Reading on Godel, Escher, Bach: Chapter 2 – Meaning and Form in Mathematics

This chapter will bring a wider view of formal systems by introducing those that have an infinite number of axioms. We shall later see the consequences of such systems and the power of proof against an exhaustive truth-value check (as introduced here) of all the theorems, usually infinite in number, that are generated by the production rules of the system. The discussion is set into motion by an interesting formal system called a pq-system consisting of a definition and a rule of production:

• DEFINITION: p-q- is an axiom, whenever is composed of hyphens only.
• RULE: Suppose and all stand for particular strings containing only hyphens. And suppose that pq is known to be a theorem. Then p-q- is also a theorem.

Since the definition above conveys the symbol as a string consisting only of hyphens of any arbitrary amount, there does exist an infinite number of axioms. Examples of some of the axioms of the system are and . If one tries to produce a sizable amount of possible axioms and theorems, then a pattern emerges that gives a criteria for theoremhood. A string qualifies as a theorem when it obeys the pattern such that the first two hyphen groups add up to the third. For example, the strings and are both qualified as theorems since the number of hyphens before and after p is equal to the number of hyphens after q.

As a bit of a digression, the chapter discusses two ways of how formal systems are constructed or extended. The first is the bottom-up approach where one starts from the bottommost axioms then constructs the system towards higher levels by producing all the other statements of the system. The second way is called the top-down approach which is the opposite of the bottom-up method. One instead starts from a given set of theorems then makes a step-back to find a smaller set of statements that could have produced the larger initial set. This is usually done by finding the common properties that the initial set of theorems have. Once these properties are found, one can create a smaller set of theorems pertaining to the properties that were found. The process is repeated until the current set of theorems cannot be broken down further to produce more fundamental ones. The most fundamental set will then be established as the axioms of the system.

The top-down approach reminds me of the remarkable progress in the history of our understanding of the universe and the physical processes within it. It has been a common state of affairs in physics when seemingly different physical phenomena can actually be explained by a few rules governing their paradigms. Isaac Newton, in probably one of the most important discoveries of all time, concluded that the force that makes the planets move around the sun is the same force that makes everything fall back to the Earth. James Clerk Maxwell unified, what seemed to be in his time, the independent phenomena of electricity and magnetism into a set of four equations that describes all conceivable electromagnetic phenomena. Albert Einstein reckoned that accelerated reference frames are no different to ones experiencing gravitational fields. Finally, all the observed particles in physics experiments can be classified and explained by a more general theory about the elementary particles called the Standard Model.

Notice that the criteria for theoremhood given above for the pq-system gives a correspondence with the addition of integers in mathematics. This correspondence happens when we map each of the symbols in the system as follows:

• p plus
• q equals
• one
• two.. and so on.

The mappings described above are called isomorphisms. Isomorphisms play a major role in mathematics and physics because it allows seemingly different systems to be expressed and interpreted in many ways. Thus it is through isomorphisms that systems acquire meaning, just as how the pq-system illustrates the addition of integers. A formal system standing on its own and having no relationship at all with other formal systems or some aspect of reality is deemed to be meaningless. If it weren’t for Newton who brought into being an isomorphism between the rates of change and constant quantities of calculus as signifying the state variables and conserved quantities in classical mechanics, then physics as it stands today could not be done. Furthermore, if it weren’t also for the same guy whose isomorphism that the force that makes things fall down towards the earth is the same force that drives the planets in their orbits, then our understanding of the cosmos would be bleak.

An important point to make at this point is that there can be any number of plausible isomorphisms among many formal systems. This alludes to the idea that a formal system can have many interpretations. Consider the formalism of quantum mechanics governed by the language of Hilbert spaces and non-commutative algebra. Different interpretations and formulations have been found for the field as a whole. To name a few, we have the Copenhagen interpretation, the Many-Worlds interpretation, the Heisenberg matrix formulation and the Schrodinger dynamical formulation. These give different meanings for the same physical reality that quantum mechanics dominates on.

How can we be sure that all the theorems of the pq-system will be isomorphic to the operation of addition, i.e. the first two hyphen groups will always add up to the third one? As was stated in a previous post, one will have to go through an infinite amount of inconvenience checking an infinite number of theorems to prove this. But is this proof-by-evidence always necessary? Not at all. Let me illustrate a proof that any integer is either a prime or a product of primes without having to check each of the infinite number of integers. (An integer is called a prime if and if the only positive divisors of are 1 and .

