Interesting and though provoking question. Many of late nights I am sure maybe with a few beers in grad school ended up in the discussion of similar questions. We know what algebra is. It is easy to define. Similar for topology, the study of topological spaces invariant under homeomorphisms, i.e., a donut and coffee cup with a one loop handle are the same. Similar for analysis (real and complex). But geometry causes some pause in coming up with a good definition.

The article by Sir Michael Atiyah pretty much hit it on the head. Geometry is more a way of perceiving and "visualizing" and mathematical intuition of a problem than a field. In modern mathematics often geometry is related to the ability to make "measurements" on underlying spaces and determine properties of elements on objects in various categories, e.g., length or angle between tangent vectors or parallel transport, curvature, etc., in the category of Riemannian manifolds, i.e., Riemannian geometry, where the underlying structure of the space supports such concepts as length, etc. For example there is no structure that support length in the category of Topological manifolds, you need a Riemannian metric to do so.

However, there is more to it than that. Today we are trying to teach machines to have "vision" so they can autonomously navigate in the world. That requires a basic understanding both of Euclidian geometry in three space along with projective geometry since light that hits the sensors arrives from a ray. This gave rise to a new field - computational commutative algebra and algebraic geometry.

So it seems that geometry is more of a concept that spreads across all of mathematics rather than a fundamental area of mathematics. For example the geometry of attractors in a dynamic system led to things like Smale horseshoe and fractals. There are three pillars for modern mathematics, topology, algebra, analysis (real and complex). That is any problem encountered requires proficiency in these areas to solve. That's why in most graduate schools the PhD qualifying exams require proficiency in those four areas. However, there are "geometric" questions that arise in all those areas - with potentially the exception of algebra .

@Muhammed

You can find answers in the nice article by Sir Michael Atiyah:

https://www.ime.usp.br/~pleite/pub/artigos/atiyah/what-is-geometry.pdf

At the end (in the conclusion), he says:

Broadly speaking I want to suggest that geometry is that part of mathematics in which visual thought is dominant whereas algebra is that part in which sequential thought is dominant. This dichotomy is perhaps better conveyed by the words "insight" versus "rigour" and both play an essential role in real mathematical problems

Geometry is not a small mathematical discipline (like differential geometry), but rather a way of reasoning. In any mathematical question, one can find a geometrical method of resolution, like the study of vector fields and dynamical systems (for differential equations), the general topology (as a geometrical language describing continuity), ...

Good question. "Geometry" seems to be just a colloquial term.

A modest proposal: maybe it could be defined as any science that stems from Euclid's axioms, including all subsequent generalizations, such as those to N dimensions, and those that relax/replace the 5th axiom (Riemann and Minkowski's geometries, etc). Also, as I see it, geometry should somehow relate to actual spaces used and studied by natural sciences, such as Physics. In other words, unlike abstract algebraic spaces, geometry should be "applied". A term like "abstract geometry" would not make sense to me.

Mathematics can be defined as: a science about number and form based on logic, and geometry, as a important branch of mathematics, involves Eclidian space, derivation, topology, and so on.

Euclidean geometry or non euclidean?

Look up and read the article "Erlangen program" in Wikipedia.

Dear Muhammad Zubair Ahmad,

I have a little bit to add to the previous helpful answers.

Geometry is simply a set of consistent axioms called a mathematical model.

The Euclidean geometry is determined by the well-known axioms:

Quoting:

The thousands of trials to prove (5) using the other previous axioms lead to discovering hundreds of different wonderful consistent geometries, for example, the hyperbolic, the elliptic, etc, all are non-euclidean geometries.

Almost all physicists considered non-euclidean geometries to describe our universe.

Best regards

Interesting and though provoking question. Many of late nights I am sure maybe with a few beers in grad school ended up in the discussion of similar questions. We know what algebra is. It is easy to define. Similar for topology, the study of topological spaces invariant under homeomorphisms, i.e., a donut and coffee cup with a one loop handle are the same. Similar for analysis (real and complex). But geometry causes some pause in coming up with a good definition.

The article by Sir Michael Atiyah pretty much hit it on the head. Geometry is more a way of perceiving and "visualizing" and mathematical intuition of a problem than a field. In modern mathematics often geometry is related to the ability to make "measurements" on underlying spaces and determine properties of elements on objects in various categories, e.g., length or angle between tangent vectors or parallel transport, curvature, etc., in the category of Riemannian manifolds, i.e., Riemannian geometry, where the underlying structure of the space supports such concepts as length, etc. For example there is no structure that support length in the category of Topological manifolds, you need a Riemannian metric to do so.

However, there is more to it than that. Today we are trying to teach machines to have "vision" so they can autonomously navigate in the world. That requires a basic understanding both of Euclidian geometry in three space along with projective geometry since light that hits the sensors arrives from a ray. This gave rise to a new field - computational commutative algebra and algebraic geometry.

