# Abstract objects and changing properties

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?

• Wells's way of making the point is erroneous, "this month" is indexical just like "this number", and "this number is 5" will also change over time if we move our index finger. However, the point itself is valid. September is temporally situated (just as equator is spatially situated) the way 5 is not, and yet an abstract object. Beethoven's 9th symphony is an abstract object that was once created, the way 5 was not. On temporal properties of abstracta see SEP. Commented Sep 17, 2022 at 4:09
• @Conifold my only possible understanding of it could be that in mathematics the properties of numbers are kind of well defined, whereas for a month we could see 'this month is' as a property, it is definitely a proposition, but whether it's a 'property' or just an indexical property, but my thought against this is the number '5' will change if I consider 'is x' as a property and then decide actually x=6, or alternatively he sees the fact that 'this month is September' having a meaning greater than just a linguistic level, like we might say, the properties of '2' is that it is the first prime' Commented Sep 17, 2022 at 9:22

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.

• the idea of 'inert' is strange here, if we 'assign' september to this month does it change it's properties (even outside of language) just the fact that 'this month' in time is september? For example if I assign Elizabeth II to the role of queen does this change the properties of Elizabeth II? Or is this just 'indexical' language to be ignored? One other thing, if most strings are constants is 'x is john' also a 'constant expression'? Is there 'variable' expressions in written languages, because in math usually a 'variable expression' is one whose meaning changes like 'this month is september' Commented Sep 18, 2022 at 9:49
• @user1007028 "we 'assign' september to this month" We are not free to assign a value to "this month". It is part of the language we use that "this month" refers to the month of the year we are in, and we cannot choose what it is. The value of "this month" just varies with the time of year we are in. However, this is only true because of what we mean by "this month". We decide that it means something else, but then we would no longer be speaking English. Commented Sep 18, 2022 at 10:47
• @user1007028 "inert" I have to guess that the idea of "inert" is that mathematical objects only vary according to our whims. By triangle, we may mean this or that triangle, and different triangles may be very different from each other. But since the triangle as the mathematical object is not any actual triangle in the so-called real world, it is no going to vary unless we say so. So, unlike "this month", it is "inert". But this is just my interpretation. Commented Sep 18, 2022 at 10:52
• I agree, just for my own understanding of written languages, an expression is constant, but it's meaning can change? Is there 'variable' written expressions? Commented Sep 18, 2022 at 10:57
• @user1007028 "if I assign Elizabeth II to the role of queen does this change the properties of Elizabeth II" This would be "Elizabeth II is the queen". This does not change the properties of Elizabeth II, who was the queen independently of whatever you may want o say. Instead, the sentence "Elizabeth II is the queen" would be true or false according to whether Elizabeth II is the queen. Similarly, x = 2 will be true or false according to whether is x is equal to 2 or not, and this would typically depend on prior assumptions. Commented Sep 18, 2022 at 10:58

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).

• I think what wells should have said is what we consider as the 'mathematical' properties of numbers don't change, because if we consider 'this month is' as an indexical property, we don't consider it, but if we consider there being something of greater importance to a month like 'being this month' than just linguistics, maybe we can consider 'being this month' similar in value to '2 is the first prime', one doesn't change the other does, however its very much open to interpretation. Commented Sep 17, 2022 at 9:30