contradictions,” *Australasian
Journal of Philosophy*, December 1999

P.
L. Raschevskii, “On the Dogma of the Natural
Numbers,” Russian Mathematical Surveys, vol. 28, Jul.-Aug. 1974, page 143 QA1.R8 [very large integers are infinite]

Jean
Paul __Van Bendegem__, “The Return of Empirical
Mathematics” (Gent, 1983)

Jean
Paul __Van Bendegem__, “Strict Finitism as a
Viable Alternative in the Foundations of Mathematics,” Logique et Analyse, 37, 1994, page 23

D.
van Dantzig, “Is 10__10____10__ a finite
number?” Dialectica, 1956, pages 273-7 B1.A15

Eduard
W. Wette, Vom Unendlichen zum Endlichen, Dialetica, 1970, pages 303-323

Eduard
W. Wette, “On new paradoxes in formalized mathematics” (abstract), Journal of
Symbolic Logic, 1971, 376-7

Eduard
W. Wette, Contradiction within pure number theory because of a system-internal
‘consistency’-deduction,” International Logic Review, 1974, pages 51-62 [claim
to furnish a formal derivation of the Gödel-sentence Con(S).]

Eduard
W. Wette, Refutation of Number Theory I (pamphlet, Würzburg, 1975)

A.S. Yessenin-Volpin, Forward to I. N. Hlodovskii,
A New Proof of the Consistency of Arithmetic [originally 1959 in Russian],
American Mathematical Society Translations, Series 2, Volume 23 (1963)

A.
S. Yessenin-Volpin, “Le programme ultra-intuitioniste des fondements des mathématiques,” Infinitistic Methods (1961) QA295.S88

A.
S. Yessenin-Volpin, “The Ultra-Intuitionistic
Criticism and the Antitraditional Program for Foundations of Mathematics,”
Intuitionism and Proof Theory, ed. A. Kino, J. Myhill,
and R. E. Vesley (1970)

A.
S. Yessenin-Volpin, “About Infinity, Finiteness, and Finitization,” in Constructive Mathematics, ed. Fred
Richman (1981)

A.
S. Yessenin-Volpin, “On the finitization of consistency proofs for ZF-like systems,” in Constructive Mathematics
[alternative title for the preceding?]

__unpublished
__

author?
“Beware of the Gödel-Wette Paradox” (2001)

James
Geiser, Review of “The Ultra-Intuitionistic Criticism,” Mathematical Review
#4938 (September 1973)

James
Geiser, Commenting Proofs, MIT, 1974

C.
C. Hennix, Appendix to “On the Main Problem in the
Foundations of Mathematics”/The Main Problems of the Foundations of
Mathematics, N.D. [refers to 12 March 1974?]

C.C. Hennix, A Finitistic Generalization of Recursiveness, 1979

C.C. Hennix, Computable Functions with Computational
Types, 1982

C.C. Hennix, Ultra-Recursion, the Theory of Methods and the Splitting of the Notion of Effectiveness,
1984

C.
C. Hennix, Hilbert’s Program, July 25,1985

[a
commentary on *From Frege to Gödel* and *The Souslin Problem* (1974)] — H.F. notes on a private lecture

C.C. Hennix, The D-Paradox: report on a new paradox discovered by A. S. Yessenin-Volpin, MIT, 1986

C.C. Hennix, Progress Report On Finitization of Metamathematics, 1987

C.C. Hennix, note on “core consistent theory,” 4 December
1993

C.C. Hennix, Volume I of Introduction to the Gödel-Wette
Paradox (1996)

C.C. Hennix, Introduction to the Gödel-Wette Paradox,
1996-7 (drafts?)

C.C. Hennix, Some Results in Ultra-Recursion Theory, N.D.

C.
C. Hennix, Quod Decet Bovem Dedecet Jovem,
N. D.

C.C. Hennix, Constructive Generalized Recursion Theory,
N.D.

The
NSF Grant proposal of **Isles** and **Yessenin-Volpin**,
May 28, 1980. The referees’ reports.

Eduard
W. Wette, On the Gödel-Wette Paradox (one-page comment on Volpin-Hennix,
June 27, 1996)

A.
S. Yessenin-Volpin, Prototheories (manuscript lectures written after 1970)

A.
S. Yessenin-Volpin, On the Main Problem in the
Foundation[s] of Mathematics (Boston, March 12, 1974)

A.
S. Yessenin-Volpin, Postscriptum to: The Main Problem, April 12, 1974

unlabeled
"Preface” to Yessenin-Volpin’s report (the 12
March 1974 paper?)

A.
S. Yessenin-Volpin, Paradoxes of Finitization and Consistenty Proofs, 1984

A,
S. Yessenin-Volpin, Formula or Formuloid? 1997

__correspondence by date
__

Robert Tragesser to Yessenin-Volpin,
March 29, 1978

[re Gödel’s Paradox of Geometric Intuition]

Flynt,
letter to Stig Kanger and
Dag Prawitz, January 30, 1978

Stig Kanger, letter to Flynt, February 11, 1978

Remko Scha to Flynt, February 8, 1980

Remko Scha to Flynt, April 26, 1980

Hennix memo to Robert Cohen, July 7, 1981 [proposal for a book by Yessenin-Volpin]

Flynt
to Wette, 29 January 1984

Wette
to__ __Flynt, 7 March 1984__
__

Wette
to__ __Flynt, 29 March 1984__
__

Hennix to Wette, 10 July 1984

Wette
to Hennix, 25 July 1984

Flynt
to Wette, 30 Nov. 1984

Wette
to Flynt, 15 December 1984

Dick Hoekzema to Flynt, 1 November 1991

Wette
to Hennix, 23 June 1996

Wette
to Hennix, 26-27 June 1996

Wette
to Hennix, 4 July 1996

Hintikka to Hennix, September 5, 1997 [Gödel’s
incompleteness results] is [sic!] simply a true combinatorial theorem of
elementary arithmetic.

Hennix to Hintikka, September 28, 1997

Wette
to Hennix, 15 March 1997

Wette
to Hennix and Yessenin-Volpin,
24 April 1997