So it seems that geometry is more of a concept that spreads across all of mathematics rather than a fundamental area of mathematics. For example the geometry of attractors in a dynamic system led to things like Smale horseshoe and fractals. There are three pillars for modern mathematics, topology, algebra, analysis (real and complex). That is any problem encountered requires proficiency in these areas to solve. That's why in most graduate schools the PhD qualifying exams require proficiency in those four areas. However, there are "geometric" questions that arise in all those areas - with potentially the exception of algebra .

I believe that in the beginning the geometry was rather the Euclidean studies (angles, lines, forms, lengths, surfaces, areas ... two and three dimensions) in nature and in the ambient space,

but with modern geometry : differential geometry, algebraic geometry, ... etc. it is more theoretical than vesuel, and even we lack examples for all this theory!

(exp: I do not find how to explain the vector fields for my students in a visible way for a dimension greater than three !!, or explain a distence on a Riemannian without having a reference on the manifold !!!)

Yes, indeed. In the old (golden) times geometry was a toolkit for the venerable trade of land surveyors. Then scientists started meddling with it, and now we do not know what it is! Very typical :-)

Geometry is a branch of Mathematics which studies the size and shape of objects on the plane or in the space . simply we can say Geometry is the science of space. the most required of a system of mathematical objects and propositions(axioms) so that they may be termed a geometry are points, lines and planes. The geometry reveal about the objects of their sizes and shapes.

I would add some more insights to previous answers.

I start with a broader question(s): What is mathematics? What is science? The later question has a generally accepted answer: The science is the activity of obtaining and extending objective (i.e. repeatable and testable) knowledge about the subject of study. If new knowledge already exists and the researcher only finds it out, this new knowledge is called "discovery". When new knowledge is the opera of researcher, it is called "invention". Still, what is the subject of study of mathematics? It is "mathematical models", which do not exist outside of researcher's mind. It would seem like mathematics is rather invention then discovery, but it is not. Regardless of invented methods the mathematical results are always the same for all people, which cannot issue other mathematical laws than what is discovered, e.g. no one can just postulate another value of Pi constant than the only possible one.

So, what is geometry? It would be a mistake to make a confusion between the subject of study of some science and its methods. Still Euclid's "Elements", geometry was studied in axiomatic way. Today almost all mathematical disciplines (and not only) are studied this way, including number theory and set theory, which stay as basis for almost all domains of mathematics. Does it mean that number theory or set theory are actually the geometry, because they use the same methodology? Of course not. Does it mean that geometry is useless because there are other theories constructed axiomatically? Again, in no way so.

From Descartes on, the geometry is often studied using linear algebra language. It means, linear vector space (or affine space in this respect) has one particular interpretation to be also geometric space. Does it mean, all the geometry is actually linear algebra? Of course not. There are also axiomatic and differential geometry apparatus in geometry, that are out of scope of linear algebra. This only mean, geometry can be modeled using linear algebra methods.

As stated above, geometry studies spaces for which are relevant the notions of points, lines, planes; the relations of their incidence and quantities of distances and angles between them. Does it exist in reality? Only some approximation of it. Does it have interpretation in other mathematical discipline? Yes. Does it mean, geometry is not a stand-alone mathematical discipline? No, geometry is a well-established domain (or more recently, domains) of mathematics.

And lastly, 2 important theorems show that axiomatically constructed structure (e.g. Euclidean geometry) will never be the same mathematical object as some object constructed as a model (e.g. Euclidean space).

These 2 theorems show that the study of an axiomatic system is distinct subject from the study of some its model in both methods used and results obtained.

Mathematics is not a science. It is not a branch of science. It does not progress through experimentation. Russell and Whitehead laid out an ambitious plan (Principia Mathematica or PM for short) when they tried to develop a set of axioms and inference rules in symbolic logic from which all mathematics truths could be proven within that logical system. Of course Godel's theorem showed that was too an ambitious undertaking. Then later Russell himself discovered a paradox (Russell's Paradox) in the fundamentals of set theory. We later discovered that such simple sounding axioms such as the axiom - which on first blush seem entirely reasonable - has equivalence which seem bizarre at first glance, i.e., Zorn's lemma. However, Zorn's lemma or some other variant of the axiom is critical in most of the foundational theorems of analysis, topology and algebra.

