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I'm trying to wrap my head around the relationship between truth in formal logic, as the value a formal expression can take on, as opposed to commonplace notions of truth.

Personal background: When I was taking a class in formal logic last semester, I found that the most efficient way to do my homework was to forget my typical notions of truth and falsehood, and simply treat truth and falsehood as formal, abstract values that formal expressions may be assigned. In other words, I treated everything formally, which from the name of the course is presumably what I was supposed to do while solving problems.

On the other hand, I came out of the course more confused than enlightened about what truth refers to in mathematics. For example, on what levels are each of the following two statements "true"?

  1. There are infinitely many prime numbers.
  2. The empty function $f\colon \emptyset \to \mathbb R $ is injective.

For the first one, I can obviously see that there would be a contradiction if there were only finitely many prime numbers. To me, the classic proof by contradiction is not a "formal proof" or anything; it is merely a natural language argument that proves (in the everyday sense of the word "proves") why the statement must be true (in the everyday sense of the word "true").

On the other hand, I run into trouble when I try to think about the second one. The very concept of the "empty function" doesn't even feel like it makes sense, but if I think about it as the relation between $\emptyset$ and $\mathbb R$ containing no elements, and then try to write out the statement formally, I get (if I did it correctly)

$ \forall x \forall y ( ((x\in \emptyset) \land (y\in \emptyset) \land (f (x) = f (y))) \implies (x=y)) $

which I think has to be true in a formal sense (since the antecedent is always false?). But to be honest, I don't really know how to think about "truth" here; the situation feels much more confusing than with the first statement.

So, in conclusion, my questions are:

In what sense is each of the above statements "true"? And, more generally,

Is the notion of truth in (mathematical) logic just a formal value assigned to expressions? Or should I think of it as encompassing, but also generalizing, the intuitive notion of a true statement?

(Any insightful comments or answers are appreciated, even if they don't address all of my questions directly.)

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    $\begingroup$ They are true because they are theorems of "established" mathematical theories. This means that they are proved (more or less "formally" does not matter) from axiosm of the respective theories. Of course, they are true if the axioms are. Mathematicians usually believe that the axioms of their theories are true, unless some contradiction has been discovered. $\endgroup$ – Mauro ALLEGRANZA May 12 '16 at 7:47
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    $\begingroup$ I will bet somebody is going to comment that the "classic" (as in classical Greek culture) proof of the infinitude of prime numbers is (was) not by contradiction. I would like to add that it was not a proof of any infinitude either, because the existence of infinite collections was considered absurd at the time. Rather the proof was that any [sic] collection of prime numbers can be extended by new ones (and it was a constructive proof). $\endgroup$ – Marc van Leeuwen May 12 '16 at 8:22
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    $\begingroup$ Don't confuse your confusion about truth in mathematics with your confusion about the empty function and its injectivity! These are really quite different matters. I can give an intuitive, natural language proof for (2) just as easily as for (1): To say that a function is injective is to say that no two distinct elements of the domain get sent to the same element of the codomain by the function. Since there isn't any pair of distinct elements in the domain of the empty function, this condition clearly holds, so the empty function is injective. $\endgroup$ – Alex Kruckman May 12 '16 at 8:36
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    $\begingroup$ @justin And by the way, I emphatically disagree with you on the cell phone example. Vacuous truths do (often) make intuitive sense in non-mathematical contexts. Let's say you're on an airplane and the pilot says "For safety reasons, before we take off, all cell phones on board must be turned off". Just because it's 1995 and no one on board happens to have a cell phone doesn't mean it's any less safe to take off... the condition "all cell phones on board are turned off" for a safe take off has been satisfied - it's true! $\endgroup$ – Alex Kruckman May 12 '16 at 18:31
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    $\begingroup$ @AlexKruckman: I think that, in everyday English, "all cell phones on board must be turned off" presupposes that there are cell-phones on board. The allowed-to-be-vacuously-true version is "any cell phones on board must be turned off". $\endgroup$ – ruakh May 12 '16 at 22:54
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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.

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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 $f(a) = f(b)$?

