Reetesh Mukul’s Blog

Posted by: Reetesh Mukul | January 25, 2009

The Natural Numbers

On any day, at any time, usually thoughts keep running. Some thoughts are silent, some are very vocal; some of them are active, some of them just keep on playing in the background; some fuse, some switched on or off by others. They may have a commonality, a unifying pattern. They may not have. The role of a poem is to add some scripts which mostly are commentary on the subject. Poems do not tell story, and even if they do so, — the story is known, the essence of poem lies in commenting on the philosophy of the subject matter. And the role of a Metaphor is to bind several seemingly unconnected scripts; its role is to bring some unifying pattern on surface. Using Metaphor, we speak about an object by narrating the course of other objects. Shelley’s, –

The Moon

AND, like a dying lady lean and pale,
Who totters forth, wrapp’d in a gauzy veil,
Out of her chamber, led by the insane
And feeble wanderings of her fading brain,
The mood arose up in the murky east,
A white and shapeless mass

Art thou pale for weariness
Of climbing heaven and gazing on the earth,
Wandering companionless
Among the stars that have a different birth,
And ever changing, like a joyless eye
That finds no object worth its constancy?

The Metaphor, here, is dying lady, the description is about “Moon”, yet the hint is towards a third person, whose state is the Poem’s plot. The resemblance of things with each other need not be a replica, their contexts echo same tone. If poem talks about an object, their is a mapping which gets created along with, for a third object. So when the poet says, “dying lady“, he is visibly talking about “Moon”, but the eventual theme of poem hints towards the state of a third person. The words – dying, lady, – do not themselves carry all meaning. In fact even in normal usage, words, are just hints to an idea. Words switch on contexts. It is one matter to get the meaning of word by consulting Dictionary, but it takes observations and time to comprehend the meaning of word used in a context. Things may look naive, but apply rigour to get their precise meaning … things fall apart.

Counting is the process of assigning symbols to the idea of enlisting objects, it is iterative in nature. An implicit mechanism is the assertion of similarity between objects. So when we are counting Books on a table, we are little bothered about what their sizes or content are. Yet it may become imprecise to define the human notion of a book. Two books share a metaphor, a context which makes them similar. In usual casual talks we may not ascertain this “context”, though in case of poems searching for metaphor is a duty. How precise is our this idea of Counting? Can we count everything ? Can we claim that we can count all objects confirming to different ideas we have ? If counting is the process of mapping symbols to objects, then is it powerful enough to capture essence of every objectives involved in the precise description of the object. Thus can we claim that with all possible scientific knowledge to illustrate the features of a Mango, we are sure we can count all its features. And further can we enlist features of two Mango and so on.

Numbers are symbolic. But are there any limitations of such Symbolism requires us to go through pages from Gödel, Turing and likes. To think numbers are naive ideas, will be too deceptive.

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Posted in Number Theory, Philosophy

Posted by: Reetesh Mukul | November 23, 2008

Meta-Post – I

Thoughts over thoughts can be suitably called meta-thoughts. And so does this post. It’s a post over posts. Apparently a post is one of the typographical formats of a thought ! So far the four posts of this blog had to do with a bit of number theory, a very rough thought on time and basic pieces of algebra. Quite minimal and disconnected. Mathematical Realism is the subject matter of this blog. The fact that prime numbers exist, and no body invented them, that they are not just an artificial construction, should not be overlooked. In this blog it is not the intention of the author to jump at once and assert loudly that “Mathematical Realism” is a valid point, neither author has any biased mindset about them; rather the wish is to investigate thoughts in a mature way, which normaly, naturally becomes indifferent from Mathematics. So the nature of the subject matter of this blog requires quite a good discussion on Mathematics with deep concentration on its foundation.

