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Is there a possible way to put every assumption into mathematical language?

You can do some, for example:

some constant c = 0, etc.

But, I mean in a more generic way, can everything be converted into mathematical language?
I do not think this is the case, but I am currently not able to put my thoughts into words.

My current thoughts: Mathematics already makes plenty of assumptions, like the ZFC axioms. So any assumption more basic than that cannot be put into a mathematical context, or anything outside of that scope.

In a sense what is the source of assumptions? Since mathematics is directly under the ambit of a few axioms or assumptions, is there something more basic than mathematics, under which all assumptions can be put into?

How are we even able to say this axiom should be held true, and another should not be held true. Sure, that usability and power are two factors, but I think that axioms being held true, is dependent on something better than just these two.

Again, I already have some thoughts on the matter, but unable to put them into concrete words right now.

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    "every mathematical assumption" or "every assumption" ? – Mauro ALLEGRANZA Sep 22 '17 at 14:26
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    God exists. I exist. You could only put these in mathematical notation if you could give a mathematically precise definition of God and I, and I'm pretty sure you can't do that. – barrycarter Sep 22 '17 at 14:54
  • @MauroALLEGRANZA every assumption. Even mathematical assumptions are written using formal logic? – user12196 Sep 22 '17 at 15:00
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    No. Even if we assume that natural language can be converted into a formal theory, which it can not be, you'd still need indexical names and material predicates to express even the most elementary statements. Such as "statue of Liberty", "Sun", "red", "dead", "moves", etc. None of these is expressible in mathematical vocabulary, and given the role they play in the language (for direct reference and empirical access) they can not be axiomatized either. On top of that, natural language is deliberately vague and often inconsistent, both traits are absent in formal theories. – Conifold Sep 23 '17 at 0:36
  • You probably mean "logically" instead of "mathematically". Even if you used something like ASCII to represent a natural language sentence as a string of numbers, that isn't stating that sentence "mathematically", that is just using different symbols to represent the natural language statement. I'm pretty sure you mean "logically", i.e. some sort of logical calculus like predicate logic, and not "mathematically", because it seems trivial to show that mathematics only has the syntax and semantics to talk about mathematics. – Not_Here Sep 25 '17 at 17:31
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This is an interesting one because the answer involves digging into what you mean by "stated mathematically." It turns out to be more nuanced than you think.

One can make the statement that every assumption which can be put into words can be stated mathematically. This is actually boringly simple to prove. If you can write your assumption down in words, broken down into letters from an alphabet, you can encode any statement as a number. For example, "God is great" might be translated into the number 22,108,099,297,835,273,666,055,987,572 simply by converting that string into ASCII and then making a number out of it as a 2s complement integer. However, I expect this falls quite short of what you intend. However, this is important to build on. Important enough to give a notation. Let's define #(p) to be a number which encodes a statement in your language. For example, we can say #(God is great) = 22108099297835273666055987572. We don't have to define a notation like this, but it's easier to keep track of things this way. More importantly, its provable that there exists a numbering scheme like this to encode every written phrase.

I think an important part of something being stated mathematically is the idea that you can manipulate it symbolically, and the result is a true statement. For example, you can take the statements P->Q and Q->R in First Order Logic, and combine them to make the statement P->R. You can do this without even knowing what P Q and R are. So a meaningful mathematical statement must also support some sort of manipulation like this.

Formally, we phrase this using a functional notation. We write φ(n) to be a function which evaluates to true if n encodes a true statement. The φ function, of course, is different for every language and interpretation of that language. We might define φ(1345142354) to be true in our language, because 1345142354 happens to be the ASCII->2's complement integer conversion of the string P->R. In other words, φ(#(P->R)) is true. This makes φ and # somewhat of oposites: # converts a phrase into an associated number, and φ converts a number into the truth value if its associated phase.

But what about this φ? Surely we can describe it using our language as well, right? It turns out it's harder than it looks. Tarksi's Undefinability Theorem states that if my language tries to define some φ* that is its description of its own meaning, there is no formula for φ* such that one can prove φ*(#(A)) <-> A for all statements A in the language, as long as that language includes arithmetic and logical negation (the "not" symbol).

Informally, this means no formal language can describe its own meaning completely. While we can define a language where the symbols mean anything we please, in any given language, there is no way to fully define a symbol which defines the "correct" interpretation of that language.

