Some days ago our professor during the course of his lecture wrote the following definition of a polynomial.

We say that an expression of the form a0 + a1x + a2x2 + ... + anxn is a polynomial of degree n where ai's are taken from some given set.

However when one of our classmate asked the professor, "What is x?" then he replied that x is just a meaningless symbol. Then he proceeded to give some rules on the "usual" polynomial addition and multiplication.

Now I have the following questions:

  1. When the professor says that x is a "meaningless symbol" what exactly is meant by that? Does it imply that the symbol x has no 'meta-lingual' interpretation?

  2. When we are saying that the x, x2, x3, ...'s all are meaning less symbols, shouldn't we first show that they indeed exist?

I think that the first question can be considered as a special case of the following question,

Does it make any sense to say that "symbols are independent of meaning or interpretation"?

I am tempted to think that it is meaningless to ask what is the "meaning" of this or that symbols because symbols have in general multiple "meaning" or "interpretations". But when I am saying this I am assuming that there is(are) no symbol(s) which has(have) no "interpretation" or "meaning" and which I think needs to be shown. So, one more question my be added to the last question, which I mark as 3.

  1. Does it make any sense to say that "symbols are independent of meaning or interpretation"? Can there exist a symbols which doesn't have any "interpretation" or "meaning"? In other words, what is the ontological status of symbols?
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    1. He clearly simply meant that where he wrote x, you could as well have written y or monkey. A better formulation would be that this expression is a polynomial of x (or y, ...). 2. Why would you think you need to? And why would you think the symbols don't exist, if he just wrote them on the board? In general, I'm missing the philosophical issue here, but please do enlighten me. – user2953 Jan 19 '16 at 13:27
  • @Keelan: 1. If "you could as well have written y or monkey" then isn't what the professor is saying essentially is that "the expression of the polynomial is independent of the 'meaning' (whatever this may mean) rather that simply 'meaningless'? 2. Can you elaborate a bit? – user13627 Jan 19 '16 at 13:37
  • 1. Sure. But it's a professor of mathematics, not of philosophy (I suppose). 2. No - I'm essentially asking you to elaborate :) – user2953 Jan 19 '16 at 13:39
  • @Keelan 1. You are right. 2. When I asked the same thing to the professor he told me to look over the matter as follows: Suppose we have a 'collection' of meaningless symbols. Then we simply say that the symbols are members of the 'collection'. If we don't prove the existence of that 'collection' (provided it is well-defined) how can we be sure that it willn't lead us to paradox(es) like Russell's Paradox. – user13627 Jan 19 '16 at 13:50
  • Russell's paradox considers a set of sets, while you consider a set of symbols. I don't see how a set of symbols can lead to something like Russell's paradox. – user2953 Jan 19 '16 at 14:39

What your professor has stated is a common approach to trying to avoid a mathematical trap that occurs in many student's minds. In such equations, the actual symbol, "x", doesn't matter. It could by a "y" or a "z" or a picture of an orangutan. I like the word "arbitrary" over "meaningless," myself.

For someone who "gets it," a statement like "X is a meaningless symbol," seems strange. However, it is very easy for students to start to attach meaning to letters. Then, they have trouble solving the same equation when they see "x" replaced by a "y." Your professor is simply trying to head them off at the pass. (I tutor mathematics from time to time. It always pains me when I tell a student "solve for x," and they simply cannot. Then I write the exact same equation, switching "x" with "y" and tell them to "solve for y," and they can now do it, because the shapes drawn by my pencil are now the shapes they expect to see)

Later, this statement will be relaxed. Once everyone gets it, you can start to recognize that variables often have conventional meaning. If you see someone doing math on a right triangle, "c" is the hypotenuse. If they're doing physics, "c" is the speed of light. However, if your professor didn't warn the students that, in reality, the symbol choice is arbitrary, they could be really bothered by trying to do physics on a right triangle, and substituting speeds of light into place because they saw the same symbol, "c", in both environments.

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I don't particularly like your professor's use of the word meaningless. The x's in your formula are conventionally called variables. Quine's view was that variables in mathematics are adequately captured by the way we use them in logic, i.e. as a kind of abstract placeholder, similar to a pronoun in a natural language. Variables are explicitly or implicitly quantified over, and the quantifier may be existential, meaning that the expression holds for some particular value, or universal, meaning that the expression holds for all values.