But where to begin with ? One can think that Mathematics essentially bootstraps out of basics of Formal System, Mathematical Logic, Set Theory and Algebra. Any analytics will involve Number Theory. In fact Number Theory guides Mathematical processes. Another interesting thing is Geometry. The Geometrical vision, whether it has to do with any physicality or not, has some of the grand original thoughts. It will be injustice, if we bring Geometry very late in this blog. Though it may be true that Set Theory forms the essential foundation of whole of Mathematics alongwith Mathematical Logic, still the ideas of Geometry and Number Theory are essential to unify many an ideas. Along philosophical lines, illustration of Formal Systems is most fundamental activity. Our discussion of Mathematical Realism will be based on it. To sum up, a discussion on the subject matter of Mathematical Realism will require precise framing of many ideas. My plan for now is to effectively blog on Number Theory. Once the core aspects of Number Theory is explored, another meta-post will guide the course.

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Posted in Mathematics, Philosophy | Tags: Add new tag, Mathematics, Philosophy

Posted by: Reetesh Mukul | October 12, 2008

Magma and Monoids – II

Formal Systems will be the atomic system for the analysis of subject matter in this blog, however I have not talked about them in this blog so far. Nevertheless, I have no intention to jump into the intricacies of it at this time, and I assume that the readers of this post have some understanding of it. Let us play with a Formal System, which will be described about the law of composition illustrated in last post. The Magma, $E$ , here is the set $\{x_1, x_2, x_3\}$ . For brevity, the infix symbol for law of composition, $\top$ , is dropped. Such type of dropping of infix symbols are typically done when law of composition is multiplication; though here we are dropping it for any law of composition.

Symbols: $x_1, x_2, x_3$

Axioms: $(x_1x_1),(x_1x_2),(x_1x_3),(x_2x_1),(x_2x_2),(x_2x_3),(x_3x_1),x_3x_2),(x_3x_3)$ are compositions.

Rule of Composition: If R and S are composition or symbol, then (RS) is a composition.

Thus in this formal systems we can manufacture following compositions:-

$(((x_2x_1)x_2)x_3)$
$((((x_2x_1)(x_1x_3))(x_1((x_2x_3)x_1)))((x_1x_3)(x_1x_2)))$
$(((x_3x_3)(((x_1x_3)(x_3x_3))(x_3x_2))) ((((x_3x_1)x_3)((x_3x_3)(x_2x_1)))$ $((x_2x_1)(x_1(x_3(x_1x_1))))))$
$(((x_2(x_3x_1))(((x_3x_3)(x_1x_1))(x_3x_1)))((x_2x_3)(x_2x_1)))$

and many others. But many strings, like $x_1x_2x_3$ , $(x_1x_2)x_3x_3$ ; are not a composition. When law of composition of a given magma is used many a times, the inner pattern of doing composition, is isomorphic to one of the composition strings produced by this Formal system. Consider another Formal System, whose Symbols and Axioms are the same as that of above one, while the Rule of Composition, which we will call Rule of Ordered Sequential Composition (the cause of such title will be clear, soon) has modification:-

Rule of Ordered Sequential Composition: If R is a symbol and S is a composition or symbol, then (RS) is a composition.

Thus we get following compositions:-

$(x_1(x_2x_3))$
$(x_1(x_2(x_1x_3)))$
$(x_3(x_1(x_2(x_3))))$

An observation of patterns of these strings gives a mechanism of creating them:-

Lay some x’s sequentially. For example, as such, $x_1 x_2 x_3x_1$
Put left parenthesis in between each of the x’s. For above mentioned example as such, $x_1(x_2(x_3(x_1$
For as many left parenthesis, put the equal number of right parenthesis on extreme left side. For above mentioned example it becomes, $x_1(x_2(x_3(x_1)))$
Enclose the formed string, in parentheses. Thus for the example, we get, $(x_1(x_2(x_3(x_1))))$ .