Theorem. Every integer n > 1 is either a prime or a product of primes.
Proof: We use induction on n. The theorem holds trivially for n = 2. Assume it is true for every integer with . If is not prime, it has a positive divisor with . Hence , where . Since both and are less than , each is a prime or a product of primes; hence is a product of primes and is therefore not a prime. This proves our initial assumption and thus proves the theorem.

The proof above applies to all the integers by using the symbols , , and as placeholders for any integer. There was no need to verify the theorem for every integer to arrive at its truth, and the proof does this well. This is why it is called proof rather than just hard evidence. I would just like to mention that in physics, the proofs of theories are done by having their predictions agree with experiments. This manner of proof isn’t as clear cut and exact as those in mathematical proofs where pure logic is used. We have to introduce errors and statistical outliers in our measurements, but nevertheless, the average data must agree with the values predicted by theory.

At the bottom of it all, we are guided by rules which we never make explicit. Our brains fire a huge number of neurons in order to execute even the simplest thought processes. Clearly, our reasoning seems to be governed by an utterly complex formal system that we have yet still to find or read about further in the book. The chapter ends by an outrageous and humorous dialogue of Achilles talking to himself. Most of the statements are barely comprehensible since I think the author is trying to show two Achilleses that are trying to interpret themselves and their thought processes by conversing with each other, but to no avail. Finding the reasons behind one’s reasoning is indeed a difficult, if not impossible, intellectual undertaking. We hope to learn more about this in the next chapters.

References:
Hofstadter, Douglas R. (1999) [1979], Gödel, Escher, Bach: An Eternal Golden Braid, Basic Books.
Apostol, Tom M. (1977), Mathematical Analysis, 2nd ed., Addison-Wesley.

Reading on Godel, Escher, Bach: Chapter 1 – The MU Puzzle

This chapter opens up with the MU puzzle as a prototype for us to better understand the underlying structure of formal systems. The MU puzzle is in essence about attempting to produce the string “MU” from an initially given string “MI” as one follows a certain set of rules that can be applied given some conditions. Here are the four rules of the puzzle (rephrased for further clarity):

1. If you possess a string whose last letter is I, you can add on a U at the end.
2. Suppose you have Mx, where x is at any point, the whole set of strings following M, then you may add Mxx to your collection.
3. If III occurs in one of the strings in your collection, you may make a new string with U in place of III.
4. If UU occurs inside one of your strings, you can drop it and link the two sides where the string was cut.

What’s the most efficient way of finding a MU within this scheme? Well first one tries all the allowed permutations of MI with the given rules then repeats the process for the newly produced strings, i.e. one inspects all the allowed manipulations for all the strings in every level of the collection where the MI string lies at the bottom-most level. As one proceeds doing this, it is then checked if given a finite amount of levels or steps, we arrive at MU somewhere within the collection. This process of trying out all the possible ways of manipulating the objects of a system using all its rules, starting from its axiom/s, is called a decision procedure. Ultimately, the answer to the question “Can you produce MU?” is “No, you can’t”. We shall see the profound implications of this answer as we dwell more into formal systems.

The point of this exercise is to get a gist of how formal systems function given the example of the puzzle above. From an earlier post , I have mentioned about axioms, theorems and proofs as more or less the elements that make up a formal system. In this context, the starting MI string in the puzzle is the axiom. It is where all the other strings will originate from, just as how all the mathematical theorems start from the primitive axioms and definitions. One does not question the truthfulness of the axioms, they are considered to be true right from the start. All the other strings that follow from the four rules and the MI string are called the theorems of the puzzle.

In any formal system, there are two types of theorems: 1.) Theorems that are produced within the system’s rules. 2.) Theorems about the formal system itself, usually statements about the system’s statements. DH makes a distinction between these two types. The former can be proven by derivation, a process of producing a theorem by merely following the rules by mechanical manipulation, through a decision procedure. For example, the string MUIIU can be shown to be a theorem within the puzzle by invoking a sequence of steps from the MI axiom. As for the proof of the latter, it is more insightful in such a way that intelligence is needed to supply both the theorem and its proof. What do we mean by this? Let us take as an example, the following theorem of the second kind: “All the theorems in the system start with the string M.” One cannot prove this by checking if each of the infinitely many theorems does start with an M. That would take an infinite amount of time and would be utterly inconvenient. Any decision procedure would fail to prove the theorem, giving us no choice but to finally invoke some form of intelligence to resolve the issue. Such intelligence would be able to interpret the puzzle from a meta-game perspective. The statement “Each of the derived theorems of the puzzle start with an M” is irrevocably a meta-statement, one that is beyond the framework of the puzzle but is nevertheless true. This leads us back to the consequence of Godel’s incomplete theorem, that provability is a weaker notion than truth.

