**Philosophy Forum**- »
**Logic** - »
**Mathematicians are in deep trouble for 2 reasons**

Get Email Updates • Email this Topic • Print this Page

Reply
Sun 11 May, 2008 06:18 pm

The australian philosopher colin leslie dean points out Mathematicians are

in deep trouble for 2 reasons

1) Godel cannot tell us what makes a statement true-thus his incompleteness theorem is meaningless

2 the skolem paradox makes set theory ZFC inconsistent

1)

mathematician have so much invested in godels incompleteness theorem

much maths is reliant on it

but at the time godel wrote his theorem he had no idea of what truth was

as peter smith the Cambridge expert on godel admitts

as peter smith kindly tellls us

http://assets.cambridge.org/97805218/57840/excerpt/9780521857840_excerpt.pdf

but Gödel didn't rely on the notion

of truth

now because Gödel didn't rely on the notion

of truth he cant tell us what true statements are

thus his theorem is meaningless

this puts mathematicians in deep trouble because all the modern idea derived

from godels theorem have no epistemological or mathematical worth for we

dont know what true statement are

Gödel's incompleteness theorems - Wikipedia, the free encyclopedia...

thus the theorem is meaningless

and modern mathematics is in deep trouble for useing a meaningless theorem

2) skolem discovered a paradox which makes set theory inconsistent

the paradox reads

Skolem's paradox - Wikipedia, the free encyclopedia

http://www.math.ucla.edu/~asl/bsl/0602/0602-001.ps.

in deep trouble for 2 reasons

1) Godel cannot tell us what makes a statement true-thus his incompleteness theorem is meaningless

2 the skolem paradox makes set theory ZFC inconsistent

1)

mathematician have so much invested in godels incompleteness theorem

much maths is reliant on it

but at the time godel wrote his theorem he had no idea of what truth was

as peter smith the Cambridge expert on godel admitts

Quote:

but truth is central to his theorem

Gödel didn't rely on the notion

of truth

as peter smith kindly tellls us

http://assets.cambridge.org/97805218/57840/excerpt/9780521857840_excerpt.pdf

Quote:

you see godel referes to true statementbut Gödel didn't rely on the notion

of truth

now because Gödel didn't rely on the notion

of truth he cant tell us what true statements are

thus his theorem is meaningless

this puts mathematicians in deep trouble because all the modern idea derived

from godels theorem have no epistemological or mathematical worth for we

dont know what true statement are

Gödel's incompleteness theorems - Wikipedia, the free encyclopedia...

Quote:

without a notion of truth we dont know what makes those statements trueGödel's first incompleteness theorem, perhaps the single most celebrated

result in mathematical logic, states that:

For any consistent formal, recursively enumerable theory that proves

basic arithmetical truths, an arithmetical statement that is true, but not

provable in the theory, can be constructed.1 That is, any effectively

generated theory capable of expressing elementary arithmetic cannot be

both consistent and complete.

thus the theorem is meaningless

and modern mathematics is in deep trouble for useing a meaningless theorem

2) skolem discovered a paradox which makes set theory inconsistent

the paradox reads

Skolem's paradox - Wikipedia, the free encyclopedia

Quote:

freankel and most mathematicians at the time saw it as an antinomy contradictionset theorywhich contains only a countable number of objects. However, it must contain the aforementioned uncountable sets. This appears to be acontradiction,since the uncountable sets are subsets of the (countable) domain of the mode

http://www.math.ucla.edu/~asl/bsl/0602/0602-001.ps.

Quote:

Neither have the books yet been closed on the antinomy, nor has

agreementon its significance and possible solution yet been reached." ""

(Abraham

Fraenkel)

and that

"most mathematicians followed fraenkels skepticiam

the wiki article says it appears to be a contradiction based on the skolem attempted solution

in regard to skolem relativism attempt at resolution - which is at

present is not accepted

skolems relativistic solution had the affect of destroying set theory -of which he himself noted

http://www.math.ucla.edu/~asl/bsl/0602/0602-001.ps

Quote:Peter Suber points out the problem with Skolems relativism IE IT DESTROYS SET THEORY"I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics; therefore it seemed to me that the time had come for a critique." - ([[Skolem]"The Bulletin of symbolic logic" Vol.6, no 2. June 2000, pp. 147

http://www.earlham.edu/~peters/cours...skol.htm#amb3]

Quote:This means that there simply are no sets whose cardinality is absolutely uncountable.For many, this view guts set theory, arithmetic, and analysis.It is also clearly incompatible with mathematical Platonism which holds that the real numbers exist, and are really uncountable, independently of what can be proved about them.

