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In linear algebra we learn about the idea of a set of 'polynomials' would this set be equivalent to a normal set such as the set of real numbers? The idea of sets (in my understanding) is that they can contain Mathematical objects (numbers, functions, other sets etc). Do we consider expressions as Mathematical objects? In which case is it more correct to say the following in terms of equality:

'The mathematical object that $x+1$ represents is the same as the mathematical object that $x+2-1$ represents'?

Instead of:

'$x+1$ and $x+2-1$ are the same mathematical object'

If expressions are objects in their own right? I would generally consider them 'syntactic objects' but the fact we can form sets with them suggests they are Mathematical Objects.

Edit: I have learnt that in dealing with Polynomials we are dealing with mappings based on indeterminates, if we can have an expression as part of a set, something like '$x+1$ is an element of A' is ambiguous, as are we referring to the expression or it's value? Is it possible to put an expression in a set?

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    $\begingroup$ Yes, we can indeed think of expressions as objects in their own right, with the "represents" relation being very different from "is the same as." Basically all of mathematical logic involves this conceptual shift, but in particular model theory is more-or-less entirely about this. I'll add a detailed answer when I have time, but I just wanted to quickly say that this is indeed a very real thing. (Meanwhile you may be interested in the latter part of this old answer of mine.) $\endgroup$
    – Noah Schweber
    Commented Aug 6, 2022 at 17:22
  • $\begingroup$ So it's not problematic to have a set of expressions? How does this work, in terms of how to we define the set using set builder notation or something. $\endgroup$
    – user37577
    Commented Aug 6, 2022 at 17:37
  • $\begingroup$ You might also find Gödel numbering interesting. It is a way of mapping between formulae and natural numbers. $\endgroup$
    – kc9jud
    Commented Aug 6, 2022 at 17:39
  • $\begingroup$ By the way, in your examples, the mathematical objects in question are functions. So you could say "the function that $x+1$ represents is the same as the function that $x+2-1$ represents". $\endgroup$
    – Lee Mosher
    Commented Aug 6, 2022 at 18:50
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    $\begingroup$ In response to your response to Alex: what are the “numbers” $1,2,3$? Do you here consider them as elements of $\Bbb N$, typically “constructed” by Von Neumann’s empty-set construction, or as elements of $\Bbb Z$, typically constructed as equivalence classes of ordered pairs of naturals, or as elements of $\Bbb Q$, equivalence classes of certain pairs in $\Bbb Z$, or as elements of $\Bbb R$, (these are all my favourite constructions, not the only ones) equivalence classes of Cauchy sequences of rationals? Et cetera. I’m not versed in foundations either, but: does the question really matter? $\endgroup$
    – FShrike
    Commented Aug 6, 2022 at 19:41

1 Answer 1

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Yes, expressions are considered mathematical objects. Examples of math objects include real numbers, compmex numbers, sets, functions, operators, algebraic structures, etc. Loosely speaking, mathematical objects are basically anything you can do math with. More formally speaking, they are objects that have a definition, follow certain rules and properties, and are the target for mathematical operations.

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  • $\begingroup$ When we define sets and rings of Polynomials would we consider them sets of expressions? $\endgroup$
    – user37577
    Commented Aug 6, 2022 at 18:36
  • $\begingroup$ If we're just talking about expressions and not functions, then yes because sets and rings of polynomials have elements. An expression is an arrangement of math symbols used to represent a mathematical object. For example, $5$, $6-1$, and $5x+6$ are expressions. An expression is basically the math version of a noun from English. $\endgroup$
    – Accelerator
    Commented Aug 6, 2022 at 21:14
  • $\begingroup$ Is it not a little bit problematic to distinguish between the expression, and the number it represents? such as $x^2+2x+1 ∈ A$ could mean the value of '$x^2+2x+1$ is an element or the expression itself is an element of A, is it just contextual? $\endgroup$
    – user37577
    Commented Aug 7, 2022 at 11:51
  • $\begingroup$ I really don't think it's that deep. I'm pretty sure what you're asking is contextual too. @user37577 $\endgroup$
    – Accelerator
    Commented Aug 7, 2022 at 20:00
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    $\begingroup$ @user37577 "Is it not a little bit problematic to distinguish between the expression, and the number it represents?" No, it is not problematic, because in 99.9% of cases, it is clear from context what you mean. In the 0.1% of cases, you simply make sure to be explicit so as to avoid confusion. No problem anywhere. $\endgroup$
    – 5xum
    Commented Aug 8, 2022 at 9:31

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