# Can all maths be represented in geometry?

I'm aware that many maths problems can be expressed and understood in geometry, for example, complex numbers can be expressed in 2 dimensions, that can be useful for some problems. Maths started in geometry.

My question is, can you map all of mathematics to geometry and solve all problems using geometry and geometric reasoning? Secondly, is that worthwhile? Thirdly, why not?

I'm aware that people don't do that, they use sets, or they use other mathematical systems defined by the rules of the system.

I only have basic 1st year university math understanding. My question comes from a few problems I've had over the time I have been a programmer and I found many solutions using geometry, for example for representation of dates and times, if they are represented with dates in the x axis and times in the y, its easier to find contained items using geometry. Many other problems seemed to map to geometry well.

One reason why I ask this question is that I had a creative though that it would be nice is there was a kind of web browser which could browse geometric structures and do reasoning in geometry and if all data was represented in this browser as geometry, graphs, charts, lines, circles, etc. As a "human" I find it easy to think about spacial relations, for example when I was programming 200,000 lines of code in a team, it was easier to imagine it spatially than in any other way. I would image the shape of the text and the logical connections.

Also there is the science fiction book Neuromancer which visualized the web as a network which integrated with the though processes of the person, it wasn't "web pages" but the idea fits well with a geometry based web.

As I said that's a creative idea, not a maths idea, but one motivate is that if I wanted to make such a "browser" I wonder if maths could fit inside it in some way.

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The problem with your question is: what do you mean by geometry? What is geometry? – Damian Sobota Nov 24 '11 at 14:33
Lines, circles, as I said, my knowledge it limited to 1st year maths at university level, I did computer science 10 years ago with just school and then 1st year math, and programmed since. My idea of geometry is what you think it is for someone at this level. – Phil Nov 24 '11 at 15:06
This is not a real question... What is all of mathematics? What is geometry? And what do you mean by mapping? You may have found "geometry" to be useful to solve a problem with dates, but "all of math" is such a immensely wide notion that you ought to have more evidence to start suspecting anything! – Mariano Suárez-Alvarez Nov 24 '11 at 15:17
Good point, in the question I have not defined "all of math", or "geometry" or "mapping". I'm not a mathematician, either tell me I'm stupid or tell me something that a layman can understand, because I won't be able to define these terms in mathematical terms. Its a good chance to communicate maths to a layman or ignore him. – Phil Nov 24 '11 at 15:19
@Phil, you are probably not stupid! But in order to ask an answerable question, you need to define its scope more precisely. Beying a layman, you probably have in mind something more specific than "all of math", for example: tell us what. In particular, there is no need to define anything in mathematical terms: but you need to tell us what you have in mind if you want someone to be able to say something relevant to that. – Mariano Suárez-Alvarez Nov 24 '11 at 16:01

I am pretty sure some problems in maths don't have any reasonnable interpretation in geometric terms.

If I had to give examples : delicate questions of regularity in analysis, existence and uniqueness of solutions to some PDEs, and really abstract algebra (no point into getting into details in you're in your first year).

That said I think that a lot of "basic" ideas and problems can be thought of in geometric terms (it's something of a miracle for some more delicate problems). This is a nice thing since as Dimitar said most people tend to have geometric intuition.

Examples of such things : the evolution of any system over time may be seen as a path in some configuration space (it's a deep idea leading to a branch of math called dynamical systems). For the "miracles" I mentionned, some abstract problems in ring theory are deeply linked to zeros of polynomials, which are geometrical objects. The resolution of the famous Fermat conjecture about the integer solutions of $x^n + y^n = z^n$ also involves geometrical tools.

So to summarize my points :

1. there are lots of deep and sometimes unexpected connections to geometry
2. however it does not work for everything
3. pro : it usually gives an intuitive point of view
4. con : it might be harder to work properly with that approach (Dimitar mentionned the algebrization of geometry : the reason for that is that it is much easier to work with equations than with geometrical objects, but you don't always see where you're going). This is why mathematicians love it when they have multiple point of views on the same mathematical object.
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While I agree with you opening sentence in sentiment, I have to pick bones with your examples: "delicate questions of regularity in analysis, existence and uniqueness of solutions to some PDEs, and really abstract algebra". For the third, there is this entire field called algebraic geometry. For the second, it can often be phrased it terms of sections of certain jet bundles (see works of Mikhail Gromov and so on). For the first, if one really wanted to, one can study Hilbert and Banach manifolds (infinite dimensional Morse theory has occasionally proven its worth). – Willie Wong Nov 25 '11 at 13:43

This question reminds me of a professor I had in the university (My major was Applied Mathematics) 3 years ago.

He explained that most of the mathematicians community's work in the last 200 years went into algebrazing everything they could get their hands on. That means that everything is explained by equations. The pinnacle of this process is the construction of the computer - a machine that operates with algebra exclusively, and does that well and fast.

However our brain (and animals brain as well) doesn't work in this way. It works with 'geometric' abstracts, shapes, sets, etc. For example our brain could much easier understand graphical representations of data rather than its numerical representation.

According to my professor the geometrization of mathematics is a process, currently ongoing, that would give us an understanding of how our own brain actually works and would one day pinnacle in the creation of a true A.I.

@Nguyễn Duy Khánh Sorry I could not comment answers yet :(

Well, traditional geometry does not go above 3D and is mostly in 2D. It works with axioms and theorems that are not numerical but analytical. If you extend geometry to higher dimensions including infinity you could have a set of axioms and theorems that are applicable to wider set of problems.

How that is possible (or not), I couldn't imagine.

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can you map all of mathematics to geometry and solve all problems using geometry and geometric reasoning? Secondly, is that worthwhile? Thirdly, why not?

Thirdly, why not?

Because of the answer to the next one :

Secondly, is that worthwhile?

No it is not, not all questions have a simple geometric interpretation, by geometric I mean a simple line + circle + angle to be easily seen as the answer e.g. transcendentality of $\pi, e$ etc.

can you map all of mathematics to geometry and solve all problems using geometry and geometric reasoning?

No, look up conductibility of trisecting an angle, doubling/halfing a cuve, squareing a circle etc. Basically Galoise theory. Where geometry can not answer geometric problems how can it answer all the mathematical problems?

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the main point is that what is ''your geometry''? I do not think we can use only geometry to solve all number theory's problem, though sometimes, they have beautiful and astonishing connection.

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My question is, can you map all of mathematics to geometry and solve all problems using geometry and geometric reasoning?

In a certain sense yes.

You can express all mathematics using some kind of formal notation. You can turn those symbols into numbers, and the manipulation of the symbols while reasoning into elementary arithmetic operations, as Gödel did (at least in theory). You can turn elementary algebraic operations into geometric constructions, which is attributed to von Staudt. So for example you could say that a certain point on a certain line (or rather its position relative to some other points which fix a basis) symbolizes a certain proposition. You could say that a proof is some construction where, starting from a given set of points (the basis and the axioms) you construct new points using a certain set of constructions (which encode valid deduction steps), eventually ending up at the point indicating the conclusion.

Secondly, is that worthwhile? Thirdly, why not?

The constructions would have very little visual relation to the stuff they are talking about. Turning concepts first into notation, that into numbers and that into geometry just isn't accessible to the mind. So the main benefit is that of showing the different systems to be equivalent in their expressive powers. Gödel could make some very important deductions based on the fact that elementary arithmetic is essentially just as powerful as logical reasoning. And geometry is as powerful as that arithmetic. So there are some points which represent true statements but don't allow for a proof construction. Theoretically interesting, but not something you'd show explicitely in a browser.

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