  • Are you sure you are simply not encoding any "statement" into numbers? I meant every "assumption". Things which can be held true. There is a subtle difference. Sorry, just read your entire answer. Very interesting. – user12196 Sep 24 '17 at 22:58
  • @novice That is the definition of statement used here. It's a string of characters which has a truth value when interpreted according to the rules of the language. Now it is possible to have unstatable assumptions, assumptions that can never be written down, no matter how hard you try. In that case, it's trivial to show that the also cannot be written down mathematically. – Cort Ammon Sep 24 '17 at 23:00
  • But, again basically you are saying that every assumption must have something which makes it to be evaluated to be "true". So you are basing every assumption on something else, which makes that evaluation. – user12196 Sep 24 '17 at 23:02
  • @novice And the statement you make is true. It is meaningless to talk about the truthood of a statement (such as "Twas brillig, and the slithy toves Did gyre and gimble in the wabe") without defining the language with which you are using to interpret that string of characters into something which can have a truth value to it. It's that interpretation step where everything gets interesting. – Cort Ammon Sep 24 '17 at 23:03
  • Yes, I am trying to get to the source there. But, things as you say definitely get interesting. Yours is definitely the best answer here. Thank you ! – user12196 Sep 24 '17 at 23:04
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Math is the process of deriving theorems from axioms. The axioms don't have to be true. I could use Euclidean axioms to derive theorems, or Riemannian axioms which assume space is a curved manifold. What ever is the case in the real world, the truth value of all possible theorems are fixed given a set of axioms. In other words, there are no theorems 'underlying' ZFC. Those are the axioms used for some domains of math, and you can define a theorem as being true or false under those axioms by doing what we call 'math'.

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More basically, can any assumption at all be put into purely mathematical terms? I would contend not.

Things can be bound to mathematical formalisms by theories, but math itself is not about reality, and cannot really be rendered so. It can only codify interpretations, it cannot make them or express them.

The actual assumptions are made in the framing of the theory that makes the math seem to say something, and they cannot really be captured precisely at all, because of the theory-embeddedness of language. The math can pull out and summarise the systematic parts of a theory, but you already have to understand the linkage between reality and the measures involved. Those are the assumptions. There is no content in the mathematics not implicit in the unstated bond between the math and the science involved.

Edit: I should be explict about the framing. I adopt a kind of specific neo-Intuitionist model of mathematics as the creative process of exploring and elaborating the contents of shared intuitions about thinking that are not optional.

By reality I mean external, shared reality, not including mental contents. If mathematics is all the built-in machinery necessary for us to apply thought to reality, then it is all about us, and not about external reality.

And by assumption, I mean something that could be decided otherwise and matter. The axioms of mathematics are not assumptions in this sense, they could be made otherwise, but to the extent they were, they would not matter anymore, at least to the problems at hand. They would apply to a different set of problems.

In that sense math is not made of assumptions we make, it is made of assertions that are made for us, that we inherit by virtue of having our minds, or that we creatively contrive by observing those basic assumptions.

The linkage that actually makes the application of mathematics to reality is not mathematics, it is made by the assumptions of some theories, and the mathematics would be there whatever assumptions you did make or theories you held instead.

To me, the language metaphor is good here -- English is not a discourse, it does not have real story or argumentative content, and it would be there no matter what discourse you chose to have in it. It is a part of our communications other than any discourse. Likewise, math is a part of all out descriptions other than any theory or decision...

So theory and mathematics are pretty much two independent aspects of a single process.

  • You are then implying a linkage between math and reality. Truthfully I already knew the answer to be no, but wanted to go deeper as to why it was a no. So more generically you are implying that assumptions come from reality? That we hold them true or not from something more basic than math? I want to reach the source of it! – user12196 Sep 24 '17 at 22:51
  • I think reality is involved, but not in the way you are currently thinking, for example some axioms in math might not have anything to do with reality. They are used for power and usability, to build something bigger. – user12196 Sep 24 '17 at 23:05
  • @novice I would claim all axioms in mathematics have nothing to do with reality, they are attempts to model causal relationships -- and they do not actually describe any real process, except that we pair models with observed phenomena by observation. – jobermark Sep 25 '17 at 16:12
  • Exactly, I would make the same claim as well. They are just statements arrived at through trial and error to enable us to make further causal relationships or deductions and inferences. – user12196 Sep 25 '17 at 23:24
  • @novice The notion of trial and error gets a little weird because from a modern standpoint there are no errors. There are no incorrect mathematical axioms, only uninteresting ones. Usually they are interesting because they become part of a model of something else (e.g. surveying or trade), but some are interesting only because we like them and the feel right (e.g. The Axiom of Choice). – jobermark Sep 26 '17 at 16:11

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