(John von Neumanns states

Quote:of which a few mathematician also agreed]"At present we can do no more than note that we have one more reason here to entertain reservations about set theory and that for the time being noway of rehabilitating this theory is known."

now rather than solving the paradox before moving on with set theory

mathematicians just ignored it and used set theory for all sorts of

proofs

Now mathematicians are in deep trouble for there is now so much invested in

set theory that the skolem paradox threatens the very foundations of

mathematics

so some mathematician now try to argue away the paradox by saying it is

not a contradiction

but

skolems paradox want go away it is at present unable to be disproved

and modern maths is buried so much in trouble for useing set theory they cant

get out

so we have

either the paradox means set theory ZFC is inconsistent

or

set theory is destroyed

Arjen

Reply
Mon 12 May, 2008 02:04 am

@pam69ur,

Say pam69ur, why are you repeating these complaints in every topic you open, but not arguing them anywhere?
pam69ur

Reply
Mon 12 May, 2008 05:15 am

@Arjen,

Quote:

Say pam69ur, why are you repeating these complaints in every topic you open, but not arguing them anywhere?

surely the evidence presented is argument enough

one dose not have to wrie 20000 word when a few quote do it

Quote:

Gödel didn't rely on the notion

of truth

thus this is meaningless

Quote:

For any consistent formal, recursively enumerable theory that proves

basic arithmeticaltruths,an arithmetical statementthat is true,but not

provable in the theory, can be constructed.1 That is, any effectively

generated theory capable of expressing elementary arithmetic cannot be

both consistent and complete.

you dont need 200000 words of argument

similarly you dont need 200000 words of argument

Quote:

set theorywhich contains only a countable number of objects. However, it must contain the aforementioned uncountable sets. This appears to be acontradiction,since the uncountable sets are subsets of the (countable) domain of the mode

to see that set theory ZFC is inconsistent

Arjen

Reply
Mon 12 May, 2008 05:53 am

@pam69ur,

Is not the presumption of thruth of one's own arguments (as opposed to the study thereof) a strange way of discussing things?
pam69ur

Reply
Mon 12 May, 2008 06:14 am

@Arjen,

Quote:

Is not the presumption of thruth of one's own arguments (as opposed to the study thereof) a strange way of discussing things?

if you disagree with the claims feel free to give your 200000 word rebutale

Arjen

Reply
Mon 12 May, 2008 07:45 am

@pam69ur,

I have done so in a number of your topics, with far less than 20000 words I might add, but you never argue your point. You merele state it and the move on to another topic to state the same thing. What are you trying to do?
Aedes

Reply
Mon 12 May, 2008 12:53 pm

@Arjen,

Arjen,He's gone all over the entire internet under the names pam69ur, pam666, nightdreamer, and daymare, saying the same thing repeatedly on about a million different forums. He's either Colin Leslie Dean himself or he's Dean's press agent.

Interestingly I did a journal article search for Colin Leslie Dean in JSTOR, which is one of the largest academic journal search engines, and did not find a single publication.

And the irony is that such a huge contribution to mathematics would certainly be heralded in academic circles, not on net forums for laypeople.

Finally, mathematics is applied. I don't think mathematicians will wake up feeling any differently irrespective of what people think about Godel's theorems. Math will still send rockets to the space station and engineer new cars.

Arjen

Reply
Mon 12 May, 2008 01:07 pm

@Aedes,

I was not aware of these facts, thank you Aedes. Although not being published does not mean not brilliant or not right and I was interested in the discussion, but the information you provided does shine a totally different light on the matter.
Aedes

Reply
Mon 12 May, 2008 01:14 pm

@Arjen,

Arjen wrote:

Of course, and it might be in press or have been presented at meetings. Still, considering that Godel singlehandedly ended the effort to unify logic with math (and threw Bertrand Russell into a depression, and caused Russell to completely change his career), I'd think there would be a great deal of interest in this subject, which makes one wonder why Dean's only audience seems to be internet forums.Although not being published does not mean not brilliant or not right.

