The Instigator
shakuntala
Pro (for)
Losing
0 Points
The Contender
Enji
Con (against)
Winning
6 Points

Godel cant tells us what makes a maths statement true-thus his theorem is meaningless.

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Post Voting Period
The voting period for this debate has ended.
after 1 vote the winner is...
Enji
Voting Style: Open Point System: 7 Point
Started: 6/3/2014 Category: Science
Updated: 2 years ago Status: Post Voting Period
Viewed: 810 times Debate No: 56012
Debate Rounds (3)
Comments (5)
Votes (1)

 

shakuntala

Pro

Australia's leading erotic poet colin leslie dean points out Godel cant
tell us what makes a mathematical statement true-thus his theorem is meaningless
http://gamahucherpress.yellowgum.com...

Godel says there are true statements which can be proven

http://en.wikipedia.org...

"Any effectively generated theory capable of expressing
elementary arithmetic cannot be both consistent and complete. In
particular, for any consistent, effectively generated formal theory
that proves certain basic arithmetic *truths*, there is an
arithmetical statementthat is *true*,[1] but not provable in the
theory."

note Godel says
.".. there is an arithmetical statement that is *true.."

but Godel cant tell us what makes a maths statement true
Peter Smith the Cambridge expert on Godel says

http://groups.google.com...
.
quote

"G"del didn't rely on the notion
of truth"

thus godels incompleteness theorem is meaningless as Godel cant tell us what true statements are
thus his theorem is meaningless
Enji

Con

I've previously debated this topic with my opponent, but not very well. I'll attempt tackle this again, hopefully with more clarity. My opponent claims that "Godel cant tells us what makes a maths statement true-thus his theorem is meaningless." [sic] His argument, then, appears to be as follows: P1: A meaningful theorem must tell us what makes a maths statement true. P2: Gödel's theorem does not tell us what makes a maths statement true. C: Gödel's theorem is not meaningful.

P1 is absurd and my opponent has not presented any argument to establish its truth. Since no mathematical theorem seeks to prove what makes a mathematical statement true, and instead seek to deduce logical consequences given certain true statements, P1 is the equivalent of assuming that all of mathematics is meaningless. Admittedly if all of mathematics is meaningless then so is Gödel's theorem, but this means my opponent's argument is begging the question.

Either a statement or its negation is true


Mathematics uses logic to deductively prove mathematical statements from more basic statements which are accepted to be true. In logic, the law of excluded middle states that any statement is either true or false. This is an important aspect of Gödel's proof; if neither a statement nor its negation can be deductively proven using the axioms of a mathematical system, then that system is incomplete. Similarly, the law of non-contradiction states that no statement can be both true and false; if both a statement and its negation can be proven in a mathematical system, then that system is inconsistent. [1]

 Gödel proves that for many mathematical systems there are statements which cannot be proven true or false; hence they are incomplete

As my opponent quotes, Gödel proves that in any mathematical system which includes some number theory (which is a vast variety of mathematical systems) there are statements which cannot be proven true or false. This means that neither the statement nor its negation can be proven to be true, or that there are true statements which cannot be proven. Therefore, any such mathematical system must be incomplete. He also proved that any mathematical system which can prove all true statements must be inconsistent.

It's relevant to note that my opponent's source, Peter Smith, is correct; Gödel did not rely on the notion of truth - he relied on the notion of logical consistency and arithmetical capablilty. This means that Gödel's theorem is meaningful independent of what makes a mathematical statement true. [2]

Significance of Gödel's theorem

Gödel's theorem is considered to be one of the most important mathematical proofs of the 20th century, with relevance to both mathematical logic and the philosophy of mathematics. Gödel demonstrated that a proof establishing the consistency and completeness of arithmetic (an answer to one of David Hilbert's famous questions, and an important aspect of mathematical formalism) is impossible. His theorem shows that mathematics isn't absolutely certain and that such absolute certainty is unattainable, despite the certainty and logic of mathematics often being considered an ideal for philosophers or scientists to strive for. Further, it provides interesting insights into philosophical questions such as whether absolute truth can be known, or the plausibility of artificial intelligence and the limits of computer science. Gödel's theorem extends far beyond mathematics; its conclusion itself is meaningful, as are its implications for mathematics, philosophy, and science: the resolution is negated. [3]

 References:

[1] http://www.stanford.edu...
[2] http://www.math.hawaii.edu...
[3] http://www.miskatonic.org...

Debate Round No. 1
shakuntala

Pro

the debate is very simple
just tell us what Godel says makes a maths statement true

if you cant then the term "true " in this statement is meaningless
"...there is an arithmetical statement that is *true*..."
thus his theorem is meaningless
Enji

Con

In the previous round I argued that Gödel proves there are true statements which cannot be proven by showing that neither a statement nor its negation is true. Since one or the other must be true, he proved that all mathematics using a small bit of number theory is incomplete. I argued that this result has had important impacts on philosophy in general, the philosophy of mathematics, and mathematics itself - having been called one of the most important results proved in the 20th century. My opponent has dropped all arguments.

Definition of "True"

My opponent now argues that truth as used by Gödel is ambiguous or undefined, and so his theorem is meaningless. But Gödel doesn't use truth differently than any other philosopher or mathematician. Fundamental laws of truth state that either a statement or its negation must be true and Gödel proved that for some mathematical statements neither the statement nor its negation can be proved. Thus he proved that there are some true mathematical statements which cannot be proven to be true. An example of such a statement is the continuum hypothesis, which can neither be proven nor disproven with the axioms of set theory. [4]


I have shown that Gödel's theorem is accurate and there exists examples of its truth, and that it has meaningful applications to philosphy and mathematics; Gödel's theorem is meaningful - the reslution is negated.

