Mathematics is nothing but an ad hoc discipline

Philosophy Forums Science, Technology, and Math
1

The australian philosopher colin leslie dean points out that mathematics is not a rigourous discipline but is no more than an ad hoc sham

3 examples

Colin leslie dean-the first person in 76 years - points out that Godel in his incompleteness theorem used the ad hoc axiom ie axiom of reducibility

the standford encyclopdeia of philosophy says of AR

plato.stanford.edu/entries/princ … thematica/

“many critics concluded that the axiom of reducibility was simply too ad hoc to be justified philosophically”

From Kurt Godels collected works vol 3 p.119

books.google.com/books?id=gDzbuU … #PPA119,M1

“the axiom of reducibility is generally regarded as the grossest philosophical expediency “

in Burali-fortis day there was a set of all ordinals( as they came up with the Burali-fortis paradox) -did Cantor accept such a set

this set has been outlawed in set theory -because it sends it into self contradiction- by the introduction of an ad hoc axiom called the Axiom schema of specification

3)In Russells day there was a set of all sets
which destroyed naive set theory-sent it into contradiction-so to avoid it set theory just introduced an axiom

Axiom schema of specification

modern set theory just outlaws this paradox by the introduction of the ad hoc axiom the Axiom schema of specification

which wiki says
"The restriction to z is necessary to avoid Russell’s paradox and its variants. "

so we have two sets - which at one time did exist-which send maths into contradiction just being dissalowed due to an ad hoc axiom

all this arbitarly defining away problems go right back to the Greek who defined irrational numbers as not being numbers as they destroyed their maths

all in all Mathematics is nothing but a ad hoc discipline and a sham–EVEN THOUGH IT WORKS- it is philosophically absurd

2

No I don’t. And I’m a garbage man, not a philosopher.

3

Well, if Standford encyclopedia already noticed the ad hoc justifications, and the authors of the book brought it up, and if even Wiki acknowledged it, then Mr. Dean can’t really claim that he’s the first person who pointed it out, nor can he claim that mathematicians are being hypocritical.
It’s possible that we might have missed something as we went along and simply chose to “sweep it under the carpet” for the time being (hence, ad hoc justifications), but apparently it was not big enough of a deal that it caused everything to come tumbling down. I really think you are just being nitpicky here, ladyjane.
Mathematics is a man-created language, just as English and the grammatical structure that it rests upon. If you don’t believe in accuracy of language and the logical principles within it (and based on your writing style I assume you don’t) then I don’t see how you can even claim that mathematics is “philosophically absurd”.

4

as i have said before every one knew and knows AR is invalid
but
only colin leslie dean was the forst to point out -first in 76 years -that godel used it in his proof and as such his proof is invalid-no one has said that or pointed it out before

5

Hi to Ladyjane,

I am curious why, if you are frustrated with ZF and ZF with C, you don’t simply use the standard definition of a set? Namely, a set is a collection of well defined objects.

Further, just because there are problems with formal set theory, why would you say that all mathematics is flawed? For example, why can not you simply use arithmetic as you were taught, along with the Peano axioms?

Finally you seem to constantly complain that the Godel’s proof of the Godel theorems are flawed. But you never seem to mention the alternative Turing machine proofs of these theorems. Do you not realize that there is an alternative proof?

6

Hi to Pandora,

You wrote, “It’s possible that we might have missed something…”

I am curious if you have a mathematical background? I always like to meet people with a similar background.

Thanks Ed

7

you say

Peano axioms are not valid either as they are based on impredicative definitions

you say

the point is about WHAT GODEL DID and not what others have done
and what godel did is invalid as he uses invalid axioms

gamahucherpress.yellowgum.com/bo … GODEL5.pdf

8

No

9

Hi to Ladyjane,

From Wiki we have:

“Peano’s fifth axiom states:

  • Allow that; zero has a property P;
  • And; if every natural number less than a number x has the property P then x also has the property P.
  • Therefore; every natural number has the property P.

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.“

The bold copy and the blue copy are my modifications.

Peano’s axioms in their original order (though the axioms are written in their modern form, exclusive of their later appeasement to the algebraists) are as follows:

1 1 is in N
2 if a is in N then a = a
3 if a and b are in N, then a = b if and only if b = a
4 if a and b and c are in N, then if (a = b and b = c), then a = c
5 if a = b and b is in N then a is in N
6 if a is in N then a + 1 is in N
7 if (a and b are in N) and (a + 1 = b + 1), then a = b
8 if a is in N, then a + 1 is not equal to 1
9 for any set K, If (1 and a are in both N and K), and (a+1 is in K), then N is contained in K

This is from “The Search for Mathematical Roots 1870 - 1940” by I. Grattan Guiness. With a bibliography of 75 pages, this is the best historical research I have ever seen.

