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I'm a bit confused on the nature of formal sums and what authors mean by them when they use them. For example, I've read up on a definition of a polynomial as an infinite formal sum. It's unclear to me whether the intended concept is that the polynomial is the symbolic expression itself, so the string of characters themselves, or if it's supposed to be thought of as representing a mathematical object that lives in some well-defined space.

Anne Bauval
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jvf
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  • @jvf I am afraid "a definition of a polynomial as an infinite formal sum" should have been "a definition of a polynomial as a finite formal sum" or "a definition of a power series as an infinite formal sum". – Apass.Jack Mar 18 '23 at 07:33
  • @jvf "I've read...". Please add references to what you have read. – Apass.Jack Mar 18 '23 at 07:35
  • @AnneBauval I'll be honest: most of what was.going on in that question went right over my head; I don't really have the mathematical baggage for that discussion. As to how it relates to my question: it certainly seems relevant, since it's centered on the definition of a polynomial, though I don't think it answers my own. I touched on polynomials, but mostly as an example. It doesn't look like that question deals with the nature of a formal sum itself. Or have I misunderstood it? – jvf Mar 18 '23 at 07:36
  • @AnneBauval Philosophy might be an excellent tag for this question. – Apass.Jack Mar 18 '23 at 07:44
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    Formal sums are mathematical objects. You can do algebraic operations on them. Say, $\left(\sum_{n=0}^\infty a_nx^n\right)\cdot\left(\sum_{n=0}^\infty b_nx^n\right):=\sum_{n=0}^\infty \left(\sum_{k=0}^na_kb_{n-k}\right)x^n$. In the above case you can identify formal sum $\sum_{n=0}^\infty a_nx^n$ with a sequence $(a_n)_{n\in\mathbb N}$ and view the above multiplication just as a peculiar operation on sequences. –  Mar 18 '23 at 07:46
  • @Apass.Jack I believe this definition of a polynomial is the one given by Fraleigh in his intro book on abstract algebra. I'm sorry I can't be more specific right now, I don't have my computer on me, I typed this on my phone. I'm fairly certain he uses those words, "infinite formal sum". – jvf Mar 18 '23 at 07:51
  • There is also a context in which (finite) algebraic expressions are viewed as pure sequences of symbols, which is the initial step of building free algebraic structures, which in turn is an extremely important construction in abstract algebra. See: Free object. –  Mar 18 '23 at 07:54
  • @Apass.Jack honestly I don't think so. It rather seems that the OP is mystified by words and does not have the mathematical education to deflate the balloon. – Anne Bauval Mar 18 '23 at 07:58
  • Formal sums are understood to be expressions together with rules for manipulating them, e.g. by adding or multiplying them, so that's the "intended concept". However, if that's uncomfortably abstract, you can always interpret them as more concrete mathematical objects, such as sequences, with corresponding rules for addition & multiplication. But in practice, any interpretation is a distraction, so we mostly don't specify one. – David M Mar 18 '23 at 08:03
  • @jvf You would have been unambiguously correct had you said "a polynomial is defined as an infinite formal series with finite support in Fraleigh's abstract algebra". (Am I hairsplitting?) – Apass.Jack Mar 18 '23 at 08:22
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    @AnneBauval Philosophy is in the eyes of a beholder. As I see it, the comment above by David has a strong flavor of philosophy. On the other hand, your opinion could be helpful for OP, too. – Apass.Jack Mar 18 '23 at 08:30
  • @StinkingBishop I might've worded my question wrong. Instead of posing it as a "is a mathematical object" vs. "is not a mathematical object", maybe I should've gone with: are the symbols that make up the expression of a formal sum, in that particular order, to be thought of as the mathematical object of interest, or do they represent another mathematical object? Similar to the distinction between $\mathbf{v}$ as a character in itself and as a symbol that represents a vector from some vector space. I understand there is an identification between a formal sum and sequences with a (continues...) – jvf Mar 18 '23 at 16:07
  • @StinkingBishop sequence and a specific operation on those sequences, but isn't that sequence a distinct mathematical object? If the object of interest is the string of characters itself and the rules to manipulate them, shouldn't $x + x^2$ and $y+y^2$ be regarded as different objects, even though they ought to be identified with the same sequences, when analyzed in separate contexts? – jvf Mar 18 '23 at 16:10
  • @Apass.Jack I can see the point you're making, as it is the same reason why I worded it as "a definition", not "the definition", but without mentioning where it came from. I don't think it's hairsplitting (unless "a definition" is what you're debating, maybe?). – jvf Mar 18 '23 at 16:12
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    Re: syntax vs. semantics, meaning of the symbols in a formal sum, e.g. what is "$x$" and "$+$" in $ax^2+bx+c$, what is "$i$" and "$+$" in $a+b:!i$ etc, see here for some historical discussion, esp. Hankel's scathing critique of Cauchy's informal definition. Nowadays we can make these expressions rigorous using formal languages (term algebras), e.g. see textbooks on universal algebra. Such term algebras are used in every computer algebra system (Macsyma, Maple, Mathematica, etc). – Bill Dubuque Mar 18 '23 at 20:00
  • @jvf I am not a philosopher so maybe I don't appreciate the distinction you want to make. In mathematics, two algebraic structures which are isomorphic (i.e. indistinguishable except for the "nature"/"identity" of their elements) are for all practical purposes the same structure. We take it for granted that any proof that "identifies" them can be rewritten as a slightly more rigorous proof that does not identify them but tracks that bijection that links them. At the end of the day, what if you write a formal sum using Roman numerals and the word "plus" written, say, in Chinese? (TBC) –  Mar 18 '23 at 23:27
  • (Cont'd) You've immediately switched to a totally different formal sum, with seemingly totally different properties. Or, wait, have you really? In Mathematics, we choose to not worry too much about that distinction. –  Mar 18 '23 at 23:29
  • @StinkingBishop That's an interesting example (Roman numerals + Chinese). I really don't have a coherent view formed on that. At the same time I want to say yes and no. It might be that I'm just not mathematically mature, but, even though I appreciate isomorphisms, it just doesn't sit right with me to think of all of them as the "same" object. The vector spaces of oriented line segments in a 2D plane, of the linear combinations of the $\sin$ and $\cos$ functions, and of the polynomial functions up to order 1 are all isomorphic, but I don't feel comfortable saying they're all the same. – jvf Mar 19 '23 at 02:29

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