Most mathematicians accept the axiom of choice as a fundamental axiom of set theory. Russell and Whitehead developed PM because at the time there was no accepted logical and foundational standard for mathematics. A lot of mathematics was ad hoc, e.g., the Italian school of algebraic geometry - particularly their work on algebraic surfaces. When Hilbert proved the what is now known as Hilbert's Basis Theorem (polynomial ring over a Northerian ring is Northerian), Paul Gorden cried out "tis is not mathematics, this is theology. Hilbert had used an argument that depended on the law of the excluded middle or a contrapositive argument while the standard at that time was one had to use a constructive proof. PM did not accomplish it's full goal but did set a standard for the foundations of mathematics. In the early 1900's Andre Weil, W. L. Chow and Oscar Zariski, laid the logical and algebraic foundations of algebraic geometry with Chow modernizing intersection theory with his introduction of Chow Ring and Zariski laying a solid foundation to the theory of algebraic surfaces, both validating the intuition of the Italian school and correct some of their errors. Weil introduced the concept of "manifold" and defined the abstract Riemannian surface.

Mathematics will exist independent of the existence of science. Mathematics is really a game of logical tautology - expanding the catalog of true propositions that follow from the basic axioms. I think many discussions get somehow off the rails since the difference between the goals of science and mathematics are often confused. From his famous speech on the topic by Paul Dirac,

http://www.damtp.cam.ac.uk/events/strings02/dirac/speach.html

" Pure mathematics and physics are becoming ever more closely connected, though their methods remain different. One may describe the situation by saying that the mathematician plays a game in which he himself invents the rules while the physicist plays a game in which the rules are provided by Nature, but as time goes on it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen."

Today that is even more true when one considers the work on string theory. However, even if one experiment wipes away the hope for super symmetry thus nullifying string theory - the beautiful mathematics related to the Yang Mills equations and gauge connections will still be beautiful, will still be true and will still have mathematical utility. That is mathematics is not dependent on any scientific theory - although most scientific theories are dependent on mathematics to describe predictions that can be tested by experiment.

As to geometry - one could suggest that geometry exist when geometric constructs are used. For example when a problem ends up being about an infinite dimensional projective space (a state in quantum mechanics is a unit vector in some Hilbert space ). It may very well be that functional analysis is required to analyze such things. The work of John von Neumann on the foundations of quantum theory gave rise to some unique subject areas going by the name of geometry since at the end of the day von Neumann algebras based on unit vectors in the underlying state, i.e., projective spaces. We have continuous geometry,

https://en.wikipedia.org/wiki/Continuous_geometry

https://en.wikipedia.org/wiki/Noncommutative_geometry

Today algebraic geometry has been generalized to address deep problems in number theory (and has been extremely successful) and to the point that it is often difficult to see the roots of the subject.

See J. P. Serre, famous GAGA paper:

https://en.wikipedia.org/wiki/Algebraic_geometry_and_analytic_geometry

http://www.numdam.org/article/AIF_1956__6__1_0.pdf

It is difficult to see geometry as a fundamental discipline of mathematics in the same way as topology, algebra and analysis. However, geometric intuition is an important part of mathematics. Today statements about which sound geometric are often a complex interplay of analytics, algebraic and topological concepts. For example the beautiful results of Sir Michael Atiyah and I. M. Singer known as the Atiyah-Singer Index Theorem which ties in the functional analysis of elliptic differential equations on a manifold, the underlying topology and geometry of the manifold and the Riemann-Roch theorem.

https://en.wikipedia.org/wiki/Atiyah–Singer_index_theorem

Another example is the Sampson-Eells Theorem, where the curvature condition (non positive sectional curvature) on the Riemann manifold ties the geometry of the manifold in with the existence of harmonic mappings which are the critical point of the energy functional. The original proof used by Sampson and Eells was classic functional analysis and the properties of elliptic operators to boot strap the smoothness.

https://en.wikipedia.org/wiki/Harmonic_map

http://www.math.utah.edu/~chu/Talks/EellsSampson/EellsSampson.pdf

My initial thesis topic was related to the Sampson-Eells theory but about six months in Sampson dropped in my office and told me he had gotten a call from Eells in which he found out the problem I was working on had been solved. So two steps forward - one steps backward.

So today it is difficult to say - what is geometry. However, geometric questions permeate much of mathematics. A more interesting question may be "what is a geometric proof." I don't know if I have an answer to that but "I know one when I see one."

It reminds me of another very long discussion about "What is Physics?". The final verdict was "Whatever the Physicists do". Open, living methodologies do not really have any fixed definitions.

Stan's answer reminds me of a comment in Leon Lederman's wonderful book, "The God Particle." Lederman a Nobel winning experimental physicists, comments about the pecking order in physics. He goes on to say something like:

"The experimental physicists is at the bottom of the food chain, always having to answer to the theoretical physicists. The theoretical physicists more interested in the rules mother nature lays down than mathematics, answers to the mathematician. The mathematician caring more about mathematical "beauty" than reality and answers only to God." Interesting fact is every famous physicists had his very own math guy, Einstein had Marcel Grossman - someone had to explain the concept of a Riemannian manifold and work out the mathematical details. Hawkins had Roger Penrose - who was on Hawkins' thesis advisory board. Without Norbert Wiener, Richard Feynman would have stumbled in his path integral formulation of QED and Feynman did admit that von Neumann's functional analytics foundations to quantum theory was superior to his - of course after he won his Nobel. A physicists is interesting in exploring the rules laid down by mother nature. The mathematician is interesting in the logical beauty of mathematics. But over time both subjects grow, change and as you say are living.