Clearly, the two theorem types above are telling us something about the advantage of intelligent intervention over any mechanical mode of working within systems. Yet in the manner of proving a theorem of the second type, we must initially accept the statements of logical deduction to be true. It is here that we come to the dialogue “Two Part Invention” where the Tortoise opened a discussion to Achilles regarding three statements, these are (rephrased for clarity)

A.) If , and are any three things and if is equal to , and is also equal to , then is equal to . (Mathematically, if and , then
B.) Let M and N be the two sides of a triangle. Let O be a certain line segment. The two sides M and N are both equal to O.
Z.) M and N are equal to each other.

Achilles readily embraces the logical deduction that if M and N are both equal to O, then those two sides of the triangle are equal to each other. But the Tortoise dissected the argument further by saying that in order to make the necessary logical deduction, then one must believe the statement (denoted by the symbol C) “if A and B are true, then Z must be true”. The Tortoise goes beyond by noting that in order to believe C, one must again believe a meta-statement of C (denoted by D) given by “if A, B, and C are true, then Z must be true”. In other words, D literally means that if one assumes that A and B are true, and that the logical induction “if A and B are true, then Z must be true” is also believed to be true, then Z must be true. Yet one can still peel further the meta-logical induction by going to the metameta-logical induction that is needed to deem the statements of those on the lower levels to be true. There is no end to this. Here we see a concept that is called an infinite recursion. Fields that are grounded on logic have rarely considered this when they operate or understand things since an inquiry into such a recursion does not provide us with new knowledge per se. They have proceeded unhindered in their operations without the need to get through the inductions that lie at the bottom of it all. But it is this actual mechanism that will be employed over and over again throughout the book for within it might (I say this because I haven’t read the whole book yet) lie the power to understand how inanimate objects can form themselves into self-conscious entities such as the one who wrote this article and the one who is reading it now.

References:
Hofstadter, Douglas R. (1999) [1979], Gödel, Escher, Bach: An Eternal Golden Braid, Basic Books.

Reading on Godel, Escher, Bach: Introduction

The introduction starts with a story behind the conception of J.S. Bach’s Musical Offering. Bach went for an impromptu visit to Frederick the Great, King of Prussia, and was given the chance to try out the King’s collection of pianos while improvising on a theme composed by his Highness. After the visit, Bach undertook the creation of Musical Offering which eventually became one of his most notable works. The book’s author, Douglas Hofstadter, then discussed about the musical structures that Bach’s Musical Offering contained. He defined canons as those having similar sets of notes played in harmony with each other by the use of varying tones, tempos, time of attacks or even by inversion of notes. As for fugues, they are just like canons but having more freedom for musical improvisation.

What caught DH’s attention was that canons and fugues have similar properties with the central concept of the book, that of strange loops. A strange loop is a hierarchy of complexities such that as one moves up or down the levels, it will find itself back to the same level of complexity it initially assumed. Now a canon or a fugue behaves like strange loops because of the recurring set of notes throughout the musical piece. The two are differentiated such that a canon strictly obeys the structure of a strange loop, a fugue has more freedom to play around other notes thus instilling it with creativity within its own strange loop. Another example that illustrates strange loops is that of Epimenides’ paradox: The following statement is false. The preceding sentence is true. This is a strange loop that goes back to its initial level of hierarchy in a series of two steps. Other complex systems that exhibit strange loopiness will be discussed further in the book.

Strange loops can also be understood with the help of the mathematically-inspired artist Maurits Cornelis Escher. Below is his art entitled Drawing Hands where it has the same two-level hierarchy as that of Epimenides’ Paradox discussed above.

Another of the strange loopiness is Escher’s Waterfall depicted below. It undergoes six levels of hierarchy before looping back on itself.