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  • $\begingroup$ It's quiet interesting for me, that in the comments on Hurkyls answer you are taking a rather formalist approach. While here you argument with game semantics, a product of pure intuitionists. What is interesting is, that I don't feeld that this is a contradiction in this special case, while otherwise I presume formalism and intuitionism quiet far from each other. Maybe it is the favour of rigour that is common in their attitude. $\endgroup$ – hase_olaf May 13 '16 at 11:19
  • $\begingroup$ @hase_olaf: What I describe here is the part that is missing in Hurkyl's answer, namely the intuitive justification for the rules of the formal system in use. Once the justification is done, we can then proceed entirely formally. That said, students always understand quantifiers better when explained this way, and most mathematicians too never engage in purely formal proofs, because often it is easier to grasp an argument that isn't presented in purely symbolic form. $\endgroup$ – user21820 May 13 '16 at 11:25
  • $\begingroup$ I agree, especially with the second part. Actually I don't think ther is any proof (except of some in mathematical logic), that is written down in a purly formalistic style. So for being really exact, there is no such thing as formailsm in mathematical praxis. But somehow we know (and we are even able to achieve interpersonal konsensus about), what non-formal argument could be translated into formal ones. Do you know, if game semantics (in one or another flavour) is equivalent with classical (= aristotelic) logic? Especially: Is the law of the excluded middle part of game semantics? $\endgroup$ – hase_olaf May 13 '16 at 11:53
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    $\begingroup$ @hase_olaf: That issue didn't appear in my answer (I think), but one can choose whether or not to include LEM (the law of excluded middle) in this game semantics. First note that the BHK (Brouwer-Heyting-Kolmogorov) interpretation of proofs as programs works perfectly for intuitionistic logic, precisely because of its constructive nature. If one adds LEM then one is essentially adding some kind of oracle (it's no longer normal TMs). If you think about it that makes sense, because to get from $A \rightarrow B$ and $\neg A \rightarrow B$ to $B$ in game semantics we need to know $A \lor \neg A$. $\endgroup$ – user21820 May 13 '16 at 12:03
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    $\begingroup$ @hase_olaf: So the game semantics actually clarifies what LEM means, namely that whenever the prover uses it he is asking the refuter which case it is. Is $A$ true or false? Whatever the refuter answers, he chooses the appropriate conditional branch to go down, which both intuitionistic and classical logic agree on. Intuitionistic logic thus corresponds to the refuter saying "I don't know!?" and the game gets stuck! $\endgroup$ – user21820 May 13 '16 at 12:06
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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.

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    $\begingroup$ Eh.. Your first sentence does not fairly describe you.. Most people who call themselves formalists think that mathematics is purely a game like chess where the rules are arbitrary and don't have any meaning to the real world whatsoever! You, on the other hand, are essentially saying that we adopt the rules of classical logic precisely to model what we believe our intuitive notion of real world truth satisfies, which I fully agree with. This viewpoint is quite platonic. $\endgroup$ – user21820 May 12 '16 at 11:15
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    $\begingroup$ @user21820: Maybe I misunderstand your point, but I would say the opposite -- I try to adapt my notion of 'real world truth' to reflect lessons learned from formal logic. I try to move away from believing my knowledge reflects truths about the real world, and more towards acknowledging that the knowledge came from playing games. $\endgroup$ – Hurkyl May 12 '16 at 12:50
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    $\begingroup$ @user21820: I think you interpret me at the wrong level. I am not adapting what sorts of things I believe to be real world truths to make the math 'real'. Instead, I am moving in the direction of rejecting the very notion of real world truth in favor of putting the emphasis on 'knowledge-producing games'. $\endgroup$ – Hurkyl May 12 '16 at 13:30
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    $\begingroup$ @user21820: Regarding mathematical reasoning, my take is that you're simply not allowed to say "... for real world reasons we have X ..." or anything similar. If you want to use X in a mathematical argument, your choices are to produce a valid formal argument concluding X or to posit X as an axiom/hypothesis. I'm not so bold as to disallow people from having outside reasons to take X as an axiom, but I insist such reasoning is not mathematical. $\endgroup$ – Hurkyl May 12 '16 at 13:36
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    $\begingroup$ Of course, if you're an intuitionist, deciding you're unwilling or unable to contemplate the negation is no reason to believe something is true... $\endgroup$ – Vectornaut May 12 '16 at 17:24
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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!

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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.

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    $\begingroup$ Haha! Out-of-order vending machines. Though in real life there are different instantiations of the same kind of vending machines with same input type and output behaviour, so technically the mathematical function is an abstraction of the quotient type. =) $\endgroup$ – user21820 May 13 '16 at 3:29
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The answer is both, due to the quirks of formal language. Truth is both a label assigned to expressions and also a specific concept related to your intuition. The reason for this dichotomy is most apparent in the work of Alfred Tarksi and his undefinability theorem.

Tarski's theorem tugs at the corners of the fabric of mathematics. He showed that, for any theory strong enough to prove all truths in arithmetic (i.e. 1+1=2, 3*5=15, and so on) and had negation, it could not define its own semantics. The specific statement he proved was that it could not define a function True such that True(encode(s)), where encode(s) was an encoded expression in that language gave the truth value for s. He used a really twisted little trick known as the diagonalization lemma to basically wedge the system against itself and prove that there must be a case where the True function breaks down.

In many cases, we are not considering such powerful self referential systems. Usually we're using a powerful system to prove something smaller and specific. For example, we may use first order logic and ZF set theory to prove a statement about the cardinality of large sets. In this sense, we are using the idea of "truth" as a label that can be applied within that system.

However, when we start exploring deeper "truths" which start to stretch the limit of our proof system, we start to find that the simple concept of "truth as a label" ceases to be as useful as it was in the simpler cases. We start relying more on the semantic meaning of "truth" rather than the syntactic one. Thanks to Tarski's work, we can prove that that semantic meaning must come from elsewhere. It must have a deeper meaning.

This eventually does push its way into the philosophy of mathematics, where we have to define what mathematics calls "truth." At this point, we can define "truth" any way we want, it's just a word, but there is an intuitive benefit for "truth" to have a meaning closely related to the intuitive concepts we all use.