When I got out of grad school there was not such thing as mathematical physicists. I only find out later that I was one.

Truman Prevatt, I cannot agree with all your statements.

" Mathematics is not a science. It is not a branch of science. It does not progress through experimentation. " - Mathematics is sure not an experimental science. But there are also theoretic sciences, and mathematics is such a science. Each time someone achieves PhD in Mathematical Sciences, the academic society acknowledges the mathematics as a science.

" It is difficult to see geometry as a fundamental discipline of mathematics in the same way as topology, algebra and analysis. " - Unfortunately, I have to admit that this is the current state of geometry in mathematics. Geometry deserves much more attention in my personal opinion, and the history of geometry shows it was the oldest fundamental domain of mathematics by present day standards of rigorousness.

One can argue that topology, algebra and analysis are built on from the more fundamental notions than geometry - number and set rather than point, line, plane etc of geometry. Still just as there exist non-Euclidean geometries constructed from different axioms, there are also different number theories constructed from alternative axioms than Peano's. Namely, when one uses the negation of Archimedean axiom, the construction of natural, integer, rational, real and complex numbers is quite different with many levels of nullity and infinity. Interestingly, this alternative number theory is more appropriate for differential analysis, and Lagrange used very similar apparatus to construct it. When speaking about the set theory, one has to admit some axiom set. One possibility is axiom of choice (among others). This axiom seems intuitive, but leads to very counter-intuitive consequences, like Banach-Tarski paradox. There are alternatives to it, e.g. the axiom of determinacy, which itself is very counter-intuitive, but leads to apparently more expected consequences. Logic is also constructed axiomatically. Some logicians deny the law of the excluded middle (axiom), which leads to non-constructive proofs. So, there are many set theories, many number theories, many geometries, many logics and many mathematics, which all are incompatible with each other. The most interesting part of it, in any alternative theory (logic, number theory, set theory, geometry) it is usually possible to construct the models of all its alternatives. So, each alternative is at least as fundamental as others. What have we choose and why? There is no objective answer to this question.

" So today it is difficult to say - what is geometry. However, geometric questions permeate much of mathematics. A more interesting question may be "what is a geometric proof." " - The same can be said about all mathematics. There is the "standard" mathematics, but it is rather a historical choice then scientific one. As of geometric proof, it does not differ from any other mathematical proof, except it is usually associated with some geometric drawing. In geometry, this drawing is not used as basis or source of a proof, rather as illustration of the proof concept. The proof itself is done exclusively deductively.

I am entering this discussion somehow in the middle.

I would bring two technical answers about the question what a geometry is:

A geometry in lattice theory according to Graetzer, General lattice theory, second edition, p. 240 corresponds to a geometric lattice which is a semi-modular, algebraic lattice such that the compact elements are exactly the finite joins of atoms of L.

The second technical answer is: A geometry is an object of an elementary topos.

If the first answer might be considered restrictive, the second might be considered to general in the sense that any presheaf would be a geometry for instance.

I am not sure the philosophical question is solved by these answers. In fact it is much a question of definition.

" Mathematics is not a science. It is not a branch of science. It does not progress through experimentation. " - Mathematics is sure not an experimental science. But there are also theoretic sciences, and mathematics is such a science. Each time someone achieves PhD in Mathematical Sciences, the academic society acknowledges the mathematics as a science.

Of course there is a continuum of what is called mathematics. There is mathematics, mathematical sciences, applied mathematics and many universities have departments in both mathematics and mathematical sciences or applied mathematics often one in the school of arts and sciences and the other in the school of engineering, e.g. Johns Hopkins, Harvard, etc.

But applied mathematics normally referred to mathematics that is often found applicable to solve problems in science and/or engineering.

The question of "is mathematics a science" can be debated forever and it makes for interesting debate. However, while mathematics can be applied to sciences and engineering, however, mathematics is not about evidence, predictions, and/or falsification of theories. It is about logical truths. That the results of these logical truths can be applied is nice but somewhat irrelevant.

http://euclid.trentu.ca/math/sb/misc/mathsci.html

As far geometric proof, the proof of the Riemann-Roch theorem which uses sheaf cohomology (sheafs are a way to capturing local properties and gluing them together in the geometry of the space). using sheaf cohomology there are slick proofs which are "geometric in nature." Some might call that an analytics proof I guess, see the book by Hurwitz-Courant. However, that's not how Riemann proved it as he used periods of intervals (complex analysis). Then there is a strictly algebraic proof contained in Lang's book "Algebraic and Abelian Functions.

