“True” itself is defined as “in accordance with fact or reality” or “accurate or exact”. “Truth” is quality of being true.
The essence boils down to “fact” and “reality”. Both of these terms seem to point this discussion to a philosophical direction. What do they have anything to do with math or computer science?
Broadly speaking, there are two kinds of truth. The first kind is what I would call “reality truth”. This is the kind of truth that relates to reality. Such truths can range from scientific theories to statistical conclusions.
For example, is photon, as a subatomic particle, massless? Well, either it is, or it is not. The statement “a photon has no mass” is true or false. We discover that this statement is false because light does bend around objects with a strong gravitational pull (such as our own sun). In other words, by observing and experimenting, we can discover the truth of a statement.
However, there is another kind of truth. For example, in logic, math and computer science, it is established that “true and true is true” (as a part of the definition of conjunction). Such a statement is “by definition” true.
In summary, there are two kinds of truths: truth by discovery and truth by definition.
What is the purpose of truth by definition? If truth is not by discovery or observation, how does it make any sense?
Just to make this discussion a little more interesting, let us consider the following statement that is always true:
In English, this is literally “for all x, P(x) is true means exactly the same as it is not the case that there exists an x such that P(x) is false.”
This is pretty abstract and not observable. What is the purpose of this truth?
Truth by definition serves two important purposes. The first one we kind of know already, the second one is somewhat less expected.
Most people know truth by definition is used for reasoning. Using axioms (each one being a particular truth by definition), we can work with equations and solve for variables. That can be quite useful when you need to compute mortgage payment based on a variety of factors!
However, a even more elemental importance by truth by definition comes from the “definition” part. Truth by definition allows people to communicate as they have the same understanding of such definitions. For example, if a fast food chain says “we are giving away a free superhero action figure to one eligible customer for each week, being at least 45 years old or having natural purple skin color qualifies for this giveaway.”
To understand this sentence, one must already understand many truths by definition. The word “and” has a definition. Even the words “each”, “one”, “at least” and etc. have definitions.
This means that a framework by truth by definition that is understood by people allows the communication of concepts among these people.
First of all, computer science is not a science. Computer science is a branch of math as it has nothing to do with the discovery of natural laws, or as we have already used the term, truth by discovery. All the elemental concepts of computer science are definitions, just like math. It is best to view computer science as a practical branch of math.
That being said, math, specifically computer science, is practical and useful in real life. How can the studying of artificial truths be useful in reality?
There is no need to look very far to see how math and computer science contribute to science. There is an entire branch of physics called “computational particle physics” to utilize computational means to predict, model and discover particles.
This means that while math is about made-up truth, such made-up-truth are consistent with discovered truth. As a result, when applied correctly, made-up truth can be used to model discovered truth.