Is formal truth in mathematical logic a generalization of everyday, intuitive truth?

I like to think that mathematical truth is a mathematical model of "real world truth", similar in my mind to the way in which the mathematical real number line $\mathbb{R}$ is a mathematical model of a "real world line", and similarly for many other mathematical models.

In order to achieve the level of rigor needed to do mathematics, sometimes the description of the mathematical model has formal details that perhaps do not reflect anything in particular that one sees in the real world. Oh well! That's just the way things go.

So yes, the empty function is injective. It's a formal consequence of how we axiomatize mathematical truth.

And, by the way, yes, there are infinitely many primes. The classical proof by contradiction that you feel is a natural language proof and not really a "formal proof" is actually not very hard to formalize at all. Part of the training of a mathematician is (a) to use our natural intuition, experience, or whatever, in order to come up with natural language proofs and then (b) to turn those natural language proofs into formal proofs.

Your main issue here seems to be that you are wondering how all the following statements:

If the Earth is flat, then the Earth exists.

If the Earth is flat, then the Earth does not exist.

If there is life on Europa, then the Earth exists.

could possibly be meaningfully assigned the same truth value in the real world. These are called vacuous truths, the first two because the falsity of the condition means that the consequent is irrelevant, and the third because the truth of the consequent means that the condition is irrelevant. One could interpret logic as a game of some sort, where the prover tries to convince the refuter of his claim. If the prover makes a claim of the form:

If A then B.

then the refuter must try to refute it. How? She must convince the prover that A is true but yet B is false. Back to our vacuous examples, the refuter must convince the prover that the Earth is flat. Nah... That's not going to happen, which is why the refuter can't refute the prover. In the third case, the refuter must convince the prover that the Earth doesn't exist. Again, no way...

On the other hand, the prover can prove the first two claims by showing that the Earth is not flat! (Come, follow me around the globe in eighty hours.) After doing this he can convince the refuter that he can always keep his promise because it can't be broken; the Earth is not flat, so the condition of his promise will never come to pass. The consequent part of his promise is irrelevant. In the third case, the condition part is immaterial because the prover can convince the refuter that no matter whether she can show that there is life on Europa, he can convince her that the Earth exists.

This is exactly the same as when you talk about an empty function being injective:

Any function with empty domain is injective.

which expands to: $\def\none{\varnothing}$

Given any function $f$ such that $Dom(f) = \none$, and any $a,b \in Dom(f)$, if $f(a) = f(b)$ then $a = b$.

Well, what does the prover have to do to convince the refuter? He says, give me any function $f$ such that $Dom(f) = \none$, and give me any $a,b \in Dom(f)$! The refuter simply can't! There isn't any object in $Dom(f)$!

But wait, you say, how about the also true statement:

Any function with singleton domain is injective.

which expands to:

Given any function $f$ such that $Dom(f) = \{x\}$ for some $x$, and any $a,b \in Dom(f)$, if $f(a) = f(b)$ then $a = b$.

This time the refuter can continue the game. She gives the prover a function $f$ and provides an $x$ such that $Dom(f) = \{x\}$, and also gives him $a,b \in Dom(f)$. But then the prover now tells her: See? You assured me that every object in $Dom(f)$ is equal to $x$, so you've to accept that $a = x$ and $b = x$, and hence by meaning of equality $a = b$. Now I can convince you that if $f(a) = f(b)$ then $a = b$. (This is exactly the third kind of vacuous statement that we discussed at the beginning.) Indeed, haven't I already convinced you that $a = b$, so you don't need to even bother to show me that $f(a) = f(b)$?

Disclaimer: I am a formalist.

In my opinion, the right lesson is to question the everyday, intuitive notion of truth — i.e. I boldly assert that the everyday, intuitive notion of truth is also just a means of assigning a value to informal statements.

We like to think there is some deeper meaning to it, and there might even be such, but in practice, a "truth" is simply a label we put to statements that we arrive at in some sort of 'acceptable' manner, such as

• the result of a sufficiently plausible argument with accepted hypotheses
• the result of a scientific study
• the result of our brain processing external stimuli

or even things like

• deciding we are unwilling/unable to contemplate the negation
• wishful thinking

(of course, the latter two are rarely done consciously in those terms).

So the everyday, intuitive notion of truth really is similar to the formal mathematical notion of a truth valuation: it's just a means of assigning values to certain informal statements. And just as how mathematical truth valuations must respect (formal) logical deduction, we like the everyday, intuitive notion of truth to respect our various informal means of gaining knowledge.

From a philosophical point of view this is really a big question to ask! Actually one would need a good notion of all day truth as well as of mathematical truth, to answer this question. What truth in an all-day sense could possibly mean is discussed virtually all the time throughout the history of philosophy, and I will skip this here.

The question what truth in mathematics could mean was one of the main subjects in the Foundational crisis in the early 20th century and is widly discussed ever since. The second (and highly related) question was (and is): what are the objects of mathematics? Back then there were mainly three points of view: The platonist, who thinks there are some kind of blueprints of mathematical objects and all one does, is discovering some truth (in a rather all day sense of truth) about their properties. The formalist, who does not think, mathematical objects exist at all, they are rather implicitly defined by some axioms and then truth is nothing else then consistency with these axioms. And last the intuitionist, who thinks mathematical objects are creatures of human mind. The starting point for the intuitionist is the ability to count, which amounts into the natural numbers. Everey other object and every true statement then has to be constructed out of these first objects and some basic logic, wich does not include the law of excluded middle, i.e. no proof by contradiction. Astonishingly enough it is possible to reconstruct a quite rich part of mathematics even in a pure intuinitionist flavour.

