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So far I have used the word "constant" to mean something that remains unchanged. Recently I started to read the book - Set theory and the structure of Arithmetic, by Hamilton and Landin, and I landed on the topic of constants and variables, in that book.

It says (p. 20)-

Definition 1. A constant is a proper name. In other words, a constant is a name of a particular thing. We say that a constant names or denotes the thing of which it is a name.

So is constant a name, or something that doesn't change? Or is it like there are two different notions? or is it like these notions overlap somewhat?

Harshit Rajput
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    It is the name of an object: $0$ for the number zero, $\emptyset$ for the empty set. – Mauro ALLEGRANZA Dec 30 '22 at 17:51
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    Interesting; I would have thought that a constant symbol is the name for a constant, the same way a function symbol is the name for a function (eg "$\log$" is the name of the logarithm function) and a relation symbol is the name for a relation (eg "$>$" is the name of the greater-than relation). What's the context? – Akiva Weinberger Dec 30 '22 at 17:55
  • @MauroALLEGRANZA So it indeed is a name. Perhaps called constant because the name of something once we give it, is unchanged? – Harshit Rajput Dec 30 '22 at 18:38
  • @AkivaWeinberger I would think the same. Here are some lines right before this definition occurs - Image. If you have time, you can browse through the book, I have added the link. – Harshit Rajput Dec 30 '22 at 18:46
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    A name refers to an object: Napoleon, New York, etc. In formal language we have terms: variables and constants. – Mauro ALLEGRANZA Dec 31 '22 at 08:24
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    In formal proofs, I think of a constant as an object that is free in an axiom. It's scope is the entire proof, e.g. $0$ in the Peano Axioms. Unlike a premise or assumption, axioms are not intended to be discharged. – Dan Christensen Dec 31 '22 at 11:42
  • I have a perspective here. – ryang Feb 15 '23 at 16:52

1 Answers1

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(A) Mathematics is all about "Definitions + Axioms" which give "theorems + theories" , in general.
There are various Definitions which , being very useful , have become "Standard" , Eg Natural Number ${0,1,2,3, \cdots}$ where sometimes $0$ is excluded by Definition. When Non-Standard Definition is used by some Authors , that will be given early in the Article.
You are seeing such a Situation in that Book.
( I see that your latest comment echoes this Point ! )

(B) When we have a variable "man" , it can take various values like "Prem" or "Harshit" Etc. Now , these values are Constant , meaning "Prem" can not take some other value ( It can not vary ) which matches the Concept of Constant.
Likewise , we can have a variable $x$ which can take various values like $1$ or $2.3$ Etc. These values are Constant , $1$ is always $1$ , never equalling $2.3$ , which matches the Concept of Constant.
It may occur that $y=x$ ( two variables , which can "store" Names , are "storing" the same Name ) , though it will never occur that $1=2.3$ ( Constants or Names ) , in general.

(C) We might imagine some abstract ( imaginary / invisible / hypothetical ) thing which "is" $1$ to which we have given the "Name" like "אחד" (HEBREW) or "एक" (HINDI) or $1$ (DECIMAL) , Etc.
No matter what names we have given , the "rules" of manipulation ( Eg addition ) will be consistent.

We might then imagine some variable $x$ which can take various values , including $1$.
It will always be true that $x+y=y+x$

Hence this Definition matches what we want to achieve , regarding Constant & Variable.

(D) The Dictionary meaning of "Constant" is "Unvarying ; A quantity that does not vary" which , being based on "Variable" , is not very useful in Mathematics or logic.
The Dictionary meaning of "variable" is "Something that is likely to vary ; Something that is subject to variation" which , though intuitive , is not rigorous.
Hence the authors have made attempts to achieve mathematical rigor with these Definitions.

Prem
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