… where exp is defined as what? If you use (1) as your definition, 2 can be used same as e… even easier, since it is more easily defined. ðŸ™‚ Of course, if you use (2), then it is easier to define than 2^x, but that is just the (Taylor expansion of) the condition f’=f. And in (3), that condition is even explicitly stated. (4) is the most interesting approach, but it also hides the condition f’=f.

“And secondly, I would say that the difference between an exponential function and a non-exponential function is more important than the difference between two different exponential functions.”

Of course. But I thought you set out to define _the_ exponential function, not _an_ exponential function. If you just want some exponential function, then f(x+y)=f(x)f(y) is of course the most important property. But in that story, the number e is really not very important.

A pedantic correction: in Definition 2′, a vector space is not merely a set V, but is a set V equipped with some additional structures (a zero element 0, an addition map , and a scalar multiplication map , where k is the underlying field.) (This point is usually glossed over in informal presentations of “natural” definitions, Definition 1 and Definition 2, as one often simply assumes that the various vector operations are externally provided and do not require definition.)

]]>Definition 1. A *vector* is an element of a vector space.

Definition 2. A *vector space* is a collection of vectors obeying the following axioms ….

However, one cannot use both of these definitions, as they depend circularly on each other. Usually, what happens in practice is that one begins with an inferior version of Definition 2, and then cleans it up later, like so:

Definition 2′. A *vector space* is a set obeying the following axioms …

Definition 1. Elements of a vector space are known as *vectors*.

Remark 2. Thus, one can view a vector space as a collection of vectors obeying the following axioms …

In this particular case, there is no great distinction between the preliminary definition and the final definition, but in some cases the difference can be quite perceptible. For instance, the “right” definition of a smooth manifold is “a manifold which is locally diffeomorphic to an open subset of Euclidean space”, but one cannot adopt this definition initially, because in order to define “diffeomorphism” one needs to define smooth manifolds (or smooth structures) first. So instead we have to settle for the preliminary definition involving equivalence classes of atlases of smoothly compatible coordinate charts, even though we never use this definition again once we are able to state the “right” one.

]]>A := B, means that the left hand side is defined by the stuff on the right hand side. For example, I may write

A := {x : x is prime}

which means that the symbol ‘A’ is defined as the set in the above manner. Of course the colon appearing inside the braces stand for “such that”. I hope that clears.

]]>Hmm … I see what you mean.

Anyone who wants to see just how arbitrary the formal definition of ordered pair is could do worse than reading this Wikipedia article on ordered pairs.

]]>I’m not sure I agree with you. To begin with, is most easily defined to be . And secondly, I would say that the difference between an exponential function and a non-exponential function is more important than the difference between two different exponential functions.

]]>“A positive integer n not equal to 1 is prime if its only factors are 1 and 0^0”

*Thanks — I’ve now swapped round the dollar sign and the full stop.*

I really think the most important property we want to get is f’=f (and f(0)=1 for normalization). It seems more complicated than adition|->multiplication equation, but it is really the property that distinguishes exp from all other exponential functions.

]]>More seriously, there is more than one function so that since if is such a function then so is for any .

*Oops, that was careless of me. I’ve reworded things and I hope eliminated all the false statements.*