6

I apologize if a similar question has been asked here, but I haven't found it.

Are mathematical statements necessary truths?
By 'mathematical statements', I mean both mathematical axioms as well as statements such as 2+2=4, but I'm not referring to statements such as 2+2 alone.

  • 1
    The philosophical debate about the nature of mathematical objects [i.e.there "exists" - in some sense of the term - abstract objects like numbers and sets ?] and the status of mathematical truth [are the truth of mathematics independent from experience, i.e. a priori and "absolute", i.e. necessary, i.e. impossible to deny ?] is one of the oldest in the history (at least) of western philosophy : it dates back to Plato and Aristotle. See at least Philosophy of Mathematics and related entries. – Mauro ALLEGRANZA Jul 2 '14 at 7:23
  • @mauroallegranza I thought so, which is why I was surprised to see that it isn't on this site yet. That should mean that I can get a good answer, right? – This lad Jul 2 '14 at 11:35
  • @mauroallegranza despite the usefulness of SEP, I thought it's still worth having an answer tailored to this site's format – This lad Jul 2 '14 at 11:46
  • 1
    You cannot get a "good answer" form me, sure ... I'm not saying that someone cannot. Personally, I think that the "issue" is still open, due to the fact that some of the best mind of the last 2,5 millenia have not yet definitively answered this basic problem. This does not mean that we cannot discuss it in order to deepen our understanding. MY humble opinion is that, regarding this topic, if you ask for a half-page answer, you will never get something worth to be read... :) – Mauro ALLEGRANZA Jul 2 '14 at 13:17
3

I am sorry, I can't comment yet.

In order to get a useful answer, you will need to define what a "truth" is. I assume, "mathematical statement" means "statement, which can be proved by some given axioms".

Anyway, if you prove by Peano's axioms that 2+2=4, this does not imply that if you take two apples and add two apples, you end up with 4 apples. To see this, just note that if you prove by the axiom "2=3" that 2=3, this does not imply that two apples is the same thing as three apples.

  • 1
    We usually assume that we have an "intuitive" understanding about the truth of arithmetical stetments. We assume that there is some sort of "abstract structure" made of numbers or we assume that "external reality" possess some properties whch can be described by "number statements". Thus, we choose arithmetical axioms in order to be true of it (the abstract structure or the said aspect of reality) and we deduce theorems about numbers with sound rules of logic (i.e. rules which do not prove falsity from previous true principles). Having said that, we are able ... 1/2 – Mauro ALLEGRANZA Jul 2 '14 at 7:15
  • 2
    "to prove by Peano's axioms that 2+2=4", and a lot of school-children through centuries has been able to check that "if we take two apples and add two apples, we end up with 4 apples". This is the "precise" meaning of "truth" for arithmetical statement. – Mauro ALLEGRANZA Jul 2 '14 at 7:17
  • Well, maybe this works for number theoretical statements. But the question speaks of mathematical statements. For example the continuum hypothesis is a mathematical statement. Here, every "intuitive understanding" fails. Is the continuum hypthesis "a truth"? – Jakob Werner Jul 2 '14 at 10:47
  • According to your statement : "which can be proved by some given axioms", it is not: we are not (yet ?) able to prove it. Maybe one day we will prove (or disprove) it. The "universe" is full of thing we do not yet know ... This does not mean (for me) that we cannot meaningfully speak of truth of a statement. 2+2=4 is true and we know that if "add apples" we do not contradict arithmetical laws. – Mauro ALLEGRANZA Jul 2 '14 at 11:13
  • No, as Gödel and Cohen showed, it is impossible to either prove or disprove CH in ZFC. So if we work in ZFC+CH, there will be nothing "intuitive" about CH, although it is provable in ZFC+CH and although ZFC+CH is consistent (if we assume that ZFC is). – Jakob Werner Jul 2 '14 at 12:08
3

Truth is a property of propositions. In the simple case of propositonal logic this means that any proposition is either true or false.

