# Why is the Fourier transform called a 'transform', and not a 'transformation'?

Why are the Fourier transform, Laplace Transform, etc called transforms, and not transformations?

This is about linguistics or terminology in mathematics. I feel there should be a reason why the word 'transform' used for such mathematical objects.

Is 'transform' as a noun, an invented word (coined word) for mathematics?

I have to name a concept in the context of functional programming, and want to know which name is suitable for it:

transform, transformation, map, homeomorphism, morphism, etc.

My name needs to be defendable for people in management and BAs (which are not Mathematicians).

I think operation which assigns to the function $$x \mapsto f(x)$$ another function $$p \mapsto \int f(x) e^{-ipx} dx = \widetilde {f}(p)$$ should be called Fourier transformation, as you suggest. On the other hand the result of this operation, i.e. the function $$\widetilde f$$, is called the Fourier transform (a noun) of $$f$$.

• So, is this word coined only to distinguish two different meaning that you described? Is it a linguistic derivative to create a noun from the verb 'transforming'? My question is perhaps more about linguistics: transformation is also a noun created from a verb. – Sohail Si Feb 2 at 19:39

Is 'transform' as a noun, an invented word (coined word) for mathematics?

Apparently so. The Oxford English Dictionary has "transform" (noun) first used by Sylvester in 1853.

SYLVESTER in Philos. Trans. (Royal Soc.) 143 i. 544 Covariant, a function which stands in the same relation to the primitive function from which it is derived as any of its linear transforms do to a similarly derived transform of its primitive.

Since 1965, "transform" (noun) is also used in geology.

• Why was that word used by Sylvester? and from which context is it borrowed? Perhaps geology? – Sohail Si Feb 2 at 19:38
• Etymology: transform (noun) comes from transform (verb). – GEdgar Feb 2 at 22:22