From here we arrive at the third and probably the most important in the three personas that make up GEB, the mathematician/logician Kurt Godel and his celebrated incompleteness theorem. To aid the reader in understanding the theorem further, the book took a bit of a recourse to expound on the history of mathematical logic and the motives behind Russell and Whitehead’s logical magnum opus, the Principia Mathematica. The goal was to make logic, set theory and number theory free from self-reference; free from the perils of strange loops. To be able to prove any statement within a system by use of other statements belonging to the same system was the ultimate quest for mathematical consistency and completeness. This was to put mathematics on a firm foundation, its inductive power envisioned to be free from senselessness. But not for long. It was Kurt Godel who put a stump on Russell and Whitehead and all the other mathematicians who put their hopes in the Principia Mathematica. Godel’s merit was due to his incompleteness theorem which, in layman’s terms, is as follows:

“All consistent axiomatic formulations of number theory include undecidable propositions.”

Why number theory? Because prior to the incompleteness theorem, Godel first showed that any symbol, statement, or formula in some formal language can be assigned a unique natural number. This process is called Godel numbering. It is through Godel’s numbering that, in a sense, all of mathematics can be reduced to the study of number theory. That’s why it was number theory that Godel has attacked forthrightly in his incompleteness theorem. This shows that one still cannot prove all the statements of a given system even if the system had perfectly implemented the rules written in the Principia Mathematica. There is no escape to self-reference, any formal system will always arrive at inconsistencies and become incomplete. No axiomatic system whatsoever could produce all number-theoretical truths, unless it were an inconsisten system! Moreover, Godel showed that provability is a weaker notion than truth. This means that self-referential statements are not necessarily false, they are just unprovable. (There is also an analogue of Godel’s theorem in the field of computing, formulated by Alan Turing).

After the introduction of the three personas (Godel, Escher, Bach), the book proceeds to link these three together in the spirit of strange loops. DH has called this synthesis as an “Eternal Golden Braid”. It is here that he introduces the first dialogue of the book entitled Three-Part Invention. The aim of the dialogues is to stir up the reader’s familiarity with self-referring frameworks. It intends to revert the reader, in encountering strange loops, from intuitively saying “This doesn’t make sense. This is wrong.” to a humbler and logically honest stance.

References:
Hofstadter, Douglas R. (1999) [1979], Gödel, Escher, Bach: An Eternal Golden Braid, Basic Books.

http://en.wikipedia.org/wiki/File:DrawingHands.jpg

http://en.wikipedia.org/wiki/File:Escher_Waterfall.jpg

Reading on Godel, Escher, Bach: An Eternal Golden Braid

Two days ago, I started reading again “Godel, Escher, Bach: An Eternal Golden Braid” by Douglas Hofstadter. I bought this massive book a couple of years ago only to find out it was a difficult read. This was partly due to the unorthodox writing style of DH and also because some of its central topics were unfamiliar to me during that time. Some of these topics (which I now have a better understanding of) are mathematical formalisms, Godel’s incompleteness theorem, recursions and the Principia Mathematica of Russell and Whitehead. I was fortunate enough to learn more about these things in my philosophy and physics classes as well as during my self-studying. It was a barrage of eureka moments when I got back to reading it this second time around. To prevent my post from being rather dull and to delay the somewhat technical discussions later, here’s an image of the book cover (it shows the orthogonal projections of blocks of wood into the letters G, E and B. It reminds me of quantum mechanics):

The 20th-anniversary edition preface talks about the main theme of the book and the history of its conception. DH will try to explain how inanimate objects can mysteriously compose themselves into animated or conscious macroscopic entities. We should be careful not to interpret this as the possible interpretation for the emergence of life since not all forms of life are self-conscious. As will be noted, the reason behind this magical transformation lies in the notion of strange loops. A strange loop is a hierarchy of complexities such that as one moves up or down, it finds itself back to the same level of complexity it initially assumed. We will see (as the title have noted) that Godel’s incompleteness theorem paves the way for the existence of strange loops. The more complex the pattern that makes up the loop, the more we can think of the loop as having an “I”. This gives us a hint that there are different degrees of consciousness and indeed DH humorously talks about “lesser/greater souls” as a metaphorical quantifier for consciousness. As a consequence: the more self-conscious an entity is, the greater its level of creativity.

