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Mathematics:

Does a function in mathematics change anything. For instance:

take the function ()+3.

If the input is 2 and the output is 5 for this function, then is 5 'derived' from 2 and the function ()+3? Is the input 2 or the funtion ()+3 changed in any way?

or is this strictly an assignment, i.e.

2 is assigned 5
3 is assigned 6, etc.

Let's take a another example, If I change the color of an object, I really apply a function to the property of that object. For instance, say I have a red ball. I add some yellow and make the color of the ball orange. Have I changed the property of color or have I changed the ball? If I apply the function AddYellow() to the color of the ball, my input is red, AddYellow() is applied, and I get the result Orange. Is this a change or an assignment from red to orange. Specifically, does the value of the ball change or the ball itself because of the function assignment of the value of the ball. How can an assignment change anything?

November 11, 2007

Response from Alexander George on November 11, 2007

One says "The value of the function f(x) = x2 changes with the value of x", but nothing is actually changing. Perhaps you can compare it to our saying that the landscape changes as one drives along the road. (One difference though: trees and hills can change over time, but numbers and other mathematical entities cannot.)

Mathematicians often look at functions as you suggest: as a collection of ordered pairs of objects <a, b> that satisfies the condition that if <a, b> and <a, c> are both in the collection, then b=c. These pairings are what you've called assignments. Functions are then just particular kinds of sets: sets that contain instructions about what is assigned to what.

You could define a function f(t) = the color of the ball at time t. It too could be viewed as a set of ordered pairs. But what such unchanging functions help us to describe is a physical object, the ball, that is changing.

I wonder whether at the root of your question is the thought that it seems odd that we should be able to describe the changing physical world using unchanging abstract mathematics. If so, many philosophers share your perplexity.

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