And also I'd think that overturning Godel would

Aedes

Reply
Mon 12 May, 2008 01:31 pm

@Arjen,

I've read some synopses of the language and logic philosophy of the late 19th and early 20th century, and one of the major movements was to simplify logic to remove lots of the vagaries of language from it. The major players in this were Frege, Russell, and Wittgenstein. The output of this was symbolic logic.Bertrand Russell took it upon himself to assert that all of mathematics could be expressed through logic, and he set out to prove this (though he didn't exactly make it to complex, higher order math in his venture). But Godel, by showing that math was

And Russell responded quite despairingly, writing at one point that if

Russell never worked on this project again (and he lived and worked until he was around 98 years old) and basically gave up logic thanks to Godel.

Now keep in mind I haven't studied this academically, I've only read it in synopses, so I'm sure a well-trained logician would have a more complete view.

pam69ur

Reply
Mon 12 May, 2008 04:37 pm

@Arjen,

Quote:

Of course, and it might be in press or have been presented at meetings. Still, considering that Godel singlehandedly ended the effort to unify logic with math (and threw Bertrand Russell into a depression, and caused Russell to completely change his career)

The Australian philosopher colin leslie dean shows that

Godel did not destroy the Hilbert Frege Russell programme to create a

unitary deductive system in which all mathematical truths can can be

deduced from a handful of axioms

http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf

for two reasons

1) as pointed out in this thread godel does not tell us what makes a statement true -thus his incompleteness theorem is meaningless

the Hilbert idea was that

Quote:

Quote:In addition, from at least the time of Hilbert's program at the turn of the twentieth century to the proof of and the development of the Church-Turing thesis in the early part of that century,true statements in mathematics were generally assumed to be those statements which are provable in a formal axiomatic system.

The works of , Alan Turing, and others shook this assumption,with the development of statements that are true but cannot be proven within the system

Quote:

, perhaps the single most celebrated result in mathematical logic, states that:

For any consistent formal, recursively enumerable theory that proves basic arithmeticaltruths, an arithmetical statement that istrue,but not provable in the theory, can be constructed.1 That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.

but as pointed out in this thread godel does not tell us what makes a statement true -thus his incompleteness theorem is meaningless

2)

Godel is said to have shattered this programme in his paper called "On

formally undecidable propositions of Principia Mathematica and related

systems". But this paper it turns out had nothing to do with Principia Mathematica and related systems" , but instead with a completly artificial system called P Godel uses axioms which where not in his version of PM Thus his proof/theorem cannot apply to PM thus he cannot have destroyed the Hilbert Frege Russell programme and also his system P is artificial and

applies to no system anyways

Colin leslie dean shows that Godel constructs an artificial system P made

up of Peano axioms and axioms including the axiom of reducibility-

which is not in the edition of PM he says he is is using. This system

is invalid as it uses the invalid axiom of reducibility. Godels theorem has

no value out side of his system P and system P is invalid as it uses the

invalid axiom of reducibilitygodel uses the axiom of reducibility

he tell us he is going to use it

Quote:

NOTE HE SAYS HE IS USEING 2ND ED PM -where the axiom of reducibility was repudiated given up and droppedand he uses it in his axiom 1v and formular 40 "A. Whitehead and B. Russell, Principia Mathematica, 2nd edition,

Cambridge 1925. In particular, we also reckon among the axioms of PM the

axiom of infinity (in the form: there exist denumerably many

individuals), and the axioms of reducibility and of choice (for all types)"

Quote:

" Godel uses the axiom of reducibility axiom 1V of his system is the

axiom of reducibility "As Godel says "this axiom represents the axiom

of reducibility (comprehension axiom of set theory)" (K Godel , On

formally undecidable propositions of principia mathematica and related

systems in The undecidable , M, Davis, Raven Press, 1965,p.12-13)

. Godel uses axiom 1V the axiom of reducibility in his formula 40

where

he states "x is a formula arising from the axiom schema 1V.1 ((K

Godel , On formally undecidable propositions of principia mathematica and

related systems in The undecidable , M, Davis, Raven Press, 1965,p.21

" [40. R-Ax(x) ≡ (∃u,v,y,n)[u, v, y, n <= x & n Var v & (n+1)

Var u

& u Fr y & Form(y) & x = u ∃x {v Gen [[R(u)*E(R(v))] Aeq y]}]

x is a formula derived from the axiom-schema IV, 1 by substitution

http://www.math.ucla.edu/~asl/bsl/1302/1302-001.ps.

