What is geometry?

This is a question I am not really sure how to answer. So I put it to Sir Michael Atiyah after his Frontiers talk in Cardiff. In essence he told me that geometry is any mathematics that you can imagine as pictures in your head.

To me this is in fact a very satisfactory answer. Geometry a word that literally means “Earth Measurement” has developed far beyond its roots of measuring distances, examining solid shapes and the axioms of Euclid.

Another definition of geometry would be the study of spaces. Then we are left with the question of what is a space?

Classically, one thinks of spaces, say topological or vector spaces as sets of points with some other properties put upon them. The notion of a point seems deeply tied into the definition on a space.

This is actually not the case. For example all the information of a topological space is contained in the continuous functions on that space. Similar statements hold differentiable manifolds for example. Everything here is encoded in the smooth functions on a manifold.

This all started with the Gelfand representation theorem of C*-algebras, which states that “commutative C*-algebras are dual to locally compact Hausdorff spaces”. I won’t say anything about C*-algebras right now.

In short one can instead of studying the space itself one can study the functions on that space. More than this, one can take the attitude that the functions define the space. In this way you can think of the points as being a derived notion and not a fundamental one.

This then opens up the possibility of non-commutative geometries by thinking of non-commutative algebras as “if they were” the algebra of functions on some non-commutative space.

Also, there are other constructions found in algebraic geometry that are not set-theoretical. Ringed spaces and schemes for example.

So, back to the opening question. Geometry seems more like a way of thinking about problems and constructions in mathematics rather than a “stand-alone” topic. Though the way I would rather put it that all mathematics is really geometry!

5 thoughts on “What is geometry?”

  1. In my (probably naive and idealized) opinion, anything in mathematics can be expressed using anything else in mathematics. In short, every branch is just a different point of view and geometry is one of those views. I’ve always liked to “play around” in the sense of using something (simple) to do something (simple) it was not meant for (in my foggy state of mind I can’t think of an example from my own experience, but stuff along the lines of solving quadratic equations, which fall under algebra, using geometry, which from a layman’s point of view have nothing in common, or using cellular automata to [url=http://www.quinapalus.com/wi-index.html]design a simple computer[/url]) and I’ve always been extremely curious as to whether the same is also possible with more advanced mathematics.

  2. Thank you for your comments and questions.

    Shadow: Can everything in mathematics be expressed using anything else in mathematics? I think this it too general a question to get a good answer to. For sure, quite often branches of mathematics that at first seem unrelated can have deep connections.

    It might also be worth saying that there was an attempt to restructure mathematics in terms of just logic. This accumulated in the book Principia Mathematica by Whitehead and Russell. (Bertrand Russell was Welsh by the way) Hilbert was also very influential in this drive.

    Gödel’s incompleteness theorems is credited with halting this logical reduction of all mathematics. All of mathematics cannot be stated in terms of pure logic alone.

    dragonstar57: Yes, I guess that would qualify. If you are moving symbols about in some “space” in your head then you are using geometry.

    Wider than this, using ideas and intuition from “classical geometry” in algebra would also constitute geometry. In reality, this is more like what algebraic geometry and noncommutative geometry is. Doing algebra, but thinking geometrically.

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