martin ohmacht about wittgenstein's remarks about...

Martin Ohmacht

About Wittgenstein's Remarks

About Mathematical Impossibilities as Negative Settlements

And On the Inherent Limitations (gaps) of Mathematics

File 03 of 5 – Letters H to L

File 03 Letters H to L: Page 180

Ohmacht on Wittgenstein’s Remarks on math. Impossibilities as negative Settlements

And on the Inherent Limitations (gaps) of Mathematics; Lacunae Mathematicae

Data for the catalogue

3 – 9501974 – 7 – 8

Table Of Contetnts: File 3 Letters H to L: ▓A Hardy result as an object of envy for

Wittgenstein (#MIMP_07, #MIMP_17, #MIMP_21)

Table Of Contetnts: File 3 Letters H to L

Martin Ohmacht ............................................................................................................................... 179

About Wittgenstein's Remarks ......................................................................................................... 179

About Mathematical Impossibilities as Negative Settlements ......................................................... 179

And On the Inherent Limitations (gaps) of Mathematics ................................................................ 179

File 03 of 5 – Letters H to L............................................................................................................. 179

Table Of Contetnts: File 3 Letters H to L .................................................................................... 181

Letter H ........................................................................................................................................ 183

▓A Hardy result as an object of envy for Wittgenstein (#MIMP_07, #MIMP_17, #MIMP_21)

.................................................................................................................................................. 184

Heptagon construction by trisection of a single angle ............................................................. 186

Hessenberg tames Russell's contradiction (#7/, #17/) .............................................................. 187

▓Hidden warning in a historiographical text (#MIMP_23)..................................................... 189

▓Hilbert's résumé in his dialectics with Emil DuBois-Reymond (#30/) ................................. 190

▓Hobbes, the loser in mathematics (#MIMP_09, #MIMP_18)............................................... 192

▓Hofstadter's MU-puzzle (#0/, #5/) ........................................................................................ 194

▓Hope that it may be impossible to find a contradiction (#MIMP_08, #MIMP_11,

#MIMP_20) .............................................................................................................................. 196

▓hypercomplex numbers (#MIMP_16, #/MIMP_13, #/MIMP_15) ....................................... 198

Letter I .......................................................................................................................................... 200

The Ignorabimus separately discussed for the natural sciences: flight to the moon, for example

(#MIMP_14, #MIMP_32, #/MIMP_18, #/MIMP_26) ............................................................ 201

▓An impossible proof, which was apparently never attempted to establish: the Axiom of

Choice (#MIMP_26) ................................................................................................................ 204

An improper impossibility (#MIMP_04, #MIMP_18, #MIMP_19/) ...................................... 206

▓Inaccessible numbers (#18/, #30/)......................................................................................... 208

▓Incommensurability and similar phenomena (#MIMP_00, #MIMP_01, #MIMP_08,

#MIMP_11) .............................................................................................................................. 210

Incompleteness (#19/) .............................................................................................................. 211

infinity has to be ascertained by an axiom (#29/) .................................................................... 213

▓Influences on the rise of non-Euclidean geometries (#12/) .................................................. 216

De Insolubilibus (#7/) .............................................................................................................. 217

▓Inspiration from a previous failure: Hamiltonian Triplets (#16/).......................................... 219

Inspiration of al-Khayyam by the Pythagorean affair (#1/, #2/) .............................................. 222

Introduction: the Problem of Impossibilities in Mathematics (#0/) ......................................... 224

Letter J.......................................................................................................................................... 226

Placeholder for the Title of Level 3J ........................................................................................ 227




Letter K ........................................................................................................................................ 228

Khayyam-Pacioli Fallacy (#2.3/) ............................................................................................. 229

Kuhn as prepared by Wittgenstein (#0/, #1/) ........................................................................... 230

Letter L ......................................................................................................................................... 232

▓Laplace vs Poincaré (#MIMP_14) ........................................................................................ 233

A light bulb moment concerning two mutually proximate impossibilities (#1/, #19/) ............ 236

Liouville on integral (exp(x)/x) dx (#23/) ....................................................................... 237

▓Logical ambiguities (#25/) .................................................................................................... 239

▓Logical problems for mathematicians (#MIMP_26, #/MIMP_12) ....................................... 241

Letter H: ▓A Hardy result as an object of envy for Wittgenstein (#MIMP_07,

#MIMP_17, #MIMP_21)

Letter H




▓A Hardy result as an object of envy for Wittgenstein (#MIMP_07, #MIMP_17, #MIMP_21)

Wittgenstein was partly informed about the foundational crisis of

mathematics when he worked on the foundations of this subject. We can see

from Ramsey's remark on the Continuum Hypothesis in a letter that the

philosopher had at least a faint knowledge of this issue.

I have found in all three quotations by Wittgenstein, where he comes close

to making remarks about this important issue, one from the LFM and two in

the RFM. When reading these quotations, one can ask: did Wittgenstein

understand the CH? But there is a counter question as well, namely, what was

Wittgenstein's goal when coming close to the CH in his writings? Let us work

on this question by discussing the statement from the LFM, first! Here, it is a

participant of Wittgenstein's seminar, who brings in this important issue:

“Levy: Is 'Professor Hardy believes that ℵ1 > ℵ0' a mathematical

statement?

Wittgenstein: No. It is no more a mathematical statement than

'Willie said that 7 x 8 = 54' is a mathematical statement” (Unit III, page

34).

Here, Wittgenstein was lead by one of his students to the difficult issue of

the CH. (But later in his LFM he mentions Aleph 0 and Aleph 1 himself, see

below.) The reason, why I think that Levy's statement is an allusion to the CH

is the fact that his remark includes a reference to the act of believing.

Wittgenstein in his reaction chooses a proposition from elementary

mathematics, which is almost true (the error is one of the value 2, only). This

comparison may be interpreted in a way, that the proposition 7 x 8 = 54 can be

seen as false but almost true.

Later, in unit XVIII on page 171 it is Wittgenstein who refers to Aleph 0

and Aleph 1. Therefore it can be said that Levy's initiative in unit III was

successful on the long run, because he succeeded in motivating Wittgenstein

to turn his focus on the transfinite cardinals invented by Cantor. From this

passage here, it is clear, that Wittgenstein accepts the series of Cantor's infinite

cardinals as a valid concept of mathematics, which makes us think – at least –

that the infinite number of all whole numbers (ofℕ) does really exist not only

potentially, but exists in the sense of an actual infinity.

But is is certainly an over-interpretation of Wittgenstein, if we would

conclude that he made an allusion on the possible independence of the CH,

because these quotations were done in 1939. Gödel had completed his proof of

the non-demonstrability of the CH for Aleph x with x <> 1 in the year before.

Letter H: ▓A Hardy result as an object of envy for Wittgenstein (#MIMP_07,

#MIMP_17, #MIMP_21)

Although the suspicion about the independence of the CH was already under

way at that time it is unlikely that Wittgenstein was informed about Gödel's

premonition (Hao Wang writes in Reflections on Kurt Gödel on page xxi f.,

that Gödel “began to tackle the independence of CH” in 1937.)

It is a historiographical fact, that G. H. Hardy worked on an aspect of the

CH in 1906 – he proved that 2ℵ0 ≥ ℵ1, as Gregory H. Moore reports on page

83. So Hardy works on an issue, which is a sort of optimum of what can be

attained without being embroiled by the independence of the CH.

Wittgenstein's intention when coming close to the CH was a certain feeling of

envy for this result. He alludes to it by mentioning a variant of it, namely the

proposition 2ℵ0 > ℵ0in his RFM on page 135 in § 35.

But it is also a fact, that the finest proof of the uncountability of the

powerset of the natural numbers is not attributed to Cantor himself, but to

Gerhard Hessenberg in his book of 1906 (I discuss it in the section on

the↑Hessenberg tames Russell's contradiction). Before Wittgenstein's remark

on Hardy's result in §35, he might allude to Hessenberg's proof. He argues the

problem, that his proof violates the principles of the theory of types.

So summing things up, two proofs of almost the same contentual

dimension were presented in 1906 by Hardy and by Hessenberg.

Hessenberg's proof: when we want to prove that the powerset of the set of

natural numbers has more elements than the set of natural numbers, then we

can use a tamed version of the paradox of the Cretan liar. Here, the focus, on

which we look, constructs a contradiction again, but it is not an overall

(global) contradiction, which spoils the entire calculus (as it happened with

Frege's Axiom of Abstraction or Axiom of Comprehension). This was the fate

of mathematicians by Russell's inference of a contradiction in 1902. Here,

with Hessenberg, by a similar construction, we only complete an indirect

proof, which shows that2ℵ0 > ℵ0.

My argument, that Wittgenstein was possibly familiar with Hessenberg's

proof is given by referring to his mentioning the term “heterological” several

times in his RFM. (on page 206, § 79, for example)

Wittgenstein is musing about this watershed between a construction of a

contradiction, which spoils an entire theory (namely, that of Frege) and a

technically very smart proof, namely that there are essentially more sets of

natural numbers than there are natural numbers. The process in the history of

ideas here is lying therein, that a absolutely destructive idea (as Russell's was)

is changed into a technically nice and methodically orderly piece of

mathematics.

Wittgenstein mentions the two concepts ofℵ0and2ℵ0a second time in his

RFM (page 409, at the end of § 42). Here he mentions Frege, and distinguishes




between the natural numbers (ℵ0) and the real numbers (2ℵ0,

“Dezimalbrüche”). It is an open question, whether here again an allusion to

the Continuum Hypothesis is the motive for mentioning these two infinite

cardinals, but it might be, that this is the case. Anyway, I think that

Wittgenstein envies G. H. Hardy and Cantor (or his disciple Gerhard

Hessenberg) for their smart proof that there are considerably more real

numbers than natural numbers.

Bibliography:

Hao Wang (1987): Reflections on Kurt Gödel, Cambridge, Massachusetts:

MIT Press.

Hardy, Godfrey Harold (1906): The Continuum and the Second Number Class.

In: Proceedings of the London Mathematical Society. (2) 4, pages 10–17.

Reprinted in 1979.

Hardy, Godfrey Harold (1979): Collected Papers of G.H. Hardy, ed. By I. W.

Busbridge and R.A. Rankin. Oxford: Calendron vol. VII, pages 438–445.

Moore, George H. (1982): Zermelo's Axiom of Choice. Its Origins,

Development and Influence, NY: Dover Publications.

Heptagon construction by trisection of a single angle

The summary of this section reads as follows: the strategic point of the

construction of a heptagon by assuming the availability of an angle trisector is

the fact that the single angle to be trisected has no geometrical interpretation at

all, but is a purely algebraical result.

Now let us develop the calculation of the angle to be trisected:

The seventh degree formula for the heptagon can be reduced – by a simple

substitution – to a third degree formula, namely

27 ∗ 𝑣3 − 63 ∗ 𝑣 = 7. (*)

In order to prove the inconstructability of the heptagon, we have to prove

the irreducibility of this cubic polynomial; by the Eisenstein criterion, for

example.

The trisection of a general angle leads to a cubic equation, too. If we assume

(by an axiom) that we have an angle trisector at hand then we can pose the

question as to whether we can use it to construct a heptagon. The answer is

“yes”.

Letter H:

The trisection of an angle produces the equation

4 ∗ 𝑥3 − 3 ∗ 𝑥 = 𝑎 (**)

where a is the cos of the angle to be trisected. Then, x is equal to the cos of

the third part of the angle represented by a.

I chose to work with a linear substitution for the variable x in equation (**),

namely 𝑥 = 𝑐 ∗ 𝑦and a second linear substitution for the entire equation of the

trisection, so that we have:

4 ∗ 𝑐3 ∗ 𝑑 ∗ 𝑦3 − 3 ∗ 𝑐 ∗ 𝑑 ∗ 𝑦 = 𝑎 ∗ 𝑑. (***)

The point of the calculation here is that I operate with a coefficient

comparison between (*) and (***), by this transfer of the solution from (**) to

(***) and from (***) to a solution of (*). The comparison of coefficients leads

to 𝑑 = 14 ∗ √72

and 𝑐 =21

𝑑and hence 𝑎 =

1

(2∗ √72

); a is the cos of the angle to be

trisected.

It is interesting to see that the construction of the heptagon can be

accomplished through a single angle trisection.

To conclude, let me make one remark. Here, we have used algebraical

equations to find the value for the angle to be trisected. The width of the angle,

the cos of which is 𝑎 =1

(2∗ √72

), has no geometrical interpretation at all. It is a

purely algebraical result which is strongly influenced by the substitutions used.

This epistemological (philosophical) result is likewise stated by Gleason

(1988). So what we are doing here is performing calculations, done by means

of geometrical drawings (see the picture by Hans Dirnböck).

Gleason reports on the regular 13-gon, that here the angle to be trisected

contains the expression1

√132 , which makes me think that there is a connection

between the “7” in my angle and the fact, that it was a 7-gon which we

constructed.

Bibliography:

Gleason, Andrew (1988): Angle Trisection, the Heptagon, and the

Triskaidecagon, in: American Mathematical Monthly, 95, pages 185–194.

Hessenberg tames Russell's contradiction (#7/, #17/)

If we look at the foundational crisis of mathematics in a surveying kind,

then the overall structure of intellectual influence shows, that Russell's

paradox was found independently of the Cretan Liar first and only later their

similarity was rediscovered. When Russell created his paradox in 1901, he




was not inspired by the self-referentiality of this classical paradox, but Gödel

in 1931 mentions the Cretan Liar as a main source of inspiration for inventing

the statement “this statement cannot be proved.” Russell was informed about

the similarity of his contradiction with the Cretan Liar in 1908.

Seemingly, Russell informed Wittgenstein and he then taught the Cretan

Liar to his pupils in the Volksschule during the 1920s, as it is reported by

Bartley.

It is true that Hessenberg in his 1906 book Grundbegriffe der Mengenlehre

in § 24, Die Menge der Teilmengen (page 40–42) does not mention the

paradoxes of classical antiquity. So if use the contentual similarity between the

Cretan Liar and Hessenberg's argument, then we can only conjecture influence

form Russell to him. Hessenberg's argument (which is sometimes more or less

directly attributed to Cantor: see Oliver Deiser page 102f.) sounds as follows:

We want to prove that there is no bijection betweenℕand its powerset

P(ℕ). The proof is done indirectly, but it is done somehow constructively. Let

us assume that there is a bijection b fromℕonto P(ℕ). From this assumption

we shall infer a contradiction. Let us have a look at the diagonal set D, which

is the set of all members m ofℕ, for which m is not included in b(m). If the

number b–1 (D) is contained in D, then according to the definition of D, b–1 (D)

is not included in D. And conversely, if b–1 (D) is not in D, then (according to

the definition of the complement of D) b–1 (D) is in D. So we have the desired

contradiction.

There is a slight epistemological problem with this argument because it

does not conform to the basic rule of the theory of types. It can be said, that

Russell developed his theory as a reaction to his self-referential argument

which leads to a contradiction. So Russell identified the self-referentiality to

be “blamed” for the contradiction.