Many subjects can be approached from completely different directions using different methods. For example there is the algebraic geometric treatment of several complex variables as in the classic Gunning and Rossi, "Several Complex Variables" and the functional analytic approach in Hormander's book of the same title.

Some would say mathematics is more of a form of art than a science. Many mathematicians would agree.

https://krieger.jhu.edu/magazine/fall-2018-v16n1/the-art-of-mathematics/

There is the domain of synthetic geometry which tries to formalize in an abstract way the instruments which are needed to carry out a geometrical proof. More recent developments even do the same for differential geometry (Synthetic differential geometry).

With the developments of category theory it has become quasi-impossible to put clear frontiers about what technique belongs to geometry rather than to algebra or analysis etc. If one calculates the greatest common divisor of some elements of an adequate ring, he does essentially the same than intersecting geometrical subspaces and so on. I tend to prefer proofs which are carried out in the most general (and most simple) setting (Occam’s razor). Not everyone likes this idea. Historical genesis makes that there is a fixed setting for this or the other domain in mathematics. One should respect the wisdom of the ancient who built the domain and who explored many paths before retaining this or the other formalism.

If in mathematics there are no clear frontiers but those due to historical (and stability) reasons, I would be interested (less in the question if mathematics is a science) in what delimits mathematics in the philosophy of knowledge and generally in philosophy? The frontier, which for the ancient Greek (Presocratics, Plato, Aristotle) did not exist (or was created at that time(?!)) should be better understood in my opinion.

It would be a mistake to judge about usefulness, or fundamental nature, of geometry by searching the geometric ways of solving problems that arise from outside of geometry, as well as to judge about analysis, topology, etc by searching their ways of solving the geometric problems. Geometry has its own rich set of notions and problems that are best approached by geometric methods. And this is one more reason for why I consider geometry as fundamental as other mathematical disciplines: the existence of geometric notions and methods that are not covered in any other domain of mathematics.

One example is the notion of geometric space duality. The supersymmetry in string theory is partly due to this duality. If someone thinks of geometric space as set of points and lines, planes as certain subset of this set, incidence of points, lines and planes as relation element-of or subset-of, if the point is viewed exclusively as a vector which essentially means the ordered tuple of numbers, then what exactly is equivalent of the operation "dual-to"? It is known fact that hyperbolic space is dual to Anti de Sitter space. In 2 dimensions it means there exists a correspondence between points of hyperbolic space and lines of Anti de Sitter space and vice-versa, distances in hyperbolic space and angles in Anti de Sitter space and vice-versa. Any true proposition about hyperbolic space can be transformed into true proposition about Anti de Sitter space, including all equations, their deduction and proofs. Of course both spaces can be constructed using the notions number, vector, set, element and subset, but there is no such mapping (or morphism) in any kind of set theory that maps an element a of set A to a set B containing element b and at the same time maps also the set A to element b. This is, and will continue to be, the intrinsic geometric transformation until some other mathematical discipline invent a formalism for it. But if this will happen, will it mean, the geometrical reasoning is not useful because is covered by other, more fundamental mathematics? In my opinion, it will not.

Here is another example, which also makes use of duality. Discrete uniform groups were necessary to study the geometric properties of crystals. The crystallography was a branch of geometry, which studied the lattices of points using geometric methods and notions. Nowadays, these groups are studied using almost exclusively algebraic apparatus. Moreover, when algebra overtook geometry, the subject of study (crystals viewed as lattices of points) was changed to study of discrete uniform groups themselves. It was gone so far as to speak about "geometry of crystallographic groups". But whilst there is isomorphism between two groups at algebraic level, there may be no isometry between their lattices. For example, there exist isomorphism of crystallographic groups of some space and its dual space, but their lattices are never isometric, because the geometry (and not algebra) of dual spaces is different (except auto-dual spaces, such as elliptic). It means, if one knows some motion group of hyperbolic space, he can immediately write down a group of Anti de Sitter motion group, even if he has no clue what this space, or the lattice of this group, looks like. Is it fair to throw out the intrinsic geometric notions, properties and methods, and use much more poor algebraic/topological/analytic/whatever notions, properties and methods in problems specific to geometry?

to Alexandru Popa:

I don't want to object to the fact that each domain has its proper techniques and that these techniques are best adapted to the questions that they are used to solve. I just want to add a remark concerning the examples you give.

Duality is essentially a categorical notion. Also incidence geometry might have a notion of duality which could be made precise when the setting is clear (As there is a notion of duality in graph theory as an example).

But also category theory is not the only possible setting for doing mathematics by the way.

Thomas Krantz pointed out that maybe the appropriate question is not related to mathematics and science but to the very philosophical structure of modern mathematics. With the advent of topology and the desire to find sufficient algebraic invariants that would establish when topological spaces were homeomorphic, it because clear that there were relations between different "categories" of objects that were important to answering such questions. This is what led to Eilenberg and Mac Lane to introduce the concepts of categories and functors to address the questions in algebraic topology. Of course this led to algebraic invariants which can establish when to differential manifolds are diffeomorphic, e.g., the de Rahm cohomology.

