Crowe quotes at some length from a book written by Grassmann entitled: Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik: dargestellt und durchAnwendungen auf die übrigen Zweige der Mathematik, wie auch auf die Statik,Mechanik, die Lehre vom Magnetismus und die Krystallonomie erläutert (which in German means tl;dr). But he makes the following distinction:

The primary division in all the sciences is into the real and the formal. The former represent in thought the existent as existing independently of thought, and their truth consists in their correspondence with the existent. The formal sciences on the other hand have as their object what has been produced by thought alone, and their truth consists in the correspondence between the thought processes themselves.From what I'm gathering from what little bits Crowe has quoted, Grassmann thinks that his system (which from what I gather was an early form of modern day linear algebra) as being merely "formal". It's existence has been posited through pure thought but it doesn't correspond to anything in the world (outside of the correspondence to human thought). Contrast that with geometry which corresponds with space and is therefore an example of a "real" science. He even went so far as to suggest that geometry properly ought not be included within mathematics:

It had for a long time been evident to me that geometry can in no way be viewed, like arithmetic or combination theory, as a branch of mathematics; instead geometry relates to something already given in nature, namely, space.What he calls "pure mathematics", by contrast is:

... the science of the particular existent which has come to be through thought. The particular existent, viewed in this sense, we name a thought-form or simply a form. Thus pure mathematics is the theory of forms.I'm not sure how far these distinctions would go today. Consider space for example. In a post-Einstein world, space has different structure when in the presence of objects of mass. Mass actually bends or warps space. And it's not entirely restricted to 3 dimensions as it was in Grassman's day.

On the more "abstract" side of things, the lines are even more blurred. We now deal with infinitely dimensional complex vector spaces to explain such phenomenon as matter waves.

At this point there should be a cue for a joke about Hilbert spaces but I'm too lazy to write one. So I'll just get a room in his hotel and crash for the night.

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