What I like about Hessenberg's argument is the fact, that it so to say tames

Russell's argument, which demonstrated Frege's system to be flawed by a

contradiction (and, by the Ex falso quodlibet, to be flooded by contradictions).

But the effect of Hessenberg's version is by far not as disastrous as it had been

the effect of Russell's idea – Hessenberg's approach to the argument is far

more civilised, in as much as he tamed the contradiction to work in an indirect

proof only.

The beauty of this proof and its epistemological effectiveness lies therein,

that it works for all sets, namely, finite sets as well as infinite sets. It is true,

that in order to prove the existence of the larger infinite cardinals, we need the

axiom of the existence ofℕ. But once we have adjoined this axiom to be true,

like in a big bang all the larger infinite cardinals can be proved to exist.

At the end of this comment on the connection between the medieval

Letter H: ▓Hidden warning in a historiographical text (#MIMP_23)

insolubilia and modern set theory I want to ask a question: why did Cantor in

his short essay Über eine elementare Frage der Mannigfaltigkeitslehre

(Ges.Abh. 278–281) not present the above proof of Hessenberg's but displayed

a proof, which is a little bit cumbersome? In this publication, the diagonal

argument carries over from the real numbers to the set of all subsets of natural

numbers in a contentually interesting way.

Cantor suffered a hostile atmosphere during all of his lifetime by many of

his critics, so that it might be, that he didn't want to be too clear in the

development of infinities. Zermelo in his comment on page 280 says that the

proof showed by Cantor can be seen as the classical proof. Yet I think, that the

proof by Hessenberg is the classical proof, indeed, because it does not make

use of the index functions, but of the diagonal set D as defined above.

▓Hidden warning in a historiographical text (#MIMP_23)

I have found the following passage about the problem of functions, the anti-

derivative of which cannot be expressed in elementary functions. For finding it,

however, I had to go through a lot of text in the book edited by Jean Dieudonné.

It was hard work to find the hidden passage in which the creators of the text

refer to this impossibility. Despite the epistemological importance of this issue,

it is not mentioned in the initial summary of the chapter.

I am convinced that impossibility results should be given a central position

in mathematical and historiographical texts, and the warning to the reader (and

learner) should be placed right in the focus at the beginning. The warning (at a

meta-level) should also be written in large, bold, bulky letters, because of the

very importance of its effect on the attitude of the reader. In my opinion, it

makes no sense to hide the confession of a gap in the realm of formulas in the

very middle of concrete results by calculations. And this placement of the truth

at the beginning should be chosen not despite its frustrating character, but

because of it.

“Das [ein Beweis über Legendresche Normalformen] beweist

insbesondere, dass man die elliptischen Integrale nicht mit Hilfe

algebraischer Funktionen oder mit Hilfe von Logarithmen solcher

Funktionen berechnen kann. Liouville zeigte (1833), dass sich die

elliptischen Integrale nicht durch 'elementare' Funktionen ausdrücken

lassen, wovon die Mathematiker seit langem überzeugt waren, (ohne es

allerdings bewiesen zu haben.) ” (Christian Houzel, page 437)

In English, this quotation runs as follows:

“This [a proof concerning Legendrian normal forms] proves in




particular that one cannot calculate the elliptic integrals with the help of

algebraic functions or with the help of the logarithms of such functions.

Liouville showed (1833) that the elliptic integrals cannot be expressed

through 'elementary' functions – something concerning which

mathematicians were long convinced (without however having proved

it.)”

In order to offer a model for how such a sustainable warning for the naive

reader could look, I want to present one here:

Warning to the reader:

There are derivatives y',

such that for all elementary functions z,

its derivative z' is unequal to y'!

The humble strategy of placing such a warning within the text and without

any emphasis by means of extralarge letters is bound to result in

misunderstandings.

Bibliography:

Houzel, Christian (German 1985): Elliptische Funktionen und Abelsche

Integrale, pages 422–540, in: Dieudonne, Jean (German 1985): Geschichte

der Mathematik 1700–1900. Ein Abriss. Braunschweig: Vieweg & Sohn.

▓Hilbert's résumé in his dialectics with Emil DuBois-Reymond (#30/)

Hilbert enlarges on his acceptance of negative settlements of mathematical

problems at the conclusion of his much-read 1899 text on the Foundations of

Geometry.

“However, as I have already remarked, the present work is rather a

critical investigation of the principles of euclidean geometry. In this

investigation, we have taken as a guide the following fundamental

principle; viz. to make the discussion of each question of such a character

as to examine at the same time whether or not it is possible to answer

this question by following out a previously determined method and by

employing certain limited means. This fundamental rule seems to me to

Letter H: ▓Hilbert's résumé in his dialectics with Emil DuBois-Reymond (#30/)

contain a general law and to conform to the nature of things. In fact,

whenever we encounter a problem or suspect the existence of a theorem

in our mathematical investigations, our reason is satisfied only when we

possess a complete solution of the problem or a rigorous demonstration

of the theorem, or, indeed, when we can clearly see the reason for the

impossibility of the success and, consequently, the necessity of failure.

Thus, in the modern mathematics, the question of the impossibility

of solution of certain problems plays an important role, and the attempts

made to answer such questions have often been the occasion of

discovering new and fruitful fields for research. We recall in this

connection the demonstration by Abel of the impossibility of solving an

equation of the fifth degree by means of radicals, as also the discovery

of the impossibility of demonstrating the axiom of parallels, and, finally,

the theorems of Hermite and Lindemann concerning the impossibility of

constructing by algebraic means the numbers e and π.

This fundamental principle, which we ought to bear in mind when

we come to discuss the principles underlying the impossibility of

demonstrations, is intimately connected with the condition for the

“purity” of methods in demonstration, which in recent times has been

considered of the highest importance by many mathematicians.” (Hilbert

1899, English translation by Townsend 1938, pages 130 f.)

This final statement is Hilbert's epistemological reaction to, on the one hand,

Du-Bois' opinion that there are insoluble problems and, on the other, to the

situation around the axiom of parallels and related phenomena. Hilbert

resolutely stresses the positive effect, if one accepts negative results in research.

However, there is a second occasion where Hilbert talks about the

acceptability of negative result:

In the year 1900, Hilbert stated his point about negative results a second

time. The obvious reason for this strategy was that his aim was to make the issue

absolutely clear, because it appeared genuinely important to him. Hilbert's

statement is not a mathematical argument, but is on a meta-level: is the glass

half full or is it half empty? Should mathematicians, in the event of a negative

result, complain about having searched in vain for something impossible for

such a long time (as in the case of the proof of the axiom of parallels) or should

the attitude towards such an issue be of a more positive, optimistic nature? A

question in which it was possible to achieve a negative settlement is finished

with: mathematicians can now deal with new questions! A definite negative

answer is not a catastrophe in research (as seen in the issue of the

incommensurability of the diagonal of the unit square) but bears within it an

element of enlightenment, because it implies the clarification of a question after

all. Hilbert:




“Occasionally it happens that we seek the solution under insufficient

hypotheses or in an incorrect sense, and for this reason do not succeed.

The problem then arises: to show the impossibility of the solution under

the given hypotheses, or in the sense contemplated. Such proofs of

impossibility were effected by the ancients, for instance when they

showed that the ratio of the hypotenuse to the side of an isosceles right

triangle is irrational. In later mathematics, the question as to the

impossibility of certain solutions plays a preeminent part, as we perceive

in this way that old and difficult problems, such as the proof of the axiom

of parallels, the squaring of the circle, or the solution of equations of the

fifth degree by radicals have finally found fully satisfactory and rigorous

solutions, although in another sense than that originally intended.” (page

1101 f. in the pagination of Ewald)

In the text printed by Ewald furthermore (page 1102), Hilbert argues that the

philosophical twist which he applies to the impossibilities offers reason for his

optimism: here (in the year 1900 again) he even dares to express publicly (and

in print!) the idea of an availability of a proof for the solvability of all

mathematical problems (including negative clarifications) and therefore I

discuss it in the section on ↑A confrontation between Hilbert and Gödel.

Just how fruitful Hilbert's approach to negative clarifications was is evident

in the fact that the first two problems of his list turned out to be of this type.

Hence, the impossibility of proving consistency of arithmetic can be seen as an

answer to Hilbert's second 2nd problem to “investigate consistency” (Constance

Reid, page 82). Hilbert in 1900 speaks of an “inverted question” (page 1102).

Bibliography:

Hilbert, David + Townsend, E.J. (transl.) (1938): The Foundations of Geometry.

Authorized Translation. La Salle, Illinois: The open court publishing

company.

Hilbert, David + Winston Newson, Mary + Ewald, William (transl.s) (1996):

From Mathematical Problems. In: Ewald, William (1996): From Kant to

Hibert: A Source Book in the Foundations of Mathematics. Vol II. Oxford:

Calendron Press.

▓Hobbes, the loser in mathematics (#MIMP_09, #MIMP_18)

Letter H: ▓Hobbes, the loser in mathematics (#MIMP_09, #MIMP_18)

Thomas Hobbes the philosopher was intrigued by the geometrical problems

of the duplication of the cube and the squaring of the circle. He maintained that

he had solved them both constructively, and this affirmation drove him into a

long-term conflict with the mathematician John Wallis, which did not end until

his (Hobbes') death. The two men fought furiously and mercilessly by

publishing texts, which were mainly directed against each other, but which have

also been preserved until today and can thus serve as material for a

reconstruction of that fight. Thomas Hobbes’ performance was rather

unfortunate here, as he remained convinced that he was the legendary solver of

these famous problems until his death. John Wallis, for his part, thought it

necessary to speak out against this amateur mathematician’s dabbling in

geometry and to prove the falseness of Hobbes' construction, in the interests of

geometrical truth.

After the publication of the letters by Hobbes in the 1990s, it became

possible to write an in-depth historiographical investigation on the unhappy,

warring relationship of the two men: it was written by Douglas M. Jesseph and

published in 1999. It is not possible here to provide a full, in-depth insight into

what Jesseph writes, so instead of offering an exhaustive summary of Jesseph's

account, I shall focus only on what Hobbes wrote concerning the Pythagorean

Theorem.

Now, just to make one point clear in advance, Wallis held the position of

university professor at Cambridge and is today considered to be the first great

authority on English mathematics (see Gottwald+Ilgauds+Schlote, page 480).

Therefore, from today's viewpoint, Hobbes did not have the slightest chance of

gaining acceptance with his affirmation that he had “breviter demonstrata” (=

“breviter constructa”) the problems which had remained unsolved for such a

long time. (I really wonder when it became evident in the history of ideas of

mathematics that squaring the circle is even more impossible than duplicating

the cube, inasmuch as π is transcendental while the third root of 2 is irrational

of degree 3 only.)

Perhaps the great (and almost overwhelming) ambition of this particular

amateur was the problem which led him to his lasting error as “Hobbes

unrepentant”, to quote the title of a chapter by Jesseph (page 273).

When the two men argued with each other in print, Wallis apparently drove

Hobbes into a corner, which made him abandon even the Theorem of

Pythagoras in a letter of March 1664. This shows relatively clearly that there is

no excuse for Hobbes' failure in his argument, because today some 400 proofs

for this theorem can be found in the literature and so Hobbes' “doubt” (pages

133 and 273 in Jesseph) will hardly find any adherents in this cause, because it

is too strange. Hobbes apparently had to surrender on this issue, because Wallis

had proved to him that, from his arguments concerning the construction

maintained to be correct, it would follow that there is “doubt” in the validity of




the Pythagorean Theorem.

Today, Hobbes has even found his way into a textbook of Galois theory, but

only as a counter-example of correct reasoning. Ian Stewart presents his

construction of π and simply requests the reader: “Find the mistake”! (page 78)

Although the book by Jesseph is well written and historiographically

satisfying, I really must ask whether this material is of relevance to the history

of mathematics. I myself have adopted the impossibilities in the history of

mathematics and its history of ideas as the central theme of my writing, which

provides a clear criterion for selecting material. Books on the history of

mathematics as such, like the book by Victor J. Katz, for example, are doomed

to present a potentially endless task, because there is no selective filter to reduce

the mass of material. May I venture to voice the question as to whether it would

be possible to write an encyclopaedia on Mathematics and its History (as

Stillwell’s title runs) without any limitation on the material? I think it would be

more realistic to write the Historiography of Historiography (as it is given by

Dauben + Scriba 2002), in which books like Jesseph’s are not integrated, but

can be mentioned. Such a meta-book would in any case be voluminous, yet

should, on the other hand, be comprehensive.

Bibliography:

Dauben, Joseph + Scriba, Christoph W. (2002): Writing on the History of

Mathematics: its historical Development. Basel: Birkhäuser.

Gottwald, Siegfried + Ilgauds, Hans-Joachim + Schlote, Karl-Heinz

(eds.,1990): Lexikon bedeutender Mathematiker. Frankfurt am Main: VEB

Bibliographisches Institut Leipzig.

Hobbes, Thomas (1669): Quadratura Circuli, Cubatio Sphaeriae, Dublicatio

Cubi, Breviter Demonstrata. London: Andrew Crooke.

Jesseph, Douglas M. (1999): Squaring the Circle. The War between Hobbes and

Wallis. Chicago+London: University of Chicago Press.

Stewart, Ian (1973, 1982): Galois Theory. NY: Chapman and Hall.

Stillwell, John (3rd ed.2010): Mathematics and Its History, NY: Springer.

▓Hofstadter's MU-puzzle (#0/, #5/)

Stuart Shanker writes, that the book Gödel Escher Bach – an eternal golden

Braid is ignored by logicians. I have read parts of it and the only passage, which

I find worthwhile to reproduce here is Hofstadter's MU-Puzzle, because it gave

me a light bulb moment some thirty years ago. The contents of this introspective

Letter H: ▓Hofstadter's MU-puzzle (#0/, #5/)

insight was the process of an ascent to a meta-level.

Hofstadter presents a little formal system. It consists of strings, which are

constructed from the letters M, I and U only. These strings are the well-formed

formulas. (See chapter I, pages 33–41)

There is only one axiom, namely the string MI.

There are four rules of inference, namely the following ones:

Rule 1: If a string,which has already been inferred, ends with an I, then a

string may be inferred by attaching a U at the end.

Rule 2: If for some string x, Mx has already been inferred, then the string

Mxx can be inferred.

Rule 3: If in an inferred string the pattern III occurs, then it can be replaced

by a U.

Rule 4: If in an inferred string the pattern UU occurs, then it can be

deleted.

Now, in my mind it would now be necessary to formulate a completeness

rule. Hofstadter obviously means that these four rules are the only ones

producing (inferring) new strings. For example, there are several methods to

infer MI (the axiom), namely

By applying rule 2, infer: MI --> MII

By applying rule 2, infer: MII --> MIIII

By applying rule 1, infer: MIIII --> MIIIIU

By applying rule 3, infer: MIIIIU --> MIUU

By applying rule 4, infer: MIUU --> MI

It is rather questionable, whether there is a relevance of inferring the axiom

a second time. But the construction shows, that it is not necessary to apply rule

3 before we can apply rule 4.