Either it rains or it does not rain, at a fixed date and time at a fixed point of Manhattan.

The most widespread definition of a proposition being true is due to Tarski: A proposition is true if and only if the actual situation, which the propositon is about, is a matter of fact. The proposition

It rains in Manhattan at a fixed date and time and at a fixed point

is true if and only if it actually rains. Otherwise the proposition is false. Hence Tarski's truth definition compares a proposition with a real situation.

Because mathematics does not deal with reality - it is not a natural science like physics - one cannot apply Tarski's definition to mathematical statements. Instead, one has two related, but different concepts:

A mathematical statement of an axiomatized theory is provable iff and only it derives by a logical conclusion, i.e. a syllogism, from the axioms of the theory.

A mathematical statement is true if and only if it holds in any model of the axioms.

Any provable statement is true. The converse holds in simple theories like propositional calculus. But it does not hold in more general mathematical theories.

  • Taking your question literally, the answer is NO: There exist false mathematical statements by trivial reasons, e.g. There is a maximal prime number is a false mathematical statement.

  • There exist mathematical statements, the truth of which is undecidable, e.g. the continuum hypothesis is undecidable within ZF set theory.

  • There exist true mathematical statements. They are true in all possible worlds where our logic is valid, which means necessarily true.

0

The only even number that is prime is the number two, and all other prime numbers are odd.

It is true because an evil demon, as Descartes proposed, could not make me go wrong in believing this incorrect.

It's, dare I say, categorically true.

0

I'll keep it short as it is rather difficult to answer the question in all his aspects. Take your example: 2+2=4. Why and when should you say that 2+2=4 is true? Using axioms of the natural numbers you has to end with the number four, hence being 'necessarily true' means here that you have applied the axioma's in a correct way to get the a priori known answer. And if 2+2=4 stands for this whole process of counting then one can say that 2+2=4 is true. I don't. I have given you 2 apples and shortly after that again 2 appels, hence I have given you four apples. So it is necessarily true that I have given you four apples? What is meant by saying this? Suppose you only know the axioms and only the numbers 0 and 1. How would you get 2+2=4?

0

Thanks to Godel, you're shown you will never have ultimate answers to the questions you can ask about your system.

Mathematical systems have some axiomatic characteristics (yep, Real space is a vectorial space with 10 basic axioms and 3 order axioms) you cannot prove because of two things:

  • Godel's crap.
  • Math systems are defined fitting human needs. Real vectorial space was defined to fit the natural world. Binary base (1+1=10 is true) to fit the computational world, and modular algebra (2+2=0 mod 4) to fit stuff like cryptography. So you cannot prove, in the same system, what actually is your starting point for the system: you take them as facts as the base building blocks on your system (this is specially true when you make a Prolog script).

So, in summary, you take them as necessarily true because, in some sense, you want them. Sounds like crap, but at least in the math/IT world it is like that.

Have a look at this topic and you will see that algebraic systems can be defined arbitrarily (according to some rules you err... accept). Even the game of rock-paper-scissors can be defined like this:

  • Define three input symbols {R, P, S} and three output symbols {W1 is wins 1, W2 is wins 2, T is tie}.
  • Define a binary (two operands are involved), non-closed (closed operations take only elements in the same set and yield elements in such set), non-commutative (you can invert the binary operands and have the same result, always), non-associative (you can have a & b & c as (a & b) & c or even a & (b & c) but this is a nonsense for non-closed operations), non-transitive (transitivity only is meaningful if the result is a truth value)... blah blah blah... operation called PLAY : IxI -> O (represented by & in this case) as a function: (R, P) -> W2, (R, R) -> T, (R, S) -> W1, ...

In this case, (R, S) -> W1 is necessarily true.

Disclaimer: I understand that I did not answer about Peano, addition, and ordering. My answer goes even beyond that: you can define any system you need, and it will somehow fit part of the reality you want to measure or understand, and so the core facts will be necessarily true in order to you be able to work with that system.

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.