Your x's are no more meaningless than the pronoun "who" in the following sentences:

There is someone who is ahead of me in the queue. (An existentially quantified statement that can be 'solved' for a particular value of "who".)

He who hesitates is lost. (A universally quantified statement that holds for all "whos".)

Quine's exposition can be found in two papers: "Variables Explained Away" in Proceedings of the American Philosophical Society, Vol. 104, No. 3 (Jun. 15, 1960), pp. 343-347; and "The Variable" in Ways of Paradox and Other Essays.

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Philosophically, the professor obviously was being causal - there is obviously no such thing as a "meaningless symbol" as a "symbol" by definition "a thing that represents or stands for something else, especially a material object representing something abstract". Here, x is a symbol that stands for something, the question then is what does it stand for?

In this abstract setting, x is called an indeterminant, with no value in particular (as opposed to x being a variable, that is, an unknown quantity to be solved).

What he probably meant to say is that x, being an indeterminant, has no particular value, but the point is that this is a polynomial in one variable. This polynomial could just as easily be expressed (as they often are) in t, but it is distinguished from a polynomial in two variables.

To address the question of whether the thing x points to exists - it clearly does, but it's hard to say what that thing is, other than using a circular definition as the thing which lets us talk about abstract polynomials.

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  • The link for symbol definition also states "an arbitrary or conventional sign used in writing or printing relating to a particular field to represent operations, quantities, elements, relations, or qualities". In this sense, symbol does not have to have an intrinsic meaning. It does not have to represent anything (meaningful), but, of course, you would go with the defined entities (for which the meaningless symbol, beside other things, is used) by, say, narrowing down / giving meaning/value/placeholderness to the symbol. – mami Jan 20 '16 at 10:42

I think that what is meant is that x is not a variable, but rather pure syntax. It's just like when mathematicians define complex numbers as ordered pairs (x, y) of real numbers, and then write them as x + yi, where the "+" and "i" are purely syntax. (Later notation is relaxed to permit, say, i instead of 0 + 1i, justified with theorems showing that if the symbols are defined correctly the syntactic "+" gives the same result as the operation +.)

In this case it means that x^2 - 3 is just a collection of coefficients, (-3, 0, 1), and different such collections can be added and multiplied according to various rules.

The general idea, I think, is to move from polynomials as functions ('give me x and I'll give you a number') to polynomials as objects. But this can be made concrete. Suppose you're looking at polynomials over the group Z/2Z in which there are exactly two numbers, 0 ("even") and 1 ("odd") with 0 + 0 = 1, 0 + 1 = 1 = 1 + 0, and 1 + 1 = 0. Now notice that whichever value you choose for x, x^2 + x + 1 is equal to 1. But as polynomials over an indeterminate ("meaningless symbol") x, 1 and x^2 + x + 1 are different (because the lists (1) and (1, 1, 1) are different).

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  • I also think that "x is not a variable, but rather pure syntax." But then the question is: are syntax really "meaningless" as mathematicians claim them to be? – user13627 Jan 20 '16 at 13:22

When the professor says that x is a "meaningless symbol" what exactly is meant by that?

It just means that you can think of a polynomial "a + b x + c x^2 + ... + d x^n" as the ordered tuple (a,b,c,...,d), and the "x" and "+" and exponentiation notation are not part of the polynomial but merely there to give suggestive notion. The coefficients of the polynomial are then simply {a,b,c,...,d}, and you can easily define addition and multiplication of polynomials with coefficients from the same ring. By this definition, the polynomial itself does not even include which symbol is used as the indeterminate, although we use "x" in describing the polynomial. For example we have that the product of (1,1) and (1,-1) is (1,0,-1). This is conventionally described by saying that (1+x) * (1-x) = 1-x^2.

But the above is not the only definition of polynomials in use. In algebra, it is common to talk about polynomials as if they are the actual strings of symbols written down including the 'powers' of "x" and "+". There are some subtle errors in reasoning that can creep in when one uses such 'intuitive' notions of polynomials. And unsurprisingly, it will be difficult to properly formalize this notion without going back to the earlier definition. For example, we can 'imagine' that "1 + x y + x^2 + y^3" is a polynomial in both "x" and "y", but is it a polynomial in "x" alone or "y" alone?