In another similar perspective, the book will try to explain the process of how meaningless entities can never escape to have meaning once they acquire a certain form of complexity or structure. This is true for formal mathematical structures that, at face value, seems to be immune to self-reference. (Formal mathematical structures are those sets of mathematical statements that are primitively constructed by a subset of statements called axioms. Axioms are statements that are believed to be true right from the start without proof. From the set of axioms, one can create additional statements called theorems, lemmas and corollaries. These additional statements have truth values, true or false, that can be determined by the use of the pre-defined axioms or by previously constructed theorems. The mechanism by which a statement is shown to be true is called a proof.) Through a logical process known as Godel’s mapping (to be discussed later), Kurt Godel was able to show that any formal system such that its statements are isomorphic to the whole numbers will inevitably produce statements about itself, i.e. the system will eventually stumble upon statements that refers to itself. DH says, “..such twisting-back, such looping-around, such self-enfolding, far from being an eliminable defect, was an inevitable by-product of the system’s vast power”.

I shall be starting the introduction of GEB – A Musico-Logical Offering where DH will be discussing about Godel, Escher and Bach separately and then ultimately “braiding” them at the end of the section. I hope to be able to review my understanding of the book and also to provide my own insights about it through a series of blog posts.

References:
Hofstadter, Douglas R. (1999) [1979], Gödel, Escher, Bach: An Eternal Golden Braid, Basic Books.

The Scope of Geometric Algebra

Just for a short post, I’d like to talk about the latest session on geometric algebra that I had with my adviser. Last week I asked him to give me a crash course on GA before the school year ends this late March. The topic was about this equation that contained all of Maxwell’s equations:

Maxwell’s equations can be shown from this equation by expressing its scalar, vector, bivector and trivector parts into four equations using the following identity:

I was dumbfounded by how GA can further simplify Maxwell’s equations into a single equation. Then I remembered a book by David Hestenes entitled “Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics.” The equation rang a bell in me that recalled the book’s title stating GA as a ‘unified language’.

After the session, my adviser told me that he does not publish or do research outside geometric algebra. Considering that I was getting more and more attracted to the power of GA, I was thinking that I might do the same thing and become a true follower of the mathematical language. I’m not sure whether we exactly had the same reason, so I still asked him why. He replied, “Because there’s nothing outside it”, followed by a wicked smile and some sense of excitement on his part.

Despite what was said, I still have to find more reasons why I should learn/relearn all of my physics via geometric algebra. But for the moment, I am currently leaning towards becoming one of its true followers. So far I’ve been hyped up by what GA has been offering to me. I’m quite lucky that at this early stage, I somehow already have a concrete idea on what I should specialize on. Because I’ve heard of some people who have only made up their mind on what to specialize on after a long time of trying out different subfields in physics. So, thank you for this.

Geometric Algebra Formulation of the Generalized Relativistic Doppler Effect

Aside from re-learning my classical mechanics and electromagnetism. I’d like to try doing my own research before starting graduate school. One topic I have in mind is the geometric algebra formulation of the generalized relativistic Doppler effect. I got this idea from one of my former colleagues’ (http://www.astro.princeton.edu/~rreyes) undergraduate thesis about the geometric algebra formulation of relativistic Lorentz boosts. Instead of tweaking Lorentz boosts, I will do so for the generalized Doppler effect using the same language: geometric algebra.

My interest in geometric algebra came from Mr. Quirino Sugon, Jr. He is the geometric algebra guru in our university and also Reinabelle’s undergraduate thesis adviser. He has somehow convinced me that it really is a more powerful mathematical tool compared to some conventional ones that physicists are currently using. To start my research I need to learn some geometric algebra first. I plan to do so by getting a copy of the book “Geometric Algebra for Physicists” by Lasenby and by starting to read it in a few days. The book has a section about the Doppler effect after 140 pages. I should be able to start solving my problem when I reach that part of the book. Aside from this I also need to re-learn the classical Doppler effect. Then from here, I will inquire on the generalized relativistic Doppler effect that is derived using mathematics other than geometric algebra. I will make an analog of the derivation using geometric algebra.

I’ve looked it up on the internet and there seems to be no publication about this problem that I have in mind. In any case, even if there already exists a publication about this, I can still benefit from it as a practice for my research skills. It can also be treated as a healthy exercise of my problem solving skills. Of course everyone on my level needs a research adviser. I won’t actually do research alone in the sense of not seeking help. I should do so, every researcher should seek help one way or another. What I mean by doing research all by myself is that I’ll do it even if it is not a requirement in school. This can be a good preparation for graduate school and towards becoming an actual physicist who solves problems out of curiosity and pleasure.