A case study in the differences here is the difference between ZF and ZFC axioms for set theory. The axiom of choice is a remarkably intuitive axiom, but it has profoundly non-intuitive effects. In ZFC, it is possible to take a sphere, cut it into a finite number of pieces, rotate and translate those pieces, and reassemble them into two perfect spheres, each with the same volume as the original. These new spheres are not swiss cheese: they are a dense sphere, just like the original one.

In ZFC, the expressions used to describe this slice and dice operaton are deemed "true," because their syntactic description is formed from a series of valid inferences in first order logic with ZFC. However, the idea that I can double my volume of spheres without decreasing their density in any way runs so afoul of our intuitive sense of "truth" that many mathematicians refuse to accept proofs done in ZFC.

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To address your two specific questions:

  1. This is true because we can demonstrate it via a proof about natural numbers. It is rather specific, since it applied to a class of very concrete objects (one could say the most concrete objects in math: "God gave us the integers, all else is the work of man." Specifically, the natural numbers probably have the most right to being "out there" in a platonic sense - we can directly observe natural numbers in the world in terms of collections of objects. Only classical geometry comes to mind as more tied to observables.
  2. This question is much more "formal", since it is about matching an object to a definition. The object $f:\emptyset \to \mathbb{R}$, is both rigorously defined and yet vague as well (what does this function look like, how would I calculate it on a computer?). It is true in the sense that it satisfies the "check boxes" that define an injective function - $f$ need to correspond to anything observable or any of our everyday notions.

In general, formal methods, in fact, most "abstractions", are not inherently about anything. They consist in definitions and their verification. The extent to which they are about things beyond their own definitions is a philosophical/empirical matter. It is up to us to determine whether or not our notions concord with some formal system. If they do, then we can take advantage of the massive, extended analogical thinking that abstract/formal systems allow.

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    $\begingroup$ I just can't get past the claim that "there are infinitely many primes" is an empirical statement. What is the experiment that I could perform that would falsify or validate this? You say that prime numbers can be counted -- but how would one go about doing this? $\endgroup$ – Daniel Wagner May 12 '16 at 4:54
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    $\begingroup$ @DanielWagner not literally empirical...I was trying to point out a difference but lacked a good word...hence the use of quotes. The first question is about more than just the definition of a prime number, so I was looking for a way to describe this. $\endgroup$ – user237392 May 12 '16 at 5:25
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    $\begingroup$ @DanielWagner: It is empirical in the following sense: I can write a concrete computer program that given any input natural $n$ (as a string) will produce as output a natural $p > n$ such that $p$ is prime (which you can use another program to check). Why is this empirical? Because no matter what input $n$ you give this program, it will never fail to do as advertized. (Unless the computer runs out of memory or energy or something, in which case our claim is empirical because we can just continue on another better computer!) $\endgroup$ – user21820 May 12 '16 at 7:12
  • $\begingroup$ @DanielWagner: Furthermore, one concrete way to do the above is to factorize every number between $n$ and $2n$ and output the first prime. Bertrand's postulate, which can be proven in PA, tells us that there will always be at least one prime in-between, and indeed that fact is empirically verifiable because this program here always halts no matter what $n$ you give it! This program not only empirically verifies Bertrand's postulate, it gives as a consequence empirical evidence that there are infinitely many primes, since if there are finitely many then there is a largest one, an upper bound. $\endgroup$ – user21820 May 12 '16 at 7:15
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    $\begingroup$ @Bey I just wanted to mention that I personally found your answer quite insightful (so I upvoted). I feel it goes along well with Hurkyl's answer to explain that formal methods are not inherently about anything, and that "truth" in general is a philosophical concept, not a mathematical one. Thus, the question of what mathematical "truth" means belongs to the domain of philosophy. $\endgroup$ – justin May 12 '16 at 17:51
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Here is a natural language statement you can use to verify that the empty set is injective:

If more than one wireless device is transmitting on the same frequency, then they will interfere with each other.

Or, in other words, given $f:device\mapsto frequency$, if $f$ is not injective then you will get interference.

Naturally, when there are no devices transmitting on any frequency ($device = \emptyset$), none of them will be interfering with any of the others, and thus our empty function must be injective.

Going back to your example of "all cellphones in the room are turned off", this is just an example of implied context, since that statement wouldn't normally serve any purpose if there weren't any cellphones. However, if you add explicit context to it, like "all cellphones must be off or on airplane mode before takeoff", it should be clear that if there are no cell phones, the requirement is satisfied.

This is the reason that many mathematically-true things seem like they don't make sense in the real world: they are victims of implied context. In fact, a lot of what mathematics is for is getting rid of these implied contexts.

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As pointed out in some of the other answers, your question can be reformulated more concretely in terms of the natural numbers: Does $\mathbb{N}$ correspond to the ordinary (or what you call intuitive) counting numbers? The answer is that it depends on whether you accept the Intended Interpretation hypothesis, as many mathematicians do. There is a detailed discussion of this issue here.

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