Quote:

(BUT "The system P of footnote 48a is Godel's

streamlined version of Russell's theory of types built on the

natural numbers as individuals, the system used in [1931]. The last

sentence ofthe footnote

allstomindtheotherreferencetosettheoryinthatpaper;

KurtGodel[1931,p. 178] wrote of his comprehension axiom IV,

foreshadowing his approach to set theory, "This axiom plays the role of

[Russell's] axiom of reducibility (the comprehension axiom of set

theory)."

IT MUST BE NOTED THAT GODEL IS USING 2ND ED PM BUT RUSSELL TOOK THE AXIOM OF REDUCIBILITY OUT OF THAT EDITION - which Godel must have known.

The Cambridge History of Philosophy, 1870-1945- page 154

http://books.google.com/books?id=I0...WOzml_RmOLy_JS0

Quote

Quote:

quote page 14 http://www.helsinki.fi/filosofia/gts/ramsay.pdf.

Quote:

Google Book Search Russell gave up the Axiom of Reducibility in the second edition of

Principia (1925)"

Phenomenology and Logic: The Boston College Lectures on Mathematical

Logic

and Existentialism (Collected Works of Bernard Lonergan) page 43

Quote:

Godels paper is called "in the second edition Whitehead and Russell took the step of using

the simplified theory of types DROPPING THE AXIOM OF REDUCIBILITY and not worrying to much about the semantical difficulties"

ON FORMALLY UNDECIDABLE PROPOSITIONS

OF PRINCIPIA MATHEMATICA AND RELATED

SYSTEMS

but he uses an axiom that was not in PRINCIPIA MATHEMATICA thus his

proof/theorem has nothing to do with PRINCIPIA MATHEMATICA AND RELATED SYSTEMS at all

Godels proof is about his artificial system P -which is invalid as it

uses the ad hoc invalid axiom of reducibility system P is the system from which he derives his incompleteness theorem quote from the van Heijenoort translation

Quote:

Godel tells us Quote:

and "P is essentially the system which one obtains by building the logic of PM around Peanos axioms..."

system P contain the axiom of reducibility

Godel uses the axiom of reducibility axiom 1V of his system is the

axiom of reducibility "As Godel says

Quote:

http://www.math.ucla.edu/~asl/bsl/1302/1302-001.ps. "this axiom represents the axiom of

reducibility (comprehension axiom of set theory)"

Quote:

EVERY ONE KNEW THAT AR WAS NOT IN 2ND ED PM EVEN "The system P of footnote 48a is Godel's

streamlined version of Russell's theory of types built on the natural

numbers as individuals, the system used in [1931]. The last sentence

ofthe

footnote allstomindtheotherreferencetosettheoryinthatpaper;

KurtGodel[1931,p. 178] wrote of his comprehension axiom IV, foreshadowing his approach to set theory, "This axiom plays the role of [Russell's] axiom of reducibility (the comprehension axiom of set theory)."

GODEL BUT NO ONE SAID

ANYTHING

Aedes

Reply
Mon 12 May, 2008 07:55 pm

@pam69ur,

Arjen,Here are some journal references. I have all these essays in PDF form. The first one is a great short synopsis that appearad in the journal

1.

Keith Devlin

2.

John W. Dawson, Jr.

3.

Hugh Lacey; Geoffrey Joseph

4.

David Wayne Thomas

5.

John W. Dawson, Jr.; Cheryl A. Dawson

And then here are some good links. The Stanford site is far better than Wikipedia for philosophy resources on the web.

