What do we explictly refer to in mathematical expressios

Question

My friend has a theory about 'instantiation' of numbers, they believe that every time we think of a number we create an 'instance' of it in our own heads, it's the same idea, but each time we think, another instance is created, and the previous one eliminated and he believe mathematical expressions are more about using these 'instances' than the objects themselves, for example:

1+2=3 means to him that 1+2 is an instance of 3 as it is equal to another instance of 3 and + is a sort of 'combining' operation between the instances of 1 and the instance of 2, giving an instance of 3 where equality implies they are 'qualitatively identical' as opposed to 'numerically identical' as we talk about most mathematical objects in standard mathematics.

I asked him whether he's referring to a particular instance, and he said it's general like a chemical formula where Co2+H20 refers to 'any molecule of each element, you reference the element, but in the context you mean 'any molecule of', in the same way, if you combine 'any instances 1+2 you get another instance of 3 in 1+2 which will be equal to any other instance of 3'. Perhaps, in the context of chemical formulas we can consider the + and the arrow as operators on the elements themselves to form mappings (such as element A's molecules for chemical bonds with element B), which eliminates any 'implied' context to it.

Of course this is all non-standard and made up, but is there any useful part of it, could we from an entirely philosophical perspective view it like this, what really are the 'referents' in a mathematical expression, do we only ever refer directly to the 'types' (numbers) or could we see it in an alternative way like this?

Answer 1

every time we think of a number we create an 'instance' of it in our own heads

Sure, and every time you write "sandwich", it is a different instance of the same word, namely, the word "sandwich". Usually, we don't care about the instance, we care about the word, and this is because we take each instance to refer to the same idea, namely, the idea of sandwich, and the idea a word refers to is what we are usually interested in. Each instance of the same word refers to the same idea in the same sense that the spelling of one instance of the same word is seen as identical to that of any other.

So it is because we can somehow see that different instances of the same word have the same spelling that we are able take the word as referring to the same thing.

And this is necessary for us to understand each other. If each instance of the word "sandwich" was somehow allowed to mean something different, we would unable to understand to each other.

what really are the 'referents' in a mathematical expression

There is no fundamental difference between the mathematical language and natural languages. Different instances of the same mathematical symbol will be understood as referring to the same mathematical concept.

This is equally critical. Being able to compute 2 + 3 = 5 requires that we take the particular instances of the symbols 2, 3, = and 5 involved in our computation as referring to the same ideas as any other instances of 2, 3, = and 5. Otherwise, there would be no reason for 2 + 3 to be equal to 5 every time we compute 2 + 3 = 5.

Answer 2

"In this sort of predicament, always ask yourself: How did we learn the meaning of this word ("good", for instance)? From what sort of examples? In what language-games? Then it will be easier for you to see that the word must have a family of meanings."

-Wittgenstein, in Philosophical Investigations

Abstractions can go beyond concrete mental instances - in fact, I'd say that's exactly the point of them.

Visually, we are limited in our subitism, to accurately picturing or guessing without counting, around twelve objects - the sides of two dies on their highest roll. We work from that in school, using blocks and toys, towards a feeling of confidence with quantity and geometry, then get progressively more abstract. Research suggests thrown-object games help, and learning music.

The main theory of mathematical savantism, is that such people gave tapped into the brain structures evolved for social learning (see Dunbar Number implication of neocortex & social group size), and redirected them towards making mathematical relationships ready-at-hand (or rather mind), with complex sensory descriptions often given for the qualities of a given number - I think of the anecdote that led to Ramanujan's Number.

But what happens when you get to the square root of minus one - or quaternions and octonions with multiple imaginary numberlines? Or the spinor of an electron that must be rotated twice to return to the same position? Or hyperoperative tetration? Newton felt he had to make all his proofs geometrically for semi-mystic reasons, which you could say meant requiring they be visualisable - that held him back, possibly influencing his not publishing his work on differentiation and his feud after Leibniz published on it first.

So I would make the case to your friend that we begin with visualising, but as a tool to go beyond it. Another example is: Is it possible to visualize higher dimensional space? I mean, who has a mental-instance of thirteen-dimensional M-Theory physics? We find the relationships, infer patterns, look hard at observed dimensions, and then can think about more. We can visualise 4D, then maybe imagine a curved 4D brane in a 5D space. Then what? But, the math still works.

I make the case here, that math is powerful exactly because numberlines are an abstraction of geometry, specifically of continuous symmetries under transformation: The Unreasonable Ineffectiveness of Mathematics in most sciences

In Wittgenstein's terminology, we go from specific instances, examples, to generalised systems, games. But when we try to provide an exact definition of the word 'game', we find we can still work with something we do not have a clear or exacg mental picture of. I relate that to the Private Language Argument, that abstraction and communication co-arise, here: According to the major theories of concepts, where do meanings come from?

Answer 3

Your friend is wrong if he considers an 'instance' of a number to mean the same thing as an instance of a water molecule, say. The molecule has properties that an abstract number does not posses. In principle, we can take a particular instance of a water molecule and place it somewhere so that later we can consider it to be the same instance that we moved earlier. You cannot do that with abstractions. If I ask you to think about the number one today, and tomorrow I also ask you to think about the number one, why are they necessarily two separate instances of the number one? If they are, how long does each last? If I ask you to think about the number one for an hour, is that an hour long instance of the number one? If you are distracted for a moment, and then return to thinking about the number one, is that the same instance or a new one? I ask you to imagine 100,000 successive instances of the number one, what does that mean? How do you define the boundary of each instance. If I asked you yesterday to think of the number one, and today I ask you to revisit yesterday's instance of the number one, and reload it, as it were, in your conscious mind, can you, or are you creating a new instance? If I ask you to imagine a hundred number ones, which you can readily do by imagining them as a ten by ten array, have you created 100 instances in parallel? If so, how does that sit with your friend's view that every time we think of a number the previous instant is eliminated from our mind? Indeed, in what sense is it eliminated?

Your friend has more work to do if he wants to defend his suggestion that 'every time we think of a number we create an 'instance' of it in our own heads'.