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Everything in math stems from definitions. Eg: Let an 'implication' be defined as ...

But any such 'let' actually means 'if it be true that'.

So what we're really saying is 'If an implication be defined as ..., Then ...'

Hence, all definitions are of the form of an implication itself. Does that mean, we actually define an implication using an implication itself?

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  • $\begingroup$ This sounds more like a philosophical discussion. I don't agree with the statement "But any such 'let' actually means 'if it be true that'." Can you give an example of what you mean by this? $\endgroup$ – Mankind Feb 21 '15 at 11:57
  • $\begingroup$ For example, we define a 'mathematical statement' to be 'a well defined sentence which has only one truth value'. Now, we go ahead and while defining a 'predicate' we would call the definition of a 'mathematical statement'. So really, we defined a mathematical statement and said that if it be defined the way we defined it, then a predicate would be defined as ... $\endgroup$ – Onkar Singh Gujral Feb 21 '15 at 11:59
  • $\begingroup$ You can see this post for a useful discussion. $\endgroup$ – Mauro ALLEGRANZA Feb 21 '15 at 12:35
  • $\begingroup$ We simply cannot define all ... In formalizing the language of e.g. predicate calculus we have to use the (natural) language to describe it (i.e. the natural language is our meta-language). And we cannot avoid to use some "basic resources" of the natural language : to "name" objects (the expressions of the formalized language,) to perform simple deductions, etc. $\endgroup$ – Mauro ALLEGRANZA Feb 21 '15 at 12:50
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    $\begingroup$ Your question can be reduced, essentially, to the following infinite regression, "Define define", "Define define define", "Define define define define", and so on. $\endgroup$ – Asaf Karagila Feb 21 '15 at 22:06
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In mathematics, the notion of a definition is an informal one. There is no formal distinction between statements that are definitions and those that are not, so the following are only suggested guidelines.

Definitions can tell us what a word means. Such definitions are usually stated as a bi-conditional, e.g. function $f$ is defined to be continuous at point $x\in \mathbb{R}$ if and only if...

They can also list the essential features of some logical structure(s), e.g. the Peano axioms for the set of natural numbers.

A recursive definition defines one structure in terms that mention that same structure, e.g. the factorial function (the $!$-operator):

$0!=1$

$ x!= (x-1)! \cdot x$, if $x\in \mathbb{ N}^+$

Note that in the last line, we have the $!$-operator on both sides of the equality.

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