Intra-mathematical Dependencies

Recently I completed all of my undergraduate level maths courses, so wanted to sum up my understanding of mathematics in the following dependency diagram:

I imagine this like a wall, where each topic is a brick. You can bake different bricks at different times (i.e. follow your curriculum to learn these topics), but finally, this is how they should be arranged (in my opinion) to get the best possible understanding of mathematics.

As of now, I have an “elementary” knowledge of Set Theory, Algebra, Analysis, Topology, Geometry, Probability Theory, Combinatorics and Arithmetic. Unfortunately, in India, there are no undergraduate level courses in Mathematical Logic and Category Theory.

This post can be seen as a sequel of my “Mathematical Relations” post.

Understanding Geometry – 3

In some of my past posts, I have mentioned “hyperbolic curvature“,”hyperbolic trigonometry” and “hyperbolic ideal points“. In this post I will share some artworks, based on hyperbolic geometry, by contemporary artists (from Tumblr):

http://regolo54.tumblr.com/post/88984180082

http://regolo54.tumblr.com/post/153466126772/hyperbolic

http://regolo54.tumblr.com/post/160221039662/hyperbolic

http://regolo54.tumblr.com/post/160327605897/hyperbolic

http://adampizurny.tumblr.com/post/160298780745/protobacillus-collab-with-adampizurny

To explain the mathematics behind the construction of these pictures I will quote Roger Penrose from pp. 34 of  “The Road to Reality“:

Think of any circle in a Euclidean plane. The set of points lying in the interior of this circle is to represent the set of points in the entire hyperbolic plane. Straight lines, according to the hyperbolic geometry are to be represented as segments of Euclidean circles which meet the bounding circle orthogonally — which means at right angles. Now, it turns out that the hyperbolic notion of an angle between any two curves, at their point of intersection, is precisely the same as the Euclidean measure of the angle between the two curves at the intersection point. A representation of this nature is called conformal. For this reason, the particular representation of hyperbolic geometry that Escher used is sometimes referred to as the conformal model of hyperbolic plane.

In the above-quoted paragraph, Penrose refers to Escher’s “Circle Limit” works, explained in detail by Bill Casselman in this article.

Understanding Geometry – 1

When we think about mathematics, what comes to our mind are the numbers and figures. The  study of numbers is called arithmetic and the study of figures is called geometry (in very crude sense!). In our high school (including olympiad level) and college curriculum we cover various aspects of arithmetic. I am very much satisfied with that treatment, and this is the primary reason for my research interests in arithmetic (a.k.a. number theory).

But, I was always unsatisfied with the treatment given to geometry in our high school curriculum. We were taught some plane Euclidean geometry (with the mention of the existence of non-euclidean geometries), ruler and compass constructions, plane trigonometry (luckily, law of cosines was taught), surface area & volume of 3D objects, 2D coordinate geometry, conic sections and 3D coordinate geometry.  In the name of Euclidean geometry some simple theorems for triangles, quadrilaterals and circles are discussed, like triangle congruence criterias, triangle similarity criterias, Pythagoras theorem, Mid-Point Theorem, Basic Proportionality Theorem, Thales’ Theorem, Ptolemy’s theorem, Brahmagupta theorem etc. are discussed. Ruler-compass constructions are taught as “practical geometry”. Students are asked to cram the formulas of area (including Brahmagupta’s formula and Heron’s formula) and volume without giving any logic (though in earlier curriculum teacher used to give the reasoning). Once coordinate geometry is introduced, students are asked to forget the idea of Euclidean geometry or visualizing 3D space. And to emphasize this, conic sections are introduced only as equations of curves in two dimensional euclidean plane.

The interesting theorems from Euclidean geometry like Ceva’s theorem,  Stewart’s Theorem,  Butterfly theorem, Morley’s theorem (I discussed this last year with high school students), Menelaus’ theorem, Pappus’s theorem,  etc. are never discussed in classroom (I came to know about them while preparing for olympiads). Ruler-compass constructions are taught without mentioning the three fundamental impossibilities of angle trisection, squaring a circle and doubling a cube. The conic sections are taught without discussing the classical treatment of the subject by Apollonius.