The question at stake about this little formal system, is, whether the string

MU can be inferred from the axiom MI by the four rules of inference. So the

entire puzzle is named after the target word do be produced. This is, as if we

were speaking of the A series of construction steps by ruler and compasses for

squaring of the circle, which implies that, seeing it hypocritically, we are

speaking of the empty set.




I found the work on the puzzle including the insight on the necessity to

climb the or a meta-level fairly inspiring. I refrain from giving hints for the

solution of this really fine riddle. I think it would be worthwhile to invent

more of these puzzles, with different kinds of “cooperation” of the rules of

inference.

▓Hope that it may be impossible to find a contradiction (#MIMP_08, #MIMP_11, #MIMP_20)

Wittgenstein worked intensively on the problem of a consistency proof (for

arithmetic that is) about eight years after Gödel had delivered his “Lemma”, as

Grattan-Guinness calls it. The result consists of parts of the LFM (extensive

discussions on “contradiction” from unit XIV onwards) and the text from page

213 to 221 in part III of his RFM, in which he focuses on this problem. One

cannot say that Wittgenstein ignores Gödel's Lemma, but his intention is to find

out how mathematicians could have tried to tackle the problem of consistency

before 1931 if Gödel had not spoiled this problem once and for all.

It is true that Gödel inspired Wittgenstein to work feverishly on the problem

of contradictions in mathematics, but it is also true that Wittgenstein started to

work on the issue on his own. The problem concerning the question as to when

Wittgenstein started to work on contradictions is that we are confronted with a

gap in the documents, as the minutes of the first section on this issue of Sunday,

22nd December 1929 in the talks with Waismann (Contradiction I) consist only

of blank pages and thus offer no information as to what Wittgenstein thought of

the matter at that time. And the second discussion on this issue was at a time at

which Wittgenstein might already have heard about Gödel's bombshell

(Contradiction II, Wednesday, 17th December 1930).

The “corollary” or “lemma” was found in September 1930 immediately after

the Königsberg congress.

Regarding all the other impossibility results in mathematics, we may

proceed according to the following motto: if a problem is known as being

impossible to solve, then we can refrain from working on it any further and can

stop investing ever more intellectual energy in it, turning our minds to more

rewarding problems. Wittgenstein compares the question of a consistency proof

with the construction of a Heptagon (RFM, page 216, section 85) and with the

trisection of a general angle (RFM, page 219, section 87). Both problems lead

to cubic equations, and it is indeed possible to construct a Heptagon if we are

supplied with an angle trisector (or, equivalently, with a marked ruler).

Of all these problems leading to impossibility results, perhaps the question

7

Letter H: ▓Hope that it may be impossible to find a contradiction (#MIMP_08,

#MIMP_11, #MIMP_20)

of consistency is the one which is hardest to endure, as it is a meta-mathematical

result. How are mathematicians supposed to react to this catastrophe? Should

we stop enriching the state of the art of mathematical research by developing

indirect proofs? Hilbert had provoked the entire question by putting forward a

proof, which was not only indirect, but also non-constructive: the proof of the

existence of a Gordan Basis.

Wittgenstein's suggestion is to nonetheless work on consistency proofs. He

applies a perspective dating from about 1928, as he wants to steer

mathematicians into a discussion of the impossible. Here, Wittgenstein does not

compare the impossibility of constructing a consistency proof with the

impossibility of constructing a heptagon, but – conversely – seeks to compare

the impossibility of constructing a heptagon with the impossibility of

constructing a contradiction!

Wittgenstein writes: “85. Could I imagine our fearing a possibility of

constructing the Heptagon, like the construction of a contradiction, and

that the proof that the construction of the Heptagon is impossible should

have a settling effect, like a consistency proof?”

This reversal of the direction of research by using the impossibility of the

trisection and the heptagon as a faint hope that similarly to these proofs it may

be proved, that it is impossible to construct a contradiction, is reminiscent of

6.1202 of the Tractatus:

“It is clear that we could have used for this purpose [logical

investigations] contradictions instead of tautologies.”

In the LFM and the RFM, Wittgenstein is pursuing a therapeutic goal,

inasmuch as he wants to talk with mathematicians about their former (pre-1931)

desire to establish a consistency proof. In Wittgenstein's eyes, it does not make

sense to suffer but endure this impossibility silently by remaining mute about

it: instead, we must therapeutically work on the frustrations with which Gödel

confronts us.

An expression that can be found in Wittgenstein's remarks on the problem

of trisection and the problem of the Heptagon is “our fearing”. Here,

Wittgenstein offers the mathematician-reader an emotional interpretation,

inasmuch as the anxiety concerning any repetition of the Russellian

contradiction is alleviated by the hope that the impossibility of constructing a

contradiction might, like the questions concerning the Heptagon and trisection,

be proved. As I have already stated above, Wittgenstein views the question

regarding a consistency proof from a perspective dating from just before the

year 1931.




▓hypercomplex numbers (#MIMP_16, #/MIMP_13, #/MIMP_15)

I have only found a single occurrence of the term “hypercomplex numbers”

in the texts which were initiated by Wittgenstein – namely, in the conversations

recorded by Waismann on page 104, dating from 29th June, 1930. An article in

the encyclopedia edited by Guido Walz (Vol.2, page 471) states that over the

years a change in the use (and thus, the meaning) of the word “hypercomplex”

had taken place. It is therefore not absolutely clear whether or not Wittgenstein

wants to mention the Hamiltonian numbers here. I shall, however, interpret his

hint concerning the issue of arbitrariness in the definition of operations for

different kinds of numbers as a reference to the Hamiltonian numbers.

According to Alten et al. in their book 4000 Jahre Algebra, page 379), it is

possible to parameterize the multiplication of number pairs (2-tupels) – that is,

by giving free, but constant, values to the numbers γ1 and γ2:

(0, a) # (0, b) := (γ1*a*b, γ2*a*b)

This is a good opportunity to dispel the magic atmosphere surrounding the

imaginary numbers, as here they are compared to the numbers of the form

𝑥 + √2 ∗ 𝑦

by assigning γ1 := 2 and γ2 := 0. The complex numbers can be represented

by, and subsumed under, the above definition through assigning γ1 := –1 and γ2

:= 0.

The tendency of mathematical rules to be arbitrary crops up frequently in

Wittgenstein's text and here he was, of course, inspired by Hilbert. However,

the example of the Hamiltonian numbers shows us that there also exists a

necessity for a certain systematic performance, which leads to the surprising

conclusion that it is not possible to get a field-like structure for ℝ3. There are

thus certain limitations to the arbitrariness of rules in the process of establishing

them.

In his LFM, Wittgenstein admits that there is also a component of historical

(elapsed) time in the invention of rules, which is evident in the fact that we

“inherit” these rules (unit XV, page 143). A nice exercise for a creative young

mathematician would be to invent a variant of chess with different rules. This

variant, chess2,, could retain the role behavior of the wooden figures as a

property derived from the original chess. However, I don’t think it would be

easy to design such a game in such a way that people would actually be

motivated to play this kind of chess2. (A suggestion into this direction would be

to design a variant of chess with one bishop, one knight and one tower on both

sides only. This would mean that each side has five figures apart from the pawns

only.)

Letter H: ▓hypercomplex numbers (#MIMP_16, #/MIMP_13, #/MIMP_15)

My impression from reading historiography is that, when finding his

Quaternions, it was an advantage for Hamilton to be in a relatively late position

in the history of mathematics, thus enabling him to

1. exceed the number 3 of dimensions without seeing this step as a

betrayal of geometric considerations (“Dimensionstreue” as mentioned on

pages 64, 75, 171 and 279 in Alten at al.) and to

2. design the results of the multiplication of i*j, i*k and j*k freely as it

was necessary to obtain a fully-fledged coherent system without developing

a bad conscience regarding the question as to what the contentual aspect of

these definitions should be.

In the event that Wittgenstein was actually thinking of the Hamiltonian

Quaternions when he mentioned the “hypercomplex numbers”, I think he put

forward this example of the arbitrariness of a design process because he

considered this fine piece of mathematics to be a good example for the invention

(and not the discovery) of an algebraical structure. Before Hamilton enriched

the world of mathematicians through his 4-tuples including operations that

made sense, this structure definitely did not exist!

Bibliography:

Alten, Heinz-Wilhelm + Naini, Alireza Djafari + Folkerts, Menso + Schlosser,

Hartmut + Schlote, Karl-Heinz + Wußing, Hans (korr. 2008): 4000 Jahre

Algebra. Geschichte, Kulturen, Menschen. Berlin + Heidelberg: Springer.




Letter I

Letter I: The Ignorabimus separately discussed for the natural sciences: flight to the

moon, for example (#MIMP_14, #MIMP_32, #/MIMP_18, #/MIMP_26)

The Ignorabimus separately discussed for the natural sciences: flight to the moon, for example (#MIMP_14, #MIMP_32, #/MIMP_18, #/MIMP_26)

Note that this book, which I present to the reader here, is focused on

mathematics only and largely ignores the epistemological situation in physics

or astronomy and all the other natural sciences with the exception of the three-

body problem. Wittgenstein also works on the formal sciences of mathematics

and logics only and in his Nachlass, there are hardly any remarks on physics. I

have a hypothesis on his motive, why Wittgenstein proceeded in this way: the

problem is, that the epistemological situation in physics and mathematics is

absolutely different, especially when it comes to a thorough reflection on

impossibilities. If Wittgenstein would have worked on the impossibilities in

physics as well, then he would have needed to be very careful to separate the

arguments about physics and mathematics in order not to mix them up.

From the world of physics, Wittgenstein discusses the ability of the human

race to fly to the moon, which is an argument, which he derives from the

discussions by G.E. Moore (On Certainty, § 171, page 25). We know that in

1969, this dream of mankind (which was anticipated by Jules Verne in a

novel) became true and from what Wittgenstein writes about this project, it is

not clear, whether he conjectured it to be possible or impossible.

Wittgenstein writes in OC in § 106: “Suppose some adult had told a

child that he had been on the moon. The child tells me the story, and I

say it was only a joke, the man hadn't been on the moon; no one has ever

been on the moon; the moon is a long way off and it is impossible to

climb up there or fly there.—If now the child insists, saying perhaps

there is a way of getting there which I don't know, etc. what reply could

I make to him? What reply could I make to the adults of a tribe who

believe that people sometimes go to the moon (perhaps that is how they

interpret their dreams), and who indeed grant that there are no ordinary

means of climbing up to it or flying there?—But a child will not

ordinarily stick to such a belief and will soon be convinced by what we

tell him seriously.”

In mathematics, if a construction or a proof is proved to be impossible,

then this problem can be taken off the agenda of mathematicians and they can

deal with other, more rewarding problems. So there is a paradoxical situation

with mathematical impossibilities, which consists therein, that a negative

settlement is a special kind of a solution, which implies that the minds which

were preoccupied by this enigma are free to leave it (In Latin: absolvere). But

this positive, liberating effect of a negative clarification is given for the




epistemological situation within mathematics only. It does not carry over to

physics and astronomy, as the following example shows:

From the study of Laplace's Demon (which was invented by Emil DuBois-

Reymond in 1872) I gained insight into one of the problems of mathematical

impossibilities, which is not simply a mathematical issue, but which is

exported to astronomy: the three-body problem. This example (which I have

mainly discussed in the section on ↑Laplace vs. Poincaré) shows very clearly,

that the effect of the negative settlement of a problem may be very much

different for astronomers than for mathematicians. While mathematicians

could, so to speak, simply tic off the problem from their to-do list, the effect of

the negative solution by Poincaré on astronomers might be of a devastating

kind.

Therefore: when investigating impossibilities, it is worthwhile to consider

the borderlines between different disciplines. Here we have an impossibility,

which is stirred up by astronomy, proved within mathematics and re-exported

to astronomy again. (With the Axiom of Choice, it is different: it is imported

from logic into mathematics.) Now, as I have stated already, the reaction of

mathematicians to insolubilities can be of a nonchalant kind, but the reaction

of astronomers to the impossibility to prove the stability of the solar system

may be that of depression or panic-strickenness (according to the basic

character of the person of the astronomer).

It is fairly thrilling to look at the historical time, when the work on the

three-body problem took place: Henry Poincaré published the impossibility to

prove the stability of the solar system in 1893 and Emil Du Bois-Reymond

died in 1896. It would be a nice investigation to work on the question, whether

the pessimist Emil Du Bois-Reymond would have dismissed his idea of the

positive Laplacian Spirit and instead would have invented a negative Daemon

for astronomy. There is no closed formula for the three-body problem!!! Isn't

that a mathematical proof for the feebleness of the human mind in an

important astronomical question, the weakness of which in this case can be

furnished by a mathematical proof? I think it to be possible that Emil Du Bois-

Reymond would consider the negative development of the issue of the three-

body problem as being hard-coded support for his thesis of the Ignorabimus.

Succinctly speaking, there are essential differences between the

impossibility for a solution of the three-body problem in astronomy and other

impossibilities in mathematics, as, for example the squaring of the circle.

The problem which I see here is the fact, that my general thesis about the a

possible positive attitude of mathematicians towards mathematical

impossibilities does not carry over to the reaction to this phenomenon in

astronomy, for example. While for mathematicians, the missing three-body

Letter I: The Ignorabimus separately discussed for the natural sciences: flight to the

moon, for example (#MIMP_14, #MIMP_32, #/MIMP_18, #/MIMP_26)

formula is an acceptable trait of a certain calculus, astronomers may have felt

to have been left with empty hands.

I think, for an investigation of the question of an Ignorabimus in the

natural sciences (as contrasted with my investigation of the impossible in

mathematics here), it would be worthwhile to speculate on how astronomers

felt in 1893 about the Job's message given by Poincaré. It was clear from the

mathematical way of arguing by Poincaré that there was no hope about a

happy ending by the world of the “synthetic a priori”. Once an impossibility is

proved in mathematics, there is no way out by developing some trick to find

the formula after all. The negative diagnosis is given once and forever and the

situation will never be remedied by any ingenious method whatsoever.

This reaction of a harsh frustration (to which we have got accustomed by

today) to Poincaré's finding is an indicator, that for astronomers, for example,

mathematical impossibilities have quite a different relevance and status than

for mathematicians themselves. And the essential insight won by him, that the

solar system theoretically may behave much more chaotic than it actually does

(see Szpiro, page 58) is the frightful idea of a collision of two planets, which

started to plague us since 1893: hopefully the empirical orderliness

perpetuates from the past to the future! Never will anybody repeat Laplace's

optimistic fallacy.

P.S. On November 6th, 2014, I had a scientific talk with a friend, who holds

a degree in mathematics, but is well acquainted with mathematical physics and

astronomy as well.

This man maintained, that astronomers do not fear a collision of planets in

our solar system, because they rely on a peaceful continuation of the last

5*109 years in which no crash of two planets has occurred. This empirical

evidence results in enough trustworthiness of astronomers into the behaviour

of the planets, although, from a mathematical point of view, this is an

empirical, incomplete, induction only.