Kurt Gödel (Stanford Encyclopedia of Philosophy)

Bertrand Russell (Stanford Encyclopedia of Philosophy)

pam69ur wrote:

Well, Russell certainly thought he had.The Australian philosopher colin leslie dean shows that Godel did not destroy the Hilbert Frege Russell programme to create a

unitary deductive system in which all mathematical truths can can be

deduced from a handful of axioms

Nonetheless, I'm sure you'll look forward to some critical response to Colin Leslie Dean's essay if it gains readership in academic circles. I wouldn't be so quick to assume that Dean's own essay is flawless, considering Godel's theorem has been the subject of inordinate study for a good 75 years. I find it more likely that Dean's essay has a hole, at least until it gets its own share of critical review.

pam69ur

Reply
Mon 12 May, 2008 08:23 pm

@Aedes,

Quote:

Well, Russell certainly thought he had

1)russell knew godel used his invalid axiom AR ramsey knew it

they all knew it and said nothing

for 76 years every one has been saying godel showed there are true statements which cant be proven- ie destroying hilbert/russell programme

but

no one has bothered to ask [untill colin leslie dcean] "what did godel means by true statement"

and it turns out he had no bloody idea what makes a statement true

thus he cannot have destroyed the hilbert/russell programme

as

his theorem is meaningless

Aedes

Reply
Mon 12 May, 2008 08:33 pm

@pam69ur,

pam69ur wrote:

That's utter nonsense (as Russell's own correspondance would show you -- his letters are published in the academic literature). Russell was perhaps the most arrogant, obstinate, self-promoting philosopher in the entire 20th century. There is absolutely no way he would be silent about it, because he wasn't silent about anything else. Under no circumstances would Russell quietly lie that Godel had crushed and invalidated his life work (up to that point). And yet that is 1)russell knew godel used his invalid axiom AR ramsey knew it

they all knew it and said nothing

Quote:

I don't think that matters to either his hypothesis or his proof, because that was NOT the purpose of his theorem. His theorem was that axioms cannot be proved within their own closed logical system. So thanks for nothing, Colin Leslie, the only thing that's meaningless is this particular critique.for 76 years every one has been saying godel showed there are true statements which cant be proven- ie destroying hilbert/russell programme but no one has bothered to ask [untill colin leslie dcean] "what did godel means by true statement"

Quote:

yes, we know, as you've said (in identical language) in about 26 of your 39 posts here.and it turns out he had no bloody idea what makes a statement true thus he cannot have destroyed the hilbert/russell programme

as his theorem is meaningless

And you're also COMPLETELY missing the point. The point is not whether Godel's incompleteness theorem was flawed, flawless, whatever. The REAL question is whether there can be such a thing as a true, mathematically sound incompleteness theorem irrespective of whether Godel was the author or not. Colin Leslie Dean, who is not so secretly all over the net loudly promoting his cute little attack on Godel (rather than a meaningful attack on incompleteness), has not actually addressed the real question. If he's the philosopher he wants everyone to think he is, he can stop obsessing over Godel and actually take on

pam69ur

Reply
Mon 12 May, 2008 09:16 pm

@Aedes,

Quote:

just read the proof dated 1932That's utter nonsense (as Russell's own correspondance would show you

it states it uses AR either russell did not read the bloody thing

or he kept quite

just as ramsey either did not read the bloody thing

or kept quite

if what you say is true how do you explain russell keeping quite when godels tells the reader he is using russels rejected 2nd PM with the abandoned invalid axiom of reducibility -which russel read and kept quite or he did not read it

Quote:

"A. Whitehead and B. Russell,Principia Mathematica, 2nd edition,Cambridge 1925. In particular, we also reckon among the axioms of PM the axiom of infinity (in the form: there exist denumerably many individuals), andthe axioms of reducibility and of choice (for all types)" ((K Godel , On formally undecidable propositions of principia mathematica and related systems inThe undecidable, M, Davis, Raven Press, 1965, p.5). NOTE HE SAYS HE IS USING 2ND ED PM- WHICH RUSSELL ABANDONED REJECTED GAVE UP DROPPED THE AXIOM OF REDUCIBILITY.

you harp on about godel

but blindly ignore what colin leslie dean has pointed out

1) he used an invalid axiom ie axiom of reducibility

Quote:

Quote:

"IV. Every formula derived from the schema

1. (∃u)(v ∀ (u(v) ≡ a))

on substituting for v or u any variables of types n or n + 1 respectively, and for a a formula which does not contain u free.This axiom represents the axiom of reducibility(the axiom of comprehension of set theory)" (K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965,p.12-13)

Quote:

2)constructs impredicative statementsQuote:

Ramsey says

Such an axiom has no place in mathematics, and anything which cannot be

proved without using it cannot be regarded as proved at all.