The good news is category theory address in a general setting constructs that arise in all of mathematics, such as maps "factoring through," "universal objects," direct and inverse limits," etc. However, one might ask, is the benefit worth the this "abstract nonsense," a term coined by Steenrod during the development of category and homological algebra. But it is clear that some subject matter, the categorical approach has proved to be a necessary evil, both associated with subject matter that is geometric and even analytic. For example the study of the geometry by the de Rahm cohomology, the properties elliptic differential operators between vector bundles on manifolds via tying it to the algebraic topology on the manifold. Maybe the categorical connection between complex algebraic geometry and communicative algebra might be the greatest success or the success of cohomology of sheafs.

The concept of an affine algebraic variety, a projective algebraic variety, a differential manifold, analytic manifold, analytic space have been generalized to the concept of ringed spaces where the underlying topological space and the structure sheaf define specific geometric object. Like most abstractions many common properties can be established in great generality. However, to establish additional fine structure, more detailed analysis is required. Just as functional analytic approaches can be used to establish solutions to many problems in partial differential equations and differential geometry, at some point the hard analysis of establishing the estimates and bounds are required for such an approach. For example while the Tychonoff fixed point theorem or Schauder fixed point theorem might be used to produce a solution to many complex non-linear functional differential equations, in reality they tell you little about the solution and additional analysis is needed.

Category theory unspecific and abstraction in general - independent of how one feels has raised the level of abstraction that has proved very useful in some problems - geometric, algebraic and analytic. Of course this comes at the expense at obscuring some of the underlying structure of the solutions. However, that has always been true in modern mathematics. Hilbert's original proof of the basis theorem (the polynomial ring over a Northerian ring is Northerian and every ideal is finitely generated) using a counter-positive argument caused quite a stir and didn't help in actually constructing such a basis.

Truman Prevatt I would add this observation:

When I was a student, I put much effort to learn different theories. After some time I became comfortable with this or the other theory. The fact that matroids had something to do with closed spaces in a geometry sounded to me as a sort of "miracle". I did not see yet that there are categorical notions behind all this. I felt uncomfortable with category theory, and found it was a sort of "abstract nonsense" much to difficult to be useful in practice.

After some time I became used also to category theory. I think differently about it now: I could have put the initial effort as well as in learning different stuff into learning what relates this stuff.

I suggest to introduce during mathematics study a sort of "baby category theory" when different branches are taught. May be at the beginning full generality is not yet needed, say monads could be closure operators to start with.

I agree that shifting to a mathematics teaching-model insisting on what is common to mathematical structures is not easy to realize due to different reasons: Category theory is not yet very familiar to a lot of mathematicians at least in Europe. In France for instance Bourbaki preferred to start with set theory and then to root the mathematics tree in sets, there is a lot of resistance. As a consequence mathematics appears at a collection of weakly related theories.

To summarize: In my opinion the notion of adjunction of functors and of T-algebra for instance is very fundamental and should be teached very early, even before or together with linear algebra. To think in these terms would become for young students as natural as to speak of generating family of a vector space etc.

Thomas Krantz , thank you for correcting me. I completely missed the category theory, whose duality is what I meant.

My argument was that I see geometry as more than just one possible application of "true mathematics".

Alexandru Popa , all right, but you make use also of set theory, mappings etc. My suggestion is to be flexible about the setting in which something is proved. It could be logics, order or lattice theory, geometry or whatever.

There is this nice vectorization theorem saying that for a geometry satisfying some basic requirements like Pasch axiom, Desarguian axiom, ... there is a field for which the geometry consists in subvectorspaces with respect to this field (I can look this up if needed.) But not every geometry satifies these axioms (see Cayley plane on wikipedia). And even other geometries are possible.

Finally we get back to the point that geometry is what a geometer does...

Comment on Thomas Krantz's comment on my post.

While category theory has proved invaluable in some cases, the utility with the exception of language, is somewhat topic dependent. When I was a lowly grad student the year long algebraic topology course was offers in alternate years by two people. One approach the development as a subset of category theory. The other approached the subject more classically or less along the classic text of Spanier. My office mate and I took the course in alternate years and it was like we didn't take the same course. This is the early 1970's.

While each approach had advantages for the application of algebraic topology such as proving the existence of solutions of differential equations by using sheaf cohomology and showing the equivalence of vanishing of higher cohomology groups and then showing they do vanish, the classical approach to algebraic topology served me better. My office mate ending up having to develop the feel for chain complexes, CW complexes, etc. to apply the techniques to his problems. Even at that the dirty nasty estimates had to be ground out to get to the point that the "atomic bomb" of sheaf cohomology could be used. So in reality a mathematician cannot remain in the rarified air of category theory - unless that is his/her field. And after that point more dirty estimates needed to be ground out to address the properties of the solutions.