Many people will find above pattern creating procedure, and the Formal systems, too plain, dry; but then for a pure mechanical system this is the basic requirement for any further action. Also, who knows that Formalizing a system may give deeper unreached perspective? Above mechanism gives a naive thought – under the “Rule of Ordered Sequential Composition”, the compositions created can also be created by an array of x’s; the information about the placement of parentheses is global. A precise set theoretic name for this array is “Ordered Sequence”, which basically is a finite family $(x_{\alpha})_{\alpha \in A}$ , $A$ being an indexing set which is totally ordered. The set $A$ maps into set $E$ . And the total ordering $A$ gets because of an increasing bijection of $A$ onto an interval $\left[0, n\right]$ of $N$ . The Formal System (corresponding to “Rule of Ordered Sequential Composition”) is analogous to Nicolas Bourbaki’s definition for Composition of an ordered sequence of Elements:-

Let $(x_{\alpha})_{\alpha \in A}$ be an ordered sequence of elements in a magma $E$ whose indexing set $A$ is non-empty. The composition( under the law $\top$ ) of the ordered sequence $(x_{\alpha})_{\alpha \in A}$ , denoted by $\underset{\alpha \in A}{\top}{x_{\alpha}}$ is the element of $E$ defined by induction on the number of elements in $A$ as follows:

if $A = \{\beta\}$ then $\underset{\alpha \in A}{\top}{x_{\alpha}} = x_{\beta}$ ;
if $A$ has $p > 1$ elements, $\beta$ is the least element of $A$ and $A^{\prime} = A-\{\beta\}$ , then $\underset{\alpha \in A}{\top}{x_{\alpha}} ={x_{\beta}}{\top}{(\underset{\alpha \in A}{\top}{x_{\alpha}})}$

The compositions produced by the formal system corresponding to “Rule of Composition” is a superset to that of compositions produced by the formal system corresponding to “Rule of Ordered Sequential Composition”.

In the next post on “Magma and Monoids”, there will be discussion on Associativity and Commutativity. I will end this post with the note that all we have done here are some constructions. A digressive question — Is Mathematics all about constructions ? Constructions are important; constructionism is an ideology, a thought. What is your take?

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Posted in Algebra, Algebraic Structures, Mathematics | Tags: Algebra, Algebraic Structures, Mathematics

Posted by: Reetesh Mukul | October 11, 2008

Magma and Monoids – I

Algebraic Structures are important for the creation or study of a model for a subject matter under analysis. Moreover the Algebraic processes are one of the chief elements for the foundation of whole Mathematics; and the conception of Algebraic Structures surfaces just with the initial organization of Mathematics through the introduction of the ideas of Set, Mappings. In this post and other following posts, I will survey these Algebraic Structures; mostly guided by Nicolas Bourbaki’s books. The style of analysis of these Algebraic Structures will be done taking into cognizance the patterns present in them, the Formal Systems, and the expressiveness of these structures. The purpose of these posts is just to appreciate the methodology of organization of ideas that Mathematics provides through Algebra, and the author will pay little effort to give a formal introduction of elements of Algebra.

Nicolas Bourbaki have (the plural have is in context of the fact that “Nicolas Bourbaki”, in Algebraic realm, denotes group of people rather than a single person) defined Magma as such:-

Let $E$ be a set. A mapping $f$ of $E \times E$ into $E$ is called a law of composition on $E$ . The value $f(x,y)$ of $f$ for an ordered pair $(x,y) \in E \times E$ is called the composition of $x$ and $y$ under this law. A set with a law of composition is called a magma.

Thus if the underlying set for the model of a subject matter under study has a law of composition, then the abstraction which contains this set with the mentioned law of composition is eligible to be called a Magma. Since law of composition works on a single set, it is the simplest binary operation pattern on the set. We can think of another law of composition, $g:S \times S \mapsto S$ , such that $g(x,y)=f(y,x)$ . The $x$ and $y$ in the ordered pair $(x,y)$ are just the placeholders, for a given mapping. How do $x$ and $y$ quantify $f(x,y)$ is subject to the illustration of $f$ . Here in the case of $g$ , the role of $x$ (and respective $y$ ) is same as that of $y$ in $f$ (and respective $x$ ). We have a definition here:-

Let $E$ be a magma and $\top$ denoted its law of composition. The law of composition $(x,y) \mapsto y \top x$ on $E$ is called the opposite of the above. The set $E$ with this law is called the opposite magma of $E$ .

