# Truth for logicians, mathematicians, and philosophers

How does the logician define truth?

What is the precise definition of truth for mathematicians?

How does a philosopher define truth?

What are the similarities and differences between these concepts?

There is an old saying: "Truth is eternal"; would Mathematicians agree with that saying?

• there is some analysis of this in the book Godels proof by Nagel & Neumann – vzn Mar 12 '14 at 3:28

Truth is indefinable & ineffable. Nobody bothers to define it for that reason. Those that do are aiming at limited objectives, or are engaging in chasing circular definitions which has its own amusement.

After Godel published his famous incompleteness theorems in formal logic, Tarski, a logician, showed that truth was undefinable using the same formal apparatus.

Euclid, a mathematician, based geometry on the first fully and aesthetically pleasing axiomatic system. Axioms are traditionally held to be self-evident truths. But sometimes the truths are so self-evident, or that the purpose of an axiomatic system is so inevident that no-one bothers to form axioms for them - for example arithmetic wasn't put into this form until the 20C by Peano.

Wittgenstein might say that we're using the word 'truth' in many different and significant ways, and each one should be labelled ie truth-1, truth-2 etc. As truth-for-me is different from truth-for-you in subtle ways for the same concept, one should label these truths by individuals which multiplies them endlessly. Truth as language in this sense ramifies spectacularly. Despite this, one might suppose that he did think truth was one but chose to remain silent about it.

Mathematicians learn about mathematical truth by learning their tradition, and it being verified by their peers - in this perspective truth appears to be socially conditioned.

There is an old saying: "Truth is eternal"; would Mathematicians agree with that saying?

Plato would have said yes - as a form of the Good. Some modern day philosophers of Number suppose it all might be some kind of fiction, but not a hoax. Others suppose both.

This is my answer according to my fields of knowledge and personal beliefs. I hope it appears objective enough.

How does the logician define truth?

A logician may define the truth as a fact. In logical propositions we have constructs as predicates which are evaluated against certain conditions (or circumstances) and produce a result which is either true or not true. So, we have a clear separation of both states (truth/untruth). Logical constructs such as deduction or induction represent operations that determine if certain circumstances are true, based on an existing result, or deriving the circumstances themselves knowing that a result is true or not. Logicians make use of logical operations (such as conjunction or logical AND, disjunction or logical OR), which are rules for evaluating and combining predicates in order to produce a statement that is true or untrue.

What is the precise definition of truth for mathematicians?

According to mathematics, truth can be defined as the reliability of core mathematical constructs. These would be:

• an axiom - that is an obvious fact which does not need a proof for accepting it to be true. Thus an axiom is always reliable.
• a theorem which is a set of rules that are seen to apply in certain circumstances. In order for a theorem to be scientifically valid, it has to be proven. Usually a theorem is proven by applying other proven theorems and axioms in such a way that either directly lead to the conclusion for the theorem to be true, or show that it is impossible for the theorem to not bet true. The latter is achieved by assuming that the theorem is not true and proving that this assumption leads to a paradox.

Considering the above, a truth for a mathematician would be something that can be verified as being correct via axioms and theorems.

In computer science, which can be viewed (in regard to your question) as a subset of the discrete mathematics (especially the binary algebra), a truth is defined by two values `0` and `1`, respectively `false` and `true`. These are the basis of various mathematical and computational models which can be evaluated or used by the computer to make decisions. Most of the constructs and operations used by logicians are re-used by this area of mathematics as well as computer science. In this context, truth would probably be an expression that evaluates to `true`, which is a quite technical definition.

How does a philosopher define truth?

And finally there are the philosophers. Unlike mathematicians or logicians, the philosophers seem to not evaluate the truth by a fixed set of commonly valid formulas. Instead, a truth is something more subjective and personal. It may conform to specific rules of the society, mathematics, physics, logic and even religion, but still it is the individual's interpretation of the facts. A truth for one may not be a truth for someone else, still both could be right as this is their own subjective belief. It could be possible for a particular person to change his beliefs, so something that s/he considered once true is now not true. One's personal experience may influence the evaluation of his/her philosophical truth.
In addition, the philosophical truth is the result of the individual's attempt to explain (for himself) paradoxical or unsolvable (for him) matters. For instance, a religious person may explain many of the phenomenon s/he may witness as caused by the divine intervention of God. One with sufficient knowledge may not, as s/he could explain the same with scientific knowledge. So, both will have their truth that helps set their minds at peace and not wondering how some event or observation was possible. In regard to psychology, this is a convenient way of the consciousness to adapt to reality and not to waste an individual's energy for thinking about non-essential matters and keep attention focused on important ones (like once survival was for the early human).

An aside from the above, the truth can be itself an interesting topic when regarded as a philosophical term. It is often viewed as eternal, single for all of us and fundamental for our existence (spiritual and conscious), but is yet too obscure and a subject to the individuals to reveal it according to thier understandings.

There is an old saying: "Truth is eternal"; would Mathematicians agree with that saying?

In regard to your comment (I am not a mathematician, but I feel quite close to the subject), I would agree that the mathematical truth is eternal, as it would be valid as long the appropriate circumstances are present. So if you take a proven theorem, it would always evaluate true whenever its conditions are satisfied. In this case I am seeing the truth not as an always valid fact, or the fact itself, but rather as the reliability of the conclusion - the conclusion will always be valid for valid circumstances and always not for invalid ones, so it is forever trustworthy in layman's terms.

What are the similarities and differences between these concepts?

In conclusion, I'd say that logicians and mathematicians define truth in a similar fashion, as both rely on a set of rules that they use against given circumstances in order to determine the truth. It could be valid to say that they attempt to measure it.
The philosophers, on the other hand, would attempt to categorize truth according to commonness of individual beliefs, rather than directly measure it. They are also more liberal in terms of that an individual is allowed to have his own and different truth, so the real truth remains abstract and obscure.

In a formal system of logic, "truth" is a technical term given an absolute definition based on the structures of the system. Generally, true statements are those which are fully consistent with the axioms of the system and the givens of the argument.

In as much as one speaks of "truth" in mathematics, it is much the same as in the world of logic. Mathematical truths are statements within the world of mathematics that are fully consistent with the previously determined mathematical truths, all the way back to the basic mathematical axioms.

Among philosophers, however, "truth" has many different definitions, depending on whom you ask. For many philosophers, truth expresses a correspondence with the factual details of the world, but for someone like Plato, truth relates not to the everyday world as we experience it, but to a deeper level of reality beneath the one we know.

• Consistency of a statement with the rest of mathematical theory is insufficient to establish the statement's truth. This is very important because for any rich enough mathematical theory T there are mathematical statements S such that both S and "not S" are consistent with T. – Michael Dec 2 '13 at 19:17
• This answer avoids the question that is being asked. – Kevin Holmes Dec 2 '13 at 20:40
• Michael, would you mind to explain or relate your comment. Is it based in Godel's incompletion theorem or else other? – Jose Torres Torres Dec 3 '13 at 13:06