Truth : what is it exactly?

[BrilloPad]

Truth : what is it exactly?

[Mich the Tester]

It´s a bit like gravity. If you doubt gravity, throw yourself off a bridge. You´ll discover it´s true.

[snaw]

"To crush your enemies, have them driven before you and hear the lamentation of their women."

conan

[Menelaus]

There are two main approaches to truth in mathematics. They are the model theory of truth and the proof theory of truth.

Historically, with the nineteenth century development of Boolean algebra mathematical models of logic began to treat "truth", also represented as "T" or "1", as an arbitrary constant. "Falsity" is also an arbitrary constant, which can be represented as "F" or "0". In propositional logic, these symbols can be manipulated according to a set of axioms and rules of inference, often given in the form of truth tables.

In addition, from at least the time of Hilbert's program at the turn of the twentieth century to the proof of Gödel's theorem 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 Kurt Gödel, Alan Turing, and others shook this assumption, with the development of statements that are true but cannot be proven within the system.

Two examples of the latter can be found in Hilbert's problems. Work on Hilbert's 10th problem led in the late twentieth century to the construction of specific Diophantine equations for which it is undecidable whether they have a solution, or even if they do, whether they have a finite or infinite number of solutions. More fundamentally, Hilbert's first problem was on the continuum hypothesis. Gödel and Paul Cohen showed that this hypothesis cannot be proved or disproved using the standard axioms of set theory and a finite number of proof steps. In the view of some, then, it is equally reasonable to take either the continuum hypothesis or its negation as a new axiom.

[Mich the Tester]

There are two main approaches to truth in mathematics. They are the model theory of truth and the proof theory of truth.

Historically, with the nineteenth century development of Boolean algebra mathematical models of logic began to treat "truth", also represented as "T" or "1", as an arbitrary constant. "Falsity" is also an arbitrary constant, which can be represented as "F" or "0". In propositional logic, these symbols can be manipulated according to a set of axioms and rules of inference, often given in the form of truth tables.

In addition, from at least the time of Hilbert's program at the turn of the twentieth century to the proof of Gödel's theorem 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 Kurt Gödel, Alan Turing, and others shook this assumption, with the development of statements that are true but cannot be proven within the system.

Two examples of the latter can be found in Hilbert's problems. Work on Hilbert's 10th problem led in the late twentieth century to the construction of specific Diophantine equations for which it is undecidable whether they have a solution, or even if they do, whether they have a finite or infinite number of solutions. More fundamentally, Hilbert's first problem was on the continuum hypothesis. Gödel and Paul Cohen showed that this hypothesis cannot be proved or disproved using the standard axioms of set theory and a finite number of proof steps. In the view of some, then, it is equally reasonable to take either the continuum hypothesis or its negation as a new axiom.

Jolly good stuff

Trouble is that in many discussions of cause and effect in a social setting, such as the causes of crime among young people or the causes of economic crises, people often seem to take a binary stance where it just can´t be sufficient to state one cause and one effect.

The right will say that crime is caused by grotty little bastards who should go and get jobs and call the lefties whingers for saying it´s caused by poverty; both factors and many others may be contributing in various degrees. Both are also making the fault of seeing a simple cause and effect scenario where there are actually multiple feedback loops, making cause and effect rather difficult to distinguish. For example, high crime rates in a particular area can lead to poverty and/or the other way round.

So perhaps the challenge now is not to define truth, but find a system for determining completeness of truth.

[TimberWolf]

Truth is what you define it to be.

[Mich the Tester]

Truth is what you define it to be.

No that's belief.

[Peoplesoft bloke]

To answer this question you must put the problem (youth crime, whatever) to CyberTory. Whatever he says, the opposite will be the truth HTH

[expat]

There are two main approaches to truth in mathematics. They are the model theory of truth and the proof theory of truth.

Historically, with the nineteenth century development of Boolean algebra mathematical models of logic began to treat "truth", also represented as "T" or "1", as an arbitrary constant. "Falsity" is also an arbitrary constant, which can be represented as "F" or "0". In propositional logic, these symbols can be manipulated according to a set of axioms and rules of inference, often given in the form of truth tables.

In addition, from at least the time of Hilbert's program at the turn of the twentieth century to the proof of Gödel's theorem 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.
It is important to note that this idea of "truth" does not carry any moral weight: it is not asserted that a provably true statement is any better, only that the rules of inference allow the inference that it has the same truth value as the axioms.

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.

Two examples of the latter can be found in Hilbert's problems. Work on Hilbert's 10th problem led in the late twentieth century to the construction of specific Diophantine equations for which it is undecidable whether they have a solution, or even if they do, whether they have a finite or infinite number of solutions. But those are insoluble equations, not undecidable-but-true statements. More fundamentally, Hilbert's first problem was on the continuum hypothesis. Gödel and Paul Cohen showed that this hypothesis cannot be proved or disproved using the standard axioms of set theory and a finite number of proof steps. In the view of some, then, it is equally reasonable to take either the continuum hypothesis or its negation as a new axiom. In which case it is not what you promised, an example of "statements that are true but cannot be proven within the system". It is neither true nor false within the existing system; and in the new systems with added axiom, either it or its negation is true and provable.

But a fine contribution

[TimberWolf]

No that's belief.

It is belief too.