In the classic “Geometry and Imagination“, the first chapter on conic sections is followed by discussion of crystallographic groups (Character tables for point groups),  projective geometry (recently I discussed an exciting theorem related to this) and differential geometry (currently I am doing an introductory course on it). So over the next few months I will be posting mostly about geometry (I don’t know how many posts in total…), in an attempt to fill the gap between high-school geometry and college geometry.

I agree with the belief that algebraic and analytic methods make the handling of geometry problems much easier, but in my opinion these methods suppress the visualization of geometric objects. I will end this introductory post with a way to classify geometry by counting the number of ideal points in projective plane:

• Hyperbolic Geometry (a.k.a. Lobachevsky-Bolyai-Gauss type non-euclidean geometry) which has two ideal points [angle-sum of a triangle is less than 180°].
• Elliptic Geometry (a.k.a. Riemann type non-euclidean geometry) which has no ideal points. [angle-sum of a triangle is more than 180°]
• Parabolic Geometry (a.k.a. euclidean geometry)  which has one ideal point. [angle-sum of a triangle is 180°]

Real vs Complex numbers

I want to talk about the algebraic and analytic differences between real and complex numbers. Firstly, let’s have a look at following beautiful explanation by Richard Feynman (from his QED lectures) about similarities between real and complex numbers:

From Chapter 2 of the book “QED – The Strange Theory of Light and Matter” © Richard P. Feynman, 1985.

Before reading this explanation, I used to believe that the need to establish “Fundamental theorem Algebra” (read this beautiful paper by Daniel J. Velleman to learn about proof of this theorem) was only way to motivate study of complex numbers.

The fundamental difference between real and complex numbers is

Real numbers form an ordered field, but complex numbers can’t form an ordered field. [Proof]

Where we define ordered field as follows:

Let be a field. Suppose that there is a set which satisfies the following properties:

• For each , exactly one of the following statements holds: , , .
• For , and .

If such a exists, then is an ordered field. Moreover, we define .

Note that, without retaining the vector space structure of complex numbers we CAN establish the order for complex numbers [Proof], but that is useless. I find this consequence pretty interesting, because though and are isomorphic as additive groups (and as vector spaces over ) but not isomorphic as rings (and hence not isomorphic as fields).

Now let’s have a look at the consequence of the difference between the two number systems due to the order structure.

Though both real and complex numbers form a complete field (a property of topological spaces), but only real numbers have least upper bound property.

Where we define least upper bound property as follows:

Let be a non-empty set of real numbers.

• A real number is called an upper bound for if for all .
• A real number is the least upper bound (or supremum) for  if is an upper bound for  and for every upper bound of .

The least-upper-bound property states that any non-empty set of real numbers that has an upper bound must have a least upper bound in real numbers.
This least upper bound property is referred to as Dedekind completeness. Therefore, though both and are complete as a metric space [proof] but only is Dedekind complete.

In an arbitrary ordered field one has the notion of Dedekind completeness — every nonempty bounded above subset has a least upper bound — and also the notion of sequential completeness — every Cauchy sequence converges. The main theorem relating these two notions of completeness is as follows [source]:

For an ordered field , the following are equivalent:
(i) is Dedekind complete.
(ii) is sequentially complete and Archimedean.

Where we defined an Archimedean field as an ordered field such that for each element there exists a finite expression whose value is greater than that element, that is, there are no infinite elements.

As remarked earlier, is not an ordered field and hence can’t be Archimedean. Therefore,   can’t have least-upper-bound property, though it’s complete in topological sense. So, the consequence of all this is:

We can’t use complex numbers for counting.

But still, complex numbers are very important part of modern arithmetic (number-theory), because they enable us to view properties of numbers from a geometric point of view [source].

Division algorithm for reals

You must have seen long-division method to compute decimal representation for fractions. Astonishingly, I never pondered about how one would divide an irrational number to get decimal representation. Firstly, this representation will be approximate. Secondly, we have been doing this in name of “rationalizing the denominator” stating the reason that division by irrationals is not allowed. But, in fact, this is the same problem as faced while analysing division algorithm for Gaussian integers.

