Abstract objects and changing properties

Question

I like to use This website to explain some of the simple ways of mathematical thinking, but in the linked article by Wells, he gives his ideas on how mathematical objects are inert, but in this he describes how 'September' and the idea of a schedule are changing because the property that 'this month is September' can change, and the idea that 'my schedule' can change', however I feel we can consider either 'unchanging' if we consider that 'this month' and 'my schedule' are variable terms in our language, just as 'is not x' is not a property of the number itself, it's simply that x does not refer to that number. Can we consider a 'name' as a property of the object', or in this case is 'being my schedule' a property that a schedule can gain, and another lose?

I think the issue here is that these are sort of 'association' properties, we associate 'September' with 'this month' we can 'assign' a role to the month 'September', (would this be a property) or we don't, similarly for a schedule, however, we 'associate' a number with a variable, in the same way, but this is not a property of the number, yet we can associate the -(-1) with 1 it's just that we consider this a true 'mathematical' property, is this a valid way to see it?

Answer 1

The expression "This month is September" is a constant. In written languages, any string of characters is a constant because a string of characters identifies itself. If one character in it is changed, then it is no longer the same string. A variable has to have something constant identifying it if anything else in it is to be allowed to vary.

What is variable in the expression "This month is September" is its truth value. It is true when we are in September and false when we are not.

The truth value of the expression "This month is September" is variable because the phrase "this month" has itself a semantic which allows its reference to range over twelve different values, from "January" to "December". Depending on the time of year, "this month" will refer to one or the other of these twelve values.

This is essentially the same situation for mathematical variables. If n is defined as a natural number, its value ranges over all the numbers which are members of the set of natural numbers. Typically, and by analogy to "This month is September", we can write for example "n = 2". This could just as well be written "n is 2", an expression which parallels "This month is September". '2' has a fixed reference just like "September" has, and 'n' has a variable reference just like "This month" has.

The mathematical language is nothing but an extension of natural languages (interestingly, mathematicians with different mother-tongues nonetheless use the same mathematical expressions, although they may write their proofs using their mother-tongue, not necessarily English). As such, its semantic obeys essentially the same logic as natural languages. There is no substantial difference between natural languages and mathematics in this respect.

Some mathematical objects are abstract objects, i.e., concepts. Some are "inert", for example the number 2, but others are variable, for example the triangle. And here again, this is similar to the situation in natural languages, where "Elisabeth II" is fixed but "the Queen" is not.

Answer 2

As you've noticed, there a lot going under the table here (and its not completely clear to me what exactly it is).

First, they say that "September's properties change over time". Its not completely clear what conception of property they have in mind, and a quick skim of their earlier article doesn't reveal anything useful. He says "this month is september" is sometimes true and sometimes false, but this is a proposition (or it maps us to one, if you allow double indexed semantics) about september. In general, propositions are not properties (rather properties of some type K are often modeled as maps from K whose target domain are the type Proposition).

He also claims that it "affects what people do", although the nature of this relation is completely unclear and does not at all seem to be causal. So I would say that you should find some writing where he makes his committments clear.

To answer your question: in philosophy, we typically do not consider a name- where a name of X is at least a signifier whose denotation is X- to be a property of X. It is true that we could consider it a property in the barest sense of the word- it picks out some attribute about the object X, namely its name, and perhaps this would suffice for a computer science usage. The philosopher it typically not concerned with properties in this sense, though, since there is not much philosophical work to be done with this bare sense of property- note that this does not mean there is not much philosophical work to be done about names (Kripke, for instance).