This shuffling of the placeholders to create an Opposite Magma, is the simplest of the tweaks on a law of composition. That for a given magma $M$ , we are looking into opposite magma $M^\prime$ , necessarily means that we are interested in a relation between $f(x,y)$ and $f(y,x)$ ; and of course the simplest of such relations will be $f(x,y)=f(y,x)$ . This hints of Commutativity, which we will discuss later on.

How we can link two Magmas ? That there exists some mapping between elements of sets of these Magmas tells about relation between the underlying sets, the wider question of relation between Magmas must talk about respective law of composition also. If we denote $\top$ as the law of composition in each of these Magmas, say $E$ and $E^\prime$ ( $\top$ usually will have different meaning for each Magma), and $f$ is the mapping from the first set to the second one, then if $f(x) \top f(y) = f(x \top y )$ , we have the simplest relation between these two Magmas.
A mapping $f$ of $E$ into $E^\prime$ such that the relation $f(x) \top f(y) = f(x \top y )$ holds for every ordered pair $(x,y) \in E \times E$ is called a homomorphism, or morphism, of $E$ into $E^\prime$ ; if $E=E^\prime$ , $f$ is called an endomorphism of $E$ .

The homomorphism will be called isomorphism, if $f$ is a bijective mapping. Homomorphism asserts stronger relation between two Magmas; the calculations for the laws of composition in second magma, $E^\prime$ can be effectively done in first magma, $E$ . Also any special property of law of composition in first magma will be reflected in second magma. Isomorphism asserts more. The two Magmas will have same behavior except for, possibly, the different names of elements and the law of composition. Of course some extra calculations may be needed( for example, of mapping function $f$ ), yet the structure of these two magmas are the same.
The exploration of Opposite Magma, Homomorphism, Isomorphism is some of the central activities in Algebra. In the next post, an extension of law of composition over a family of elements of a magma will be presented along with description of a formal system about it.

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Posted in Algebraic Structures, Mathematics | Tags: Algebra, Algebraic Structures, Mathematics

Posted by: Reetesh Mukul | June 25, 2008

Time and Motion

Our senses prompt us about the notion of motion. Movement of stars, automobiles, elements in an animation, wind, water, etc. build up the idea of motion. So central is the observation of motion, that our study of any physical phenomenon necessarily begins with it. In common dialect, we usually talk about time taken by certain thing to move from one place to another one. Quite spontaneously, we assert the thought of “time taken” and “change in position” whenever we describe motion. A mathematically inclined person quickly builds up a mental picture of “Euclidean Geometry” and “Time” when it comes to the description of motion. The visuals, the imaginations about motion are so intermixed with our senses that whole Classical Mechanics appears as an application of Differential Geometry. But doing so, we ignore many questions, answering them is not going to be easier, modeling them may lead to mutual inconsistencies; in fact different entities can lead to paradoxes!

For example, if we are going to describe motion in terms of time, then a question arises — what time is ? And if we are going to describe time in terms of ticks, be it of our local quartz clock or on the basis of vibrations in Caesium atom or in terms of length of shadow of a tower, we are effectively defining ticks in terms of some motion. So, Motion requires Time to describe it, but Time in itself requires some Motion to describe it! Circularity of definitions, descriptions. Probably at this juncture, someone can assert that Time is an absolute quantity, and every change in nature has a time parameter $t$ , associated with it. It will be job of human beings to ascertain this time as precisely as he can. But how ? If he tries to measure time in terms of some motion then there is the problem of circularity and otherwise if he uses some other basis for measurement then the fundamental idea about time will get changed. The later in turn can possibly alter our notion of motion. In Physics significant changes in equations may not happen, approximations can still work, … but the vision, the idea may get sea change.