Axiom #9 can be shown to imply:

  • Allow that; 1 has a property P;
  • And; if every natural number less than a number x has the property P then x also has the property P.
  • Therefore; every natural number has the property P.

Any axiom of a system can not be proven by that system. This is why it is “difficult to prove”. If it were possible to prove an axiom, then it would fail to be an axiom; and it would become a theorem. The copy in blue is simply a psychological rationalization of why this axiom should be valid and why that rationalization fails. However, again, you can not prove an axiom.

My personal experience is that attempted proofs or justifications of any axiom almost always end up being circular just as the above example.

In conclusion I think that the whole paragraph beginning with “This is the principle of complete induction“ is complete jibberish.

I realize that Russell and Poincare are two of the greatest minds of all time, but I will stand by my statement even though a reasonable person might think that the chance that I am right and they are wrong would be nill.

10

Hi to Ladyjane,

From Wiki we have:

“Peano’s fifth axiom states:

  • Allow that; zero has a property P;
  • And; if every natural number less than a number x has the property P then x also has the property P.
  • Therefore; every natural number has the property P.

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.“

The bold copy and the blue copy are my modifications.

Peano’s axioms in their original order (though the axioms are written in their modern form, exclusive of their later appeasement to the algebraists) are as follows:

1 1 is in N
2 if a is in N then a = a
3 if a and b are in N, then a = b if and only if b = a
4 if a and b and c are in N, then if (a = b and b = c), then a = c
5 if a = b and b is in N then a is in N
6 if a is in N then a + 1 is in N
7 if (a and b are in N) and (a + 1 = b + 1), then a = b
8 if a is in N, then a + 1 is not equal to 1
9 for any set K, If (1 and a are in both N and K), and (a+1 is in K), then N is contained in K

This is from “The Search for Mathematical Roots 1870 - 1940” by I. Grattan Guiness. With a bibliography of 75 pages, this is the best historical research I have ever seen.

Axiom #9 can be shown to imply:

  • Allow that; 1 has a property P;
  • And; if every natural number less than a number x has the property P then x also has the property P.
  • Therefore; every natural number has the property P.

Any axiom of a system can not be proven by that system. This is why it is “difficult to prove”. If it were possible to prove an axiom, then it would fail to be an axiom; and it would become a theorem. The copy in blue is simply a psychological rationalization of why this axiom should be valid and why that rationalization fails. However, again, you can not prove an axiom.

My personal experience is that attempted proofs or justifications of any axiom almost always end up being circular just as the above example.

In conclusion I think that the whole paragraph beginning with “This is the principle of complete induction“ is complete jibberish.

I realize that Russell and Poincare are two of the greatest minds of all time, but I will stand by my statement even though a reasonable person might think that the chance that I am right and they are wrong would be nill.

11

nothing wrong with questioning authorities-there should be more of it

all i am doing is pointing out that from accepted authorites peano axioms are impredicative and thus circular

thus philosophically invalid

as ramsey etal have said the axiom of reducibility that godels uses in his proof is also invalid

you see maths is rubbish-even though it works
for some of the reasons listed in the post

12

Hi Ladyjane,

I have some additional thoughts on the apparent Russell and Poincare objections to the Peano axioms.

From:

en.wikipedia.org/wiki/Impredicativity
“In mathematics, impredicativity is the property of a self-referencing definition. More precisely, a definition is said to be impredicative if it depends on a set of things, at least one of which is the thing it defines.
Russell’s paradox is a famous example of an impredicative construction: the set of all sets which do not contain themselves. The paradox is whether such a set contains itself or not — if it does then by definition it should not, and if it does not then by definition it should.
The rejection of impredicatively specified objects (but the acceptance of the natural numbers as classically understood) leads to the position in the philosophy of mathematics known as predicativism, taken by Henri Poincaré and (in Das Kontinuum) Hermann Weyl. Poincare and Weyl argued that impredicative definitions are only problematic when they depend on an infinite set.
As Frank P. Ramsey showed, “impredicative” definition over finite sets is absolutely necessary. For instance, the definition of “Tallest person in the room” is impredicative, since it depends on a set of things of which it is an element, namely the set of all persons in the room. Concerning mathematics, an example of an impredicative definition is the smallest number in a set, which is formally defined as: y = min(X) if and only if for all elements x of X, y is less than or equal to x, and y is in X.
Classically, the greatest lower bound of a set is considered a generalization of this concept; it is defined as y = glb(X) if and only if for all elements x of X, y is less than or equal to x, and any z less than or equal to all elements of X is less than or equal to y. But this definition is also impredicative, and quantifies over the infinite set of lower bounds of X, which contains the glb itself. So it is rejected by predicativism.”