Bottom line: Numbers are just symbols. We tend to assign meaning to them as we grow up. Since the set of real numbers, rational numbers and integers  form an Euclidean domain, we can write a division algorithm for them. For example, we don’t have special set of symbols for 3 divided by π, but 3 divided by 2 is denoted by 1.5 in decimals.

Hilbert Effect

Human beings tend to measure the influence of a person(s) on a particular field of study by associating their name to cornerstones. For example: Urysohn lemma, Tychonoff theorem, Gauss Lemma, Eisenstein Criterion, Chinese Remainder Theorem, Hensel Lemma, Langlands program, Diophantine Analysis, Clifford algebra, Lie Algebra, Riemann Surface, Zariski Topology, Banach–Tarski paradox, Russell paradox, Bernstein polynomial, Bernoulli Number ……

In mathematics their have been some fights about naming the cornerstones, which ended up creating a compund-name. For example, Lobachevsky-Bolyai-Gauss geometry (in textbooks it is generally referred as hyperbolic geometry), Bolzano–Weierstrass theorem (Bolzano prove it in 1817, later Wierstrass proved it again rigorously and popularized it), Schönemann–Eisenstein theorem (in textbooks it is generally referred as Eisenstein Criterion), ……

But, David Hilbert influenced mathematics at a whole new level. Apart from terms like Hilbert Cube (and many more..) named after him, he introduced exotic words in mathematics which are very popular in (research-level) mathematics. Following are some of the terms:

• Eigen: This word troubled me a lot when I came across the term “eigen-vector” and “eigen-values” a couple of years ago. Hilbert used the German word “eigen”, which means “own”, to denote eigenvalues and eigenvectors of integral operators by viewing the operators as infinite matrices. You can find more information about the history of introduction of this term in mathematics in this web-page by Jeff Miller.
• Entscheidungsproblem: It is german word for “decision problem”, but still mathematicians tend to use this particular term. For example, the famous paper by Alan Turing titled “On computable numbers, with an application to the Entscheidungsproblem“.
• Syzygy: Interestingly, “syzygy” is greek word used in astronomy to refer to the nearly straight-line configuration of three celestial bodies in a gravitational system. In Hilbert’s terminology,  “syzygies” are the relations between the generators of an ideal, or, more generally, a module. For more details refer to this article by Roger Wiegand titled “WHAT IS…a Syzygy?“.
• Nullstellensatz: It is german for “Set of zeros” (according to google translate). But today, just like syzygy, it has whole new meaning in mathematics. For more details, refer to this MathOverflow discussion: What makes a theorem *a* “nullstellensatz.”

Apart from the terms used in mathematics, Hilbert popularized the term “ignorabimus” in philosophy during his famous radio address. For more details read this short Wikipedia article.

It appears that mathematicians (sometimes) tend to use their creativity in naming theorems like Snake Lemma …

 

Introspection

I have written many philosophical blog posts motivated by the idea of existence of reason for everything.  So far I have written (and revised) my point of view regarding topics like Being Alive, Loving Mathematics,  Becoming Mathematician and Dreaming Big.

A more fundamental and more disturbing question is:

Since each and everything is made up of atoms, what is the borderline between a living organism and non-living object.

Clearly, mathematics will fail to answer this question since there are no absolute quantifiers involved. But still we can try deducing an answer from logical arguments. Following video by Kurzgesagt – In a Nutshell illustrates my question:

We generally describe living organism as something capable of reproduction, growth and consciousness. So, can we convert anything into living organism by somehow adding artificial intelligence to it? Is internet itself a living organism?

Once you call yourself a living organism, the immediate question is about the purpose of your existence (since we believe that there is reason for everything). So we can use this as a quantifier to classify something as living and non-living. Many people have tried (and failed) to answer this question. I came across a possible answer for this question in the film Kubo and the two strings:

We live to write a story and then become immortal in memories of others in form of our stories.

Kubo and the two strings (© 2016, Focus Features)

I really liked this point of view. Being alive is all about being able to create memories. But this view point is very much human centred since we don’t know how other organisms (like other animals, plants, cells, organelles…) interact. Moreover, non-living objects also have stories associated with them (like monuments, paintings,…). So this view point also fails to capture the central idea for classification of something as living or non-living.

I will be happy to know your viewpoint of being able to classify something as living or non-living.