So I think it really depends on the problem at hand as to how much "abstract nonsense" is useful.

However, you are right the language of category theory needs to be developed early onion the mathematics education starting in the undergraduate years.

So, finally, what might the answer to Muhammad Zubair Ahmad 's question? It looks like it is hopeless to offer any answer beyond just a fuzzy feeling. Another question is what are the roots of that feeling; how does it emerge?

Truman Prevatt , in fact I did not so much have "pure" category theory in mind which quickly gets very difficult to represent mentally. I am more in favor of proving theorems by omitting hypotheses that are not helpful. I am in favor of most simple proofs even if they are not most general, but simplicity and generality often coincides. The hypotheses (or axioms) should be adapted to the problem set that is intended to be solved with.

About your point on "dirty estimates": Yes, I agree that, when applying theorems, one needs sometimes to justify that hypotheses are met, but again the arguments should be as easy to retrace as possible.

I suddenly notice that the topic of fractal geometry has not been raised yet...

Or any of the geometry of strange attractors of dynamical systems or horseshoe maps, etc. But one might ask is it so much geometry or topology? The more I think about it the more I think that the term geometry is too vague to have a concrete answer. We know what Algebra is. We know what topology is. We know what real and complex analysis is. Geometry seems to me to be a mixture the three directed to problems associated with properties of sets of points. In looking at algebraic curves over the complex numbers we can use algebra to answer our questions or we can use complex analysis to get to the same answers. However, the questions are geometric questions. No one has brought up either the interplay between geometry and physics, i.e., relativity theory, Yang Mills theory, string theory, etc.

> ,,,application of algebraic topology such as proving the existence of solutions

> of differential equations by using sheaf cohomology...

Talk about killing a flea with a sledgehammer...

Truman Prevatt , there is also the question of non-commutative geometry related to physics. And, Schwartz' distribution theory, could it be considered to be a geometrical theory? Also think of this famous article "Can one hear the shape of a drum?"

Returning to original question, and speaking informally, let me illustrate the different domains of mathematics. If the whole mathematics would be a body, then:

- Logic would be the brain,

- Topology would be the imagination,

- Geometry would be the eyes,

- Algebra would be the tongue,

- Analysis would be the hands.

Please feel free to complete / tune this illustration or just laugh at it.

Felix Klein was trying to answer this same question in his Erlangen program (https://en.wikipedia.org/wiki/Erlangen_program). For 1872, that was a truly formidable effort,

I give an informal introduction to the subject I addressed yesterday:

Non-commutative geometry roots in the observation made by Gelfand(?) that instead of considering the space itself one can also consider its "dual": the space of functions on the space. The corresponding category is in a sense dual to the category of the spaces.

Instead of considering commutative algebras of functions (which is the case for ordinary spaces) one can also consider non-commutative ones and we enter the subject of non commutative geometry.

A35. The eight geometries of William Thurston

Answer 35, posted on April 2, 2019

========================

To the long list of different meanings of the word “geometry”,

we must add the eight geometries of William Thurston:

https://en.wikipedia.org/wiki/Geometrization_conjecture#The_eight_Thurston_geometries

========================

Additional pieces of information:

========================

1. William Thurston was awarded the Fields Medal in 1982:

https://en.wikipedia.org/wiki/Fields_Medal#Fields_medalists

========================

2. Also in 1982, William proposed the geometrization conjecture:

https://en.wikipedia.org/wiki/Geometrization_conjecture

========================

3. The proof of the geometrization conjecture

was completed in 2006 through the hard work

of several mathematicians, notably:

Huai-Dong Cao

https://en.wikipedia.org/wiki/Huai-Dong_Cao

Richard Hamilton

https://en.wikipedia.org/wiki/Richard_S._Hamilton

Grigori Perelman

https://en.wikipedia.org/wiki/Grigori_Perelman

Shing-Tung Yau

https://en.wikipedia.org/wiki/Shing-Tung_Yau

Xi-Ping Zhu

https://en.wikipedia.org/wiki/Zhu_Xiping

========================

4. The geometrization conjecture contains

the Poincaré conjecture as a special case:

https://en.wikipedia.org/wiki/Poincaré_conjecture

========================

5. It is a surprise far beyond anything we can imagine

that one of the most difficult problems of mathematics,

namely, the Poincaré conjecture,

was solved by solving a much bigger problem,

namely, the geometrization conjecture.

It is like to move a cup of coffee 10 centimeters to the left,

we can instead fix the cup of coffee,

and move the whole universe 10 centimeters to the right.

