Questions about mathematical models of the real world

Question

I'm just starting to learn about mathematical modelling but i'm getting stuck understanding how real world processes and objects are modelled by maths.

The way i'm thinking about at the moment it is that a variable represents an unspecified object of its type, so when i say 'let x be an integer', i mean that x can be any one of the members of integer. And each individual members of the integer set represent individuals in the real world. So if we have 'let b be an integer' and 'let b represent a (any nonspecific) bank account', the members of the integer set represent members of the bank account set.

But I am getting hung up on a couple of points with the example:

1. Why don't mathematicians we create a 'Bank Account' type (like programmers do) and define it as the integer type, so we would have 'let b be a bank account' instead of 'let b be a number'?
2. At school when i applied maths we always used it to represent quantities of things like 'the age of a person' or 'the speed of a car' and never objects like 'Bank accounts', why is this?

Thanks for reading my question, sorry if i am not clear enough, any help is really appreciated, thanks

Answer 1

Representing a brick in maths is done differently depending on which attribute of the brick you wish to build into your model that has bricks in it.

For example, if you wish to build a model which predicts the weight of a building made of bricks, then you use the weight of a single brick as the representation.

If you wish to build a model which predicts the number of truckloads of bricks needed to construct a certain building, then you use the weight of a single truckload of bricks as the representation.

If you wish to build a model which predicts the material cost of a certain building, then one of the inputs to the model will be the cost of a single brick, to which you will add the cost of the mortar needed to glue two bricks together, plus the labor cost, etc.

Answer 2

Mathematicians typically define as little as possible, so that their results are as generic as possible. If they found a need to have a "BankAccount" type, it would be because we needed some very particular behavior of bank accounts which is not present in integers in general. If we don't need any properties that integers don't have, there's no reason to try to create a new type and prove all of the properties we need for that type when we can just stick with integers. You will find that mathematicians do speak to the semantics of the integer representing a bank account in prose, and perhaps use variable names like "b" for bank or "a" for account. But those semantics don't affect the underlying math.

There will be exceptions. Two that I'm familiar with are category theory and description logic. Category theory operates on what they call 'objects' which are lumped into a class: a containers of objects sharing some property. If you wanted a Category of bricks, you could indeed have it (so long as you can make good on the scant few requirements placed on such categories). Description logics also focus on classes of objects (they call them "concepts"). In description logics, classificaiton is very important, so identifying that one has a bank account rather than an arbitrary integer is something they care for.

And, of course, if you are taking a modeling approach based on modern science, modern science is greatly entwined with the idea of measuring things, so it builds models which heavily rely on numbers rather than things.

Answer 3

The obvious answer is that numbers are used in mathematical models to represent properties that can be quantified. Take a brick, for example. You cannot quantify a brick, per se. You can quantify the length of a brick, its height, width, position, orientation, mass, hardness, age, cost, and so on, but you cannot quantify the brick as an object in itself. You can quantify the number of bricks under consideration, and out of an ordered set of bricks you might meaningfully consider the nth brick, where n is an integer, but a brick itself does not have an intrinsic numerical character. Mathematical models typically quantify properties that can be a) quantified individually, and b) have interrelationships that can be quantified.