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I have read online and personally believe that every statement has some degree of ambiguity to it. With this in mind, I was wondering how any propositions can be true. For example, I have heard some people say that math is absolute truth (because math is based on a system of rules/axioms) -- e.g. 2+2=4 is absolutely truth (assuming you are using proper axioms, like the Peano axioms). How can this be the case, though, if all statements are ambiguous? Wouldn't this mean that are multiple ways to interpret the axioms, and only one of those ways can be correct? Are the axioms completely unambiguous? How could you prove that they are unambiguous?

On a related note, why, after reading the Peano axioms (for example), can all humans seem to agree on the same properties? Why can all humans agree that 2+2=4? Even if there is a level of ambiguity to the axioms, how are humans able to converge on the same meaning? Thank you!

Clarification: I think the real point I was going at was ambiguity. Yes, I understand that 2+2=4 is true, and I understand that this can be shown with pebbles. However, thinking about the English language, for example, I could tell you that something is red. The red we have in our minds, though, may be different (I might be thinking of a crimson and you may be thinking of a scarlet). However, with math, I can ask you what 2 plus 2 is and you will say 4. There is no ambiguity, at least for humans. Why is this? You may say that we have defined a set of rules for the word plus: if I have 2 of something and I plus 2 to that something, I have 4. But how have these rules been formulated in such an unambiguous manner that no one will dispute them? Why can't the English language work the same way? I am looking for why/how, fundamentally, something can be totally unambiguous.

closed as unclear what you're asking by Swami Vishwananda, Mauro ALLEGRANZA, christo183, Eliran, virmaior Dec 10 '18 at 1:37

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  • Why do we have the word 'truth'.. Truth is slippery but it exists. One person kills someone and blames the other. The other claims they're being falsely accused. Who is telling the truth? – Richard Dec 8 '18 at 22:59
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    Can you explain why you think "2+2=4" is ambiguous? – Eliran Dec 8 '18 at 23:01
  • @Eliran I think it's confusion about the fact that on can mathematically prove that 0.9999 recurring actually equals 1. And if that's the case.. exactly how true is any arithmetic equation. This is a hugely interesting but absolutely pointless question since one can simply 'do' a proof of 2+2=4 with pebbles. – Richard Dec 8 '18 at 23:16
  • @Richard: That relies on the decimal place system and the real numbers. None of that is necessary to see that 2+2=4. – Mozibur Ullah Dec 9 '18 at 0:00
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    To summarize and answer your questions directly. Natural language is ambigous since it directly maps reality leading to a demand for continously improving the concepts while maintaing stability of concepts to safeguard that communication can take place. This is not the case for math since it is an abstracted mapping that directly refers to stable aspects of some concepts rather than reaility directly. Fundamentally unambiguosness gets created by abstracting by only refering to stable aspects of concepts. This can be understood as defining new discrete concepts. Hope I could help. – CaZaNOx Dec 11 '18 at 11:28
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Ambiguity does not rule out truth. An ambiguous statement is one with two meanings. You may not know which meaning to apply but whichever it is the statement, given either or both meanings, may be true.

For example, 'She attends the small girls' school'.

This is ambiguous between :

  1. She attends the school for small girls.

&

  1. She attends the small school for girls.

You may not know which meaning is intended but whichever meaning applies, 1. and 2. capable of truth - or falsity.

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2+2 = 4 has been known since well before the Babylonians. Most likely it was known by man before he stepped out of Africa. In no sense were they using a system of axioms like the Peano axioms. It's true, because one can see that it is true. It's what Descartes means by 'clear and distinct ideas'.

As for the second question in your edit: because colour is a different kind of concept from number ...

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Mathematics abstracts human assumptions about reality, none of which are true. It can itself be true, but not applicable, or applicable but not exactly true.

If I pile up apples 1 at a time to a billion, by the time I am done, some of them are no longer apples. Some will no doubt already be trees. So if I then set about removing them, I will run out of apples before I have removed a billion, and I will have a lot of stuff other than apples left. I can draw all the triangles I want, and the odds of their internal angles adding up to exactly 180 degrees, is about zero...

Statements in any other sort of language have the same problem as that language approaches a level of useful abstraction. To the degree they are meaningful and precise, they will have exceptions or omit cases. To the degree they are absolute and still apply to reality, they are going to be vague. Apples are red. Well, are they? Apples are between red and green. Exactly what do you mean by green, there? Some of the reflected wavelengths that make up the green are quite blue.

So this is what is really meant by the proposition of Quine's that all statements are ambiguous. We do not have to find that 2 + 2 is itself unclear, only that we have never seen two truly interchangeable things anywhere in real experience, so you can't really apply it to anything real.

The fact that simple statements simply do not apply to reality in an absolute way does not stop reality from existing, and it does not stop us from talking about it. Nor does it stop us from having an internal abstraction mechanism that we largely share with other humans. If we are counting, we know that counting is a perfect process that we can teach our children, and the vast matjority of them will understand it in exactly the same way we do.

But when we set about counting actual things, we have to make up rules to conquer various ambiguities as we go along. If we are counting cows, maybe some calves are too small to matter. Do we count the dead ones? How about the pregnant ones -- one cow or two? The bulls may count for some purposes, and not for others... There will always be unconquered ambiguities farther in the future, and the process does still remain ambiguous even when we reach the level of a published military specification of counting cows.

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