About a quarter of an hour later in our talk, he made a remark about the

belief of human astronomers, that in some time (estimation: 5*109 years

again), our solar system will collapse. The sun will swallow the earth. Of

course, this knowledge of a major accident after such an absurdly long time

does not affect “us”, nor our children and it does not affect our grand-children

as well. But from the point of view of the question, weather we rely in the

stability of our solar system, this argument of a major accident is an important

contribution, although it is predicted for the far future only.

We also talked about perturbation calculation in astronomy (“Störungs-

rechnung” in German), which is based on the fact, that the orbits of the planets

are attractive, i.e., , that if there is a perturbation, the forces of the other




planets are “designed” in such a way, that the planet in question is pushed

back into his former path again. This fact of the attractiveness of the orbits is

an important counter argument of consolation for people like me who are

worrying about the consequences of the impossibility to calculate a simple

three-body system.

A truly philosophical consideration about an accident involving the entire

earth runs as follows: the responsibility of the human race in this question is

considerable, because it may be we are the only planet with such a broad

variety of life on it.

Bibliography:

Szpiro, George G. (2007): Poincaré’s Prize. The hundred-year quest to solve

one of math’s greatest puzzles, NY: Dutton.

▓An impossible proof, which was apparently never attempted to establish: the Axiom of Choice (#MIMP_26)

It is a rewarding task to reconstruct the historiography of the Axiom of

Choice from 1890 until 1938, as do Gregory H. Moore extensively, and Bar-

Hillel+Fraenkel and Levy succinctly (pages 53–58 and 80–86, if one focuses on

the history of ideas). The beginning of the work on the Axiom of Choice in 1890

is recorded in a footnote in Bourbaki's note historique on Foundations of

Mathematics; Logic; Set Theory on page 36. It contains a passage by Peano,

who was thus the first person to be reported as having commented on the Axiom

of Choice.

This axiom was called Multiplicative Axiom by Russell and Whitehead and

this name carries over to Wittgenstein's texts. In the Principia Mathematica and

the axiomatic version of set theory by Zermelo (1904 and 1908), the axiom is

made explicit. This step was made very much in the sense of Hilbert's axiomatic

approach, because here things became clearer, inasmuch as an unintentional use

of the axiom was replaced by a deliberate one. The term “inadvertent” use is

coined by Bar-Hillel, Fraenkel and Levy in 1972 (page 57).

The two short texts by Zermelo are probably altogether the most greatly

hated texts in mathematics, since he fulfilled his task, as Cantor and Hilbert

wanted him to do, by proving the well-ordering principle and the trichotomy of

cardinals from the Axiom of Choice (Oliver Deiser, page 57f.). This must have

been fairly satisfying for Cantor, who had for so long sought this proof. I think

the core issue concerning this proof is the fact that the axiom of choice seems

Letter I: ▓An impossible proof, which was apparently never attempted to establish:

the Axiom of Choice (#MIMP_26)

to be far more harmless contentually than the well-ordering principle.

Therefore, although it seemed inappropriate to assume the well-ordering

principle itself as an axiom, it therefore appeared to require a proof. And Hilbert

served the discussion in 1922 by claiming that the AC seemed as clear as the

proposition “2+2=4” (see Moore, page 253).

A very important step in history is the proof that the theorem of trichotomy

of cardinals (i.e. that two sets can always be compared as to size) is equivalent

to the Axiom of Choice. This means that the orderly existence of the Alephs in

a totally ordered fashion is indirectly assumed to be an axiom, if we adjoin the

Axiom of Choice! This proof was established by Friedrich Hartogs in 1915, yet

has barely found its way into the textbook representation of set theory (however,

this milestone is mentioned by Gregory H. Moore, page 170).

If one does something, then it is always best to know what one is doing. And

it is clear from Hartogs' proof: if we use the AC, then we postulate the fact that

all the Alephs are well defined as an axiom indirectly, because the two are

equivalent. This means that if we look at this procedure in a radically critical

manner, then we can accuse mathematicians of establishing the escalating

infinities directly by choosing the Axiom of Choice as an axiom with the truth

value true. It emerges that what happens here is a petitio principii on a very

large scale, but it seems that Cantor and Zermelo have achieved acceptance

through this decisive step.

In a somewhat banal manner, the equivalence of the AC with the total

ordering of all sets (and therefore the existence of Alephs) is simply given in a

side-comment in the textbook by Ebbinghaus on page 124.

P.S.: At the beginning of the 20th century, it was assumed in set theory that

in addition to the sets, there is a (countably) infinite number of Urelements,

which do not contain elements themselves, but are not equivalent to the empty

set. Under this assumption of this existence of Urelements, Fraenkel proved

the independence of the Axiom of Choice in 1922.

Bibliography:

Bar-Hillel + Fraenkel + Levy (1958, rev. ed. 1973): Foundations of Set Theory.

Amsterdam & London: North Holland Publishing Company.

Bourbaki, Nicolas (1994): Foundations of Mathematics; Logic; Set Theory. In:

Elements of the History of Mathematics. Berlin: Springer, pages 1–44.

Deiser, Oliver (2002): Einführung in die Mengenlehre. Die Mengenlehre Georg

Cantors und ihre Axiomatisierung durch Ernst Zermelo. Berlin et al.:

Springer.




Ebbinghaus, Heinz-Dieter (2003, 4th ed.): Einführung in die Mengenlehre.

Heidelberg+Berlin: Spektrum Akademischer Verlag.

Fraenkel, A. (1922): Über den Begriff „definit“ und die Unabhängigkeit des

Auswahlaxioms, Sitzungsberichte der Preußischen Akademie der

Wissenschaften, Physikalisch-mathematische Klasse, pages 253–257.

Hartogs, Friedrich (1915): Über das Problem der Wohlordnung. In:

Mathematische Annalen 76, pages 436–443.

Moore, Gregory H. (1982): Zermelo's Axiom of Choice. Its Origins,

Developments, and Influence. New York: Springer.

Zermelo, Ernst (1904): Beweis, dass jede Menge wohlgeordnet warden kann,


Zermelo, Ernst (1908): Neuer Beweis für die Möglichkeit einer Wohlordnung,


An improper impossibility (#MIMP_04, #MIMP_18, #MIMP_19/)

It may be that Wittgenstein was not just interested in Fermat's Last Theorem

for the reason given in the section on ↑Fermat's Last Theorem – namely, that an

open problem causes certain mathematical objects to be fuzzy and unclear. He

also considered the possibility that Fermat's Last Theorem could be a formally

undecidable proposition in the sense of Gödel. The long series of unsuccessful

attempts to solve the problem may have been a reason for Wittgenstein – among

others – to raise the suspicion that maybe no proof for either of the directions

exists: for neither the one in which Fermat expressed it, nor for its negation.

The conjecture on the independence of Fermat's Last Theorem is explicitly

expressed in the book on many-valued logic by the Russian mathematician

Alexander A. Sinowjew (1968). I found a reference to this remarkable detail on

current mathematics in Karel van het Reve’s Dr Freud and Sherlock Holmes

(page 107). See also the remark on the suspicion “that the problem might be

impossible” in the book by Simon Singh. (page 118)

However, two impossibilities are linked with Fermat's Last Theorem: one is

the question as to whether it can be clarified at all, or whether we can use the

original direction of the conjecture as stated by Fermat himself to express a

number theoretical impossibility.

Before 1996 it may have been said: it may be impossible to find numbers

x, y, z, n>2,

Letter I: An improper impossibility (#MIMP_04, #MIMP_18, #MIMP_19/)

such that 𝑥𝑛 + 𝑦𝑛 = 𝑧𝑛 (*)

Let us now, for the rest of this section, consider this number-theoretic

impossibility, rather than the possible logical impossibility of non-existence of

a proof for both directions.

I have, however, the impression that this statement is not seen as an

impossibility by the majority of mathematicians (much less by all of them), but

that here a way of looking at matters was applied in which the statement was

regarded in a neutral light. In the case of the Fermat conjecture, no one

expressed the yearning for the impossible (to quote the title of the book by John

Stillwell) – namely, that people wanted to find such numbers x, y, z, and n by

any means possible. There is the image of a dog, which barks at the moon.

However, it was clear from the very beginning that it may be the case that there

do not exist numbers that make (*) a true proposition.

So we have a phenomenon of propositions in the historiography of

mathematics, which are impossibilities formally, but not in the prevailing

attitude of researchers. The best example for this status of a proposition (or a

conjecture) is the transcendence of the Eulerian number e, which was proved by

Hermite in 1873. For the number e, the result was regarded in a neutral light

(and not as a negative result; i.e. without any disappointment about its non-

constructability) whereas for π, the prevailing interpretation of the result was

that of a non-constructability – i.e. it was seen as a negative way of settling the

problem, resulting in a mixture of relief and frustration.

What I want to make clear here is the fact that we can subsume the fact of

the non-constructability of e as a formal impossibility, although this kind of

view was not the dominant one held by mathematicians. The same procedure

can now be used for Fermat's conjecture expressed in what is called his Last

Theorem: namely, that we can neutrally state that it is impossible to find

numbers x, y, z and n>2 to satisfy the above equation (*). This result given by

Andrew Wiles lacks the atmosphere of negation such as surrounds, for example,

the non-constructability of π.

Bibliography:

van het Reve, Karel (1994): Dr.Freud und Sherlock Holmes. Frankfurt am

Main: Fischer Taschenbuch Verlag GmbH.

Singh, Simon (1997, 1998): Chapter 3: A mathematical disgrace, in: Fermat's

Enigma: The epic quest to solve the world's greatest mathematical problem.

New York: Anchor Books, pages 71–120.

Sinowjew, Alexander Alexandrowitsch (Zinov’ev Alexandr Alexandrovic)

(1968): Über mehrwertige Logik, Berlin: Deutscher Verlag der

Wissenschaften.




Stillwell, John (2006): Yearning for the Impossible, The surprising truths of

Mathematics, Wellesley: A K Peters.

▓Inaccessible numbers (#18/, #30/)

Wittgenstein suffers a great deal of philosophical hardship concerning

variants of the number π. He constructs a number π', which emerges from the

original, by changing all occurrences of the pattern “777” to “333”, which very

probably results in a transcendental number again. This variant has two

drawbacks as compared to the original – namely:

1) It lacks a geometric interpretation

2) It does not involve a great deal of interest or effort in research. (It does not

lie within the focus of research.)

However, this variant is a number which can be effectively computed. By

this, I mean that there exists a finite chain of symbols (including the tokens

“lim”, Σ,∞and the replacement operator) which represents this number exactly.

I have a hypothesis as to why Wittgenstein works so hard on this issue of

variants of π: he is here, in my opinion, caught in a paradoxical situation –

namely, that he wants to construct a number (with an infinite number of digits

after the decimal point) which is symbolically inaccessible; but let us examine

this issue step by step.

The number of reals is uncountable. The number of reals which can be

written down in any language is countable only. Therefore there are real

numbers which cannot be written down when employing a fixed language L.

The set of all numbers which can be written down is a closed set (a sub-field)

of . The fixed language L contains the following parts:

1) We need the token ∞.

2) We need the brackets ( and ).

3) We need variables xi to be bound by the operators lim and Σ.

4) The number “1” is a real number, which can be written down by ordinary

means. All ten digits can be written. But the rules of our fixed language L say

that we are only allowed to write down a finite chain of digits explicitly.

5) If the two numbers a and b are both symbolically accessible numbers, then

it follows that a + b, a – b, a*b, a/b and ab are symbolically accessible numbers.

6) The most powerful tokens allowed are by the symbols “lim” and Σ.

7) We need case distinctions, i.e. replacements of certain finite combinations

Letter I: ▓Inaccessible numbers (#18/, #30/)

of digits by alternative finite combinations of digits.

8) The numbers symbolically accessed by (1) through (7) are all symbolically

accessible numbers. (It is this last, the completeness rule, which arouses

Wittgenstein's doubts.)

From these eight rules, it follows that (as I have already stated) the number

of symbolically accessible real numbers is countable. If, by some means, we

were able to write down a single number z which cannot be accessed

symbolically by (1) through (8), this would result in an increase in the number

of symbolically accessible real numbers, though they would nonetheless remain

countable. (See the argument in the entry “Fehlschluss” in Mittelstrass.)

This Wittgensteinian paradox, which I have reconstructed from the trouble

he has with variants of the number π, can be stated as follows: it is not possible

to find a real number which is symbolically inaccessible, because if we were

able to pin down such a number, we would obviously have found some way to

write it down, and so it would therefore no longer have the character of a

symbolically inaccessible number.

This π' paradox is similar to the Emil Du Bois Paradox concerning an

insoluble mathematical problem ( = #30/): if we are looking for a problem which

cannot be solved, and we find one, then we can perhaps prove that it is insoluble.

However, the problem has then been settled negatively and thus no longer

occupies our minds. Therefore, it no longer has the character of an insoluble

problem.

And if we cannot prove the character of a problem which cannot be solved,

then Emil Du Bois' opponents might argue that a day may come on which it is

solved or proved to be unsolvable and clarify it that way.

Wittgenstein wants to specify an inaccessible real number using symbols,

but does not manage to do so. It is not only possible to describe this situation

(at a meta-level) by using the term “symbolical inaccessibility”, but also by the

term “ineffable”, which is used for certain trans-trans-finite cardinals (See

Pierre Basieux: Abenteuer Mathematik, page 117).

Had Wittgenstein been acquainted with the Skolem Paradox then he would

have used this knowledge to argue that symbolically inaccessible real numbers

exist in any language – as a passage from 106 296 in his Nachlass testifies:

“Es ist also denkbar, dass es zwei Klassen von irrationalen Zahlen

gibt: die eine durch ein Gesetz bestimmt, also alle die wir kennen

können, und eine durch keine Gesetze bestimmte, also die Gesamtheit

derer, die wir nicht kennen können!”

I do in fact have an idea why Wittgenstein includes some of the algebraical

numbers here by making a statement concerning irrationals (and not concerning

transcendental numbers): this is an indication that, from the perspective of 1826,




only tautological (circular) descriptions of the “roots” existed for most of the

quintic equations – namely, that x is the solution of some quintic equation.

Bibliography:

Mittelstrass (ed., 2004): Fehlschluss, in: Enzyklopädie der Philosophie und

Wissenschaftstheorie, Vol. 1, page 634.

▓Incommensurability and similar phenomena (#MIMP_00, #MIMP_01, #MIMP_08, #MIMP_11)

This section presents a view on my code to Wittgenstein's philosophy of

mathematics.

The shock which the Greek mathematicians suffered when they encountered

this essential gap in their ideology (“all is number”) was the basis for the

insights into the nature of irrational numbers two centuries later. Wittgenstein

mentions this issue in his LFM on page 90. On this page, he also writes about

the series of constructible regular polygons. Here, Wittgenstein imagines

teaching someone about this dangerous question – and this issue is risky,

because if one chooses to work on a polygon with a certain number of corners

(7 or 13, for example), then one may be caught in an insoluble task.