Towards a Philosophy of Trust

This has been in my mind since the time I watched the movie “Babel” almost 2 years ago. Afterwards, different situations came up to me that lead me to more reflections about the concept of trust. Let me briefly explain to you what I have in mind. But my ideas are far from being complete so this post will not be a complete exposition of the philosophy I am currently forming.

The following statements sum up what I have in mind: I have arrived at a conclusion that trust grounds all reason, and for that matter, all knowledge. Trust must also encompass every relation (social, political, economic) in order to arrive at democracy, i.e. in living a life in common.

How can I say that trust grounds all reason? First, all of language (images, ideas and words) is not perfectly accurate. We might talk about a tree, but all of us doesn’t have one exact definition or image of a tree in mind. One might think of an apple tree that has 6 branches while the other thinks about a mango tree with no regard to the number of branches. Now if we try to be more specific, say thinking about an apple tree with 4 branches covered in snow, our images in mind will still be not entirely similar. No matter how specific or certain our ideas are, there will always be differences between each person’s mental projections. Even if we situate ourselves in looking at the same drawing of a tree, no two observers can have the same perception about it. The perceptions still differ by the angle or lighting that affects our own sight of the drawing.

Although language is not relatively certain, we still arrive at knowledge that accurately describes nature. Amidst all our own unique ideas and definitions of things, the general definitions work well enough for us to produce knowledge. Please keep in mind that when I say ‘definition’, it doesn’t only pertain to the dictionary/word definition of a thing. ‘Definition’ can also mean a certain picture, equation or any sort of mental projection or physical perception. These general definitions are grounded by trust. Without trust, there can be no knowledge. We will just have an endless quarrel of first arriving at some perfectly accurate definitions of things. By writing this post, I trust that you will be able to understand my point because we both have the same general definitions of the words I am currently using. You also trust me that I am using words that mean the same to you in the sense of their general definitions. This is how trust grounds all reason and knowledge.

I still haven’t started upon a clear exposition of how trust must encompass every relation in order to arrive at democracy. The ideas are quite clear for me, but I haven’t undertaken the task of translating them into understandable language. But trust me, I will be able to pull off a very clear and concise argument about this. What’s going on in my mind is that it has a certain truth in it that I want to communicate to you in the near future. Through this philosophy, maybe I can also be able to put trust in the context of the meaning of being human, i.e. what it means to be human. Human knowledge and relations cannot function as well as they do without trust at its core. Thus, trust entirely surrounds a moral and rational life in its entirety.

Hegel’s Take on Clarification

It is oftentimes a case that we end up confused due to some concepts that are delivered to us through reading or listening to a statement. Concepts that are new to us challenge the definitions of the older ones, leading us to inspect and reform them. Different concepts get entangled up in our minds, and it seems that the only way to reconcile with this difficulty is to ascribe to all of them the same meaning. It is here that synonyms play a useful role. This gives us much ease in understanding the text or conversation. But the state of confusion becomes worse when the discussion at hand becomes more complicated. There is a need to break down the synonymous aspects of different similar-looking concepts. How can this dilemma of conceptual clarification be solved? It is here that the philosopher Georg Wilhelm Friedrich Hegel comes into play.

Hegel, from his conception of logic, introduces the term differentiation. This is defined as the activity in knowing the differences among the meanings of all the concepts that make up our knowledge. Simply put, it is the ability to make the sharpest distinctions between seemingly similar concepts. In my opinion, differentiation is the most important aspect of our reflection that we must pursue in order to clarify our understanding of concepts.

There are stages that define the mechanism towards Hegel’s differentiation. First, our understanding ascribes definitions to concepts and fixes them. We then apply our definitions by going into further inquiry with new knowledge; it is here that new concepts arise. The more concepts given to us, the more probable the fixed definitions will intersect. It is here that confusion arises and hence the need for strict differentiation becomes apparent.