Throughout the 20th century there were a lot of mathematical and philosohical writings around these question (see Hilbert, Brower, Whitehead, Putnam, Günther, Turing, Gödel and many others). Luckily this philosophical uncertainity didn't keep mathematicians from doing mathematics. What I think is really interesting is the capability of switching between the different states: Attacking a problem requires often (at least in my field of interest which is differntial geometry) first to get a good intuition about the objects (this is the platonist part), some strong constructive work inside the mind (this is the intuitionist part) and finally putting everything down into a rigorous proof (that is the formalist part). And most often this is not a linear process, rather a continous going forth and back...

As Reuben Hersh put it: "“The working mathematician is a Platonist on weekdays, a formalist on weekends. On weekdays, when doing mathematics, he’s a Platonist, convinced he’s dealing with an objective reality whose properties he’s trying to determine. On weekends, if challenged to give a philosophical account of this reality, it’s easiest to pretend he doesn’t believe in it. He plays formalist, and pretends mathematics is a meaningless game.... Does it matter? Yes. Truth and meaning aren’t recondite technical terms. They concern anyone who use or teaches mathematics."

If by chance you understand german: There is a really nice book of a sociologist, who did some research on how the mathematical community works and what are the main characteristics of mathematics compared to other scientific fields. It gives a good bases on some philosophical background and quiet interesting insights into the mechanics of mathematical research. Bettina Heintz: Die Innenwelt der Mathematik.

Another good question enters the game: What is the connection between mathematics and "reality"? More precicly: How does mathematics and it's applications to science, social science, engineering and so forth interact? If mathematics was just a technical game, or some construct in human mind, how is it possible, that it has such a rich field of applications? On the other hand: Do all this applications imply, that mathematical objects really live in the physical world, and if so, what does this imply for beeing able to make true statement? See for example papers of Putnam, which include deep thoughts in this direction.

Not much of an answer, more of a loos collection of some thought. Hope it gives some input though!

Lee Mosher has already given an excellent general answer, and user21820 has given a nice explanation of vacuous truth. I'd like to expand on their answers by describing how I think about empty functions, which I find just as concrete and intuitive as other functions.

Like many people, I like to think of a set as a type of stuff.^ For example, there's

• a set $\mathbf{Fruit}$ which every fruit qualifies as an element of,
• a set $\mathbf{Papayas}$ which every papaya qualifies as an element of,
• a set $\mathbf{Mammals}$ which every mammal qualifies as an element of,
• a set $\mathbf{OwlPellets}$ which every owl pellet qualifies as an element of,
• a set $\{\text{Vectornaut}\}$ which only I qualify as an element of, and
• a set $\varnothing$ so exclusive that nothing qualifies as an element of it.

Observe that $\mathbf{Papayas} \subset \mathbf{Fruit}$, because every papaya is a fruit. Similarly, $\{\text{Vectornaut}\} \subset \mathbf{Mammals}$, because I, the only thing that qualifies as a $\{\text{Vectornaut}\}$, happen to be a mammal.

A function $A \to B$ is like a vending machine: if you drop an $A$ into the slot, the machine will spit out a $B$ for you to enjoy. If you drop in something that's not an $A$, the machine will reject it, because it only accepts $A$s. If you drop a banana into a $\mathbf{Fruit} \to \mathbf{Papayas}$ vending machine, for example, the machine will spit out a papaya. (This is a useful vending machine if you like papayas more than any other fruit.) If you drop a mouse into a $\mathbf{Fruit} \to \mathbf{Papayas}$ vending machine, the mouse will just roll out the coin return, because it's not a fruit. If you drop a mouse into a $\mathbf{Mammals} \to \mathbf{OwlPellets}$ vending machine, on the other hand, the machine will rattle around for a bit and then spit out an owl pellet. (I think I can guess what's hiding in there.)

A $\varnothing \to \mathbf{Fruit}$ vending machine is so selective that it won't accept anything as payment. You can dump in dolphins or owl pellets or gold dubloons, but it all just rolls out the coin return, and you never get any fruit. This is a stupid kind of vending machine, but it's useful to be able to talk about, because vending machines like this really exist—in everyday language, we call them “out of order.”

It's impossible for a $\mathbf{Fruit} \to \varnothing$ vending machine to exist. The reason is that some things really do qualify as fruit—I have one right here on my kitchen counter—and if you dropped one of those things into a $\mathbf{Fruit} \to \varnothing$ vending machine, the machine would have to spit out something that qualifies as a $\varnothing$. Since nothing qualifies as a $\varnothing$, the machine can't function as adversised.

On the other hand, it is possible for a $\varnothing \to \varnothing$ vending machine to exist. Although there's nothing the machine could give you in return for a payment, there's also nothing it will accept as a payment, so it will never fail to work as advertised.

^ The rules of set theory are designed to capture this intuitive picture.