The essential words in this quotation from Wittgenstein are the words

“similarity” and “similarly”:

“If he follows the method I teach him, he will get more things looking

like regular polygons. But it is not merely that. Similarly, it is not merely

the fact that by messing about with ruler and compasses he will hardly

ever get a trisection of the angle which makes him give up trying. --

There are reasons connected with the single steps of the proof and their

similarity to other proofs he has made. So with the proof that the diagonal

is incommensurable with the side of the square 1x1.” (LFM, unit IX,

page 90)

What Wittgenstein wants to convey to his listeners here is the fact that he is

working on a general concept that includes several phenomena which are

different from a mathematical viewpoint but similar from a philosophical

viewpoint. Look at the regular Heptagon on the one hand and the issue of angle

trisection on the other. They both lead to cubic problems and there is a single

proof by Pierre Wantzel which shows that both problems are insoluble (using

an unmarked ruler and compasses).

The problem of representing the number 2 as the square of a fraction of two

Letter I: Incompleteness (#19/)

whole numbers is – as seen by mathematicians – absolutely different from the

problem of angle trisection, but in Wittgenstein's philosophy of mathematics,

they are both subsumed under the concept of a gap in the calculi of

mathematicians. Thus, in his philosophical view, they have a great deal in

common – namely the fact that, in the case of mathematicians willing to solve

them at all costs, both problems lead to dead ends.

What makes the history of mathematics a somewhat cynical business here

is the misleading nature of the problem and the necessary (a priori) futility of

past work. But this insight is included in the view of the generations who are

later born into a world after the solution of the problem at a meta-level.

In his philosophy of mathematics, Wittgenstein's code is to form a single

concept of impossibility and to incorporate in it such apparently different things

as non-existent geometric constructions, impossible proofs, non-existent

algorithms and impossible formulae (See: ↑Introduction). He wants to

investigate these phenomena of calculi in mathematics, which are seemingly

widespread issues, within a unified investigation of impossibilities.

His general view on impossibilities can be regarded as being pessimistic,

because mathematical calculi do possess the property of containing gaps. Yet,

on the other hand, once such a gap has been clarified then mathematicians can

tick off that question and set to work on other problems. This makes room for

an optimistic view. This dichotomy of a pessimistic versus an optimistic view

is only one of the common epistemological traits of impossibilities which make

generalized philosophical work on these problems worthwhile.

Incompleteness (#19/)

This article is on a creative Statement by Wittgenstein on Incompleteness.

For Wittgenstein, Gödel's incompleteness results of 1931 are of central

importance in his philosophy of mathematics. He investigates the non-

constructability of the heptagon in order to get a clear view of Gödel's

impossibility results, since for him it is obvious that they do have a eminent

philosophical (and mathematical) relevance, because they look at mathematics

from the outside, as well.

Now, Wittgenstein states that “Mathematics cannot be incomplete” (Phil.

Rem. XIII, page 188). Why does he say this? Is it only his private, idiosyncratic

view of the state of affairs, or can this expression be reconciled with what

logicians think about Gödel's results? After all, Wittgenstein said in his LFM

right at the beginning (unit I, page 13): “it will be most important not to interfere

with the mathematicians.”




I have brooded at length over this statement of Wittgenstein’s, and I have

come to the conclusion that an interpretation of it can be achieved if we interpret

the word “mathematics” in a wider sense, rather than only taking a single formal

system consisting of a logical calculus plus the Peano Axioms, as is done in

Principia Mathematica. Here in Wittgenstein's statement, the word

“mathematics” is used in the sense of “all formal systems, that have been

developed so far”. So here we have the historical dimension, that at a specific

point of time in the history of mathematics – say, the beginning of the year 2011,

we have a certain collection of formal systems, whereby some of them are meta-

systems for others. And a logical calculus plus Peano axioms is the starting

system for the entire business, but this one system does not exhaust mathematics

because, after Gödel, mathematics is the additive sum of several formal systems.

Since we consider the set of formal systems which could be accomplished

up to a certain point of time, their number is always finite. Also, there exists

only a finite number of open questions, which might – in the worst case – lead

to the same number of undecidable propositions.

Now let us consider a concrete problem and let us assume that relevant

mathematicians have expressed the suspicion, that it might be undecidable.

What then needs to be done is to prove that it is undecidable. My question here

is: could it be that, although the question (take the Goldbach conjecture, for

example) is undecidable, there is no proof that this is the case? I have called this

a towered undecidability. The problem here is, that we cannot proceed to work

formally on meta level #2, because the concept “all meta-calculi at level #1 for

the object problem” is not a formal system, but an enumerative sum of formal

systems.

So the philosophical, extra-mathematical question is: can we be optimistic

that in all cases of undecidable propositions at level #0 there exists a proof at

meta level #1 that the object level proposition is undecidable? I think we may

find that this epistemological question can be answered by saying Yes, because

the concept of an enumeration of all meta-systems at level #1 is not (I repeat) a

formal system, but this concept includes the idea of mathematical “creativity”

(this word is used by Brouwer, for example). Here, imagination becomes

relevant.

We have to look at the object system (focused around the Goldbach

conjecture, for example) from an appropriate angle. I would like to explore this

concept of the angle of viewing by a metaphor which seduces us to hold the

optimistic view that for each non-decidable proposition in the object system

there is a suitable system among the (creatively numerous) meta-systems, which

proves the non-decidability of the object proposition.

Once, some years ago, while doing my homework for school as a young

Letter I: infinity has to be ascertained by an axiom (#29/)

pupil, my rubber fell off my desk onto the floor. I looked for it and saw that it

had fallen on a colourful carpet of my parents’. I thought of the fact that one

half of my rubber was red and the other was blue. The multicoloured carpet

included these two colours in many patterns. There was a certain danger that I

would not be able to see my rubber contrasted against the wild carpet pattern -

but then I had an idea.

I knelt down on the floor at the edge of the carpet and looked, with my

viewing angle parallel to the plane of the carpet. This “way of looking at things”

(Wittgenstein, among others in PU, part I, §144 and § 401) was a productive

one, because I could see my rubber against the background of the rest of the

room. My search was not disturbed by having the colourful patterns on the

carpet as a background, because I could hardly see the carpet through my way

of looking.

So what I want to point out here is that taking an optimistic view is

reconcilable with the Gödelian incompleteness theorems, and that for each

(concrete and single) undecidable proposition of a mathematical system, there

is a meta-system in which proof of this undecidability is possible.

At the end of this interpretation of Wittgenstein's statement “Mathematics

cannot be incomplete”, I must confess that a source of inspiration for this

optimistic possibility was Abel's motto, “Turn the tables”: “Once such a

question has been posed, an answer had to be found; the answer had to be either

yes or no.” (Stubhaug, page 302) Kuhn uses the phrase “taking the stick by the

other end” (page 85).

This form of argument, a singular Tertium Non Datur for a concrete and

specific question, means: if the formula exists, then I want to find it, or if it does

not exist among the root expressions, then I want to prove that. So my raising

the possibility of optimism is not a linear propagation of Hilbert's motto, that

there is no ignorabimus. The essential point here is that the set of all meta-

systems for an object system is not itself a formal system, but an informal

system.

To summarise: Since the concept of all meta-systems of level #1 for an

object level problem is nourished by creativity, it may be the case that for each

undecidable object problem there exists a level #1 undecidability proof. But

even if this is so, it certainly won't be possible to prove this optimistic situation

at level #2.

infinity has to be ascertained by an axiom (#29/)

About a fortnight ago (in October 2014), I discovered that I have – in my

plan to write on all known impossibilities – forgotten to write on the




impossibility to prove the existence of an infinite set. But this issue is probably

the most important issues of this kind. It is closely connected with Russell's

contradiction.

I read some years ago in the encyclopedia by Mittelstrass, that Frege had

indeed thought to have found a proof of the existence of the set of natural

numbers, i.e. the smallest of all infinite entities.

This alleged proof by Frege of the existence of the natural numbers was

available by Frege's version of the axiom of comprehension. The predicate,

which is a prerequisite for constructing a set can be chosen so that the natural

numbers emerge out of the axiom. And the axiom of comprehension looks so

innocent and simple!

For each predicate P, there exists a set MP, so that for all x, x is a member of

MP, if and only if P(x).

Here, the formulation of the predicate P contains x as a free variable, which

is then bound by the universal quantifier. Now let us specify P as follows:

The empty set is a member of P

and

For all n, which are members of P, also {n} is a member of P.

Therefore the old original form of the Fregean axiom of comprehension

implies the existence of the set of natural numbers and hence an infinite set.

Furthermore, from the original version of Frege's axiom, the existence of a

power set for any set can be inferred as well (see Deiser, page 278). From this

we can see how powerful this axiom was.

But we know about the fate of the original version of Frege's axiom of

comprehension: Russell found out that this old and original form of the axiom

of comprehension is flawed by contradictions and as a consequence of this

catastrophe, the fine and elegant proof of the existence of an infinite set by Frege

was torn to pieces.

Today we can prove that we have to assume the existence of an infinite set

by an axiom, because this axiom is independent from the other Zermelo axioms.

This proof of independence is done by the model of the heriditarily finite

numbers (sets) as they are presented by John Stillwell in his book Road to

Infinity (chapter 7.1, Set Theory without infinity, pages 165ff.)

In the modern version of axiomatic set theory by Zermelo, which is mainly

a reaction to the Russellian contradiction there has to be an axiom of infinity

(see also Hoffmann, page 151).

Letter I: infinity has to be ascertained by an axiom (#29/)

There is no such thing as a proof of infinity, as Frege thought.

If once the existence of an infinite set is assumed (without giving rise to a

contradiction, hopefully), then many of the further escalating infinities can be

inferred. By this decision we have managed to make a step into the door, which

can be opened then to get many of all the other escalating infinities. To achieve

existence of the “hyper-hyper-inaccessible cardinals “(Barwise, page ZZZ) we

have to assume further axioms of infinity, though. Barwise calls them

“ridiculously large cardinals”, which is not a technical term, but a humorous

expression for the cardinals found by further axioms. The decision to adjoin

them to usual set theory has an effect on the decidability of some number

theoretic questions, as Gödel writes (GW II, 267–269).

We can sum things up by saying, that Frege's axiom of comprehension

simply was too powerful. That is why logicians and mathematicians have

developed a more humble strategy and assured the existence of an infinite set

by an axiom. The proof of the existence of the initial and starting infinity turned

out to be not possible.

What is interesting when speaking about infinity is Gödel's Compactness

Theorem, by which it is possible to conclude the existence of countable models

from the existence of arbitrarily huge finite models. See Bernt Buldt in Köhler

page 35.

Note that from a philosophical point of view, there is a connection between

the infinite and a belief in the existence of God. Georg Cantor thought his

escalating infinities to be a staircase to Him and from this, he was convinced,

that they exist and are not altogether flawed by contradictions.

Bibliography:

Buldt, Bernd (): Kompaktheit und Endlichkeit in der Formalen Logik.

Deiser, Oliver ():

Mückenheim, Wolfgang (2006): Die Mathematik des Unendlichen.

Aachen: Shaker Verlag.

Stillwell, John:




▓Influences on the rise of non-Euclidean geometries (#12/)

It is not easy to pinpoint the crucial moment in the historical development

out of which the non-Euclidean geometries emerged. It is, however, possible to

determine a central person, who was initially part of the German circle

surrounding Carl Friedrich Gauss and afterwards changed to the very university

where Lobachevski studied. The person I am talking about is Johann Martin

Christian Bartels, who features in many encyclopaedias, such as the one edited

by Guido Walz (Vol. 1, page 168) and the purely historiographical dictionary of

relevant mathematicians, published by Gottwald + Ilgauds + Schlote (page 39).

It is absolutely clear that Bartels was at first doubtless in contact with Gauss

in Germany and later on with Lobachevski, who was one of his students in

Kazan. There is therefore a possibility that Bartels may have passed on

information about curved spaces from Gauss to Lobachevski. Although there is

no historiographical evidence that this actually happened, it is nonetheless

possible. Historiographical sources are vague and unclear on this point, and this

situation was already reflected upon by an early biographer of Lobachevski,

namely Aleksandr V. Vasiliev (1852 – 1929), who remarked that this hint may

possibly have been passed from Gauss via Bartels to the “Copernicus of

Geometry” in 1914.

Since this biography has apparently not yet been translated from the Russian

into English or German, I must rely on Herbert Meschkowski's Mathematiker-

Lexikon, in which this crucial passage in Vasiliev's text is paraphrased in

German. Although his text was “the first scientific biography of Lobachevskij”

(Demidov, page 181), Vasiliev states that the question as to whether knowledge

was transferred from Germany to Russia will probably remain unclear forever,

and thus makes it clear that historiographers of mathematics must make up their

own minds to whom they give the credit for pioneering non-Euclidean

geometries. I have already called Lobachevski the “Copernicus of Geometry”

(he was the first to publish a printed text on this new kind of geometry) and have

thus revealed the end of the story – namely, that historians of geometry have

arrived at a consensus not to assume that such a transfer of information from

Gauss to Kazan university ever took place (see Hans Reichardt, page 67 for

clarification).

A little semiotic experiment concerning this question might be done by

developing extremely brief and succinct verbalisations which might suffice as

hints from Germany to Kazan. Perhaps a hint concerning “curved spaces”, or, a

Letter I: De Insolubilibus (#7/)

little more specifically, concerning the pseudo-sphere with constant negative

curvature? Such extremely condensed messages might suffice as the grain of

sand from which a pearl develops in an oyster. One such tiny piece of

information might have had a catalytic effect on the sort of eager and extremely

witty man that Lobachevski certainly was. Even one such small element of

strategic information might suffice in the case of someone who was himself on

the verge of inventing the entire concept of curved space.

My intention in focusing on the situation surrounding Bartels here is to

reflect on the decision reached by early historiographers of non-Euclidean

geometries, as can be seen from the text by Hans Reichardt, who writes that

there is no evidence for any such conjectured influence, although it is not so

very far-fetched to suppose one.

Bibliography:

Dauben, Joseph + Scriba, Christoph W. (2002): Writing on the History of

Mathematics: its historical Development. Basel: Birkhäuser.

Demidov, Sergei S. (2002): Vasiliev, Aleksandr Vasilievich (1853 – 1929): In:

Dauben, Joseph + Scriba, Christoph W. (2002), page 552.

Demidov, Sergei S. (2002): Russia and the USSR. In: Dauben, Joseph + Scriba,

Christoph W. (2002). pages 179–197.

Meschkowski, Herbert (1973): Lobatschewsky, Nikolai Iwanowitsch, in:

Meschkowski, Herbert (2nd ed., 1973): Mathematiker-Lexikon. Mannheim:

Bibliographisches Institut/BI Wissenschaftsverlag, pages 184–186.

Reichardt, Hans (1976): Gauss und die nicht-euklidische Geometrie. Leipzig:

B.G.Teubner Verlagsgesellschaft.

