Question

In this blog post, we find the following passage:

This connects with something Thomas Forster said, when he rightly highlighted the distinctively modern conception of a function as any old pairing of inputs and outputs, whether we can define it or not — this is the ‘abstract nonsense’, as Thomas called it [...]

But isn't that definition of a function still productive (theoretically), even if it encompasses functions that we may not be able to construct? After all, it seems at first that anything we will say about functions defined that way will stay true even for functions we could not construct? Or is the difficulty that for functions that we could not construct (for example because they would require an infinite extensional enumeration that would be suspicious), that definition may lead to further difficulties? What is really the issue with that definition?

Answer 1

A function is a mathematical concept that describes a relationship between inputs and outputs. It can be expressed as a mathematical equation or formula, and it allows for the consideration of functions that may be difficult or impossible to construct. The difficulty with this definition arises when dealing with functions that cannot be constructed.

Step-by-step explanation:

A function is a mathematical concept that describes a relationship between inputs and outputs. It is often used in economic models and expressed as a mathematical equation or formula.

For example, if we define a function f(x) = 2x + 1, this means that for any given input value of x, the output will be double the input plus 1.

The definition of a function as any pairing of inputs and outputs, whether we can define it or not, allows for the consideration of functions that may be difficult or even impossible to construct. The main issue with this definition is that it may lead to further difficulties when dealing with functions that cannot be constructed due to reasons such as requiring an infinite extensional enumeration. However, from a theoretical standpoint, this definition is still productive in studying functions.