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Is all math just a description of space and it's properties? Every mathematical concept I have learned can be represented in space. There is a lot left for me to learn in the field of mathematics but I have recently taken on a very helpful way of thinking about and learning new concepts which involves modeling the concept spatially first. Being a visual / spacial person I can easily see patterns and understand new concepts when I do this. e.g. adding numbers is just moving along a number line, linear transformations is just skewing lines on a grid, etc. This has made me wonder if I should have been thinking about math in this way all along instead of foreign symbols in a textbook. Are these symbols fundamentally just a description of something spacial? Can I generalize all of math and correctly say 'Math just a description of space and its properties'?

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closed as unclear what you're asking by Did, Matthew Towers, Nils Matthes, Mostafa Ayaz, Cesareo Jul 10 '18 at 18:28

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  • $\begingroup$ I would not say that math is only about (3D-)space, but because the human is a visually driven animal, we can get best intuition from this sense. I think that many mathematicians find it helpful to have a geometrical understanding of a topic. But this is no necessity. You can always only operate on symbols on paper. Whatever is helpful to you. But saying that infinite dimensional spaces or large cardinals are basically about (3D-)space is a bit far-fetched. Also, what does it mean for a math concept to "be about space"? Are binary numbers about space? You can visualize them on a paper. $\endgroup$ – M. Winter Jul 10 '18 at 11:46
  • $\begingroup$ no, because whatever mathematics you are doing, even if it describes space, you could change the axioms of it such that it no longer describes space. $\endgroup$ – Cato Jul 10 '18 at 13:13
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I too prefer pictures over symbols -- or at least, I like to start with a picture if I can, and then use symbols to speed up my work once I have the right ideas.

It's quite traditional to see mathematics as divided into "Algebra" and "Geometry". One thinks of algebra as being about moving symbols around according to rules and geometry as being about understanding physical space by means of constructions, which are pictures made according to strict rules.

If you take this view, most mathematics looks like Algebra, not Geometry. Open any mathematics textbook (especially an advanced one) and most of what you see will be symbols, not pictures.

But it became apparent (at various historical moments, but most obviously in 17th century Europe) that these two can actually be seen as two perspectives ont the same "objects". For example, you can put two stright lines end-to-end to make a longer line; but if you label them $A$ and $B$, it looks like the longer one could be called $A + B$. And in fact you can do all of arithmetic by drawing with a straightedge and compass (John Stillwell's book The Four Pillars of Geometry includes a good explanation of how this works).

This is not just "using algebra to help us study geometry" -- there is a much deeper sense in which the two really are the same thing. An example of this is Hilbert's Nullstellensatz, which establishes a perfect correspondence between a certain branch of high-powered algebra and a very large class of geometric objects. This discussion may or may not make sense to you. Yes, this subject is called "algebraic geometry", but really all geometry is algebraic and all algebra is geometric.

Even advanced fields, such as geometric group theory, leverage this by translating back and forth between pictures and symbols in fruitful ways. I've heard it said that category theory was at first derided as "comic book mathematics" because it contains so many drawings and so little algebra.

It was long believed that geometry must be the study of the one real physical space in which we live -- you can find this idea as late as the philosopher Immanuel Kant writing around 1800, who was no fool. But we now know that can't be true, because there are many incompatible geometries that can be studied mathematically.

Certainly these spaces usually have nothing to do with the (physical) 3D space or 4D spacetime we seem to live in. But that doesn't mean they have no connection with real spatial phenomena. For example, a flying aircraft can move in any of three directions or rotate in any of three ways, all independently. If we consider a particular motion of the plane as a point in a space, that space must have three infinitely extended flat dimensions and three more that are curled up into circles: a six-dimensional space. Yet this exotic geometric object corresponds to something every child with a paper aeroplane instinctively understands. Such "configuration spaces" are very important in many areas of science and technology.

So as Mees de Vries said, the answer to your question is "yes" as long as you are willing to accept an almost absurdly expanded definition of the word "space".

I strongly recommend having a look at Stillwell's book, you might like it.

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  • $\begingroup$ Having read as much of Immanuel Kant's "Critique Of Pure Reason" as I could stand, I disagree with your assessment of him as "no fool". $\endgroup$ – DanielWainfleet Jul 10 '18 at 18:03
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I think this is fundamentally going to depend on how far you are willing to stretch the word "space". E.g., is the class of languages $\mathsf{NP}$ a 'space'? It consists of particular infinite sequences of zeros and ones, so you could see it as a subset of the interval $[0, 1]$ or as a set of branches in an infinite binary tree. But that spatial intuition doesn't immediately tell you much -- in particular from just looking at it, it's indistinguishable from the probably very different class $\mathsf{P}$.

Is a Turing machine a 'space'? You can visualize the tape as a line, but what about the program? Do you visualize it as a sort of graph? Does that make a Turing machine a 'space'?

In both these cases, I would say spatial intuition doesn't give a very good picture (although it can help for particular properties, of course).

In general you could say that you can build all of mathematics on the back of set theory, and a set can be seen as a sort of "discrete space", just a collection of points. But that doesn't generally match with your spatial intuition: you obviously don't think of the number line as just a bunch of loose points related in non-obvious way.

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