De Insolubilibus (#7/)

This section examines a connection between, on the one hand, Russell and

Gödel’s uses of the idea of a variant of the Cretan Liar in 1902 and 1931

respectively and, on the other, the medieval literature on this topic.

Russell does not reveal whether he used the Cretan Liar as a source of

inspiration, but instead claims that he was later informed about the similarity

between it and his contradiction. See his Collected Papers, vol. 8, page 228.

The Cretan Liar was used constructively several times during the time of the

Foundational Crisis of mathematics; one was the Russell Paradox (a

contradiction), while another way in which this idea was used was Gödel’s

proposition, which is neither provable nor disprovable. However, the idea of the




Cretan Liar was not at all new, but in part a rediscovery of something already

known. What was new at the beginning of the 20th century was the fact that, in

1902, it became clear that it could be used to construct something utterly

contradictory.

During the Middle Ages, this issue was not at all clear to philosophers;

however, from the viewpoint of the thinkers of that time, the “Insolubilia” might

well have turned out to be solvable after all.

In the Mittelstraß encyclopedia, we read:

“Trotz der Wortbedeutung von 'Insolubilia' wurden die Insolubilia

von den meisten mittelalterlichen Autoren für schwierig, aber lösbar

erachtet.” (Vol. 2, page 248)

Which translates in English as:

“Despite the meaning of the word 'Insolubilia', the Insolubilia were

regarded by most mediaeval authors as difficult but soluble.”

I would venture to assert that it was to the credit of the cooperative work

around the turn of the century (from the 19th to the 20th) that things became clear.

Russell's Paradox, which was based on Frege's axiom schema of comprehension

(the term “axiom of abstraction” appears to be outdated), makes the question of

the solubility of the Cretan Liar absolutely clear. A text of only about one and a

half pages was necessary to settle this question, which had, in part, been left

open by most mediaeval thinkers. That was the length of Russell's letter to

Frege, in which he informed him of the catastrophe. Thick volumes like the one

written by Paul of Venice, who had been unable to settle the problem perfectly,

were no longer necessary.

In 1499, at the close of the Middle Ages, Paul of Venice had written the

hitherto most comprehensive account of the topic, which included suggestions

for solutions. This contribution was rediscovered by Hermann Weyl and

mentioned in his Handbook of Mathematics and Science. Paul of Venice is

mentioned right at the beginning of the second part of this work, published in

English in 1949. The passages on Paul of Venice can be found on pages 229 and

220 (where Weyl quotes 192r. B et seq. of De Insolubilibus).

To me, it remains an open historiographical question as to whether Cantor

knew about Paul of Venice's book, or whether he won his knowledge of the

paradox of the largest cardinal independently. However, it has been alleged that

Cantor had a generous attitude concerning contradictions, and was therefore not

taken aback by Russell's discovery of the matter – unlike Gottlob Frege.

In my view, the connection between the modern logical-mathematical

research shows the strength of the axiomatic method, inasmuch as through the

Letter I: ▓Inspiration from a previous failure: Hamiltonian Triplets (#16/)

cooperation by Cantor, Frege and Russell, it was possible to settle a question

which had embroiled logicians for so long throughout the Middle Ages. While

it is true that Frege was shocked by being confronted with the fact that his

system allowed to infer a contradiction, if however Frege had not developed his

system, including the axiom of comprehension, then this clarification would not

have been possible. The entire axiomatisation of the theory of sets by Zermelo

took on its shape (its Gestalt) through the process of striving to avoid sets like

the Russellian set.

I consider that the discovery of a set flawed by a contradiction was a

necessary intermediary stage through which it was possible to show that the

Cretan Liar could not, if formalised appropriately, be salvaged from a

contradiction. On the contrary, the Insolubilia from the Middle Ages could, as

their name suggests, be transformed into something absolutely unsolvable. Yet

it is also possible to construct the Gödelian proposition from it, which

constitutes quite fruitful work, inspired by the thinkers of the Middle Ages.

Bar-Hillel, Yehoshua + Fraenkel, Abraham A.+ Levy, Azriel (2nd, rev. ed. 1973):

Foundations of Set Theory. See § 3.1, page 32.

Hijenoort, Jean van ()

Ohmacht, Martin (2003): Wittgenstein's Critique of Gödel's Incompleteness

Results. In: Löffler, Winfried + Weingartner, Paul (2003, eds.): Wissen und

Glauben / Knowledge and Belief. Beiträge / Papers of the 26th International

Wittgenstein Symposium, Kirchberg am Wechsel, 3rd – 9th, August, 2003

Russell, Bertrand: The Philosophy of logical Atomism. The Collected Papers of

Bertrand Russell 1914–1918 Vol. 8.

Weyl, Hermann (1949):

▓Inspiration from a previous failure: Hamiltonian Triplets (#16/)

We forgo the search for a richer structure on R3 than a vector space.

Wittgenstein frequently uses the phrase “the way of looking at things”. For

example, we may look at a circle and interpret it as being the image of a sphere

or we may discern a cone (seen from below) in it. The example I want to discuss

here is the set of quadruples of real numbers. We may see in it Hamilton's




Quaternions, or we may interpret it as a vector space.

Hamilton's Quaternions are an algebraically rather refined or strong

structure which is special for the case of a four-dimensional space. Vector

spaces can be constructed for any dimension from the natural numbers, but their

drawback is that the linear algebra that can be discerned in them is

comparatively simple.

The kernel argument that I want to present to the reader here is a

consideration of R3, which is the most important special case. Gauss had

extensively used the complex numbers and Hamilton interpreted them as pairs

of real numbers with the multiplication rule (0, 1)2 = (–1, 0). The

epistemological value of this way of seeing complex numbers is the avoidance

of the magical number i, which should be equal to the magical expression “root

of –1”. My argument concerning R3 here is the fact that a vector space is the

most common way of looking at it, which is – without being pejorative – a

relatively poor structure.

I would now like to ask a question: when some mathematician is working

on R4, he has available to him its formalisation in the Quaternions. Is there any

value in considering R4 as a vector space? Or is this formal model futile, since

the formal model of the Quaternions, which is far richer and stronger than a

vector space, is available to us?

The history of the invention of the Quaternions is traced by Bartel L. van

der Waerden in his book A History of Algebra on pages 179 ff. The germ of the

story is the fact that Hamilton wanted to construct a structure on R3 almost as

rich as that on C. This turned out to be a failed endeavour, but he found that he

could construct a structure of such a kind on R4 – which was the transgression

of the magical number 3 for the number of dimensions considered. At the same

time (in publications 1844 and 1862), the Polish-German Grassmann took the

same step in his construction of vector spaces. (See Katz, pages 862–865)

What we can see in Hamilton's self-imposed task of constructing a powerful

structure for R3 is the fact that he was trying to do something impossible – which

was thus bound to be a failure, as we know today. However, the result was

nonetheless a really breath-taking new mathematical structure (in the sense of

Bourbaki). Hamilton invented the Quaternions. I ask the reader to refer to the

text by van der Waerden, which gives a detailed report on the history of this

attempt, including the letter which Hamilton wrote to his son at the end of his

life, published in 1963.

There are, in all, four possible “candidates” for recognition as relatively full

impossibility proofs – namely:

Letter I: ▓Inspiration from a previous failure: Hamiltonian Triplets (#16/)

The first impossibility proof is the example of the number 63: its factors

21 and 3 have a representation of the form n = x12 + x2

2 + x32, but the number

itself does not. Legendre presented this classical example in 1830. The property

is shared by the number 15, because it is also of the form 8*m+7. But when we

look at the number 15, some of the xi are equal to zero, which is not the case

with 63.

The second impossibility proof was supplied by Georg Ferdinand

Frobenius, published in 1878. It is impossible to have Octions with an

associative multiplication.

The third impossibility proof was by Weierstrass: he proved in 1884 that

for the Quaternions, a commutative multiplication is impossible.

The fourth and final impossibility proof is a paper by Adolf Hurwitz

(1898), which shows that Hamilton's law of moduli

(a12 + a2

2 + ... + an2) * (b1

2 + b22 + ... + bn

2) = (c12 + c2

2 + ... + cn2)

is possible for n = 1, 2, 4 and 8 only. The title of this paper is Über die

Composition der quadratischen Formen von beliebig vielen Variablen

(On the composition of quadratic forms of an unlimited number of

variables). A limiting condition has to be put on the ci; van der Waerden

calls them “bilinear forms”. From this limitation of possibilities it follows

that there is no continuation in a solution for the case n = 16, as might be

conjectured.

For a survey of the last three impossibility proofs, see Stillwell's

Mathematics and its History, page 396.

It is not quite as clear which of the last three publications supplied the

momentum of a negative settlement for the dimension 3. Hamilton was an

autodidact and had obviously not read the paper by Legendre. But though

Hamilton worked on his own and did not realize that the question could

potentially already have been clarified by Legendre in 1830: he used the term

must for the insertion of a fourth dimension (see the letter to Graves reproduced

in B.L. van der Waerden, page 181).

Thus, the all-important step of introducing one more dimension than those

given by R3 (which represented an innovation in exceeding the number of

dimensions used since the times of the Ancient Greeks) was obliged by this

impossible task to design a differentiated structure for R3. Here we can see that

stumbling upon an impossibility, with all the shame accompanying the

occurrence of a failed endeavour, can come very close to being a great success.

My epistemological conclusion does not concern the Quaternions but the

fact that, as a conclusive result, we have a vector space for R3 as the most

commonly used model. We have to endure the fact of this relatively weak




structure, which is a word used by mathematicians in oral communication.

However nobody would be willing to dispense with, for example, almost as

strong a structure for R3 as is given for C, if we did not have a definitive proof

of the impossibility of constructing such a structure.

The consequence of the proven impossibility of having a stronger structure

than a vector space on R3 is the use of this relatively weak, formalised model as

a prevailing formal structure. It can be used in R3 as well as in all other

dimensions from N.

We owe to these five men – Hamilton, Legendre, Weierstrass, Frobenius and

Hurwitz – the fact that we definitively know that we cannot see R3 as being

similar to C. This is almost possible for R4, though in the case of R3 we have to

be humble and remain content with a vector space, which is the dominant formal

system as an image of our ordinary, three-dimensional space. It’s no use crying

over spilt milk or mourning the lack of strength in our formal representation of

R3; however, through the impossibility conjecture by Hamilton 1843, the

impossibility proof by Legendre 1830 and a full negative settlement by Hurwitz

in 1898, we know that mathematicians have done best by modeling R3 as a

vector space.

P.S.: I have quoted Hamilton with the word “must” for the creation of a

fourth dimension. Now, there is a really fine and absolutely elementary proof

for this step in Stillwell's Yearning for the Impossible (page 136). He simply

assumes that there are three numbers 1, i and j, the vectors of which are

mutually perpendicular. From this he proves that the vector i*j is

perpendicular to each of the three other vectors. We call this vector k and

suddenly we have the fourth dimension!

Bibliography:

Stillwell, John (2006): Yearning for the Impossible. The surprising truths of

Mathematics. Massachusetts: A K Peters.

Inspiration of al-Khayyam by the Pythagorean affair (#1/, #2/)

The establishing of al-Khayyam's conjecture concerning the impossibility

of solving cubic equations by ruler and compass lies very early in history,

namely in the Dark Ages. There arises the question as to how he was guided to

the idea that there might be a gap in geometry, and one answer is that he may

have been inspired by the Pythagorean impossibility of finding two whole

numbers such that the square of their quotient would make 2.

Yet there is a powerful argument against this influence of the outcome

Letter I: Inspiration of al-Khayyam by the Pythagorean affair (#1/, #2/)

concerning the Pythagorean paradigm as, in this impossibility before al-

Khayyam, the geometrical calculus is richer than the algebraical calculus (of the

whole numbers), and for al-Khayyam it is the geometrical calculus which is

defective. For the Pythagoreans, geometry seemed to be stronger and superior

to algebraical thinking, so that it may be doubted that al-Khayyam learned from

the Pythagoreans. On the other hand, if we choose the abstract paradigm of a

gap in mathematics as a focus for analysis, then it is still possible that al-

Khayyam took the Pythagoreans’ catastrophe as a model for his own conjecture.

It is rather cynical to call the outcome of the Pythagoreans’ catastrophe a

“paradigm” because, a short time after the innovative discovery of this gap, it

was not regarded in a neutral manner by contemporary mathematicians, but as

a destructive type of discovery, inasmuch as the ideology “all is number” had

been destroyed. This term that something is “discarded” is used by Gillies (page

11) in offering a criterion for a shift from one paradigm to a consecutive

paradigm. Today, we know that it is possible to construct the “root of two”

(through the Dedekind cuts) and we also know that there are formulae for cubic

equations (the so-called Cardano Formula), yet the impossibilities discovered

by the Pythagoreans (in a proof) and by al-Khayyam (in a conjecture which

came true) still hold.

We know today not only that certain calculi contain inherent limits, but also

that, in the time of al-Khayyam, it was rather audacious and courageous to assert

the existence of this considerable gap. Al-Khayyam received some consolation

for the negative impact of his statement, inasmuch as he did construct solutions

for the cubic equations (by enhanced means: conic sections), so that he did not

face the world empty-handed.

The general concept behind this section is the result of an overview (or

survey) of the early history of impossibilities. We are well aware of the

possibility that later mathematicians may have learned from the earlier ones,

which means here that al-Khayyam may very well have been inspired by the

Pythagoreans. In al-Khayyam's time, it was not yet clear that gaps would

become apparent in most calculi of a certain minimal complexity, but in his

work on the concrete problem of cubic equations, he dared to postulate (in a

conjecture) that all of his later colleagues would definitively and necessarily fail

in looking for a solution.

This fact of collective learning amongst mathematicians at an abstract level

is a tendency which possibly started with al-Khayyam and continued at least

until Gödel’s Incompletability Theorems, which produce an attitude of mature

pessimism in mathematicians. Al-Khayyam's conviction of the negative

discovery of an eternal impossibility of solving cubic equations by ruler and

compass turned out to be correct, and after Wantzel's proof in the matter (1837),

this impossibility has acquired the character of an eternal negation.




Introduction: the Problem of Impossibilities in Mathematics (#0/)

It was Ludwig Wittgenstein under whose guidance I embarked on the issue

of impossibilities in mathematics. But this book can also be read as a

contribution to the topic of applying Kuhn's ideas in his famous book on the

natural sciences, The Structure of Scientific Revolutions. Some of the

impossibilities incorporate revolutions in their historiography, others do not.

Wittgenstein proposes the concept of logical, grammatical impossibilities

(BlB, page 56) and the concept of a “geometrical, logical impossibility” (RFM,

3rd ed., 1978, page 92, § 141). His work on this concept has had a lasting

influence on me at least. Already in his TLP, he mentions that propositions

which are never true (impossibilities) form a duality with necessities, i.e. with

propositions which are always true:

“§ 6.375 As there is only a logical necessity, so there is only a logical

impossibility”. (The article “a” reads as “eine” in the German original).