========================

6. Huai-Dong Cao and Xi-Ping Zhu were the firsts

to complete the proof of the geometrization conjecture,

in 2006, by accomplishing the super-human enormous task

of assembling a huge number of small pieces of proof

published in chaos order,

into a huge well-organized complete proof:

https://web.archive.org/web/20120514194949/http://www.intlpress.com/AJM/p/2006/10_2/AJM-10-2-165-492.pdf

========================

7. Huai-Dong Cao and Xi-Ping Zhu were very unfairly

treated as criminals by some mathematicians,

because they committed the abominable crime

of giving full credit to some mathematicians

in the introduction of their article,

and not very precisely for every small contribution.

https://en.wikipedia.org/wiki/Manifold_Destiny

========================

8. The Riemann conjecture is the only

of the 7 Millennium Prize Problems

to have been solved:

https://en.wikipedia.org/wiki/Millennium_Prize_Problems

========================

With best regards, Jean-Claude

As one is used to work (since the very influential Euclid's axiomatics) with points, lines, planes etc. I would be interested if there is also a dual axiomatics for their dual notions? What distinguishes both approaches? Thanks in advance for any comments on this question.

(If you know something on closed subspaces in topological vector spaces and their duals, it would be also very interesting for this purpose.)

To Thomas Krantz : Consider 2-dimensional case. Then, there is the following dual notions / relations / quantities:

• Point P (is dual to) Line l
• P ∈ l (is dual to) p ∋ L
• P ∉ l (is dual to) p ∌ L
• a = PQ (is dual to) A = p ∩ q
• |PQ| = d (a number) (is dual to) ∡(p, q) = φ (a number)
• p ∩ q = ∅ (lines p and q do not intersect) (is dual to) P is not connectible with Q (there is no straight line containing both P and Q)

Now, you can construct the several axioms, dual to well-known Hilbert's ones:

• "For every two points A and B there exists a line l that contains them both." Its dual is: "For every two lines a and b there exists a point L that belongs to them both." (this axiom is valid in e.g. elliptic geometry)
• "For every two points there exists no more than one line that contains them both" is dual to "For every two lines there exists no more than one point that belongs to them both".
• "There exist at least two points on a line" is dual to "There exist at least two lines that contain a point".
• "Of any three points situated on a line, there is no more than one which lies between the other two." Its dual is "Of any three lines that intersect in a point, there is no more than one which lies between the other two." (this axiom is valid in e.g. Minkowski geometry).
• "If A, B are two points on a line a, and if A′ is a point upon the same or another line a′ , then, upon a given side of A′ on the straight line a′ , we can always find a point B′ so that AB ≅ A′ B′" This axiom is dual to "Let an angle ∠ (h, k) be given and let a line a′ be given. Suppose also that a definite side of the straight line a′ be assigned. Denote by h′ a ray of the straight line a′ emanating from a point O′ of this line. Then there is one and only one ray k′ such that ∠ (h, k) ≅ ∠ (h′, k′) and at the same time all interior points of the angle ∠ (h′, k′) lie upon the given side of a′."
• "Let l be any line and P a point not on it. Then there is at most one line, that passes through P and does not intersect l." The dual of this axiom is "Let L be any point and p a line not passing through it. Then there is at most one point, that lies in p and is not connectible with L." (this axiom is valid in e.g. Galilei geometry).

It is important to mention that incidence of elements and equality of quantities in dual spaces always hold.

I just would like to speak about a related question which is: Could there be only one truth value, i.e. instead of {true,false} only one: true-false so to say?

It is related to geometry based on the one element field F_1 which is very studied these days.

(See the discussion on RG: Can logic exist of only one concept: truth or falsehood?)

A39. Geometric Algebra

Answer 39, posted on April 16, 2019

========================

To the long list of different uses of the words “geometry” and “geometric”,

we must add “Geometric Algebra”:

https://en.wikipedia.org/wiki/Geometric_algebra

========================

The following document shows that Geometric Algebra

has good chances to produce important improvements

in mathematical theoretical physics:

========================

Clifford Algebra to Geometric Calculus

A Unified Language for Mathematics and Physics

David Hestenes, Garret Eugene Sobczyk, and James Marsh

DOI: 10.1119/1.14223

https://www.researchgate.net/publication/258944244_Clifford_Algebra_to_Geometric_Calculus_A_Unified_Language_for_Mathematics_and_Physics

========================

See also the following book:

========================

Clifford Algebra to Geometric Calculus

A Unified Language for Mathematics and Physics

David Hestenes and Garret Sobczyk

Fundamental Theories of Physics

314 + xviii pages, Springer, 1984

DOI: 10.1007/978-94-009-6292-7

Hardcover ISBN-13: 9789027716736

https://www.springer.com/us/book/9789027716736

Book review:

American Journal of Physics

Volume 53, issue 5, pages 510—511, April 1985

https://aapt.scitation.org/doi/10.1119/1.14223

http://aapt.scitation.org/doi/pdf/10.1119/1.14223?class=pdf

================

With best regards, Jean-Claude