There are fundamental assumptions right from the beginning of description of motion. When we study geometry of a nearly fixed shape ( like pyramid, automobile’s wheel, etc. ) we begin confidently with a well thought structure. Of course to gain more precision we modify our structures, nevertheless we always have something concrete to begin with or at least we think concretely. But the description of motion begins with experiments. We loosely begin with a mathematical structure/framework, try to find laws using this framework, and we find that — some times new laws are inconsistent with older one, often the framework implicitly takes assumption about behavior of components of the system, some times new laws do not favor the underlying framework; and then we redevelop our framework. Reference frames are the elementary term in the process of identification of Space and Time; however reference frame in themselves presume idea of motion. For example, in Classical Mechanics the basic assumption was that changes are propagated instantly; i.e., if a charge appears at $(x,y,z,t)$ , then it affects the Electric field everywhere at time $t$ itself. Changes happen instantly. That changes occurs instantly and same changes in two different reference frames occur at equivalent time intervals is central idea practiced in Classical Newtonian Physics. But things went wrong ! Things went wrong because there were many presumptions, the idea of “Time” being a primary one.

Does Motion Exist ? Is space quantized or time is quantized ? How we can assert coexistence of two instants in Nature ? What instants are ? Since motions are often defined in terms of ratios of infinitesimal quantities ( $\vec {v} = \frac {d\vec r}{dt}$ ), is there any physical existence attributed to these infinitesimal quantities ? What is the core principle behind motion of a particle when it follows a given path ? Does particle exist ? Does paths exist ? And most importantly, why should Nature follow a Mathematical Law ? How Mathematics is linked to Nature ? Why Nature likes Mathematics ? Many questions ! Answer for each of them is linked to answers of the rest. Many answers are yet to be found. Ideas change in one go. Physics is meant for brave-hearts ( and restless minds ).

Heraclitus ( 540 BC ) said — “Everything flows”. Zeno of Elea ( 490 BC ) said — “all motion is illusion”. In the Greek Era, this debate never ended … for almost 250 years. And the debate is still on. It is the description of “Time and Motion”, that is what constitutes Man’s ultimate mental voyage.

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Posted in Motion, Philosophy, Physics | Tags: Motion, Philosophy, Physics

Posted by: Reetesh Mukul | June 14, 2008

Stirling Numbers of First Kind

Vikalpa, is a Sanskrit/Hindi word, often used in context of choice making. The ancient Indian Scripture, Sthananga Sutra, uses this word as a topic name for a text on Permutation and Combination. For the study of subject matter such as, “Vikalpa“, I will like to use the term “Combinatorics“, mainly because the terms, Permutation, Combination are Mathematical quantities, and the word Combinatorics captures the wider essence of Mathematics of “choice making”. So whenever we deal with problems which involve enumerating possible choices, often conforming certain patterns, we will be doing Combinatorics. Combinatorial problems are the theme of many of the every day calculations, for example, the possible decisions to traverse from one place to other place. And unless some patterns are not discovered, calculations can become pretty tedious. But when patterns are found, and effectively used, things can become fun. A very fine approach for Combinatorial proofs are given in Proofs that really count: The art of Combinatorial proof, by Arthur T. Benjamin and Jennifer J. Quinn.

Consider this question — “How many ways $n$ persons can be seated across a round table having $n$ chairs around it, where only relative order of persons matter?”. This question is a part of the problem statement of the plan for seating people across a round table at dinner, where situation like who should/shouldn’t sit adjacent to a person are often dealt. If we consider a person as an element of set,

$\{P_0,P_1,...,P_{n-1}\}$ , we need all possible arrangements

$(P_{p(0)}, P_{p(1)}, ..., P_{p(n-1)})$ ,

where $p$ is a bijective mapping on and into the set $\{0,1,...,n-1\}$ and arrangements are considered equivalent when,

$p_{1}(i) \equiv p_{2}(i)+c\hspace{9 pt} (modulo\hspace{4pt}n)$ , $c\hspace{4 pt}$ being an integer constant.