I use the term “impossibility” as a summarizing term covering the following

four phenomena in formal problems:

1) The impossibility of constructively solving a specific problem, the

solution of which can nonetheless be clearly specified.

2) The impossibility of a specific geometrical construction in which a clear

set of operations are allowed (compass and unmarked ruler, in most cases).

3) The existence of a well-formed statement, which can be neither proved

nor disproved.

4) The non-existence of an algorithm with specified properties.

I have discovered that one fruitful approach to the process of the history of

mathematics is to take the concept of impossibility as a basic concept and to

also work on those 30 occurrences that do exist by analyzing the history of

ideas, as they emerge from the minds of mathematicians.

Letter I:

My view of the history of mathematics shows its prevailing trait to be its

difference from the history of the natural sciences. Kuhn does not write a great

deal about mathematics, but in my impossibility approach to this hardest of all

sciences, I can make use of his dominant idea of a Gestalt switch.

Mathematicians have to turn the tables on the problem (In German: ‘sie müssen

den Spieß umdrehen’.) They have to back the right horse.

There already exists some literature on the phenomenon of impossibilities,

but this topic is not discussed very frequently. I shall work only a little on the

existing literature, because I shall at once embark on the history of mathematics,

as seen from Wittgenstein's way of looking at things. My plan is to compile a

complete list of the known impossibilities (insolubilities) in mathematics,

spanning everything from the Pythagorean impossibility to the concrete

incompleteness in Peano arithmetic, as discovered by Paris+Harrington in 1977.

Klagenfurt, 22nd September 2008 Martin Ohmacht




Letter J

Letter J: Placeholder for the Title of Level 3J

Placeholder for the Title of Level 3J




Letter K

Letter K: Khayyam-Pacioli Fallacy (#2.3/)

Khayyam-Pacioli Fallacy (#2.3/)

This article concerns an alleged impossibility (an erroneous consensus of a

gap in the world of algebraical formulas), which was passed on from the 11th to

the 13th century. I shall discuss the historiographical sources here and the

philosophical relevance of the incidence in ↑deceived by and overcoming a

fallacy.

The poet-mathematician Omar Khayyam and the Italian mathematician

Luca di Pacioli emerge as twins in the history of mathematics on cubic

equations, as they both held a conjecture on them. Since Omar Khayyam lived

before Pacioli, I looked for a passage in a history book on him alone. Carl Boyer

and Uta Merzbach write:

“For general cubic equations, he believed (mistakenly, as the

sixteenth century later showed), arithmetic solutions were impossible;

hence, he gave only geometric solutions.” (1989, page 269)

Boyer's remark is confirmed by Alten et al. on page 251 and I think this text

is an independent source.

Now I want to present an argument that shows why this alleged impossibility

is so important for the historiography of mathematics. Omar Khayyam lived

long before the end of the Middle Ages and therefore it is all the more

remarkable that he had the courage to state such a negative vision. From today's

viewpoint, in order to reconcile us with his error we could say that he simply

wrongly guessed the parameter value as being 3 (instead of 5), which would

make a surprisingly modern statement out of his conjecture.

Now Luca di Pacioli enters the stage. Peter Pesic writes concerning him and

Omar Khayyam:

“Pacioli thought them [cubic equations] quite unsolvable by algebra,

an opinion he derived from the Arabic poet-mathematician Omar

Khayyam” (page 30).

The difference between the two researchers lies in the fact that the scope of

Khayyam's conjecture is not clear, whereas the book by Pacioli was apparently

widely read in the Renaissance world. It had great influence, as is shown by a

passage in the Encyclopaedia Britannica (1977, Vol. 11, page 662g), which I

quote in full in ↑deceived by and overcoming a fallacy.

Now, there definitely exists a problem in interpreting this Khayyam-Pacioli

fallacy, because it’s possible to construct verbalizations of such a hypothesis,

which are opaque. Sometimes it isn’t clear whether the authors think it will

never be possible to find a formula for cubic equations or whether they only

state that it has not yet been possible to find such a formula. Van Waerden has

studied the original 1494 text by Pacioli and translates the passage at the end of




his book as follows:

“It has not been possible until now to form general rules” (page 47).

Victor J. Katz’s examination of Omar Khayyam's original text (pages 287—

290) unfolds a similar result.

My conclusion on the question “Did Khayyam and Pacioli really fully

express this conjecture?” is the impression that although they probably did put

forward weaker versions of an absolute impossibility hypothesis in the form of

a hitherto impossibility, so many contemporaries attributed (ascribed) an

absolute impossibility hypothesis to them that mathematicians could clearly

anticipate the philosophical issue of the possibility of an impossibility at a

relatively early stage in the history of mathematics.

Kuhn as prepared by Wittgenstein (#0/, #1/)

Kuhn mentions Wittgenstein (page 44) and the oscillating picture of the

duck-rabbit (page 114), which Wittgenstein in his PI gains from the Polish-

American psychologist Jastrow.

I have read in detail what is said in Ewald Craig's Encyclopaedia of

Philosophy concerning incommensurability (Vol. 4, pp.732-736), but I was

disappointed because here the historiographical affair of the Pythagoreans is not

mentioned, and the term is analysed in a purely Kuhnian way. The expression

“incommensurable” has two meanings – one within mathematics, and the word

is then extended by Wittgenstein in such a way as to mean an incompatibility or

irreconcilability at the level of the sociology of communication. In his use of

the word, Kuhn acts as an adherent of Wittgenstein. Wittgenstein uses the word

in both meanings, the mathematical term in his LFM (Unit IX; page 90) and the

sociological term in his PR (page 310) and in PP Vol. 1, part I (page 62, § 314).

Kuhn in his essay does not mention the Pythagorean aspect of the word.

Let us take a deeper look at the sociological variant here: I think the central

feature of the situation when alternative ways of viewing reality become

incommensurable is the fact that there is no compromise between the two

positions. This is well demonstrated by the duck-rabbit head, because the two

ways of looking at it (the two different aspects) do not allow a third position

which could serve as a compromise. This is the application of the principle of

the excluded middle in the sociology of communication.

The missing third position can be seen in the change from Ptolemaian

astronomy to Copernican astronomy, where it does not make sense to seek for

Letter K: Kuhn as prepared by Wittgenstein (#0/, #1/)

a third view of things by, for example, choosing a point in the empty space

between the earth and the sun as a midpoint for the planetary system. A point

which offers itself is the one point where the gravitational forces of the earth

and the sun add up to zero and thus form an equilibrium.

As Kuhn argues on in his postscript on page 199: if, in a dispute, one side is

forced to concede an error, then the other side must be right – as assertion which

is not necessarily true from a logical point of view, but which, from the

perspective of the sociology of conflicts, is indeed true.




Letter L

Letter L: ▓Laplace vs Poincaré (#MIMP_14)

▓Laplace vs Poincaré (#MIMP_14)

This article is about Laplace's radical optimism concerning the calculability

of the orbits of all planets and a correction by Poincaré.

In the issue of the three body problem, which emerged out of ↑Newton's

Idealizations when establishing his law for the mutual attraction of celestial

bodies, we have – as in the case of the First of Gödel's Incompletability

Theorems – a certain dialectical development (see ↑Bourbaki’s dialectical

Distance to Gödel). To put it quite bluntly, an error occurred, but it was not just

an error of an individual, namely Laplace, but here a larger group of intellectuals

was caught by an alleged proof of the stability of the solar system.

Due to the development of Newton's gravitational law mankind had, so to

speak, taken over a lot of responsibility from God concerning the question as to

whether the planets would move around the sun without crashing or fleeing into

the surrounding empty space. Intellectuals worried a lot about this question at

the end of the 18th century and when Laplace explained that he had proved

analytically that the solar system would remain stable, people were readily

willing to believe him, although, as a matter of fact, his proof was only an

alleged proof and it took almost a century to unravel that. I wonder whether this

willingness to believe Laplace and to accept his proof was motivated religiously

or anti-religiously; see what Ernst Peter Fischer writes about Newton's God on

page 317.

The historical context of Laplace's radical optimism was his success with

the Jupiter-Saturn anomaly, which had been worked on by Euler. Stillwell sums

up the historiography of this issue:

“A famous example [where Newton could only obtain results by

approximations] was the so-called secular variation of Jupiter and

Saturn, which was detected by Halley in 1695 from the observations then

available. For several centuries Jupiter had been speeding up (spiraling

towards the sun) and Saturn had been slowing down (spiraling outward).

The problem was to explain this behaviour and to determine whether it

would continue, with the eventual destruction of Jupiter and

disappearance of Saturn. Euler and Lagrange worked on the problem

without success; then, in the centenary year of the Principia, Laplace

(1787) succeeded in explaining the phenomenon. He showed that the

secular variation was actually periodic, with Jupiter and Saturn returning

to their initial positions every 929 years. Laplace viewed this as

confirmation not only of the Newtonian theory but also of the stability

of the solar system, though it seems that the latter is still an open

question.” (page 235)

So Laplace's optimism, although it was not well-founded as a whole, was

not without a reason – he had solved a serious problem and concluded that he




could solve the entire problem of the stability of the solar system.

It is fairly easy to find remarks on Laplace's optimism, which are in many

cases without critical components. For example, Katz writes:

“During the period from 1799 to 1825, he [Laplace] produced his

five-volume Traité de mécanique céleste (Treatise on Celestial

Mechanics), in which he successfully applied calculus to the motions of

the heavenly bodies and showed, among much else, why Newton's law

of gravitation implied the long term stability of the solar system” (page

654).

Similar quotations about “Laplace's triumph” can be found in the book

edited by Ivor Grattan-Guinness (Vol 2, 1050f.) and in the encyclopaedias

edited by Guido Walz (Vol 3, page 253) and Mittelstrass (Vol. 2, page 539). This

abundance of material on Laplace is an indicator that he had achieved

acceptance with his optimism.

The problem here is that in the sense of conventionalism (or mental

constructivism), this belief in an optimistic proof by a majority of intellectuals

was outweighed by a counter-proof by Henri Poincaré almost a hundred years

after Laplace's optimistic memoirs, which he, Laplace, had read to the Paris

Academy from 23 November 1785 onwards. To repeat this remarkable fact a

second time: this correction happened as much as almost a century later.

It is not necessary to study the entire story of Poincaré´s publication, as that

is related by Szpiro (pages 48–71). The essential point here is that the

Newtonian differential equations do not have solutions in a closed form, as it is

clearly stated by Mittelstrass:

“In 1893, he [Poincaré] showed that no integrable solutions … exist

for numerous dynamic problems (including the three body problem).”

(Vol. 3, page 282); see also Herrmann, page 61.

So here we again have a mathematical impossibility which is worthy of

study. To cut a long (and emotionally-fraught) story short: Poincaré's

impossibility result was submitted for a prize initiated by the Swedish King.

The task required for this prize was to solve the three body problem. Szpiro

writes of the twelve submissions: “None of them contained a solution to the

three body problem” (page 37).

Here we have the interesting case of a prize where a problem was posed that

turned out to be unsolvable. It was the genius of Poincaré, who performed the

↑problem transmutation (Hao Wang) and who dared to submit a negative

settlement of the original constructive problem to the committee. Szpiro writes:

“He proved rigorously that no analytical solutions (i.e., no elegant

formulas) exist that would describe the position of the bodies at all

Letter L: ▓Laplace vs Poincaré (#MIMP_14)

times” (page 37).

The most comprehensive description of the situation surrounding this

impossibility (including the religious aspect) is given by Ernst Peter Fischer on

pages 303f. of his book Die kosmische Hintertreppe. There is one statement in

the book by Grattan-Guinness which sums up all the dialectics between Laplace

and Poincaré:

“Poincaré showed that in general these series were not uniformly

convergent, but that they represented the dynamical coordinates in an

asymptotic sense. This result showed ... Laplace's demonstrations of

stability to be inconclusive” (page 1060).

In their chapter on The problem of three bodies and the stability of the solar

system, Victor Szebehely and Hans Mark confirm the fact that the mathematical

part of the epistemological problem with Laplace's error was given by a

convergence problem. They speak of the fact that his series were “not absolutely

convergent” (page 275).

Bibliography:

Fischer, Ernst Peter (2009, 2011): Die kosmische Hintertreppe. Die Erforschung

des Himmels von Aristoteles bis Stephen Hawking. Frankfurt am Main:

Fischer Taschenbuch Verlag.

Grattan-Guinness, Ivor (ed., 1994): Companion Encyclopedia of the History

and Philosophy of the mathematical Sciences, Vol 2. London + New York.

Herrmann (1973): DTV-Atlas zur Astronomie. München: Deutscher

Taschenbuch Verlag.

Karamanolis, Stratis (1996): Einsteins Relativitätstheorie. Eine leicht

verständliche Einführung. München: Elektra Verlags GmbH.

Kuhn, Thomas S. (3rd ed., 1996): The Structure of Scientific Revolutions.

Chicago + London: University of Chicago Press.

Laplace, Pierre S. (1799–1802, French): Traité de Mécanique Céleste vols. 1–

3, Paris: Duprat. (1805): vol. 4, Paris: Courrier. (1823–1825): vol. 5, Paris:

Bachelier. English translation by Bowditch, N. (1829–1839): vols. 1–4., NY:

Chelsea.

Poincaré, Henri (1890): Sur le Probléme des Trois Corps et les Équations de la

Dynamique. In: Acta Mathematica, 13, 1–270.

Szebehely, Victor G.+Mark, Hans (1998, 2nd ed.): The problem of three bodies

and the stability of the solar system, in: Adventures in Celestial Mechanics,

NY et al.:John Wiley & Sons, pages 263–282.

Szpiro, George G (2007): An Oscar for the Best Script. Poincaré's prizewinning




theory of the solar system's stability ... and the scandal. In: Same Author:

Poincaré's Prize. The Hundred-Year Quest to Solve one of Math's Greatest

Puzzles. New York: Dutton.

Wilson, C.: The Three-Body-Problem. In: Grattan-Guiness (1994), Vol 2, pages

1054–1062.

A light bulb moment concerning two mutually proximate impossibilities (#1/, #19/)

I want to discuss a passage by Wittgenstein from his RFM, where he

proceeds from the Gödelian gap in the proof structure of axiomatic systems (for

mathematical analysis) to the gap in the Euclidean line concerning the root of

2. Both phenomena are strategic gaps in the architecture of mathematics. They

are, however, quite different in kind: one is simply a point missing in the set of

rational numbers and the other consists of the frightful discovery that there is

no axiomatic system for arithmetic which would be both complete and

consistent.

Wittgenstein mentions the Gödelian issue in § 36 of part V of the RFM and

the Pythagorean issue in the paragraph which directly follows this passage, §

37. To my mind, his association (i.e. chain of thought) implies that Wittgenstein

holds the opinion that there is a connection between the two issues. Although

very abstract, the connection which I have found is, from the viewpoint of

impossibility research, quite clear. The point which connects the two issues is

the fact that they are both impossibilities.

