Registered: Nov 2007
Posts: 20

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;

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Registered: Dec 2003
Posts: 179

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.

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Registered: Nov 2007
Posts: 20

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.

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Registered: Dec 2003
Posts: 179

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.

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Information Science, Neuroscience, Quantum Mechanics, and Leibniz
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Registered: Nov 2007
Posts: 20

There are no absolute truths.

Last edited by damsell on 03-30-2008 at 04:46 AM

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Registered: May 2004
Posts: 27

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

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Registered: May 2004
Posts: 27

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

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Registered: Sep 2007
Posts: 20

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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Registered: Nov 2007
Posts: 5

The fundamental flaw that Godel chose was that recursion was based on natural numbers. What he failed to consider was recursion based on real numbers or complex numbers. Recursion is based on computing step x by computing step x - n where n is an integer. What if n were much closer to 0 (less than one). This leads to much more interesting ideas, such as recursive functions over reals and differential recursion. If we can apply this to set theory, then we will overthrow Godel. For example, if you have a set A at time t1 (measured), it is not the same as the set A at time t2 (measured). Even on a computer, the electrons are different. A - A is not the null set.

I think the key is to focus on concrete sets, and stay away from abstract sets. That will help clear your thinking.

What if n were imaginary?

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Registered: Sep 2011
Posts: 4

Hi everyone,

As a mathematical logician, I can respond by describing the current thinking of the math logic community on this. Mike is right in saying that truth and proof are relative to the axioms you choose. Sometimes these axioms are thought of as just a variable to play around with, the input to the function F[axioms] = "truths" (statements that follow from these axioms). [By the way, the function F here is also variable in formal logic, but almost always it is merely repeated applications of MP]. More philosophically inclined theoreticians think about their axioms in terms of their intuitive credibility, etc. In this reckoning, the axioms of (Peano) Arithmetic are considered by >90% of us to be credible. Many less believe in the axiom of infinity ("there are infinite sets"), as well as in AC: "given any collection of sets we can choose one from each" (as in "let f be a function from the real line to the real line")

The relationships we study among axioms are actually more interesting than just implication. Because of Godel's incompleteness theorem, we know that some statements are neither implied nor refuted by a given axiom system. In this case can still ask: Will assuming the statement is true lead to a stronger theory?. By this we mean, assuming our original system is free of contradiction, can we show that adding the statement as a new axiom does not create a contradiction. Such "consistency" questions are a main concern of modern mathematical logic.

about the last post: I do not understand what a recursion on reals would be. If we start with zero, and do our recursion, then after the first "recurrence" we are at the "next" number, right? what is the "next" real number? perhaps you are talking about an infintesimal amount. In that case you might be talking about ODEs.

Remember that the real numbers may be a complete fiction!

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Registered: Sep 2011
Posts: 4

There is a stir in the math logic community, and even a rare bit of press (http://www.newscientist.com/article...and-beyond.html)
for this most ivory-tower-ish of disciplines. It is due to a claim of Hugh Woodin, among others, that they may have found a formal theory which unifies all of set theory into a single framework, called "Ultimate L", for answering whether mathematical statements should be considered true or false.

Prof. Woodin really believes in the infinite. He has devoted a career to studying it, and is one of the most respected living mathematical logicians. Recently, at the Harvard seminar "Exploring the Frontiers of Incompleteness", Woodin described his recent thinking on mathematical truth. While making a point about how large infinite sets ("large cardinals") can have meaningful implications, he gave an example which I think might be of interest to this forum:

There is a Godel-like sentence that a has real-life consequence if true, yet our 'evidence' for it, while convincing, depends on the existence of things too large to possibly be meaningful for us. His point was that the phenomena of "large cardinal hypotheses" exists on the finite level as well.

The setting is a special version of set theory, ZFC_0, that has as an axiom "there is a set which contains all sets". These axioms are satisfied by all of the standard finite universes for set theory, called V_n, for any whole number n. The sentence X he contructs "says" (using the magic of Godel numbering) that "I (this sentence) am unprovable by any proof from ZFC_0 less than 10^24 in length AND the set V_N exists", where N is some very big finite whole number. Crucially, Woodin chooses N to be so big actually that there is no reason at all to believe that our universe contains that many things, and certainly so big that we could never hope to represent such an object in a digital non-quantum computer. Say N = 10^10^124.
By Godel's logic, which can be summarized as "Suppose this sentence is false. Then by what is says, it's provable, and therefore true", X is true, and we cannot find a proof of length 10^24 or less.

The amazing conclusion: that we are persuaded by Woodin's proof that our real-world computers will never verify such a proof of X (with current technology, we could in principle check if a given "proof" of length less than 10^24 were valid), yet how can we believe this and not believe that V_N also has a physical meaning, despite the lack of evidence?

Last edited by Peter Barendse on 09-26-2011 at 05:00 PM

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