This axiom there is no reason to suppose true; and if it were true, this

would be a happy accident and not a logical necessity, for it is not a

tautology. (THE FOUNDATIONS OF MATHEMATICS* (1925) by F. P. RAMSEY

Quote:

3)uses peano which is impredicativeQuote:

Ponicare Russell and philosophers argue these types of definitions are invalid Ponicare Russell point out that they lead to contradictions in mathematics

Quote from Godel

" The solution suggested by Whitehead and Russell, that a proposition cannot say something about itself , is to drastic... ((K Godel , On undecidable propositions of formal mathematical systems in The undecidable , M, Davis, Raven Press, 1965, p.63 of this work Dvis notes, "it covers ground quite similar to that covered in Godels orgiinal 1931 paper on undecidability," p.39.)

What Godel understood by "propositions which make statements about

themselves"

is the sense Russell defined them to be

'Whatever involves all of a collection must not be one of the collection.'

Put otherwise, if to define a collection of objects one must use the total

collection itself, then the definition is meaningless. This explanation

given by Russell in 1905 was accepted by Poincare' in 1906, who coined the

term impredicative definition, (Kline's "Mathematics: The Loss of

Certainty"

Note Ponicare called these self referencing statements impredicative

definitions

texts books on logic tell us self referencing ,statements (petitio

principii) are invalid

Quote:

axiom 5 peanoaxiom 3 from godels system P

Quote:

3. x2(0).x1 ∀ (x2(x1) ⊃ x2(fx1)) ⊃ x1 ∀ (x2(x1))

The principle of mathematical induction: If something is true for x=0, and if you can show that whenever it is true for y it is also true for y+1, then it is true for all whole numbers x.

Quote:

4)he cant tell us what makes statements trueQuote:

This is the principle of complete induction, it establishes the property

of induction as necessary to the system. Since Peano's axiom is as

infinite as the natural numbers, it is difficult to prove that the

property of P does belong to any x and also x+1. What one can do is say

that, if after some number n of trails that show a property P conserved in

x and x+1, then we may infer that it will still hold to be true after n+1

trails.But this is itself induction. And hence the argument is a vicious

circle.

Quote:

but Peter smith the cambridge expert on godels saystruths, an arithmetical statement that istrue, but not provable in the theory, can be constructed.1 That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.

Quote:

5)he uses a discarded rejected version of PMQuote:

Gödel didn't rely on the notion

of truth

Quote:

thus on a number of points godels theorem is invalid meaningless and rubbish
Quote:

IT MUST BE NOTED THAT GODEL IS USING 2ND ED PM BUT RUSSELL ABANDONED REJECTED GAVE UP DROPPED THE AXIOM OF REDUCIBILITY IN THAT EDITION - which Godel must have known. Godel used a text in PM that based on Russells revised version of PM in 2nd ed PM Russell had rejected abandoned dropped as stated in the introduction. Godel used a text with the axiom of reducibility in it but Russell had abandoned rejected dropped this axiom as stated in the introduction. Godel used a rejected text as it used the rejected axiom of reducibility.

The Cambridge History of Philosophy, 1870-1945- page 154

http://books.google.com/books?id=I09...Ozml_RmOLy_JS0

Aedes

Reply
Mon 12 May, 2008 09:23 pm

@pam69ur,

Colin -- may I call you that? Here is a letter that Bertrand Russell wrote in 1963. He was personally rocked by Godel's work and it altered his entire career.Quote:

pam69ur

Reply
Mon 12 May, 2008 09:47 pm

@Aedes,

Quote:

Colin -- may I call you that? Here is a letter that Bertrand Russell wrote in 1963. He was personally rocked by Godel's work and it altered his entire career

i dont doubt that

but

**Philosophy Forum**- »
**Logic** - »
**Mathematicians are in deep trouble for 2 reasons**

Copyright © 2024 MadLab, LLC :: Terms of Service :: Privacy Policy :: Page generated in 0.03 seconds on 06/19/2024 at 04:17:08