Gödel forces mathematicians to decide between incomplete and inconsistent

axiomatic systems and it is clear that the final decision leans towards

incompleteness. The repair work carried out to establish the existence of the

root of 2 was done out of a certain reaction of frustration and shock too. In both

cases, it was necessary to digest (work through) the truth about a foregoing

naïveté in the prevailing opinion of the mathematicians. In both cases, an

illusion was destroyed.

This comparison between the First Gödelian Incompletability Theorem and

the absence of a root of 2 in the rational numbers prompts the question as to just

how widespread the idea of completeness of mathematics was in Hilbert's time,

i.e. in the 1920s, for example. Was it Hilbert's responsibility to simply urge his

colleagues to believe in completeness or did he find many implicit adherents in

this radically optimistic attitude? Wittgenstein wrote the passage which I would

now like to quote some years after Gödel's discovery, and he was flexible

enough to believe it. So after writing “Mathematics cannot be incomplete” in

Letter L: Liouville on integral (exp(x)/x) dx (#23/)

1929, he was no longer trapped in the philosophical pitfall of the completeness

of mathematics in the 1940s. Here, in this passage, Wittgenstein accepted

Gödel's result.

On page 294 of his RFM in §40, Wittgenstein writes:

“Compare the two forms of definition: We say:

'lim(𝑥 → ∞)𝑃ℎ𝑖(𝑥) = 𝐿when it can be shown that ...'

and

'lim(𝑥 → ∞)𝑃ℎ𝑖(𝑥) = 𝐿means: for every ε there is a δ ...'”

The epistemological difference between the two variants of the definition

for a limit value for a mathematical function Phi can definitely be found in the

following danger: it may be that Phi (x) converges to L, but this cannot be

proved because it leads to a Gödelian proposition which is true but not provable!

Although it might seem that such a calculation of a limit value of a mathematical

function is routine work for mathematicians, Wittgenstein sees a certain danger

in this apparently simple calculation. Even in such a straightforward task, the

abyss of the independence of a proposition may open up and lead the working

mathematician (who severely underestimates the difficulty of the calculation,

regarding it as being of a standard kind) to an unsolvable problem.

As I have already stated: in § 36 (part V, page 290) of his RFM Wittgenstein

makes an allusion to this distinction of two possible kinds of definitions of a

limit value, and mentions the gap of root of 2 on the Euclidean line in the next

paragraph (§ 37). It is therefore clear, from the viewpoint of impossibility

research, that Wittgenstein wants to juxtapose these two impossibilities with one

another.

Here, the Pythagorean impossibility is compared to the First

Incompletability Theorem by Gödel. The closeness of the two impossibilities

i.e. the association of thoughts (connection of ideas) given by them once more

shows the importance of the concept of impossibility in Wittgenstein's

philosophy of mathematics.

Liouville on integral (exp(x)/x) dx (#23/)

My aim in this section is to consider the following question: which element

of Liouville's Theory of Symbolical Non-Integrability can be presented to

secondary school students in their mathematics lessons?

The model here is Wittgenstein's procedure in choosing the heptagon as a




symbol of non-constructability, carried over from geometry to algebra and thus

including non-existent desired objects of calculi, which are no longer

necessarily geometrical. This method of selecting one of the Liouvillian

functions, which lack an indefinite integral, leads us directly to the integral

.

I did not manage to find an English or German translation of his 1835 paper,

so I am relying on a quotation, in German, from the book Analysis in

historischer Entwicklung by Hairer + Wanner:

“Mit unserer Methode kann man sich sehr leicht davon überzeugen,

dass das Integral , das viele Geometer lange beschäftigt hat,

unmöglich in endlicher Form geschrieben werden kann.”

“Using our method, it is easy to find out for oneself that the integral

, which has long preoccupied many geometricians, cannot possibly

be written in a finite form.” (page 138)

Formula seekers can find a degree of consolation in the fact that, in the case

of some functions for which no indefinite integral exists, a definite integral can

be given. Lists of definite integrals constitute portions of formula collections,

for example the small one by Karl Rottmann (pages 153–170) and the big one

called Taschenbuch der Mathematik by I. N. Bronstein et al. (Chapter 21.8,

pages 1086–1090). If a definite integral existed for the function which I want to

focus on here, then it would be subsumed in Bronstein's chapter 21.8.2, pages

1087ff., but it seems that no definite integral even exists for . We must

therefore reluctantly accept the fact that, in the case of Liouville's central

function of 1835, the question concerning some definite integral once more

leads to a gap in the formula collection.

I am still looking for a perspicuous name for this function which might allow

us to discuss it. A clear, translucid name would be a prerequisite for presenting

the function to secondary school students when they are 17 years of age as an

integral part of their calculus training. To be taught the impossibility of finding

an anti-derivative of certain functions could modify the students’ impression

that integration is definitely more difficult than differentiation. In presenting

this résumé on the art of working on curves mathematically, a statement on the

facts could be helpful: it is not always the student who is to blame if no anti-

derivative can be found. Sometimes, mathematics itself is simply saying “No!”.

The material suitable for secondary school students includes work on

variants of this Liouville 1835 function – namely, a parameterised set of

functions , which cannot be integrated in a closed form for even

values of n (see Marchisotto+Zakeri, page 300, Example 4). If we assume, wíth

Letter L: ▓Logical ambiguities (#25/)

regard to mathematic lectures at the secondary school level, that students are

already informed concerning the symbolical non-integrability of the Gaussian

Error Function, then, by using the product rule of integration, we can relatively

simply reduce (with n = 2*m) to it (namely, the Gaussian Error

Function), thus expanding the number of functions for which it is possible to

prove symbolical non-integrability.

I think that this application of the product rule of integration, aimed not at

finding an integral but, rather, at proving non-integrability, would be an

epistemologically worthwhile exercise for secondary school students. It could

serve as a warning to them not to erroneously believe that an anti-derivative can

be found for all functions.

P.S. The simple proof of the fact that the integral of exp(x)/x is not

elementary, as derived from a downgraded version of Liouville's Theorem (the

Rational Liouvillian Theorem) can be found in the article by Fitt and Hoare in

The Mathematical Gazette of 1993. I do particularly appreciate the

introductory note on page 227. It gives an image of a drama, where young

students are lead from pan-solvable problems (differentiation) to those where

the outcome is “at worst complete failure.” Fitt and Hoare indicate the illusion

of pan-solvability by the students in this second case also.

P.P.S; An alternative specimen for a simplemost anti-derivative, which

cannot be integrated is the elliptic integral ∫√1 + 𝑥3𝑑𝑥.

Bibliography:

Fitt, A.D. + Hoare, G.T.Q (1993): The closed-form integration of arbitrary

functions, In: The Mathematical Gazette, pages 227–236.

▓Logical ambiguities (#25/)

In my section on ↑Ambiguity, I describe a nice pun that arises from the

double meaning of a German verb. Hence, this is an ambiguity which rests on

the structure of the semantical field of this word and it depends on linguistic

material. Here, in this section, I should like to describe ambiguities which are

radically different in the way they are constructed, since they are logical

ambiguities which carry over from German to English.

“Hermann Maier hat gestern sein erstes Abfahrtsrennen in St.Moritz

gewonnen” (“Yesterday, Hermann Maier won his first downhill race in

St. Moritz.”) (*)

This statement could have been printed on any given day in an Austrian

newspaper, as Hermann Maier is a famous Austrian skier. The statement is

ambiguous, since it is not clear whether (1) it was the very first victory of




Hermann Maier over his contestants, which took place in St. Moritz or (2)

Hermann Maier has won several races in other locations and now won the race

in this particular place for the first time.

In German, the two meanings can be distinguished from one another by

adding the expression und zwar (“in fact”, “namely”): “Hermann Maier hat

gestern sein erstes Abfahrtsrennen gewonnen und zwar in St.Moritz”. So if we

agree on the use of the phrase “und zwar” in the first of the two different

meanings above, then the meaning of the statement (*) can be narrowed down

to the second one (2).

This ambiguity is based on a logical problem – that is, the question of the

scope of the word “first”. Does it refer to all downhill races or only to downhill

races in St. Moritz? From the logical origin of this play on words, one can infer

that it likewise carries over to other languages such as English. The material of

which the ambiguity consists is independent of linguistic source.

TV advertisements are also good sources of puns. There is one example

which is frequently broadcast on the German broadcasting station ZDF.

“Es gibt 37 verschiedene Arten von Kopfweh die man selbst heilen

kann.”

This is to be translated by “There are 37 kinds of headache, which

you can cure yourself.”

In German, it is not clear whether the relative clause “die man selbst heilen

kann” (“which you can cure yourself”) refers to every kind of headache or only

to 37 kinds. Two interpretations are possible: either (1) only 37 kind of

headaches exist altogether and you can treat them all yourself, or (2) there are

more than 37 kinds of headaches, but 37 of them can be treated without

consulting a doctor.

I should like to present the last logical ambiguity in this section in English:

“Today I shall refrain from going to town, as I so often do.”

At the end of this section, I should like to state that I am clearly aware that

the term “pun” is not the optimal choice for the examples presented, but I want

to keep a certain distance to Wittgenstein's term of “language games”, by which

he means serious and adult games. At all events, to draw a connection between

these linguistic problems and the issue of impossibilities – I think that it would

be impossible to programme a computer to process the double meanings of the

statements in this section.

The absolute worst case concerning ambiguities is the negation of a

consequence, because it may be mixed up with a negation as a (positive)

Letter L:

consequence. See the example from a TV series:

“Er ist nicht nach München gezogen, weil seine Tante dort wohnt.”

In English: “His aunt lives in Munich. He did not move there for this

reason.”

Note that the statement is compatible with both the statements ”He moved

to Munich” as well as with the statement “He did not move to Munich”! To give

a more clear picture about this example, ohne may use brackets again:

“(He did not move there) for this reason” versus “He did not (move

there for this reason).”

There is another strategic problem concerning syntactical ambiguities,

which lead to the insight that natural languages are not associative (See ↑non-

associativity).

▓Logical problems for mathematicians (#MIMP_26, #/MIMP_12)

If we wish to create an idealistic image of the relationship between

mathematicians and logicians, then we could say that logic is the fundamental

supporting basis on which mathematics is built: logic is a service discipline to

mathematics. Yet this picture is too highly refined to be realistic, and we are

called upon to develop a more down-to-earth description.

There is one central issue raised by logicians which ought to cause

mathematicians a good few headaches: at least, logicians think that this question

that they have found should affect mathematical life. Yet, in reality, many

mathematicians ignore this logical problem.

I am referring to the Axiom of Choice, which contains an implication that is

even more annoying to mathematicians. Logicians tell them that it is possible

to well-order the set R. Wittgenstein mentions this issue in his Nachlass (106

274 and 106 276).

Historically, this question of well-ordering all sets is older than the issue of

the Axiom of Choice. Here logical research has revealed a problem which

severely affects mathematicians. The problem for mathematicians is that they

would never have even started to think about this axiom were it not for the

logicians who have stirred up this question.

Now, it is not necessary for mathematicians (dealing with real analysis) and

logicians to agree on a mutual appeasement as to whether or not to adjoin the

Axiom of Choice. “An important consequence of the axiom of choice in

analysis is the existence of a set of real numbers which is not Lebesgue-




measurable.” (Bar-Hillel+Fraenkel+Levy 1973, page 67) What presented an

existential annoyance for mathematicians was the fact that, in their view, this

drawback concerning the axiom was not supported by a constructive algorithm

which would show what the non-measurable set looks like. In this case,

mathematicians would be able to investigate this problem – but the Axiom of

Choice does not offer any information concerning this major “bug” in the theory

of measurement. To be informed concerning a severe problem is one thing, but

not to be informed concerning the concrete shape of the problem provokes a

deep-seated annoyance.

However, the structure of the problem is not only a subject of conflict

between mathematicians and logicians, but the problem stirred up by the

logicians (firstly, Zermelo) drives mathematicians into a corner, and pits them

against each other in a no-win situation.

The very same axiom which gives rise to problems in real analysis is

urgently needed in algebra, because it offers an assurance that that every vector

space has a basis. If the axiom of choice is not granted, then there exist vector

spaces without bases, which is an essential drawback. So the line of decision

does not lie between logicians and mathematicians: rather, the borderline of the

decision at stake (as to whether to adjoin the axiom of choice or refrain from

doing so) lies within mathematics.

Thus, the overall history of the relationship between mathematicians and

logicians in the 20th century is shaped by the fact that, with their pedantic

thinking, logicians stir up problems which – in the case of the AC – lead to a

dilemma for mathematicians. The logicians’ hair-splitting is not some sort of

service rendered to the mathematicians but, rather, a process of creating

problems that were originally unnecessary.

In this situation of dilemma for mathematicians, concrete contentual work

sometimes produces a certain relief, as was accomplished by a result achieved

by Solovay in 1970. As Gregory H. Moore describes on page 102 in footnote

24, there is a weaker variant of the Axiom of Choice, which offers hope of not

stirring up the problem with the existence of non-Lebesgue-measurable sets –

namely, the Denumerable Axiom of Choice, with a denumerable number of sets

only. So here we have a text on the historiography of logic by Moore, in which

he expresses the opinion that there is some consolation with regard to this

situation. (Solovay’s solution requires the existence of a strongly infinite

cardinal.) However, do mathematicians affirm the optimistic opinion that the

offer concerning the Denumerable Axiom of Choice is already satisfactory?

Indeed, Jean Dieudonné writes: “Many theories of classical analysis require

only a weak form of the axiom of choice, namely the statement ACD that a

denumerable product of nonempty sets is nonempty” (page 217).

Letter L: ▓Logical problems for mathematicians (#MIMP_26, #/MIMP_12)

So this solution seems to be acceptable, provided that the “ACD” grants, for

example, the existence of a basis for all vector spaces.

As we know from geometry, even contradictory axioms are assumed in

Euclidean Geometry and non-Euclidean geometry, respectively. So it ought to

be permitted for real analysts to work with the Denumerable Axiom of Choice,

whereas logicians should be able to follow Zermelo and seek to investigate the

higher Alephs. The set theorist could indulge in adjoining the fully-fledged

Axiom of Choice, while mathematicians could work with the ACD.

It may be that the hope offered by the Denumerable Axiom of Choice is its

contentually satisfying intermediary position – avoiding non-Lebesgue-

measurable sets, on the one hand, and granting a basis for all vector spaces, on

the other. The business of logicians concerning the trichotomy of cardinals is a

separate building site which is not needed by mathematicians.

Bibliography:

Dieudonné, Jean Alexandre (1982): VI Mathematical logic, noncontradiction

and undecidability, in: A panorama of pure mathematics, Orlando: Academic

Press, pages 216f.

Bar-Hillel + Fraenkel + Levy (1958, rev. ed. 1973): Foundations of Set

Theory. Amsterdam & London: North Holland Publishing Company.

Moore, George H. (1982): Zermelo's Axiom of Choice. Its Origins,

Development and Influence, NY: Dover Publications.

Solovay, Robert M. (1970): A Model of Set Theory in which Every Set of

Reals is Lebesgue Measurable, Annals of Mathematics 92, 1–56.

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