Note that the formal definition of a univariate polynomial as a tuple can be extended to multivariate polynomials, by defining a k-variate power series over R as a function from N^k to R, and then defining a k-variate polynomial over R as a k-variate power series that is zero except on finitely many input index-tuples. Note that a bivariate polynomial over R can be viewed as a univariate polynomial over univariate polynomials over R, in the same way that functions can be curried.

Does it imply that the symbol x has no 'meta-lingual' interpretation?

So your professor's answer is actually not precise enough to determine whether he is using the purely rigorous definition. There is a third possible definition, where a polynomial is a function of the form ( x ↦ a + b * x + c * x^2 + ... + d * x^n ), where "+" and "*" can be overloaded and "x^k" is just a short-form for the k-fold product of "x". Such a function will be undefined or crash if given an input x for which any of the multiplications or additions are not defined. In this sense "x" is not really meaningless but is in fact a parameter name. Of course, one can use a different parameter name (as long as it is not used elsewhere) without changing the meaning. The disadvantage of this approach is that it is very difficult to formalize with all the overloading, but it actually corresponds quite closely with how we informally reason.

It is actually not too hard to use the formal definition to capture this notion, via the evaluation map on univariate polynomials over R (where R is a ring), that maps each such polynomial p and object x in R to the value ( a + b * x + c * x^2 + ... + d * x^n ). Similarly for multivariate polynomials. The only thing is that we always have to use this evaluation map to 'apply' a polynomial to some object. People naturally create the short-hand "p(x)" for the above value, despite this conflicting with the formal definition! (By the formal definition, p(1) would be the coefficient of the linear term, but by the convenient short-hand p(1) would be the sum of all the coefficients!)

Does it make any sense to say that "symbols are independent of meaning or interpretation"?

As I hope the rest of my answer has shown, it does not actually have much to do with polynomials and your professor's statements. That said, indeed it is reasonable to say that symbols are just symbols and devoid of meaning unless you interpret them. In mathematical logic we even specifically define an interpretation to be a function on strings of symbols, which is intended to be a map to their 'meanings'.

Can there exist a symbols which doesn't have any "interpretation" or "meaning"? In other words, what is the ontological status of symbols?

Choose an arbitrary collection of 100 pixels in a 100*100 square, and that collection forms a symbol. What meaning does it have? Well, you could tell people that you want to use it to denote something, in which case it has meaning to whoever accepts your decision. Likewise we can say that the answer to your question is "No." because given any symbol I can choose to interpret it to mean whatever I like, and it will have that meaning to me.

Indeed, it is useless to ask whether a string of symbols has meaning (in itself), simply because any meaning can be given to it. What makes a particular interpretation useful is when it can be applied uniformly to a whole collection of strings (such as first-order semantics over a first-order structure applied to first-order sentences) and has non-trivial properties (like soundness and completeness).

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In your example "x" is a indeterminate of the polynomial, that's its meaning. Normally one writes "X" (capital). The expression "meaningless symbol" is misleading.

You can substitute for x an arbitrary value. Then also the polynomial evaluates to a certain value under the assumption that you know the value of the constants a_i.

Remark. Thanks due the answer of @Kames Kingsbery I changed to "indeterminate".

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    If this is the answer the OP is looking for, this question does not belong on Philosophy.SE. And in that case, you should not give this answer, but instead flag the question. – user2953 Jan 19 '16 at 13:35

Just a quick response: x is not a meaningless symbol but a meaningless sign pointing to anther sign whose connections are logically standardized. If the x stood for another object it would become a meaningful symbol. Of course the problem you are raising belongs to branch of logic called philosophy of mathematics and mathematicians themselves are usually not philosophers so they might be unprepared to answer any philosophical questions and their communication with students reflects this. To ask ontological question about mathematics, or about essential reality of all deductive systems of logic in general, is a little bit like asking whether concrete reality (created by some divine being) is essentially logical or essentially irrational but partially rationalized by us in the course of our activities. Your answer will depend on whether you believe in god mathematician who created the logical world, including mathematics, or whether you believe in humans as cultural agents who created mathematics alongside other systems of knowledge and who have been playing god since the beginning of their cultural existence as creators of their cultural reality.

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  • Would you have a reference where this distinction between symbol and sign is defined? This would strengthen your answer and guide the reader on where to go for more information. Welcome to Philosophy! – Frank Hubeny Feb 22 '19 at 17:25

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