Let me take into account the importance of differentiation with respect to mathematics. One of the most important field of mathematics is set theory. It is held that all mathematical objects can be classified into sets and subsets. Set theory deals with the concept of sets in order to draw the line between the differences and similarities of mathematical objects. The more properties a set of objects have, the more subsets it must contain. This is very important in determining whether the whole set or just a part of it can be logically applied to a host of problems and situations. It is obvious that this host of problems I am talking about also comprises another set. As can be seen, set theory not only deals in making distinctions possible, but also is a crucial field in determining the relations between mathematical sets.

But we can make an argument against this method of differentiation by the following: It is an inherent human trait that we tend to generalize or unify our knowledge according to the rules of logic. This generalization works by pointing out the similarities among different concepts. So does this mean that our natural tendency to generalize is at fault? Not really.

The fascinating thing about the previous argument against differentiation is that it can be solved by the method of differentiation itself, a counter-argument in this sense. The fault of the argument stems from the synonymous treatment of the concept of ‘property’ and the concept of ‘relations’ of properties. These two concepts seem to be alike but, with proper inspection, really point to different implications. The concept of property is used in order to gather objects into sets. Things having the property of redness all belong to the set of red objects. Objects having the same property are within the same set. On the other hand, the concept of the relation of properties is used in order to arrive at the connections between objects having different properties. Cheese can be related to milk in such a way that cheese is made out of milk. But this relation doesn’t imply that cheese and milk are the same. It is not just the distinctions between concepts that structure our thinking of the world, but also the ways in which concepts are related.

Differentiation plays an important role in all levels of our conceptual understanding. The simplest concepts must already be differentiated in order to avoid confusion arising out of additional concepts that will later come upon us. So the more complicated the discussion becomes, the more we need to differentiate. Learning to compartmentalize our thoughts plays an essential role for a deeper understanding of our knowledge.

I would like to end the discussion by posing the problem of synonyms. Based on this method of differentiation, I believe that synonyms have a role in the deficiency of our understanding of concepts. It may help us play around with concepts and to deliver them with ease on the informal level, but that is all there is to it. Synonyms will nevertheless give us difficulties when additional concepts start to come in. They only provide a lazy alternative to a rigorous inspection of our ascribed definitions to concepts. We must always be ready to reform our preconceived definitions to make way for clarity. So, I think we would be better off in avoiding the application of synonyms.

Clarification = Differentiation. Avoid making synonymous definitions.

Why God Doesn’t Show Himself

This is a post that will be of interest to the believers of the existence of God. Pardon me for using such words because I find the phrase “the existence of God” philosophically and linguistically problematic. First it is only a concept, and it contains nothing other than what it does. And second, a concept that we can utter or that makes sense to us does not imply the existence of the object of the concept, i.e. the concept of God that is very real to us does not imply the existence of God Himself.

Let me start this discussion by bringing up the concept of the infinite. The infinite is something that is incomprehensible no matter how deep we push the powers of our understanding. Again, pardon me for using the word ‘something’ because the infinite is obviously more than ‘something’. It is that which nothing greater can be thought of. No matter how we try to provide a definition for the infinite (its content and properties), it will always transcend those definitions. There is no final criteria in determining a knowledge of it. It is its own final criteria, but that level we can never arrive at.

As for men, we are finite. This finiteness, first and foremost, is in terms of our understanding. Man might be limitless in his inherent nature to learn the contents of knowledge, but understanding them is what is finite. One good example of this finiteness in understanding is the infinite number of possibilities one can interpret such a simple topic, say on life. I have always dreamed of reading all the great books in the world, but this is impossible. Statistically speaking, a person can read only about a thousand books in his lifetime, less if he were to deepen his understanding of those he read or plan to read.

At this point of the post, I would like to say that all I have been using are words, even to describe the infinite. I know this is philosophically improper, but there is nothing I can do for being trapped in language. Language is limited, but it is the best tool we have in expressing our thoughts. Words, as we know, is finite. Words cannot express everything we want to show or deliver to others. But paradoxically, in silence, we can. I will not discuss how it is possible, for this can only be understood in silence. Think about it, be silent.

Let us now pose the following question: Why doesn’t God show himself or speak to us? It is here that I assert the assumption that God is infinite. God doesn’t show himself to us because anything that we can perceive is of finite nature, and God is more than that. Take the image of a tree for example, God could have shown himself to us in the form of a tree but it would make him finite. We would reduce our perception of him to a mere tree, for this is how he showed himself to us. God, being infinite, is more than a tree. So how can he project himself to us? In such images that are incomprehensible themselves. I’d like to quote one of the instances in the bible where God showed himself as a man, but a man whom only his back can be seen. God didn’t show himself up front as a man because his front would be infinitely imperceivable to us, for his true nature is infinite. If he turned up front, those seeing him could have been swept away by such an image, an image that is infinitely perceivable. Thus, the back of the man symbolizes God’s unfathomable mystery. It also marks the disability of man to fully see him up front.

