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There are no "true statements' in mathematics
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Posted by: damsell
The Australian philosopher colin leslie dean points out that mathematicians cannot tell us what a true statement is.
Truth is a tricky problem
http://en.wikipedia.org/wiki/Truth#Truth_in_mathematics
The term has no single definition about which the majority of professional philosophers and scholars agree. Various theories of truth continue to be debated. There are differing claims on such questions as what constitutes truth; how to define and identify truth;
In mathematics the old idea of "true" was that of Hilbert who believed that
a true statement was one proven from axioms
true statements in
mathematics were generally assumed to be those statements which are
provable in a formal axiomatic system.
Now since Godel made a distinction between a proven statement and a true statement mathematicians are at a loss to tell us what a true statement is
Godel noted a true statement was independent of provability by his
distinction
Mathematicians now need a definition of a true statement independent of provability but they cant give us that definition of true independent of provability
THUS THERE IS NO "TRUE" STATEMENTS IN MATHEMATICS ONLY PROVEN ONES
Posted by: MikeHelland
True is something consistent with some given arbitrary axioms.
So yes, truth is relative to arbitrary, replaceable axioms.
That's been known for some time.
It's ok.
If you have more information to contribute to the topic, just keep it in one of the existing threads you have.
There is no need to start another one.
Posted by: damsell
you say
True is something consistent with some given arbitrary axioms.
So yes, truth is relative to arbitrary, replaceable axioms.
That's been known for some time.
It's ok.
the point is Godel is said to have shown that truth is not about deriving from axioms -that was Hilberts idea
a true statement is independent of proving from axioms
that is mathematicians dillemma they want to accept Godels distinction
but
still use Hilberts idea
"true statements in mathematics were generally assumed to be those statements which are provable in a formal axiomatic system."
but
http://en.wikipedia.org/wiki/Truth#..._mathematicsThe works of Kurt Gödel, Alan Turing, and others shook this assumption,
with the development of statements that are true but cannot be proven
within the system.[29]
Thus your reply show you are at a loss to tell us what a true statement is independent of provability -WHICH IS WHAT Godel is said to have shown
Posted by: MikeHelland
Existential dread.
There are no absolute truths.
Oh no!
Deal with it.
You have been for most of your life.
Even if you didn't realize it.
Posted by: damsell
There are no absolute truths.
i know that
i am just pointing out that in mathematics there are only proven statements
but
no true statements
as mathematician cant tell us what a true statement is
as following Godel there are proven statements and true statement
but they cant tel us what a true statement in mathematics is
they can tell us they can prove
1+1=2
but
they cant tell us why it is true
for since Godel provability is not a criteria for truth
but they cant tell us what this criteria for truth is now with the collapse of the Hilbert idea
true statements in
mathematics were generally assumed to be those statements which are
provable in a formal axiomatic system.
The works of Kurt Gödel, Alan Turing, and others shook this assumption,
with the development of statements that are true but cannot be proven
within the system.[29]
Posted by: Ray Donald Pratt
I believe that "truth" is a judgment that occurs after a truth-finding comparison between an assertion and a fact or standard taken as definitive. A 'truth' derived from axioms would only be an example of that larger process of finding truth, not the only example.
When we call a statement true or false, we are saying that we have compared that statement to a given fact or principle and either found corroboration or contradiction. For example, if we say "All apples are cubes that glow in the dark," we can only judge that statement as being true or false by comparing the statement to what we know about apples.
In the Liar's Paradox, we say "This very statement is false." The alleged paradox is that if the statement is true, then it is false as it claims, but if it is false as it claims, then it has stated the truth and cannot be false, ad infinitum.
However, to correctly judge the truth or falsity of "This very statement is false," we must compare the statement not only to itself as the statement explicitly requires, but we must also compare the statement to what we know about finding the falsity of any general statement. This is implied by the use of the term "false," much like the term "apple" would require us to compare a statement to what we know about apples.
With the Liar's Paradox, the very fact of the supposed paradox proves that the statement cannot be definitively proven false. And as such, the statement is ultimately true because it admits that it falsely asserts that it is provably false.
Similarly, in the Truth Teller, we say "This very statement is true." Although there is no alleged paradox, the statement's self-reference to its own veracity is not sufficient evidence for its truth. The informal fallacy called petitio principii, or begging the question, occurs where a questioned fact is called in as proof of that fact, and such proofs are always illegitimate (which may also apply to the Liar's Paradox). However, we can go further here and say that the Truth Teller is definitely false because it falsely claimed that it was provably true.
The supposed confusion about the truth or falsity of such statements, or of truth values in logic generally, arises solely from confusing the process of determining truth values with the mere terms of 'true' or 'false' themselves, which have no power to make something true or false. The terms are merely appended descriptions of statements found to be true or false through comparisons to definitive facts or principles.
If modern logic makes more of this than it is, it has made itself impractical.
Very Respectfully,
Ray Donald Pratt
Posted by: Roger Kowalski
I am a math major and computer science major... If there
are new truths where are they anyway.... In China 50,000
new engineers yearly, In the USA 5,000 engineers....
I am looking and waiting for some new theories from these
graduates.... I am tired of reading equations developed
from the middle ages... Isn't it about time we came up with
our own by now... I would think the great mathmaticians would
think by now the 20th and 21st century math guru's would
start thinking on their own by now... Sure, we have had
Einstein.... who was great....but now....who...Greenspan????
I am a seeker of the truth.... But there is no truth....
Never has been and never will be... Please face those
facts... If your looking for the Truth I guarantee u will never
never never find it.... It was never there in the first place...
Of course I am kidding... I wanted to find a three leaf
clover many years ago... Have not found 1 yet....
The truth is there is one out there....if I look for a long long
time.... That is how long I will have to search for the truth...
a long long long time... .......
Posted by: Ray Donald Pratt
Although Mr. Kowalski decries the lack of any great truths coming to the fore despite the great masses of those being trained in the sublime arts, I stand my ground that truth for man is a process of comparing assertions with facts or principles accepted as definitive.
See immediately that those facts and principles may later be proven wrong by reference to some deeper insight taken as a new truth (the world is round, not flat). Nonetheless, the process of truth for man remains, and the study of logic aids it.
The Truth that Mr Kowalski decries the lack of is actually beyond all of us -- namely, God's Perfect and Complete Truth. As to man's truths, however, we will see no end of them.
However, as to the processes of determining truth or falsity, we can safely say that the comparative methods are the best that we can assure ourselves of being able to do, and that only divine revelation could ever do better.
Very Respectfully,
Ray Donald Pratt
Posted by: Abby Nussey
Forgive me if I'm not as eloquent or versed in logic as some of the posters here. This has been an interesting thread, in spots, and I wanted to chime in. :)
What is being referred to as an absence of truth could be an absence of knowledge of the truth, which isn't the same thing. This is referred to above as God's Absolute Truth or somesuch, though as an atheist my tendency is to believe what we don't know is simply beyond the precision of our instruments or power of our computation, and may be, in fact, chronically beyond our possession of those things.
To say "there are no true statements in mathematics" seems to be a fundamental misunderstanding of the nature of mathematics. Mathematics is a symbolic language of expression, much like our various human languages (hence the ability to create logical paradoxes with words). Mathematics can represent a Thing, but never "be" a Thing.
Hence questioning the "truth" of mathematical statements doesn't make sense to me. But perhaps I'm just naive.
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