2

I have been thinking about how all our mathematical ideas and methods are, to a computer, a mere manipulation of symbols through algorithms with no meaning attached to them. But even though they have no meaning to a computer, all the things "add up", we don't get any contradiction by just manipulating the symbols with certain rules. My question is, suppose I have a statement about symbols and certain manipulations of them, can I be sure it is true if I can find some interpretation of the symbols for which the equivalent statement is true?

Suppose I regard the digits {0,1,2,3,4,5,6,7,8,9} as mere symbols with no meaning assigned to them. Now suppose I have the usual multiplication and addition table for 1 to 9 (in which, for example, $9*8=72$ is just a statement about symbols and * and + are binary operations between symbols). Now, suppose I have the symbols 123 and 456 and define 123*456 to be the symbol resulting from following the usual multiplication algorithm (using the tables I have). I get, of course, 56088. Here, all these symbols have no meaning and * has no interpretation. I notice that 123*456 gives the same symbol as 456*123 and I conjecture that $a*b=b*a$ where $a$ and $b$ are symbols. Now, I would like to know if this is a valid proof:

I construct the natural numbers and define addition and multiplication using ZFC axioms. I define 2=1+1, 3=2+1,..., ten=9+1, and prove that multiplication is commutative and all its other properties. Prove that every number $x$ can be written as $x=a_{0}+a_{1}*ten+...+a_{n}*ten^n$ (where $a_{i}\in{[0,1,2..9]}$) in a unique way and identify $x$ with the symbol $a_{0}a_{1}...a_{n}$ creating a one to one correspondence between the symbols and natural numbers. Now, notice that I can obtain the decimal representation of the product of the number corresponding to $a=a_{0}a_{1}...a_{n}$ and $b=b_{0}b_{1}...b_{m}$ by using the properties of multiplication, and this turns out to be equivalent to following the multiplication algorithm (i.e. the symbol corresponding to $ab$ is the same as the one obtained following the multiplication algorithm). Now, because I know multiplication to be commutative and that each number has only one decimal representation we must have that the symbol for $ab$ is the same as the one for $ba$, concluding that the symbol obtained using the multiplication algorithm for $a_{0}a_{1}...a_{n}$ and $b_{0}b_{1}...b_{m}$ must be that same as the one obtained for $b_{0}b_{1}...b_{m}$ and $a_{0}a_{1}...a_{n}$.

Therefore proving a statement about symbols by giving them the interpretation of numbers.

Thank you!

MathUser123
  • 617
  • 3
  • 13
  • Since 56,088 includes the digit 0, I think you meant to include 0 in the sets of digits in your second and third paragraphs. – mephistolotl Sep 19 '17 at 05:09
  • 1
    A proof is a proof in a formal system. What formal system are you starting with? Providing a model for a theory that validates some statement does not (in general) show that that statement is provable, it only shows that it is not refutable (assuming consistency). Just because there's some model where $*$ is commutative, doesn't mean there isn't a model where it's not. For example, if your theory is an algebraic one, there's a model where everything is mapped to the same thing and thus any equality you want holds in that model. – Derek Elkins left SE Sep 19 '17 at 05:10

1 Answers1

1

It sounds like you're describing an isomorphism $f:(\omega,\cdot) \rightarrow (S, *)$, where the binary operation of $(S,*)$ is given by pushing the digit symbols around, $(\omega, \cdot)$ is a more familiar and direct construction of natural number multiplication, and $f$ is the 1-1 correspondence of symbols. In particular, "the symbol corresponding to $ab$ is the same as the one obtained following the multiplication algorithm", sounds like it can be translated as $f(ab) = f(a) * f(b)$.

Isomorphic structures satisfy the same first-order sentences. (This statement can be found in model theory texts, for example 2.4.3 of A Shorter Model Theory. See also this question.) In particular, if you prove $f$ is an isomorphism and prove $(\omega, \cdot)$ satisfies the commutative law, then you can conclude that $*$ also satisfies the commutative law.

Hugh Denoncourt
  • 2,000
  • I want to say this is a "yes" answer to your question, but only in my informal understanding of what you mean. The question, "what is an interpreted vs uninterpreted symbol?", gives me reason to hesitate to say anything with confidence. Are proofs in ZFC "interpreted"? Anyway, you might like, if you are not yet familiar, the MU puzzle from Godel, Escher, Bach: https://en.wikipedia.org/wiki/MU_puzzle – Hugh Denoncourt Sep 21 '17 at 14:31