This is like finding all arrangements of persons, where a given arrangement should not be circular shift of another one. Since for a given arrangement there are $n$ number of circular shifts, so total number of ways in which $n$ people can be seated across a round table, is $n!/n$ , i.e., $(n-1)!$ . So when on a clear sky night, if we want to sketch straight lines between a given $n$ number of stars in such a way that all stars are on the final sketch and they are crossed only once, then we have $(n-1)!$ number of possible sketches. What if we want to make $k$ , such sketches( each sketch should have atleast one star on it ), which do not have any star in common, when we have $n$ stars? Or, suppose if we have $k$ number of identical round tables, how many ways are there to seat $n$ people across them, in such a way that at least one person should seat across a round table ?

If there are $m_i, 1\leq m_i\leq n-k+1$ , number of stars in $i$ th cycle, total number of solutions is given by,

$\sum \frac {{\binom{n}{m_1}(m_1-1)!}{\binom{n-m_1}{m_2}(m_2-1)!}...{\binom{m_k}{m_k}(m_k-1)!}}{Number\hspace{3pt}of\hspace{3pt} permutations\hspace{3pt} of\hspace{3pt} set\hspace{3pt} \{m1,m2,...,m_k\}}$ ,

where the summation ranges on all solutions $(m_1,m_2,...,m_k)$ of the equation $m_1 + m_2 +...+ m_k = n$ . The factor ${\binom {n}{m_1}}{\binom{n-m_1}{m_2}}...{\binom{m_k}{m_k}}$ , is number of ways of choosing groups of $m_1, m_2, ..., m_k$ stars out of $n$ stars. The factor $(m_i-1)!$ , is number of sketches that can be made on $m_i$ number of stars ( the same total number of seating arrangement solution ). Also, since solutions of $m_1 + m_2 +...+m_k = n$ can be permutation of each other, so I provided an expression in denominator. A simple reduction can be acheived by expanding the combinations–

$\sum \frac{n!}{(m_1m_2...m_k)(Number\hspace{3pt}of\hspace{3pt} permutations\hspace{3pt} of\hspace{3pt} set\hspace{3pt} \{m1,m2,...,m_k\})}$ .

Like the way there is no explicit formula for factorial, permutation and combination; so is the situation with the above solution, and in literature we use ${n\brack k}$ as their symbolic representation. This quantity, ${n\brack k}$ , is called unsigned Stirling number of first kind. We will now explore falling factorial power, $x^{\underline{n}}$ , to gain more insight into Stirling number of first kind:-

${x^{\underline{n}}} = x(x-1)...(x-n+1)$ .

Unsigned coefficient of $x^k$ in the above product equals:-

Unsigned Coefficient of ${x^{k-1}}$ , in $(x-1)(x-2)...(x-n+1)$

$=\sum (a_1a_2...a_{n-k})$ ,

where $a_1,a_2,...,a_{n-k}$ are $n-k$ distinct number chosen from the set $\{ 1,2,...,n-1 \}$ . The summation thus runs for ${\binom{n-1}{k-1}}$ cases which is equal to total number of solutions of $m_1 +m_2 +...+m_k = n$ , $1 \leq m_i \leq n-k+1$ . Literatures assert that, unsigned coefficient of ${x^k}$ , in $x(x-1)(x-2)...(x-k)$ is ${n\brack k}$ . Does this mean,

$\sum \frac{n!}{(m_1m_2...m_k)(Number\hspace{3pt}of\hspace{3pt} permutations\hspace{3pt} of\hspace{3pt} set\hspace{3pt} \{m1,m2,...,m_k\})}$ $= \sum \frac{(n-1)!}{q_1q_2...q_{k-1}}$ ,

where $q_1,q_2,...,q_{k-1}$ are $k-1$ distinct numbers from the set $\{ 1,2,...,n-1 \}$ while the summations on both sides are running for equal number of time ? Remember, $m_i$ ’s can have same values.