Clearly, Poincare had either changed his mind about the Natural numbers or was at worst ambivalent on the matter.
Next I would like to attack, what I believe to be Russell’s augment, based on its’ canonical nature.

Consider the 6th Peano Axiom.

If x is in N the x + 1 is in N. This is a statement about a property of N. Note that this is an inductive statement.

Now we can reapply the original Wiki statement.

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.“

This statement is just as applicable to the 6th axiom as the 9th axiom.

Now I would ask you, who believes that there exists a Natural number N such that N + 1 is not also a Natural number?

If you disallow the Peano axioms, then axioms 6 through 9 will all fall to this type of reasoning.

To avoid this type of problem all the axioms from 6 through 9 must be axioms.

Excepting axiom 1, the other axioms are required by logic and axiom 1 is required in order for the inductive statements to have any grounding.

Fundamentally, the nature of Arithmetic is inductive, and any assaults solely based on the inductive nature of the Peano axioms must be rejected for Arithmetic to stand.

If axiom 9 is true then the argument should be restated in the green copy.

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 (which is what axiom 9 allows). And hence the argument is a vicious circle ( actually a simple tautology)."

I do not know if this is convincing to anyone other than myself, but I am curious to know.

13

I’m confused. What’s original here? Math is a priori - it’s a form of language.

Arguments are invalid. Propositions are false. Such words don’t make sense when applied to systems. You can’t have an invalid axiom - against what should you compare it? It isn’t a proposition about reality - this is obvious. Consequently, of course it’s ad hoc. That’s even more trivial than its conclusions.

A formal system can be useless; axioms can produce absurdities. These are the only things you can judge a formal system on.

You can’t knock down math, ladyjane. There’s nothing there - the emperor has no clothes, and everyone knows it.

14

fyi don’t engage in communication of any foprm with ladyjane… we have tried before and she has been found to have the I.Q of a rabbit… good luck sir.

15

Hi Denali,

I liked your post on 1 = .999999.

Perhaps you could expand your thoughts here as they might pertain to the parallel postulate? It seems to me that questioning axioms has proven to be a good thing.

Hope you enjoy your time here.

Thanks Ed

Hi Wonderer,

I am a contrarian by nature and so I will take this opportunity to defend Ladyjane.

Let us assume, for the sake of argument (I strongly dislike ad hominem), that Ladyjane characteristically advances rationally weak arguments. This does not mean that the content of some or many of his OP’s are not profoundly thought provoking.

Not only that but the posts are not completely raw speculation, he has provided reference material citing, in some cases, well respected sources.

Anyway, assuming that he does not, frequently, simply recycle his posts, I hope to see more posts from him.

Thanks Ed

16

being quite the contrarian myself i also feel inclined to defend my own position…

assuming that each of LadyJanes posts are original, and assuming that each post is not a recycled citation of the same refrence, and assuming that the OP is not completly constituted of refrence material…

i still avoid communication with lady jane because her actual posts are largely unreadable which shows either a grose misunderstanding of the english language, or that there is an extreme lack of respect for the reader…

i have seen Lady Jane make sensible posts, but others have meen monstrocities…

i do cede the fact that it was wrong to publicly degrade Lady Jane… i guess you’re not much of a conclift theorist (though a contrarian :-k )

but what i do maintain is my right to decide if Lady Jane is worth the time…

17

Ed3 say

sorry just because you need maths to stand is no argument to stop that maths falling apart is meaningless
as is seen with the problem of peano axioms

time and time again we here that maths is consistent
so what happens when paradoxes are found
ie russsell
branch-tarski
buril-forti

mathematicians just change the axioms

18

justbecause something changes is no reason to assert that it is, was, or will be, meaningless

19

the changeing of the axioms
shows maths is ad hoc
and
that mathematician are just cheats-as they only keep math from being inconsistent and meaningless by cheating- ie ad hoc adding or subtracting axioms