God also doesn’t speak to us in a direct manner because this would make him finite. Speaking involves the use of language, and as I have said earlier, language is finite. If God were to speak to us through words, we would be swept away with the infinite concepts that will be coming to us in the form of words. When God communicates to us, he is incomprehensible to us. That is why it has always been the case that God speaks to us in silence. Silence, confusing as it may seem, can tell everything. As what my former Jesuit professor in physics told our class, silence is the wavelength of the soul. Only in silence can we express our deepest thoughts and feelings.

Again, why doesn’t God show himself or speak to us? Because God is infinite. Any image that we can perceive and any form of language is always finite. God transcends such things.

Therefore, God is infinite. This completes the proof.

On Thinking

Think think think!

As what an admirable colleague of mine says whenever he gets stuck in a solution to a physics problem. Nobody can hope for a miracle to come that will instantly provide them with the solution they are looking for. It is only by thinking that you can get to solve a problem, and by thinking more still in order to solve a more difficult problem. It is by the experience of thinking and arriving at plausible conclusions that we become mature.

Indeed, it is very exhausting to think. This is why people are more inclined to have their minds in an idle state rather than in an active one. It is one of the hardest moments when you have a blank paper in front of you and you have to fill it with words or symbols from scratch; a moment when you have nothing that you can refer to, like lecture notes or some open textbook. People become afraid that the products of their own self-thinking are going to be inconsistent and fallible compared to others or those already known. It is precisely this reason that makes people prefer not to think and just rely on the knowledge of another person or article that had been published/written down.

The powerful will to exceed the boundaries of already known knowledge and to be able to think for one’s self no matter how uncertain it seems is a crucial key to this problem of laziness. For those interested in such matters, a good motivational reading would be Immanuel Kant’s answer to the question “What is Enlightenment?”. Here is the introductory paragraph to this article:

“Enlightenment is man’s emergence from his self-incurred immaturity. Immaturity is the inability to use one’s own understanding without the guidance of another. This immaturity is self-incurred if its cause is not lack of understanding, but lack of resolution and courage to use it without the guidance of another. The motto of enlightenment is therefore: Sapere aude! Have courage to use your own understanding! “

One might think that the use of one’s own understanding should tell us that we must not rely much on educating ourselves with knowledge that is already known or written. This is wrong. Education and the use of one’s own understanding are two sides of the same coin that we must both consider. The experience of learning goes hand in hand with the ability to use one’s understanding. How can one think if there is nothing to think about, if nothing has been learned? Moreover, how can one think deeper if one doesn’t know much of what has already been thought of?

It is true that we must not rely solely on known or written knowledge as the sole factors of our learning. But it is also true that written knowledge must not be branded as unreliable. Certain written knowledge are the fruits of the rigorous thinking that have been done by those who already came before us. And it is up to us on how we will take such conclusions that they arrived at; a thought that will make sense to us and to our times, or one that needs rethinking.

Again, one might argue that we are easily giving authority to those thinkers who came before us. This is also wrong. Authority is given only after another person’s understanding. It is proper that we only give authority to those people whose thoughts are already closely examined and deemed valid. The thoughts of the philosophers that came before us are worthy to be learned because these have survived the rigorous criticism of later thinkers. A system of thought where flaws are perpetuating eventually die out. And a system of thought that has stood the test of reason and experience are eventually marked as worthy for our understanding.

Personally, I prefer to think in a slow but rigorous and careful manner. This is very crucial in the formation of theoretical, scholarly and academic knowledge. Knowledge such as these are in praise of slowness. That is why it takes decades for a philosopher to arrive at a new system of thought. It takes a physicist his whole lifetime in finding an exact solution to a very significant physical problem. It takes generations of mathematicians to finish a 300-page proof for an unsolved mathematical theorem. As for practical knowledge, I think it would be better to think fast but efficiently. This I cannot tackle much.

To end this post, here is a quote from the philosopher Ayn Rand: “Thinking men cannot be ruled.”