 References:

[4] http://en.wikipedia.org...
Debate Round No. 2
shakuntala

Pro

I must win this debate for 1 reason
I maintain that Godels theorem is meaningless as he cant tell us what makes a maths statement true

all con has to do to refute my claim and win is tell us what Godel says makes a maths statement true
which he has not done
thus
thus my claim stands and I must win this debate

my question to con was

"just tell us what Godel says makes a maths statement true

if you cant then the term "true " in this statement is meaningless
"...there is an arithmetical statement that is *true*..."
thus his theorem is meaningless"

con has not said what Godels says makes a maths statement true
thus my claim stands and I must win this debate
Enji

Con

Gödel's theorem should not be expected to define mathematical truth because he treats it no differently than any other mathematician. The idea that there are mathematical systems which are incomplete is hardly surprising; for example, Euclidean geometry without the parallel postulate cannot prove the vast majority of interesting geometric proofs true or false. What Gödel showed was that the incompleteness of mathematics extends far further than most mathematicians had thought, encompassing all systems which contain a bit of number theory. Alone, this is an interesting conclusion with implications for mathematical logic and the philosophy of mathematics.

My opponent has forgone every opportunity to argue why a mathematical theorem must explain what makes mathematical statements true in order to be meaningful, thus he has not fulfilled his burden of proof. As I explained in the first round, his assumption that a meaningful theorem must explain what makes a mathematical statement true is equivalent to assuming that all of mathematics is meaningless, and hence his argument begs the question. Mathematics is meaningful and is arguably the foundation of the modern world, so my opponent’s argument fails.

The law of noncontradiction states that both a statement and its negation cannot be true and the law of excluded middle states that either a statement or its negation must be true. Using these features of true statements, Gödel proves that there are true statements that cannot be proven. Further, I provided an example of such a mathematical statement. In set theory, either the continuum hypothesis is true or its negation is true, but neither can be proven (and we can prove that neither can be proven) with the axioms of the theory. In some cases it can be useful to treat the continuum hypothesis as true, while in others it can be useful to treat it as false, but it cannot be proven true or false. Thus the theorem proves meaningful results despite not defining mathematical truth. My opponent has dropped this argument.

Gödel's theorem should not be expected to define mathematical truth because he treats it no differently than any other mathematician. The idea that there are mathematical systems which are incomplete is hardly surprising; for example, Euclidean geometry without the parallel postulate cannot prove the vast majority of interesting geometric proofs true or false. What Gödel showed was that the incompleteness of mathematics extends far further than most mathematicians had thought, encompassing all systems which contain a bit of number theory. Alone, this is an interesting conclusion with implications for mathematical logic and the philosophy of mathematics. But, as I have argued, Gödel's theorem also has meaningful applications to general philosophy, discussions of artificial intelligence, and one’s ability to know truth, and it is referenced by various philosophers and writers as I cited earlier. My opponent has dropped this argument.

Since my opponent has not sufficiently supported his own argument and has dropped all arguments to the contrary, my opponent has failed to uphold his burden of proof – vote Con.

Debate Round No. 3
5 comments have been posted on this debate. Showing 1 through 5 records.
Posted by Enji 2 years ago
Enji
Whoops in reformatting my response it appears I've repeated a paragraph.
Posted by Empiren 2 years ago
Empiren
This is implying that theorems have no meaning if the theorist does not explain what makes a math statement true?

Seems like an arbitrary "hoop" that has no impact on the significance of a theory.

Kinda like "well if godel can't eat 10 gallons of chocolate, then his theory is meaningless".
Posted by Ajab 2 years ago
Ajab
I would take this debate on, but I think I will be trolled. lol
You see there is a reason in the last 50 years everyone has called Godel a genius, there is mathematical proof for it. If you would like I think there is a really easy essay on it, type Godels Theorem on academia.edu.
Posted by Samreay 2 years ago
Samreay
I invite you to just read through the pdf you linked. If you think that that work, with horrific grammar, nonsensical sentences, varying text size, that has been hacked together from a score of web pages, is credible material that would stand up in a mathematical journal, you need to perhaps reevaluate how mathematics is done these days.
Posted by Samreay 2 years ago
Samreay
With all due respect, this is not debate worthy material. You should perhaps try and take a maths course so that you can actually understand the theorem and what it pertains to, rather than just assuming that one of the most famous collections of theorems in modern mathematics is actually completely incorrect.
1 votes has been placed for this debate.
Vote Placed by Ajab 2 years ago
Ajab
shakuntalaEnjiTied
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Total points awarded:06 
Reasons for voting decision: I do not know why Con spent so much time trying to debate this, S clearly does not understand Godel's Theorem, instead of debating truth Godel showed that if a system of axioms for an arithmetic construct is complete then it is not provable , if it provable then the system will remain incomplete. So you have constructs like Goldbach's Conjecture, and all. To seek whether a system is true it has to be provable, as in it should have the potency to be proved, this cannot happen for a complete system of arithmetic axioms. E showed that rather well, gave sources. Pro kind of forfeited a round without giving any argument so I gave conduct to E for that. S's spelling and grammar was not good but it did not cause a problem in understanding what he wanted to say so that is equal. I would be happy to provide more detailed reasoning if asked.